Ternary Goldbach Problem for the Subsets of Primes with Positive Relative Densitis

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希尔伯特的23个数学问题

BULLETIN(New Series)OF THEAMERICAN MATHEMATICAL SOCIETYVolume37,Number4,Pages407–436S0273-0979(00)00881-8Article electronically published on June26,2000MATHEMATICAL PROBLEMSDAVID HILBERTLecture delivered before the International Congress of Mathematicians at Paris in1900.Who of us would not be glad to lift the veil behind which the future lies hidden;to cast a glance at the next advances of our science and at the secrets of its development during future centuries?What particular goals will there be toward which the leading mathematical spirits of coming generations will strive?What new methods and new facts in the wide and richﬁeld of mathematical thought will the new centuries disclose?History teaches the continuity of the development of science.We know that every age has its own problems,which the following age either solves or casts aside as proﬁtless and replaces by new ones.If we would obtain an idea of the probable development of mathematical knowledge in the immediate future,we must let the unsettled questions pass before our minds and look over the problems which the science of to-day sets and whose solution we expect from the future.To such a review of problems the present day,lying at the meeting of the centuries,seems to me well adapted.For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.The deep signiﬁcance of certain problems for the advance of mathematical science in general and the important rˆo le which they play in the work of the individual investigator are not to be denied.As long as a branch of science oﬀers an abundance of problems,so long is it alive;a lack of problems foreshadows extinction or the cessation of independent development.Just as every human undertaking pursues certain objects,so also mathematical research requires its problems.It is by the solution of problems that the investigator tests the temper of his steel;heﬁnds new methods and new outlooks,and gains a wider and freer horizon.It is diﬃcult and often impossible to judge the value of a problem correctly in advance;for theﬁnal award depends upon the grain which science obtains from the problem.Nevertheless we can ask whether there are general criteria which mark a good mathematical problem.An old French mathematician said:“A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to theﬁrst man whom you meet on the street.”This clearness and ease of comprehension,here insisted on for a mathematical theory,I should still more demand for a mathematical problem if it is to be perfect;for what is clear and easily comprehended attracts,the complicated repels us.Moreover a mathematical problem should be diﬃcult in order to entice us,yet not completely inaccessible,lest it mock at our eﬀorts.It should be to us a guide408DA VID HILBERTpost on the mazy paths to hidden truths,and ultimately a reminder of our pleasure in the successful solution.The mathematicians of past centuries were accustomed to devote themselves to the solution of diﬃcult particular problems with passionate zeal.They knew the value of diﬃcult problems.I remind you only of the“problem of the line of quickest descent,”proposed by John Bernoulli.Experience teaches,explains Bernoulli in the public announcement of this problem,that lofty minds are led to strive for the advance of science by nothing more than by laying before them diﬃcult and at the same time useful problems,and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne,Pascal, Fermat,Viviani and others and laying before the distinguished analysts of his time a problem by which,as a touchstone,they may test the value of their methods and measure their strength.The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.Fermat had asserted,as is well known,that the diophantine equationx n+y n=z n(x,y and z integers)is unsolvable—except in certain self-evident cases.The attempt to prove this impossibility oﬀers a striking example of the inspiring eﬀect which such a very special and apparently unimportant problem may have upon science.For Kummer,incited by Fermat’s problem,was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circularﬁeld into ideal prime factors—a law which to-day in its generalization to any algebraicﬁeld by Dedekind and Kronecker,stands at the center of the modern theory of numbers and whose signiﬁcance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.To speak of a very diﬀerent region of research,I remind you of the problem of three bodies.The fruitful methods and the far-reaching principles which Poincar´e has brought into celestial mechanics and which are to-day recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that diﬃcult problem and to approach nearer a solution.The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason,belonging to the region of abstract number theory,the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.But it often happens also that the same special problemﬁnds application in the most unlike branches of mathematical knowledge.So,for example,the problem of the shortest line plays a chief and historically important part in the foundations of geometry,in the theory of curved lines and surfaces,in mechanics and in the calculus of variations.And how convincingly has F.Klein,in his work on the icosahedron,pictured the signiﬁcance which attaches to the problem of the regular polyhedra in elementary geometry,in group theory,in the theory of equations and in that of linear diﬀerential equations.In order to throw light on the importance of certain problems,I may also refer to Weierstrass,who spoke of it as his happy fortune that he found at the outset of his scientiﬁc career a problem so important as Jacobi’s problem of inversion on which to work.MATHEMATICAL PROBLEMS409 Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely theﬁrst and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena.Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization,just as the child of to-day learns the application of these laws by empirical methods.The same is true of theﬁrst problems of geometry, the problems bequeathed us by antiquity,such as the duplication of the cube, the squaring of the circle;also the oldest problems in the theory of the solution of numerical equations,in the theory of curves and the diﬀerential and integral calculus,in the calculus of variations,the theory of Fourier series and the theory of potential—to say noting of the further abundance of problems properly belonging to mechanics,astronomy and physics.But,in the further development of a branch of mathematics,the human mind, encouraged by the success of its solutions,becomes conscious of its independence. It evolves from itself alone,often without appreciable inﬂuence from without,by means of logical combination,generalization,specialization,by separating and col-lecting ideas in fortunate ways,new and fruitful problems,and appears then itself as the real questioner.Thus arose the problem of prime numbers and the other problems of number theory,Galois’s theory of equations,the theory of algebraic invariants,the theory of abelian and automorphic functions;indeed almost all the nicer questions of modern arithmetic and function theory arise in this way.In the meantime,while the creative power of pure reason is at work,the outer world again comes into play,forces upon us new questions from actual experience, opens up new branches of mathematics,and while we seek to conquer these new ﬁelds of knowledge for the realm of pure thought,we oftenﬁnd the answers to old unsolved problems and thus at the same time advance most successfully the old theories.And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions,methods and ideas of the various branches of his science,have their origin in this ever-recurring interplay between thought and experience.It remains to discuss brieﬂy what general requirements may be justly laid down for the solution of a mathematical problem.I should sayﬁrst of all,this:that it shall be possible to establish the correctness of the solution by means of aﬁnite number of steps based upon aﬁnite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated.This requirement of logical deduction by means of aﬁnite number of processes is sim-ply the requirement of rigor in reasoning.Indeed the requirement of rigor,which has become proverbial in mathematics,corresponds to a universal philosophical necessity of our understanding;and,on the other hand,only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full eﬀect.A new problem,especially when it comes from the world of outer experience,is like a young twig,which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem,the established achievements of our mathematical science.Besides it is an error to believe that rigor in the proof is the enemy of simplic-ity.On the contrary weﬁnd it conﬁrmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended.The410DA VID HILBERTvery eﬀort for rigor forces us toﬁnd out simpler methods of proof.It also fre-quently leads the way to methods which are more capable of development than the old methods of less rigor.Thus the theory of algebraic curves experienced a considerable simpliﬁcation and attained greater unity by means of the more rigor-ous function-theoretical methods and the consistent introduction of transcendental devices.Further,the proof that the power series permits the application of the four elementary arithmetical operations a well as the term by term diﬀerentiation and integration,and the recognition of the utility of the power series depending upon this proof contributed materially to the simpliﬁcation of all analysis,particularly of the theory of elimination and the theory of diﬀerential equations,and also of the existence proofs demanded in those theories.But the most striking example for my statement is the calculus of variations.The treatment of theﬁrst and second variations of deﬁnite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations.By the examples of the simple and double integral I will show brieﬂy,at the close of my lecture,how this way leads at once to a surprising simpliﬁcation of the calculus of variations.For in the demonstration of the necessary and suﬃcient criteria for the occurrence of a maximum and minimum,the calculation of the sec-ond variation and in part,indeed,the wearisome reasoning connected with theﬁrst variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the diﬀerential coeﬃcients of the function vary but slightly.While insisting on rigor in the proof as a requirement for a perfect solution of a problem,I should like,on the other hand,to oppose the opinion that only the concepts of analysis,or even those of arithmetic alone,are susceptible of a fully rigorous treatment.This opinion,occasionally advocated by eminent men,I con-sider entirely erroneous.Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry,mechanics and physics,to a stoppage of theﬂow of new material from the outside world,and ﬁnally,indeed,as a last consequence,to the rejection of the ideas of the continuum and of the irrational number.But what an important nerve,vital to mathematical science,would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever,from the side of the theory of knowledge or in geometry,or from the theories of natural or physical science,mathematical ideas come up,the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms,that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.To new concepts correspond,necessarily,new signs.These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts.So the geometricalﬁgures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians.Who does not always use along with the double inequality a>b>c the picture of three points following one another on a straight line as the geometrical picture of the idea “between”?Who does not make use of drawings of segments and rectangles enclosed in one another,when it is required to prove with perfect rigor a diﬃcult theorem on the continuity of functions or the existence of points of condensation?Who could dispense with theﬁgure of the triangle,the circle with its center,or with the crossMATHEMATICAL PROBLEMS411 of three perpendicular axes?Or who would give up the representation of the vector ﬁeld,or the picture of a family of curves or surfaces with its envelope which plays so important a part in diﬀerential geometry,in the theory of diﬀerential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?The arithmetical symbols are written diagrams and the geometricalﬁgures are graphic formulas;and no mathematician could spare these graphic formulas,any more than in calculation the insertion and removal of parentheses or the use of other analytical signs.The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie thoseﬁgures;and in order that these geometricalﬁgures may be incorporated in the general treasure of mathematical signs,there is necessary a rigorous axiomatic investigation of their conceptual content.Just as in adding two numbers,one must place the digits under each other in the right order,so that only the rules of calculation,i.e.,the axioms of arithmetic,determine the correct use of the digits,so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations.The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical,any more than in geometrical discussions.On the contrary we ap-ply,especially inﬁrst attacking a problem,a rapid,unconscious,not absolutely sure combination,trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols,which we could dispense with as little in arithmetic as with the geometrical imagination in geometry.As an example of an arithmetical theory operating rigorously with geometrical ideas and signs,I may mention Minkowski’s work,Die Geometrie der Zahlen.1Some remarks upon the diﬃculties which mathematical problems may oﬀer,and the means of surmounting them,may be in place here.If we do not succeed in solving a mathematical problem,the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. Afterﬁnding this standpoint,not only is this problem frequently more accessible to our investigation,but at the same time we come into possession of a method which is applicable also to related problems.The introduction of complex paths of integration by Cauchy and of the notion of the ideals in number theory by Kummer may serve as examples.This way forﬁnding general methods is certainly the most practicable and the most certain;for he who seeks for methods without having a deﬁnite problem in mind seeks for the most part in vain.In dealing with mathematical problems,specialization plays,as I believe,a still more important part than generalization.Perhaps in most cases where we seek in vain the answer to a question,the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved.All depends,then,onﬁnding out these easier problems,and on solving them by means of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important leers for overcoming mathematical diﬃculties and it seems to me that it is used almost always,though perhaps unconsciously.412DA VID HILBERTOccasionally it happens that we seek the solution under insuﬃcient hypotheses or in an incorrect sense,and for this reason do not succeed.The problem then arises:to show the impossibility of the solution under the given hypotheses,or in the sense contemplated.Such proofs of impossibility were eﬀected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational.In later mathematics,the question as to the impossibility of certain solutions plays a pre¨e minent part,and we perceive in this way that old and diﬃcult problems,such as the proof of the axiom of parallels,the squaring of the circle,or the solution of equations of theﬁfth degree by radicals haveﬁnally found fully satisfactory and rigorous solutions,although in another sense than that originally intended.It is probably this important fact along with other philosophical reasons that gives rise to the conviction(which every mathematician shares,but which no one has as yet supported by a proof) that every deﬁnite mathematical problem must necessarily be susceptible of an exact settlement,either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts.Take any deﬁnite unsolved problem,such as the question as to the irrationality of the Euler-Mascheroni constant C,or the existence of an inﬁnite number of prime numbers of the form2n+1.However unapproachable these problems may seem to us and however helpless we stand before them,we have,nevertheless,theﬁrm conviction that their solution must follow by aﬁnite number of purely logical processes.Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone,or is it possibly a general law inherent in the nature of the mind,that all questions which it asks must be answerable?For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility.I instance the problem of perpetual motion.After seeking in vain for the construction of a perpetual motion machine,the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible;2and this inverted question led to the discovery of the law of the conservation of energy,which,again,explained the impossibility of perpetual motion in the sense originally intended.This conviction of the solvability of every mathematical problem is a powerful incentive to the worker.We hear within us the perpetual call:There is the problem. Seek its solution.You canﬁnd it by pure reason,for in mathematics there is no ignorabimus.The supply of problems in mathematics is inexhaustible,and as soon as one problem is solved numerous others come forth in its place.Permit me in the fol-lowing,tentatively as it were,to mention particular deﬁnite problems,drawn from various branches of mathematics,from the discussion of which an advancement of science may be expected.Let us look at the principles of analysis and geometry.The most suggestive and notable achievements of the last century in thisﬁeld are,as it seems to me,the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor,and the discovery of non-euclidean geometry by Gauss,Bolyai,MATHEMATICAL PROBLEMS413 and Lobachevsky.I thereforeﬁrst direct your attention to some problems belonging to theseﬁelds.1.Cantor’s problem of the cardinal number of the continuumTwo systems,i.e.,two assemblages of ordinary real numbers or points,are said to be(according to Cantor)equivalent or of equal cardinal number,if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one deﬁnite number of the other.The inves-tigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless,in spite of the most strenuous eﬀorts,no one has succeeded in proving.This is the theorem:Every system of inﬁnitely many real numbers,i.e.,every assemblage of numbers (or points),is either equivalent to the assemblage of natural integers,1,2,3,...or to the assemblage of all real numbers and therefore to the continuum,that is,to the points of a line;as regards equivalence there are,therefore,only two assemblages of numbers,the countable assemblage and the continuum.From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage;the proof of this theorem would,therefore,form a new bridge between the countable assemblage and the continuum.Let me mention another very remarkable statement of Cantor’s which stands in the closest connection with the theorem mentioned and which,perhaps,oﬀers the key to its proof.Any system of real numbers is said to be ordered,if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that,if a is before b and b is before c,then a always comes before c.The natural arrangement of numbers of a system is deﬁned to be that in which the smaller precedes the larger.But there are,as is easily seen,inﬁnitely many other ways in which the numbers of a system may be arranged.If we think of a deﬁnite arrangement of numbers and select from them a particular system of these numbers,a so-called partial system or assemblage,this partial system will also prove to be ordered.Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way,that not only in the assemblage itself but also in every partial assemblage there exists aﬁrst number.The system of integers1,2,3,...in their natural order is evidently a well ordered assemblage.On the other hand the system of all real numbers,i.e.,the continuum in its natural order,is evidently not well ordered.For,if we think of the points of a segment of a straight line,with its initial point excluded,as our partial assemblage,it will have noﬁrst element.The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have aﬁrst element,i.e., whether the continuum cannot be considered as a well ordered assemblage—a ques-tion which Cantor thinks must be answered in the aﬃrmative.It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor’s, perhaps by actually giving an arrangement of numbers such that in every partial system aﬁrst number can be pointed out.414DA VID HILBERT2.The compatibility of the arithmetical axiomsWhen we are engaged in investigating the foundations of a science,we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science.The axioms so set up are at the same time the deﬁnitions of those elementary ideas;and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of aﬁnite number of logical steps.Upon closer consideration the question arises:Whether,in any way,certain statements of single axioms depend upon one another,and whether the axioms may not therefore contain certain parts in common,which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms:To prove that they are not contradictory,that is,that aﬁnite number of logical steps based upon them can never lead to contradictory results.In geometry,the proof of the compatibility of the axioms can be eﬀected by constructing a suitableﬁeld of numbers,such that analogous relations between the numbers of thisﬁeld correspond to the geometrical axioms.Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of thisﬁeld of numbers.In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms.The axioms of arithmetic are essentially nothing else than the known rules of calculation,with the addition of the axiom of continuity.I recently collected them3and in so doing replaced the axiom of continuity by two simpler axioms,namely,the well-known axiom of Archimedes,and a new axiom essentially as follows:that numbers form a system of things which is capable of no further extension,as long as all the other axioms hold(axiom of completeness).I am convinced that it must be possible toﬁnd a direct proof for the compatibility of the arithmetical axioms,by means of a careful study and suitable modiﬁcation of the known methods of reasoning in the theory of irrational numbers.To show the signiﬁcance of the problem from another point of view,I add the following observation:If contradictory attributes be assigned to a concept,I say, that mathematically the concept does not exist.So,for example,a real number whose square is−1does not exist mathematically.But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of aﬁnite number of logical processes,I say that the mathematical existence of the concept(for example,of a number or a function which satisﬁes certain conditions)is thereby proved.In the case before us,where we are concerned with the axioms of real numbers in arithmetic,the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum.Indeed,when the proof for the compatibility of the axioms shall be fully accomplished,the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless.The totality of real numbers,i.e., the continuum according to the point of view just indicated,is not the totality of。

曼昆宏观经济学最新英文版参考答案第24章

Chapter 24Problems and Applicat ions1. a. Find the price of each good in each year:b. If 2006 is the base year, the market basket used to compute the CPI is 100 heads ofcauliflower, 50 bunches of broccoli, and 500 carrots. We must now calculate the cost of themarket basket in each year:2006: (100 x $2) + (50 x $1.50) + (500 x $.10) = $3252007: (100 x $3) + (50 x $1.50) + (500 x $.20) = $475Then, using 2006 as the base year, we can compute the CPI in each year:2006: $325/$325 x 100 = 1002007: $475/$325 x 100 = 146c. We can use the CPI to compute the inflation rate for 2007:(146 − 100)/100 x 100% = 46%2. Many answers are possible.3. a. The percentage change in the price of tennis balls is (2 – 2)/2 × 100% = 0%.The percentage change in the price of golf balls is (6 – 4)/4 × 100% = 50%.The percentage change in the price of Gatorade is (2 – 1)/1 × 100% = 100%.b. The cost of the market basket in 2006 is ($2 × 100) + ($4 × 100) + ($1 × 200) = $200 +$400 + $200 = $800.The cost of the market basket in 2007 is ($2 × 100) + ($6 × 100) + ($2 × 200) = $200 +$600 + $400 = $1,200.The percentage change in the cost of the market basket from 2006 to 2007 is (1,200 –800)/800 × 100% = 50%.c. This would lower my estimation of the inflation rate because the value of a bottle ofGatorade is now greater than before. The comparison should be made o n a per-ouncebasis.d. More flavors enhance consumers’ well-being. Thus, this would be considered a change inquality and would also lower my estimate of the inflation rate.4. a. Because the increase in cost was considered a quality improvement, there was no increaseregistered in the CPI.b. The argument in favor of this is that consumers are getting a better good than before, sothe price increase equals the improvement in quality. The problem is that the increasedcost might exceed the value of the improvement in air quality, so consumers are worse off.In this case, it would be better for the CPI to at least partially reflect the higher cost.5. a. introduction of new goods; b. unmeasured quality change; c. substitution bias; d. unmeasured1Chapter 24/Measuring the Cost of Living 2quality change; e. substitution bias6. a. ($0.75 − $0.15)/$0.15 x 100% = 400%.b. ($14.32 − $3.23)/$3.23 x 100% = 343%.c. In 1970: $0.15/($3.23/60) = 2.8 minutes. In 2000: $0.75/($14.32/60) = 3.1 minutes.d. Workers' purchasing power fell in terms of newspa pers.7. a. If the elderly consume the same market basket as other people, Social Security wouldprovide the elderly with an improvement in their standard of living each year because theCPI overstates inflation and Social Security payments are tied to the CPI.b. Because the elderly consume more health care than younger people do, and becausehealth care costs have risen faster than overall inflation, it is possible that the elderly areworse off. To investigate this, you would need to put together a mar ket basket for theelderly, which would have a higher weight on health care. You would then compare the risein the cost of the "elderly" basket with that of the general basket for CPI.8. When bracket creep occurred, inflation increased people's nominal incomes, pushing them intohigher tax brackets, so they had to pay a higher proportion of their incomes in taxes, even though they were not getting higher real incomes. As a result, real tax revenue rose.9. In deciding how much income to save for retirement, workers should consider the real interest rate,because they care about their purchasing power in the future, not the number of dollars they will have.10. a. When inflation is higher than was expected, the real interest rate is lower than expected.For example, suppose the market equilibrium has an expected real interest rate of 3% andpeople expect inflation to be 4%, so the nominal interest rate is 7%. If inflation turns outto be 5%, the real interest rate is 7% minus 5% equals 2%, which is less than the 3% thatwas expected.b. Because the real interest rate is lower than was expected, the lender loses and theborrower gains. The borrower is repaying the loan with dollars that are worth less than wasexpected.c. Homeowners in the 1970s who had fixed-rate mortgages from the 1960s benefited fromthe unexpected inflation, while the banks that made the mortgage loans were harmed.。

人教版吉林四平高考专题英语高考复习试卷及解析

人教版吉林四平高考专题英语高考复习1.填空题第1题.根据中文提示完成句子（1）The mother ______ her son's safety（worry）（2）His success ______ years of hard work（credit）（3）Can you ______ on that shelf ______ more books?（space）（4）Recent discoveries about corruption have ______ the company's reputation（serious damage）（5）She knew the children were up to mischief, and she found them ______ the flowers（dig）【答案】'（1）is worried about（2）is credited to（3）make space;\xa0for\xa0（4）done serious damage to（5）digging up'【解答】（1）is worried about 考查形容词短语。

句意：这位母亲担心她儿子的安全。

be worried about...意为担心……，本句的主语为The mother，故be动词为第三人称单数，故填is worried about。

（2）is credited to 考查固定用法。

句意：他的成功归因于多年的努力。

be credited to意为归功于，归因于，本句的主语His success为第三人称单数，故填is credited to。

（3）make space; for 考查动词短语。

句意：你能在那个架子上腾出些地方再放些书吗？make space for为……腾出空间，情态动词can后跟动词原形，故填make space; for。

英语四级长篇阅读匹配试题及答案

英语四级长篇阅读匹配试题及答案英语四级长篇阅读匹配试题及答案 1There are three kinds of goals: short-term,medium-range and long-term goals. Short-range goals are those that usually deal with current activities,which we can apply on a daily basis.Such goals can be achieved in a week or less,or two weeks,or possible months.It should be remembered that just as a building is no stronger than its foundation ,out long-term goals cannot amount to very munch without the achievement of solid short-term goals.Upon completing our short-term goals,we should date the occasion and then add new short-term goals that will build on those that have been completed. The intermediate goals bukld on the foundation of the short-range goals.They might deal with just one term of school or the entire school year,or they could even extend for several years.Any time you move a step at a time,you should never allow yourself to become discouraged or overwhelmed. As you complete each step,you will enforce the belief in your ability to grow adn succeed.And as your list of completion dates grow,your motivation and desire will increase.Long-range goals may be related to our dreams of the future. They might cover five years or more. Life is not a static thing.We should never allow a long-term goal to limit us or our course of action. 1.Our long-term goals mean a lot______.A.if we complete our short-range goalsB.if we cannot reach solid short-term goalsC.if we write down the datesD.if we put forward some plans2.New short-term goals are bulid upon______.A.two yearsB.long-term goalsC.current activitiesD.the goals that have been completed3.When we complete each step of our goals ,______.A.we will win final successB.we are overwhelmedC.we should build up confidence of successD.we should strong desire for setting new goals 4.Once our goals are drawn up,_______.A.we should stick to them until we complete themB.we may change our goals as we have new ideas and opportunitiesC.we had better wait for the exciting news of successD.we have made great decision5.It is implied but not stated in the passage that ______.A.those who habe long-term goals will succeedB.writing down the dates may discourage youC.the goal is only a guide for us to reach our desinationD.every should have a goal答案：adcbc英语四级长篇阅读匹配试题及答案 2If the population of the earth goes on increasing at its present rate, there will eventually not be enough resources left to sustain life on the planet.By the middle of the 21st century,if present trends continue, we will have used up all the oil that drives our cars,for example.Even if scientists develop new ways of feeding the human race,the crowded conditions on earth will make it necessary for lus to look for open space somewhere else. But none of the other planets in our solar system are capable of supporting life at present. One possible solution to the problem, however,has recently been suggested by American scientist, Professor Carl Sagan. Sagan believes that before the earths resources are compleetely exhausted it will be possible to change the atmophere of Venus and so create a new world almost as large as earth itself. The difficult is that Venus is much hotter than the earth and there is only a tiny amount of water there. Sagan proposes that algae organisms that can live in extremely hot or cold atmospheres and at the same time produce oxygen,should be bred in condition similar to those on Venus.As soon as this has been done, the algae will be placed in small rockets. Spaceship will then fly to Venus and fire the rockets into the atmosphere .In a fairly short time, the alge will break down the carbon dioxide into oxygen andcarbon. When the algae have done theri work, the atmosphere will become cooler,but befor man can set foot on Venus it will be neccessary for the oxygen to produce rain. The surface of the planet will still be too hot for man to land on it but the rain will eventually fall and in a few years something like earth will be reproduced on Venus. -1.Inte long run, the most insoluble problem caused by population growth on earth will probably be the lack of ______.a.foodb.oilc.spaced.resources2.Carl Sagan believes that Venus might be colonized from earth because _____ a.it might be possible to change its atmosphere b.its atmosphere is the same as the earthsc.there is a good supply of water on Venusd.the days on Venus are long enough3.On Venus there is a lot of ________.a.waterb.carbon dioxidec.carbon monoxided.oxygen4.Algae are plants that can____.a.live in very hot temperaturesb.live in very cold temperaturesc.manufacture oxygend.all of the above5. Man can land on Venus only when_______. a.the algae have done their work -b.the atmosphere becomes coolerc.thereis oxygend.it rains there答案：cabdd英语四级长篇阅读匹配试题及答案 3Like a needle climbing up a bathroom scale, the number keeps rising. In 1991, 15% of Americans were obese(肥胖的); by 1999, that proportion had grown to 27%. Youngsters, who should have age and activity on their side, are growing larger as well: 19% of Americans under 17 are obese. Waistbands have been popping in other western countries too, as physical activity has declined and diets have expanded. By and large, people in the rich world seem to have lost the fight against flab(松弛).Meanwhile, poorer nations have enjoyed some success in their battles against malnutrition and famine. But, according to research presented at the annual meeting of the American Association for the Advancement of Science, it is more a case of being out of the frying pan and into the fire. The most striking example actually in the poor world comes from the Pacific islands, home of the world’s most obese communities. In 1966, 14% of the men on this island were obese while 100% of men under the age of 30 in 1996 were obese.This increase in weight has been uneven as well as fast. As a result, undernourished and over-nourished people frequently live cheek by jowl(面颊). The mix can even occur within a single household. A study of families in Indonesia found that nearly 10% contained both the hungry and the fat. This is a mysterious phenomenon, but might have something to do with people of different ages being given different amounts of food to eat.The prospect of heading off these problems is bleak. In many affected countries there are cultural factorsto contend with, such as an emphasis on eating large meals together, or on food as a form. ofhospitality.Moreover, there is a good measure of disbelief on the part of policymakers that such a problem Could existin their countries. Add to that reluctance on the part of governments to spend resources on promoting dietand exercise while starvation is still a real threat, and the result is a recipe for inaction. Unless something is done soon, it might not be possible to turn the clock back.英语四级阅读模拟试题：Choose correct answers to the question:1.The first sentence of the passage most probably implies that ______.A.many Americans are obsessed with the rising temperature in their bathroomB.more people are overweighed in the United StatesC.people are doing more physical exercises with the help of scalesD.youngsters become taller and healthier thanks to more activities2.As physical exercise declines and diet expands, ______.A.other western countries has been defeated by fatB.obesity has become an epidemic(流行病)of the rich worldC.waistbands begin to be popular in other western countriesD.western countries can no longer fight against obesity3.Which is NOT the point of the example of the Pacific Islands?A.The poor community has shaken off poverty and people are well-fed now.B.Obesity is becoming a problem in the developing world too.C.Excessive weight increase will cause no less harm than the food shortage.D.The problem of overweight emerges very fast.4.Of tackling obesity in the poor world, we can learn from the passage that____A.the matter is so complex as to go beyond our capacityB.no matter what we do, the prospect will always be bleakC.it is starvation, the real threat, that needs to be solvedD.we should take immediate actions before it becomes incurable5.What is the main idea of this passage?A.Obesity is now a global problem that needs tackling.B.The weights increase fast throughout the whole world.C.Obesity and starvation are two main problems in the poor world.D.Obesity has shifted from the rich world to the poor world.英语四级阅读参考答案1.[B] 推理判断题。

Bi2Se3未考虑vdw的错误汇总

在没有考虑vdw作用之前，算Bi2Se3材料soc中出现的错误汇总V ASP自旋轨道耦合计算错误汇总静态计算时，报错：VERY BAD NEWS! Internal内部error in subroutine子程序IBZKPT:Reciprocal倒数的lattice and k-lattice belong to different class of lattices. Often results are still useful (48)INCAR参数设置：对策：根据所用集群，修改INCAR中NPAR。

将NPAR=4变成NPAR=1，已解决！错误：sub space matrix类错误报错：静态和能带计算中出现警告：W ARNING: Sub-Space-Matrix is not hermitian共轭in DA V结构优化出现错误：WARNING: Sub-Space-Matrix is not hermitian in DA V 4 -4.681828688433112E-002对策：通过将默认AMIX=0.4，修改成AMIX=0.2（或0.3），问题得以解决。

以下是类似的错误：WARNING: Sub-Space-Matrix is not hermitian in rmm -3.00000000000000RMM: 22 -0.167633596124E+02 -0.57393E+00 -0.44312E-01 1326 0.221E+00BRMIX:very serious problems the old and the new charge density differ old charge density: 28.00003 new 28.06093 0.111E+00错误：WARNING: Sub-Space-Matrix is not hermitian in rmm -42.5000000000000ERROR FEXCP: supplied Exchange-correletion table is too small, maximal index : 4794错误：结构优化Bi2Te3时，log文件：WARNING in EDDIAG: sub space matrix is not hermitian 1 -0.199E+01RMM: 200 0.179366581305E+01 -0.10588E-01 -0.14220E+00 718 0.261E-01BRMIX: very serious problems the old and the new charge density differ old charge density: 56.00230 new 124.70394 66 F= 0.17936658E+01 E0= 0.18295246E+01 d E =0.557217E-02curvature: 0.00 expect dE= 0.000E+00 dE for cont linesearch 0.000E+00ZBRENT: fatal error in bracketingplease rerun with smaller EDIFF, or copy CONTCAR to POSCAR and continue但是，将CONTCAR拷贝成POSCAR，接着算静态没有报错，这样算出来的结果有问题吗？对策1：用这个CONTCAR拷贝成POSCAR重新做一次结构优化，看是否达到优化精度！对策2：用这个CONTCAR拷贝成POSCAR，并且修改EDIFF（目前参数EDIFF=1E-6）,默认为10-4错误：WARNING: Sub-Space-Matrix is not hermitian in DA V 1 -7.626640664998020E-003网上参考解决方案：对策1：减小POTIM: IBRION=0，标准分子动力学模拟。

哈佛公开课-公正课中英字幕第三节

制作人：心舟 QQ：1129441083 欢迎交流公正课\N迈克尔·桑德尔教授主讲第三讲《给生命标价》上节课我们讨论了\Last time, we argued about女王诉达德利和斯蒂芬斯案\the case of Queen versus Dudley and Stevens,即救生艇的案例\the lifeboat case,海上食人惨案\the case of cannibalism at sea.带着对救生艇上发生事件的讨论\And with the arguments about the lifeboat in mind,即对达德利和斯蒂芬斯行为赞同与否的讨论\the arguments for and against what Dudley and Stephens did in mind,让我们再回归\let's turn back to the philosophy,杰里米·边沁的功利主义哲学\the utilitarian philosophy of Jeremy Bentham.边沁 1748年生于英格兰\Bentham was born in England in 1748.12岁进入牛津大学\At the age of 12, he went to Oxford.15岁入读法学院\At 15, he went to law school.19岁取得律师资格\He was admitted to the Bar at age 19但从没当过律师\but he never practiced law.而是将毕生精力献给了法学和道德哲学\Instead, he devoted his life to jurisprudence and moral philosophy.上节课我们开始\Last time, we began to思考边沁的功利主义\consider Bentham's version of utilitarianism.他的主要观点简单明确就是\The main idea is simply stated and it's this:道德的最高准则\The highest principle of morality,无论是个人道德还是政治道德\whether personal or political morality,都是最大化公共福利或曰集体幸福感\is to maximize the general welfare, or the collective happiness,或者说权衡苦乐将幸福最大化\or the overall balance of pleasure over pain;一句话功利最大化\in a phrase, maximize utility.边沁是这样论证这一原则的\Bentham arrives at this principle by the following line of reasoning: 我们都受到痛苦和快乐的支配\We're all governed by pain and pleasure,苦乐是我们至高无上的主宰\they are our sovereign masters,因此任何道德体系都应考虑到它们\and so any moral system has to take account of them.最好怎样考虑呢通过最大化\How best to take account? By maximizing.从而引出"为最多的人谋求最大的幸福"这一原则\And this leads to the principle of the greatest good for the greatest number.我们到底该最大化什么呢\What exactly should we maximize?边沁说应最大化幸福\Bentham tells us happiness,或更精确来说最大化功利\or more precisely, utility功利最大化原则不只针对个人\maximizing utility as a principle not only for individuals也适用于共同体及立法者\but also for communities and for legislators.边沁问到底什么是共同体\"What, after all, is a community?" Bentham asks.共同体是其成员的集合\It's the sum of the individuals who comprise it.所以在制定最优政策时\And that's why in deciding the best policy,制定法律时决定何谓公正时\in deciding what the law should be, in deciding what's just,公民和立法者应扪心自问这个问题\citizens and legislators should ask themselves the question 当用政策带来的总效益\if we add up all of the benefits of this policy减去总成本\and subtract all of the costs,正确的选择应该是\the right thing to do is the one减去苦难后幸福最大化的那一个\that maximizes the balance of happiness over suffering.这就是所谓功利最大化\That's what it means to maximize utility.今天我想听听\Now, today, I want to see你们是否赞同这点\whether you agree or disagree with it,功利主义的逻辑\and it often goes, this utilitarian logic,通常被称作成本效益分析\under the name of cost-benefit analysis,一再被企业和政府运用\which is used by companies and by governments all the time.其做法包括作出估价\And what it involves is placing a value,通常是估计出金额来代表功利\usually a dollar value, to stand for utility即分别列出成本和各项收益的金额\on the costs and the benefits of various proposals.最近捷克共和国\Recently, in the Czech Republic,有一个增加香烟消费税的提案\there was a proposal to increase the excise tax on smoking.烟草公司菲利普·莫里斯公司\Philip Morris, the tobacco company,在捷克共和国的生意做得很大\does huge business in the Czech Republic.他们资助了一项研究\They commissioned a study,进行在捷克吸烟的成本效益分析\a cost-benefit analysis of smoking in the Czech Republic,分析结果显示\and what their cost-benefit analysis found was让捷克人民吸烟能让政府获利\the government gains by having Czech citizens smoke.那政府如何获利呢\Now, how do they gain?它确实会对捷克政府的\It's true that there are negative effects公共财政产生负效应\to the public finance of the Czech government因为吸烟造成的疾病\because there are increased health care costs会增加医疗支出\for people who develop smoking-related diseases.但另一方面也有正效应\On the other hand, there were positive effects它们被记在账目的另一侧\and those were added up on the other side of the ledger.正效应主要来自\The positive effects included, for the most part,销售香烟为政府带来的\various tax revenues that the government derives各项税收\from the sale of cigarette products,但还包括\but it also included人们早逝为政府节省的医疗支出\health care savings to the government when people die early,免去的养老金\pension savings...政府不需要继续支付养老金\you don't have to pay pensions for as long还省去了老年人的住房开支\and also, savings in housing costs for the elderly.当把总成本和各项收益分别加总\And when all of the costs and benefits were added up,菲利普·莫里斯公司的研究表明\the Philip Morris study found that捷克共和国公共财政将获得\there is a net public finance gain一亿四千七百万的净收益\in the Czech Republic of $147,000,000,算上住房医疗养老金方面节省的开支\and given the savings in housing,in health care, and pension costs,政府从每个因吸烟早逝的人身上\the government enjoys savings of over $1,200 for each person赚得超过1200美元\who dies prematurely due to smoking.成本效益分析\Cost-benefit analysis.在座功利主义的支持者们\Now, those among you who are defenders of utilitarianism可能觉得这个研究不公\may think that this is an unfair test.菲利普·莫里斯公司遭到媒体谴责\Philip Morris was pilloried in the press他们为这项冷血的计算公开道歉\and they issued an apology for this heartless calculation.你也许会说\You may say这里无疑忽略了\that what's missing here is something功利主义者认为应当包含的部分\that the utilitarian can easily incorporate,即那些死于肺癌的患者本身\namely the value to the person加上其家属的价值\and to the families of those who die from lung cancer.怎么能忽略生命的价值呢\What about the value of life?有些成本效益分析\Some cost-benefit analyses确实计算了生命的价值\incorporate a measure for the value of life.其中最著名的是福特平托的案例\One of the most famous of these involved the Ford Pinto case. 有人读过吗\Did any of you read about that?当时是二十世纪七十年代\This was back in the 1970s.还有人知道福特平托是什么车吗\Do you remember what the Ford Pinto was, a kind of car? Anybody? 它是一种小型次紧凑型车风靡一时\It was a small car, subcompact car, very popular,但它有一个缺陷\but it had one problem,油箱装在车的尾部\which is the fuel tank was at the back of the car发生追尾时油箱就会爆炸\and in rear collisions, the fuel tank exploded造成了严重伤亡\and some people were killed and some severely injured.受害者一纸诉状将福特告上了法庭\Victims of these injuries took Ford to court to sue.案件审理中发现\And in the court case, it turned out福特早就知道油箱的缺陷\that Ford had long since known about the vulnerable fuel tank还进行了成本效益分析\and had done a cost-benefit analysis来决定是否值得装上一面特殊的隔板\to determine whether it would be worth it to put in a special shield以保护油箱防止油箱爆炸\that would protect the fuel tank and prevent it from exploding.该分析指出\They did a cost-benefit analysis.能增加平托安全性的隔板\The cost per part to increase the safety of the Pinto,每块成本是11美元\they calculated at $11.00 per part.这就是审判时发现的成本效益分析\And here's... this was the cost-benefit analysis that emerged in the trial.给1250万辆轿车和卡车配上11美元的隔板\Eleven dollars per part at 12.5 million cars and trucks提高安全性共需花费一亿三千七百万美元\came to a total cost of$137 million to improve the safety. 但接着又算出\But then they calculated花这些钱提高安全性能带来的收益\the benefits of spending all this money on a safer car预计可减少180例死亡\and they counted 180 deaths因车祸死亡预计每条人命20万美元\and they assigned a dollar value, $200,000 per death,可减少180例伤残每例67000美元\180 injuries, $67,000,加上车辆维修费用\and then the costs to repair,无此安全装置车会完全损毁\the replacement cost for 2,000 vehicles,所以需算上2000辆汽车的重置成本每辆700美元\it would be destroyed without the safety device $700 per vehicle.收益最后只有4950万\So the benefits turned out to be only $49.5 million因此他们没有安装该装置\and so they didn't install the device.不用说\Needless to say,当福特汽车公司的这份成本效益分析备忘录\when this memo of the Ford Motor Company's cost-benefit analysis在审理时被公之于众\came out in the trial,陪审团大为震怒判定巨额赔偿\it appalled the jurors, who awarded a huge settlement.这算是功利主义计算思路的反例吗\Is this a counterexample to the utilitarian idea of calculating? 因为福特计算了生命的价值\Because Ford included a measure of the value of life.现在就这个明显的反例\Now, who here wants to defend cost-benefit analysis有谁想为成本效益分析辩护\from this apparent counter example?有谁辩护\Who has a defense?还是你们认为它完全推翻了\Or do you think this completely destroys功利主义的演算\the whole utilitarian calculus?请说\Yes?我觉得他们犯了与前面案例\Well, I think that once again, they've made the same mistake相同的错误\the previous case did,量化了生命的价值\that they assigned a dollar value to human life,但同样的\and once again,他们没有考虑受害者家人承受的\they failed to take account things like suffering痛苦和精神损失\and emotional losses by the families.他们不但家庭收入受损还丧失了亲人\I mean, families lost earnings but they also lost a loved one 那损失远不止20万美元\and that is more valued than $200,000.没错等等说得好你叫什么名字\Right and... wait, wait, wait, that's good. What's your name? 朱莉·罗托\Julie Roteau .朱莉要是20万美金不够\So if $200,000, Julie, is too low a figure因为没有算丧失亲人\because it doesn't include the loss of a loved one和生命的损失\and the loss of those years of life,那你认为什么数目更合适\what would be what do you think would be a more accurate number?我无法给出数目\I don't believe I could give a number.我觉得这种分析\I think that this sort of analysis不应该用在人的生命这个问题上\shouldn't be applied to issues of human life.人命不能用金钱衡量\I think it can't be used monetarily.所以朱莉认为他们不是定价太低\So they didn't just put too low a number, Julie says.他们压根就不该定价\They were wrong to try to put any number at all.那好让我们听听别人...\All right, let's hear someone who...你必须考虑通胀\You have to adjust for inflation.你必须考虑通胀\You have to adjust for inflation.行啊有道理\All right, fair enough.那如今应该是多少\So what would the number be now?那是35年前\This was 35 years ago.两百万美元\Two million dollars.两百万美元你会定价两百万吗\Two million dollars? You would put two million?你叫什么名字\And what's your name?佛伊泰克\Voytek佛伊泰克说我们必须考虑通胀\Voytek says we have to allow for inflation.应该更慷慨些\We should be more generous.这样你就满意了吗\Then would you be satisfied that这样思考这个问题就可以了吗\this is the right way of thinking about the question?我觉得不幸的是...\I guess, unfortunately, it is for...有时确实需要标价\there needs to be a number put somewhere,不过我不确定具体数字\like, I'm not sure what that number would be,但我确实认同\but I do agree that人的生命也许可以被标价\there could possibly be a number put on the human life.很好所以佛伊泰克不同意朱莉的看法\All right, so Voytek says, and here, he disagrees with Julie. 朱莉认为我们不该为了成本效益分析\Julie says we can't put a number on human life给人的生命标价\for the purpose of a cost-benefit analysis.佛伊泰克认为我们别无选择\Voytek says we have to因为不管怎样我们必须做出决定\because we have to make decisions somehow.别的人怎么看\What do other people think about this?有没人来赞同成本效益分析的\Is there anyone prepared to defend cost-benefit analysis here认为它精确合宜吗你说\as accurate as desirable? Yes? Go ahead.我觉得要是福特和其他汽车公司\I think that if Ford and other car companies不使用成本效益分析的话\didn't use cost-benefit analysis,他们最后就会倒闭\they'd eventually go out of business因为他们无法盈利\because they wouldn't be able to be profitable这样就会有数百万人无法开车上班\and millions of people wouldn't be able to use their cars to get to jobs,没法赚钱养不起小孩\to put food on the table, to feed their children.所以我认为此种情况下如果不用成本效益分析\So I think that if cost-benefit analysis isn't employed,会牺牲更多人的利益\the greater good is sacrificed, in this case.很好我加一句你叫什么名字\All right, let me add. What's your name?劳尔\Raul.劳尔最近有一项\Raul, there was recently a study done关于司机开车时使用手机的研究\about cell phone use by a driver when people are driving a car, 关于是否应该禁止此行为有一场争论\and there was a debate whether that should be banned.数据显示每年有2000人左右\And the figure was that some 2,000 people因开车时使用手机而死于车祸\die as a result of accidents each year using cell phones.而目前哈佛风险分析中心\And yet, the cost-benefit analysis which was done作出的成本效益分析表明\by the center for Risk Analysis at Harvard found that如果考虑使用手机带来的效益\if you look at the benefits of the cell phone use并与生命的价值做比较\and you put some value on the life,就会得出同样的结论\it comes out about the same因为这样做经济效益巨大\because of the enormous economic benefit of可以使人们更有效地利用时间\enabling people to take advantage of their time,不浪费时间边开车边谈生意\not waste time, be able to make deals边和朋友聊天等\and talk to friends and so on while they're driving.这不就表明\Doesn't that suggest that用金钱衡量人的生命是个错误吗\it's a mistake to try to put monetary figures on questions of human life?我觉得如果绝大多数人想要\Well, I think that if the great majority of people try to从某项服务中获得最大功利\derive maximum utility out of a service,比如使用手机享受手机所带来的便利\like using cell phones and the convenience that cell phones provide,那么为了满足需求这种牺牲就是必要的\that sacrifice is necessary for satisfaction to occur. 你是个彻底的功利主义者嘛\You're an outright utilitarian.是的可以这么说\Yes. Okay.好那么最后一个问题劳尔\All right then, one last question, Raul.我也问过佛伊泰克\And I put this to Voytek,在决定是否禁止使用手机这件事时\what dollar figure should be put on human life人命应该如何定价\to decide whether to ban the use of cell phones?我不想武断地算出一个数字\Well, I don't want to arbitrarily calculate a figure,我是指马上就算出我觉得...\I mean, right now. I think that...你想要深思熟虑之后再决定\You want to take it under advisement?对我会深思熟虑\Yeah, I'll take it under advisement.但大概有多少\But what, roughly speaking, would it be?会死2300人\You got 2,300 deaths.你必须用金钱来衡量\You got to assign a dollar value to know是否需要禁止司机使用手机\whether you want to prevent those deaths by来避免此类事件发生\banning the use of cell phones in cars.那你感觉是多少钱一百万\So what would your hunch be? How much? A million?两百万佛伊泰克觉得是两百万\Two million? Two million was Voytek's figure.-这么多可以吗 -也许一百万吧\- Is that about right? - Maybe a million.-一百万 -对\- A million? - Yeah.很好谢谢\You know, that's good. Thank you.以上即为近来对成本效益分析\So, these are some of the controversies that arise these days引发的一些争论\from cost-benefit analysis,尤其是其中那些\especially those that involve认为可以用金钱衡量一切的观点\placing a dollar value on everything to be added up.现在我想听听反对意见\Well, now I want to turn to your objections, to your objections不一定仅仅针对成本效益分析\not necessarily to cost-benefit analysis specifically,因为那只是功利主义逻辑现今的实践之一\because that's just one version of the utilitarian logic in practice today,而是针对整个功利主义理论\but to the theory as a whole,针对那些认为正确之举\to the idea that the right thing to do,就是以功利最大化作为政策法律基础的观点\the just basis for policy and law is to maximize utility. 有多少人不同意\How many disagree功利主义在法律及公共利益方面的做法\with the utilitarian approach to law and to the common good? 有多少人同意\How many agree with it?看来多数表示同意\So more agree than disagree.我们来听听批判声吧请说\So let's hear from the critics. Yes?我对此的异议是\My main issue with it is我觉得不能因为一些人占少数\that I feel like you can't say that just because someone's in the minority,就断定他们的需要和欲望不如多数人的重要\what they want and need is less valuable than someone who's in the majority所以我反对\So I guess I have an issue with the idea"为最多的人谋求最大的幸福"这一观点\that the greatest good for the greatest number is okay因为还有...\because there are still...占少数的人怎么办呢\what about people who are in the lesser number?这对他们不公平\Like, it's not fair to them.他们对此没有发言权\They didn't have any say in where they wanted to be.很好这是个有趣的异议\All right. That's an interesting objection.你担心其对少数人的影响\You're worried about the effect on the minority.是的\Yes.顺便问一句你叫什么名字\What's your name, by the way?安娜\Anna.谁能回答\Who has an answer to安娜对于少数人影响的担心\Anna's worry about the effect on the minority?你怎么回答安娜\What do you say to Anna?她说少数人的价值被低估了\Um, she said that the minority is valued less.我认为事实并非如此因为\I don't think that's the case because少数人当中每个个体的价值\individually, the minority's value is just和多数人的个体价值是一样的\the same as the individual of the majority.只不过多数在数量上胜过少数\It's just that the numbers outweigh the minority.有时你必须做出选择\And I mean, at a certain point, you have to make a decision我对少数表示遗憾\and I'm sorry for the minority但有时这是牺牲小我成全大我\but sometimes, it's for the general, for the greater good.成全大我安娜你怎么看\For the greater good. Anna, what do you say?你叫什么名字\What's your name?杨达\Yang-Da.你怎么反驳杨达\What do you say to Yang-Da?杨达说必须总体考虑人们的选择\Yang-Da says you just have to add up people's preferences而其中少数人的选择其实也被衡量过了\and those in the minority do have their preferences weighed. 你能举个你所担心的类似例子吗\Can you give an example of the kind of thing you're worried about 即你所说的担心\when you say you're worried about功利主义缺少对少数的关心和尊重\utilitarianism violating the concern or respect due the minority?举个例子\give an example.我就举一个我们讨论过的案例\Okay. So, well, with any of the cases that we've talked about,比如海上食人惨案中我认为被吃的男孩\like for the shipwreck one, I think the boy who was eaten 仍然与其他人享有相等的生存权\still had as much of a right to live as the other people仅仅因为他是少数\and just because he was the minority in that case,他存活的机率可能最小\the one who maybe had less of a chance to keep living,并不意味着其他人就自然而然有权利吃他\that doesn't mean that the others automatically have a right to eat him就为了让多数人有存活的机会\just because it would give a greater amount of people a chance to live.所以可能少数人\So there may be certain rights或个体的某些权利\that the minority members have that the individual has不该为了功利最大化而被牺牲\that shouldn't be traded off for the sake of utility?是的\Yes.是吗安娜下面这个例子我来考考扬达\Yes, Anna? You know, this would be a test for you.在古罗马\Back in Ancient Rome,基督徒被扔去斗兽场与狮子搏斗\they threw Christians to the lions in the Colosseum for sport. 如果以功利主义方式演算\If you think how the utilitarian calculus would go,没错丢给狮子的基督徒\yes, the Christian thrown to the lions确实经历了撕心裂肺的剧痛\suffers enormous excruciating pain.但看看罗马人共同的心醉神迷啊\But look at the collective ecstasy of the Romans!杨达\Yang-Da.在那个时代我不... 要是如今\Well, in that time, I don't...if in modern day of time,衡量观众获得的快乐\to give a number to the happiness given to the people watching,我觉得没有任何政策制定者会认为\I don't think any policymaker would say一个人的痛苦煎熬会比\the pain of one person, of the suffering of one person is much, much... 众人因之获得的快感更...\is, I mean, in comparison to the happiness gained, it's不但你必须承认\No, but you have to admit that要是有足够多的罗马人对这种快感足够狂热\if there were enough Romans delirious enough with happiness,那就会胜过\it would outweigh even the少数几个被丢给狮子的基督徒承受的极端剧痛\most excruciating pain of a handful of Christians thrown to the lion.因此我们确实对功利主义有两点异议\So we really have here two different objections to utilitarianism.一点是关于功利主义\One has to do with whether utilitarianism是否充分尊重个体和少数的权利\adequately respects individual rights or minority rights,另一点是关于\and the other has to do with加总功利或偏好或价值的看法\the whole idea of aggregating utility or preferences or values. 所有的价值都有可能用金钱衡量吗\Is it possible to aggregate all values to translate them into dollar terms?二十世纪三十年代\There was, in the 1930s,有位心理学家试图解决第二个问题\a psychologist who tried to address this second question.他试图证明功利主义者的假设\He tried to prove what utilitarianism assumes,所有的利益价值人类的心声\that it is possible to translate all goods, all values,都可能被统一衡量\into a single uniform measure,并通过对年轻的救济金领取者的调查来证明此点\and he did this by conducting a survey of young recipients of relief,当时是二十世纪三十年代\this was in the 1930s, and he asked them,他给了他们一张不愉快经历的清单问他们\he gave them a list of unpleasant experiences and he asked them,给你多少钱你就愿意忍受以下经历\"How much would you have to be paid to undergo the following experiences?"并作了记录\and he kept track.比如给你多少钱\For example, how much would you have to be paid你才愿意拔掉自己的一颗门牙\to have one upper front tooth pulled out?抑或给你多少钱\Or how much would you have to be paid你才愿意砍掉一根小脚趾\to have one little toe cut off?抑或吃一条六英寸长的蚯蚓\Or to eat a live earthworm six inches long?抑或后半生居住在堪萨斯农场\Or to live the rest of your life on a farm in Kansas?{\an8}{\fn方正黑体简体\fs18\b1\bord1\shad1\3c&H2F2F2F&}堪萨斯位于美国西部平原\N1930年代遭受重大自然灾害抑或亲手掐死一只流浪猫\Or to choke a stray cat to death with your bare hands?你们觉得清单里的哪一项最贵\Now, what do you suppose was the most expensive item on that list? 堪萨斯\Kansas?没错是堪萨斯\You're right, it was Kansas.他们认为余生都住堪萨斯农场\For Kansas, people said they'd have to pay them至少得给他们30万美元\they have to be paid $300,000.你们觉得第二贵的是什么\What do you think was the next most expensive?不是猫\Not the cat.也不是门牙\Not the tooth.也不是脚趾\Not the toe.是蚯蚓\The worm!他们说给10万美元才肯吃蚯蚓\People said you'd have to pay them $100,000 to eat the worm.你们觉得最便宜的是哪项\What do you think was the least expensive item?不是猫\Not the cat.是门牙\The tooth.大萧条时期\During the Depression,人们愿意为了区区4500美元拔掉自己的牙\people were willing to have their tooth pulled for only $4,500.什么\What?桑代克得出的结论是\Now, here's what Thorndike concluded from his study.任何需求或满足都能有个价钱\Any want or a satisfaction which exists exists in some amount 因此能用金钱衡量\and is therefore measurable.狗猫小鸡的生命\The life of a dog or a cat or a chicken都充斥着各类嗜好渴望欲望以及满足感\consists of appetites, cravings, desires, and their gratifications.人亦如此\So does the life of human beings,只是人的嗜好和欲望更加复杂罢了\though the appetites and desires are more complicated.但桑代克的研究说明了什么呢\But what about Thorndike's study?它是不是支持了边沁的观点\Does it support Bentham's idea认为所有利益所有价值都可以\that all goods, all values can be captured用统一的方式衡量\according to a single uniform measure of value?抑或清单上那些荒谬的项目\Or does the preposterous character of those different items on the list恰恰揭示了相反的结论\suggest the opposite conclusion也许\that maybe,不论是生命堪萨斯还是蚯蚓\whether we're talking about life or Kansas or the worm,还是我们重视珍爱的东西\maybe the things we value and cherish都是不能用统一方式衡量的?\can't be captured according to a single uniform measure of value? 如果不能\And if they can't,那么功利主义道德理论意义何在\what are the consequences for the utilitarian theory of morality? 我们下次将会继续探讨这一问题\That's a question we'll continue with next time.{\an8}{\fn方正黑体简体\fs18\b1\bord1\shad1\3c&H2F2F2F&}公正课下讲预告好现在我们再投个票\All right, now, let's take the other part of the poll,哪个是最高级的体验或快乐\which is the highest experience or pleasure.{\an8}{\fn方正黑体简体\fs18\b1\bord1\shad1\3c&H2F2F2F&}第四讲《如何衡量快乐》多少人认为是莎士比亚\How many say Shakespeare?多少人认为是《挑战恐惧极限》\How many say Fear Factor?你开玩笑的吧是吧\No, you can't be serious. Really?上节课我们开始思考一些\Last time, we began to consider some objections to对杰里米·边沁功利主义的反对观点\Jeremy Bentham's version of utilitarianism.讨论中提出了两点异议\People raised two objections in the discussion we had.第一点异议是说功利主义\The first was the objection, the claim that utilitarianism,只关注"为最多的人谋求最大的幸福"\by concerning itself with the greatest good for the greatest number,没有充分地尊重个人权利\fails adequately to respect individual rights.今天我们要讨论严刑拷打和恐怖主义\Today, we have debates about torture and terrorism.假设一名恐怖主义嫌犯在9丒11慜堦揤旐曔\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}Suppose a suspected terrorist was apprehended on September 10th你桳棟桼憡怣\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}and you had reason to believe这柤寵斊彾埇椆\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}that the suspect had crucial information彨导抳3000恖嬾难揑嫲晐袭击揑廳梫忣报\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}about an impending terrorist attack that would kill over 3,000 people你撬晄开懠揑岥\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}and you couldn't extract the information.为椆漒摓忣报帶对懠严孻崏懪惀斲崌棟\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}Would it be just to torture the suspect to get the information梷埥你晄赞摨\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}or do you say no,你认为桳懜廳槩恖权棙揑绝对摴责擟\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}there is a categorical moral duty of respect for individual rights?朸种掱搙忋変们枖夞摓椆嵟弶揑问题\N{\fn曽惓综艺简懱}{\fs14}{\b0}{\c&HFFFFFF&}{\3c&H2F2F2F&}{\4c&H000000&}In a way, we're back to thequestions we started with。

研究生英语B_课后答案

新世纪研究生公共英语教材阅读B 课后答案UNIT ONE Party PoliticsP8 I Comprehension Check1-5 DCDAB 6-10 DDCABP10 Vocabulary StudyI 1-5 CBADB 6-10 CDBCDII 1.etiquette 2.looped 3.unaccountable 4.told off 5. conspicuously 6. pesky 7.let loose 8.racy 9.murky 10.ticklishP11III TranslationTo invite eminent persons to help make advertisements should be regarded as one of the best advertising strategies and could, of course, produce a spectacular(powerful) VIP effect, privided that those celebrities are perfectly willing to accept the invitation and, more importantly, the products to be advertised are genuine and of fair prices. Sometimes, while a commodity is of inferior quality, the advertisement is full of words lavishing praise on it, if a celebrity shows up as an image agent for such a product, the advertisement could, if any, be temporarily successful before it turns the brand of the product in question notorious and, more disastrously, ruins the reputation of the eminent person thereafter. So, the famous are well advised to think more than twice before they agree to appear on the commercial.P13 Key to Supplementary ReadingsA.1-5 FFFTT 6-10 FTFTTB.1-5 FTFTF 6-10 FTFTFUNIT TWO The New SinglesP29 I Comprehension Check1-5 BDBDC 6-10 ACCADP31 Vocabulary StudyI 1.neo-realist 2.neo-Nazis 3.Neo-fascist 4.neocolonialism 5. neologisms 6.neo-Darwinist7.neoclassical 8. neonatesII 1.fostering 2.reaved 3.holy grail 4.mainstay 5.twenty-somethings 6.heterosexuals 7.mandatory 8.embracing 9.meditating 10.fusionP32III TranslationNowadays in the city’s tonier residential districts there are peple named as singles, who are usually young, rich and tech-savvy professionals and choose independently their own lifesyles. The number of singles has increased dramatically over the recent years. The reasons of remaining single are various:some may be busy exploring careers without putting their marriage into the agenda, some may indulge in their jobs, travel, entertainment, physical fitness or friendship, More than 80% of them have not abandoned the value of marriage, and they say they aspire to marry or they want to be married someday, but they are patient and feel content being single until they meet the right person.Key to Supplementary Readings(略)UNIT THREE Doctor’s Dilemma:Treat or Let Die?P51 Comprehension Check1-5 BCCBD 6-10 DCDADP53 V ocabulary StudyI 1.outstrip 2.limbo 3.ceased 4. in the wake of 5. paramount 6.ethical 7.prolonged 8. thorny9.congenital 10.subsequentlyII 1.euthanasia 2.salvaged 3.deformity 4.defects 5. handicaps 6.lingering 7. grapple 8. allegedly 9.acquitted 10.frontiersIII TranslationPeople who are energetic, happy, and relaxed are less likely to catch a cold than those who are depressed, nervous, or angry. When the brain is “happy”, it sends messages to our organs that help keep the body healthy and sound. Your chance of developing the common cold, pneumonia, or even cancer may very well be decreased by keeping your brain in a healthy state. In addition, happy and relaxed people are prone to better health practices than their negative and stressed counterparts. They are more likely to get plenty of sleep and to engaged in regular exercise, and have been shown to have lower levels of certain stress hormones.P59 Key to Supplementary ReadingsA.1-5 FTFTF 6-10 TTTFT B 1-5 FFTTF 6-10 TFTFFUNIT FOUR The Cultural Patterning of SpaceP71 Comprehension Check1-5 BABCC 6-9 DDDBP73 V ocabulary StudyI 1-5 begja 6-10 hcifdII 1.anthropologists 2. Patterns 3.tangible 4. persistent 5. infringe 6. integrate 7. secular 8. spatial 9.florist’s 10.ArchitectureIII TranslationAs one travels abroad and examines the ways in which space is handled, startling variations are discovered; differrences which we react to vigorously. Since none of us are taught to look at space as isolated from other associations, feelings cued by the handling of space are often attributed to something else. In growing up people learn literarily thousands of spatial cues, all of which have their own meanings in their own contexts.当人们到海外旅游时，如果留心观察外国人如何处理空间关系，就会发现许多令人惊讶的不同之处；而这些不同之处总让我们反应强烈。

投资学精要(博迪)(第五版)习题答案英文版chapter2,4,5

Net asset value =
4,000,000
= 10.49
8. a. Start of year price = $12.00 × 1.02 = $12.24
End of year price = $12.10 × 0.93 = $11.25
Although NAV increased, the price of the fund fell by $0.99.
Distributions + ∆(Price) $1.50 – $0.99 Rate of return = Start of year price = $12.24 = .042 = 4.2%
b. An investor holding the same portfolio as the manager would have earned a rate of return based on the increase in the NAV of the portfolio:
$1000 × (1.095)4 = $1,437.66,
which after paying the exit fee will leave you with: $1,437.66 × .99 = 1423.28.
Class B is better if your horizon is 4 years.
Chapter 4:
2. The offer price includes a 6% front-end load, or sales commission, meaning that every dollar paid results in only $.94 going toward purchase of shares. Therefore,

新标准大学英语,综合教程3,第二版 Reading across cultures

c. orange
d. pens and inkstands e. stamp f. stethoscope g. a stalk of celery
6. hard-working
7. intelligent
Reading across cultures
The Guan and Ji ceremonies
Reading across cultures
Talking point
Discussing
Delivering a speech
Reading across cultures
Discussing
Discuss the topic in groups: Should Hanfu be made a national dress?
Reading across cultures
Delivering a speech
Watch a video clip and finish the task.
Delivering a high school graduation speech is a definite honor. Graduating from high school is a major milestone in a young person’s life. Suppose you were a high school graduate, how would you deliver your speech?
More
Reading across cultures
Discussing
The revival leans too far in the direction of a narrow nationalism, focused on looks rather than content. A blanket dismissal of non-Hanfu dress could lead a step further, towards a rejection of the West that goes beyond clothing matters. Hanfu as a national dress would fail to represent the other 55 ethnicities of China. Hanfu is out-dated and old fashioned. Since it is theoretically a “period costume”, it is unsuitable for wear in a modern progressive society. Wearing Hanfu is inconvenient, since it’s too formal, elaborate and should be a bit heavy.

多恩布什宏观经济学第十版课后习题答案04

CHAPTER 4GROWTH AND POLICYSolutions to the Problems in the TextbookConceptual Problems:1. Endogenous or self-sustained growth supposedly can be achieved by policies that affect anation's savings rate and therefore the proportion of GDP that goes towards investment.The neoclassical growth model of Chapter 3 predicted that long-term growth can only be achieved through technological progress and that changes in the savings rate have only transitory effects. The endogenous growth model, however, predicts that countries with a higher savings rate can achieve higher long-term growth and that a nation's government can affect the long-term growth rate by implementing policies that affect the savings rate.2. A simple model with constant returns to scale to capital alone implies increasingreturns to scale to all factors taken together, which could cause a single large firm to dominate the economy. However, such a model ignores the possibility that external returns to capital exist, in addition to the internal (private) returns. In other words, more investment not only leads to a higher and more efficient capital stock but also to new ideas and new ways of doing things, which can then be copied by others. Therefore, a single firm does not necessarily reap all of the benefits of increased output.3.In the neoclassical growth model, an increase in the savings rate does not increase thelong-term growth rate of output. However, because of the short-run adjustment process, there is some transitional gain that will lead to a higher level of output per capita. In the endogenous growth model, however, the savings rate does affect the long-term growth rate of output.4.a. Chapter 4 suggests that the key to long-term economic growth is investment in humanand physical capital with particular emphasis on research and development.4.b. (i) Investment tax credits may potentially affect economic growth in the long run byachieving a higher rate of technological progress.(ii) R&D subsidies and grants lead to technological advances that will have private andsocial returns. They are very effective in stimulating long-term economic growth.(iii) According to the endogenous growth model, policies designed to increase thesavings rate will increase the long-term growth rate of output. However, empiricalevidence does not lend much support to that notion.(iv) Increased funding for primary education has large private and social returns andis therefore an excellent means to stimulate long-term growth, even though it may take along time until these policies have their full effect.5. The notion of absolute convergence states that economies with the same savings rate andrate of population growth will reach the same steady-state equilibrium if they have access to the same technology. The notion of conditional convergence states that economies that have access to the same technology and the same rate of population growth but different savings rates will reach steady-state equilibria at a different level of output but the same economic growth rate. There is empirical evidence to support the notion of conditional convergence across countries.6. Endogenous growth theory assumes that the steady-state growth rate of output isaffected by the rate at which the factors of production are accumulated. Therefore, an increase in the savings rate would increase the rate at which the capital stock is accumulated and this would increase the growth rate of output. While this notion may be important in explaining the growth rates of highly developed countries at the leading edge of technology, it cannot explain the differences in growth rates across poorer countries.For these countries, the notion of conditional convergence seems to hold.7. Investing in physical capital will lead to a higher capital stock and to a higher level ofoutput in the short run, but often to the detriment of long-term growth unless there are significant external returns to capital. Therefore, investing in human capital is a better strategy, since it has high returns and leads to an increase in long-term growth.8.a. A country that is able to choose its rate of population growth through population controlpolicies can shift the investment requirement down, thereby increasing the level of steady-state output. With a lower rate of population growth it is possible to achieve a higher level of income per capita with a lower level of investment spending. Therefore, implementing population control policies may be an effective way to escape the so-called poverty trap.8.b. In an endogenous growth model, a lower population growth rate (n) will increase anation's long-term growth rate (∆y/y). We can see this since, in the second optional section, the per-capita growth rate was derived, as follows:∆y/y = sa - (n + d).9. The Asian Tigers (Hong Kong, Singapore, South Korea, and Taiwan) experienced a highrate of economic growth between 1966 and 1990 by concentrating on improving the education of the population and increasing the savings rate, as suggested by the endogenous growth model. However, increases in the labor forces of these countries suggested by the neoclassical growth model, were also at work.10. The decline in living standards experienced by Eastern European countries in transitionfrom centrally planned economies to free market economies cannot easily be explained by neoclassical or endogenous growth theory. The decline in GDP in these countries was largely due to disorganized markets that lacked properly assigned property rights or liability rules and an insufficiently developed banking system. In addition, the need for large-scale replacement of outdated production technology caused further disruption.11. In is unclear whether countries can actually experience indefinite increases in theirgrowth potential. However, if technological advances occur continuously and if intelligent resource management is practiced, it is conceivable that economic growth will continue for a very, very long time.Technical Problems:1.a. A production function that displays both a diminishing and a constant marginal product ofcapital can be displayed by drawing a curved line (as in an exogenous growth model), followed by a upward-sloping line (as in an endogenous growth model). Such a graph is depicted below.1.b. The first equilibrium (Point A in the graph below) is a stable low-income steady-stateequilibrium. Any deviation from that point will cause the economy to eventually adjust again at the same steady-state income level (and capital-output ratio). The second equilibrium (Point B) is an unstable high-income steady-state equilibrium. Any deviation from that point will lead to either a lower income steady-state equilibrium (if the capital-labor ratio declines) or ongoing growth (if the capital-labor ratio increases).yy = f(k)y BsyB (n+ d)ky AAk A k B k1.c. A model like the one in this question can be used to explain how some countries can findthemselves in a situation with no growth and low income while others have ongoing growth and a high level of income. In the first case, a country may have invested in physical capital, leading to some short-term growth at the expense of long-term growth, whereas in the second case, a country may have invested heavily in human capital, reaping significant social returns.2.a. If population growth is endogenous, that is, if a country can influence the rate of population growth through government policies, then the investment requirement is no longer a straight line. Instead it is curved as depicted below.yy C y = f(k)y B[n(y) + d]]kC sf(k)y A BAk A k B k C k2.b. The first equilibrium (Point A) is a stable steady-state equilibrium. It is a situation of lowincome and high population growth, indicating that the country is in a poverty trap. The second equilibrium (Point B) is an unstable steady-state equilibrium. It is a situation of medium income and low population growth. The third equilibrium (Point C) is a stable steady-state equilibrium. It is a situation of high income and low population growth. None of these three equilibria have ongoing growth.2.c. To escape the poverty trap (Point A), a country has several possibilities: First, it cansomehow find the means to increase the capital-labor ratio above a level consistent with Point B (perhaps by borrowing funds or seeking direct foreign investment). Second, it can increase the savings rate such that the savings function no longer intersects the investment requirement curve at either Point A or Point B. Third, it can decrease the rate of population growth through specifically designed policies, such that the investment requirement shifts down and no longer intersects with the savings function at Point A or Point B.3.a. If we incorporate endogenous population growth into a two-sector model in Problem 2,then we get a curved line for the investment requirement line and a production function with first a diminishing and then a constant marginal product of capital as depicted below.(Note that the savings function has the same shape as the production function.)yy = f(k)y Dsf(k)D [n+d)]ky Cy By A CBAk A k B k C k D k3.b. There should be four intersections of the savings function and the investment requirement.The first equilibrium (at Point A) is a stable low-income steady-state equilibrium. Any deviation from that point will cause the economy to eventually adjust again at the same steady-state income level (and capital-output ratio). The second equilibrium (at Point B) is an unstable low-income equilibrium. Any deviation from that point will lead to either a lower income steady-state equilibrium at Point A (if the capital-labor ratio declines) or a higher income steady-state equilibrium at Point C (if the capital-labor ratio increases).Point D is again an unstable equilibrium but at a high level of income. Any deviation from that point will lead to either a lower income steady-state equilibrium at Point C (if the capital-labor ratio declines) or ongoing growth (if the capital-labor ratio increases). 3.c. This model is more inclusive than either of the two models discussed previously, andtherefore has greater explanatory power. But now the graphical analysis is far more complicated. It may not be worth the effort to introduce such complications.4.a. The production function is of the formY = K1/2(AN)1/2 = K1/2(4[K/N]N)1/2= K1/2(4K)1/2= 2K4.b. Since a = y/k = 2, it follows that the growth rate of output isg = sa - (n + d) = (0.1)2 - (0.02 + 0.03) = 0.15 = 15%.4.c. The term "a" in the equation above stands for the marginal product of capital. If weassume that the level of labor-augmenting technology (A) is proportional to the capital-labor ratio (k), we imply that the level of technology depends on the amount of capital per worker that we have, which may not be realistic.4.d. In this model, we have a constant marginal product of capital, and therefore we have anendogenous growth model.5.a. The production function is of the formY = K1/2N1/2==> Y/N = (K/N)1/2 ==> y = k1/2.From k = sy/(n + d) = sk1/2/(n +d) ==> k1/2 = s/(n + d)==> y* = s/(n + d) = (0.1)/(0.02 + 0.03) = 2==> k* = sy*/(n + d) = (0.1)(2)/(0.02 + 0.03) = 45.b. Steady-state consumption equals steady-state income minus steady-state investment, thatis,c* = f(k*) - (n + d)k* .The golden rule capital stock corresponds to the highest permanently sustainable level of consumption. Steady-state consumption is maximized when the marginal increase in capital produces just enough extra output to cover the increased investment requirement.From c = k1/2 - (n + d)k ==> (∆c/∆k) = (1/2)k-1/2 - (n + d) = 0==> k-1/2 = 2(n + d) = 2(.02 + .03) = .1==> k1/2 = 10 ==> k = 100Since k*= 4 < 100, we have less capital at the steady state than the golden rule suggests.5.c. From k = sy/(n + d) = sk1/2/(n + d) ==> s = k1/2(n + d) = 10(0.05) = .55.d. If we have more capital than the golden rule suggests, then we are saving too much andwe do not have the optimal amount of consumption.。

多恩布什＜宏观经济学＞第八版第三章英文答案

Solutions to the Problems in the TextbookConceptual Problems:1. The production function provides a quantitative link between inputs and output. For example, theCobb-Douglas production function mentioned in the text is of the form:Y = F(N,K) = AN1-θKθ,where Y represents the level of output. (1 - θ) and θ are weights equal to the shares of labor (N) and capital (K) in production, while A is often used as a measure for the level of technology. It can be easily shown that labor and capital each contribute to economic growth by an amount that is equal to their individual growth rates multiplied by their respective share in income.2. The Solow model predicts convergence, that is, countries with the same production function, savingsrate, and population growth will eventually reach the same level of income per capita. In other words,a poor country may eventually catch up to a richer one by saving at the same rate and makingtechnological innovations. However, if these countries have different savings rates, they will reach different levels of income per capita, even though their long-term growth rates will be the same.3. A production function that omits the stock of natural resources cannot adequately predict the impactof a significant change in the existing stock of natural resources on the economic performance of a country. For example, the discovery of new oil reserves or an entirely new resource would have a significant effect on the level of output that could not be predicted by such a production function.4. Interpreting the Solow residual purely as technological progress would ignore, for example, theimpact that human capital has on the level of output. In other words, this residual not only captures the effect of technological progress but also the effect of changes in human capital (H) on the growth rate of output. To eliminate this problem we can explicitly include human capital in the production function, such thatY = F(K,N,H) = AN a K b H c with a + b + c = 1.Then the growth rate of output can be calculated as∆Y/Y = ∆A/A + a(∆N/N) + b(∆K/K) + c(∆H/H).5. The savings function sy = sf(k) assumes that a constant fraction of output is saved. The investmentrequirement, that is, the (n + d)k-line, represents the amount of investment needed to maintain a constant capital-labor ratio (k). A steady-state equilibrium is reached when saving is equal to the investment requirement, that is, when sy = (n + d)k. At this point the capital-labor ratio k = K/N is not changing, so capital (K), labor (N), and output (Y) all must be growing at the same rate, that is, the rate of population growth n = (∆N/N).266. In the long run, the rate of population growth n = (∆N/N) determines the growth rate of the steady-state output per capita. In the short run, however, the savings rate, technological progress, and the rate of depreciation can all affect the growth rate.7. Labor productivity is defined as Y/N, that is, the ratio of output (Y) to labor input (N). A surge inlabor productivity therefore occurs if output grows at a faster rate than labor input. In the U.S. we have experienced such a surge in labor productivity since the mid-1990s due to the enormous growth in GDP. This surge can be explained from the introduction of new technologies and more efficient use of existing technologies. Many claim that the increased investment in and use of computer technology has stimulated economic growth. Furthermore, increased global competition has forced many firms to cut costs by reorganizing production and eliminating some jobs. Thus, with large increases in output and a slower rate of job creation we should expect labor productivity to increase.(One should also note that a higher-skilled labor force also can contribute to an increase in labor productivity, since the same number of workers can produce more output if workers are more highly skilled.)Technical Problems:1.a. According to Equation (2), the growth of output is equal to the growth in labor times the labor shareplus the growth of capital times the capital share plus the rate of technical progress, that is, ∆Y/Y = (1 - θ)(∆N/N) + θ(∆K/K) + ∆A/A, where1 - θ is the share of labor (N) and θ is the share of capital (K). Thus if we assume that the rate oftechnological progress (∆A/A) is zero, then output grows at an annual rate of 3.6 percent, since ∆Y/Y = (0.6)(2%) + (0.4)(6%) + 0% = 1.2% + 2.4% = + 3.6%,1.b. The so-called "Rule of 70" suggests that the length of time it takes for output to double can becalculated by dividing 70 by the growth rate of output. Since 70/3.6 = 19.44, it will take just under 20 years for output to double at an annual growth rate of 3.6%,1.c. Now that ∆A/A = 2%, we can calculate economic growth as∆Y/Y = (0.6)(2%) + (0.4)(6%) + 2% = 1.2% + 2.4% + 2% = + 5.6%.Thus it will take 70/5.6 = 12.5 years for output to double at this new growth rate of 5.6%.2.a. According to Equation (2), the growth of output is equal to the growth in labor times the labor shareplus the growth of capital times the capital share plus the growth rate of total factor productivity (TFP), that is,∆Y/Y = (1 - θ)(∆N/N) + θ(∆K/K) + ∆A/A, where271 - θ is the share of labor (N) and θ is the share of capital (K). In this example θ = 0.3; therefore, ifoutput grows at 3% and labor and capital grow at 1% each, then we can calculate the change in TFP in the following way3% = (0.3)(1%) + (0.7)(1%) + ∆A/A ==> ∆A/A = 3% - 1% = 2%,that is, the growth rate of total factor productivity is 2%.2.b. If both labor and the capital stock are fixed and output grows at 3%, then all this growth has to becontributed to the growth in factor productivity, that is, ∆A/A = 3%.3.a. If the capital stock grows by ∆K/K = 10%, the effect on output would be an additional growth rate of∆Y/Y = (.3)(10%) = 3%.3.b. If labor grows by ∆N/N = 10%, the effect on output would be an additional growth rate of∆Y/Y = (.7)(10%) = 7%.3.c. If output grows at ∆Y/Y = 7% due to an increase in labor by ∆N/N = 10%, and this increase in laboris entirely due to population growth, then per capita income would decrease and people’s welfare would decrease, since∆y/y = ∆Y/Y - ∆N/N = 7% - 10% = - 3%.3.d. If this increase in labor is due to an influx of women into the labor force, the overall population doesnot increase and income per capita would increase by ∆y/y = 7%. Therefore people's welfare would increase.4. Figure 3-4 shows output per head as a function of the capital-labor ratio, that is, y = f(k). The savingsfunction is sy = sf(k), and it intersects the straight (n + d)k-line, representing the investment requirement. At this intersection, the economy is in a steady-state equilibrium. Now let us assume that the economy is in a steady-state equilibrium before the earthquake hits, that is, the steady-state capital-labor ratio is currently k*. Assume further, for simplicity, that the earthquake does not affect people's savings behavior.If the earthquake destroys one quarter of the capital stock but less than one quarter of the labor force, then the capital-labor ratio falls from k*to k1 and per-capita output falls from y* to y1. Now saving is greater than the investment requirement, that is, sy1 > (d + n)k1, and the capital stock and the level of output per capita will grow until the steady state at k* is reached again.However, if the earthquake destroys one quarter of the capital stock but more than one quarter of the labor force, then the capital-labor ratio increases from k*to k2. Saving now will be less than the investment requirement and thus the capital-labor ratio and the level of output per capita will fall until the steady state at k* is reached again.If exactly one quarter of both the capital stock and the labor stock are destroyed, then the steady state is maintained, that is, the capital-labor ratio and the output per capita do not change.2829If the severity of the earthquake has an effect on peoples’ savings behavior, then the savings function sy = sf(k) will move either up or down, depending on whether the savings rate (s) increases (if people save more, so more can be invested in an effort to rebuild) or decreases (if people save less, since they decide that life is too short not to live it up).yy 2 y * y 10 k 1 k k 2 k5.a. An increase in the population growth rate (n) affects the investment requirement, and the (n + d)k-line gets steeper. As the population grows, more saving must be used to equip new workers with the same amount of capital that the existing workers already have. Therefore output per capita (y) will decrease as will the new optimal capital-labor ratio, which is determined by the intersection of the sy-curve and the (n 1 + d)k-line. Since per-capita output will fall, we will have a negative growth rate in the short run. However, the steady-state growth rate of output will increase in the long run, since it will be determined by the new and higher rate of population growth.y y o y 1k1 k o k5.b. Starting from an initial steady-state equilibrium at a level of per-capita output y*, the increase in thepopulation growth rate (n) will cause the capital-labor ratio to decline from k* to k1. Output per capita will also decline, a process that will continue at a diminishing rate until a new steady-state level is reached at y1. The growth rate of output will gradually adjust to the new and higher level n1.yy*y1t o t1 tkk*k1t o t1 t6.a. Assume the production function is of the formY = F(K, N, Z) = AK a N b Z c ==>∆Y/Y = ∆A/A + a(∆K/K) + b(∆N/N) + c(∆Z/Z), with a + b + c = 1.Now assume that there is no technological progress, that is, ∆A/A = 0, and that capital and labor grow at the same rate, that is, ∆K/K = ∆N/N = n. If we also assume that all natural resources available are fixed, such that ∆Z/Z = 0, then the rate of output growth will be∆Y/Y = an + bn = (a + b)n.In other words, output will grow at a rate less than n since a + b < 1. Therefore output per worker will fall.6.b. If there is technological progress, that is, ∆A/A > 0, then output will grow faster than before, namely30∆Y/Y = ∆A/A + (a + b)n.If ∆A/A > c, then output will grow at a rate larger than n, in which case output per worker will increase.6.c. If the supply of natural resources is fixed, then output can only grow at a rate that is smaller than therate of population growth and we should expect limits to growth as we run out of natural resources.However, if the rate of technological progress is sufficiently large, then output can grow at a rate faster than population, even if we have a fixed supply of natural resources.7.a. If the production function is of the formY = K1/2(AN)1/2,and A is normalized to 1, then we haveY = K1/2N1/2.In this case capital's and labor's shares of income are both 50%.7.b. This is a Cobb-Douglas production function.7.c. A steady-state equilibrium is reached when sy = (n + d)k.From Y = K1/2N1/2 ==> Y/N = K1/2N-1/2 ==> y = k1/2==>sk1/2= (n + d)k ==> k-1/2 = (n + d)/s = (0.07 + 0.03)/(.2) = 1/2 ==> k1/2= 2 = y ==> k = 4 .8.a. If technological progress occurs, then the level of output per capita for any given capital-labor ratioincreases. The function y = f(k) increases to y = g(k), and thus the savings function increases from sf(k) to sg(k).yy1k1 k2k8.b. Since g(k) > f(k), it follows that sg(k) > sf(k) for each level of k. Therefore the intersection of thesg(k)-curve with the (n + d)k-line is at a higher level of k. The new steady-state equilibrium will now be at a higher level of saving and output per capita, and at a higher capital-labor ratio.8.c. After the technological progress occurs, the level of saving and investment will increase until a newand higher optimal capital-labor ratio is reached. The ratio of investment to capital will also increase in the transition period, since more has to be invested to reach the higher optimal capital-labor ratio.kk2k1t1 t2t9. The Cobb-Douglas production function is defined asY = F(N,K) = AN1-θKθ.The marginal product of labor can then be derived asMPN = (∆Y)/(∆N) = (1 - θ)AN-θKθ = (1 - θ)AN1-θKθ/N = = (1 - θ)(Y/N)==> labor's share of income = [MPN*(N)]/Y = (1 - θ)(Y/N)*[(N)/(Y)] = (1 - θ)32。

(完整word版)哈佛公开课-公正课-第五课双语字幕

公正课\N迈克尔·桑德尔教授主讲第五讲《选择的自由》上节课结束时\When we finished last time,我们讲到约翰·斯图尔特·穆勒试图回应\we were looking at John Stuart Mill's attempt to reply对边沁功利主义的批判\to the critics of Bentham's Utilitarianism.在穆勒的《功利主义》中\In his book Utilitarianism,他试图证明与批判者所言相反\Mill tries to show that critics to the contrary在功利主义的框架下\it is possible within the utilitarian framework是能区分高级和低级快乐的\to distinguish between higher and lower pleasures.是能对价值进行定性区分的\It is possible to make qualitative distinctions of worth.我们用《辛普森一家》\And we tested that idea和莎士比亚作品检验了这一观点\with the Simpsons and the Shakespeare excerpts.检验结果\And the results of our experiment却似乎让我们质疑穆勒的区分\seem to call into question Mill's distinction因为在座大多数\because a great many of you都表示更喜欢《辛普森一家》\reported that you prefer the Simpsons却仍然认为莎士比亚的作品\but that you still consider Shakespeare能带来更高级更有价值的快乐\to be the higher or the worthier pleasure.这就是我们的检验中穆勒的观点所遭遇的困境\That's the dilemma with which our experiment confronts Mill.那么穆勒在《功利主义》\What about Mill's attempt to account第五章中提到的\for the especially weighty character of个人权利和公正重要性的解释又是否成立呢\individual rights and justice in chapter five of Utilitarianism.他想说明个人权利\He wants to say that individual rights值得特别的尊重\are worthy of special respect.实际上他甚至声称\In fact, he goes so far as to say that公正是道德中最神圣\justice is the most sacred part和最不可或缺的部分\and the most incomparably binding part of morality.但穆勒的这番辩护面临着同样质疑\But the same challenge could be put to this part of Mill's defense.为何公正是道德中最主要\Why is justice the chief part最不可或缺的部分\and the most binding part of our morality?他说因为从长远看\Well, he says because in the long run,如果我们秉持公正尊重权利\if we do justice and if we respect rights,社会整体会发展得更好\society as a whole will be better off in the long run.这能令人信服吗\Well, what about that?如果有个特例\What if we have a case where making an exception侵犯个人权利\and violating individual rights actually长远来看反而让人们获益更多呢\will make people better off in the long run?那样就可以利用人了吗\Is it all right then to use people?还有另一个能更深入地\And there is a further objection驳斥穆勒有关公正和权利的观点\that could be raised against Mill's case for justice and rights.假设如他所说长远来看\Suppose the utilitarian calculus in the long run功利主义演算真能实现\works out as he says it will即尊重个人权利\such that respecting people's rights从长远来看真的能让大家都获益\is a way of making everybody better off in the long run.这理由说得过去吗\Is that the right reason?这就是我们该尊重别人的唯一理由吗\Is that the only reason to respect people?如果那位医生\If the doctor goes in偷摘走那位来体检的\and yanks the organs from the healthy patient健康人的器官\who came in for a checkup去挽救另外五人\to save five lives,这事从长远来看会有负面影响\there would be adverse effects in the long run.人们终会得知此事\Eventually, people would learn about this而不再去医院体检\and would stop going in for checkups.这理由说得过去吗\Is it the right reason?这就是唯一原因\Is the only reason让你作为医生\that you as a doctor不会偷摘取体检病人的器官吗\won't yank the organs out of the healthy patient因为你认为如果你这样利用他\that you think, well, if I use him in this way,长远来看会导致更多人丧命\in the long run more lives would be lost?还是有另一原因\Or is there another reason这其实跟在本质上尊重每个个体有关\having to do with intrinsic respect for the person as an individual?如果其中确有这一原因\And if that reason matters那隐约可以看出\and it's not so clear即便是穆勒的功利主义也考虑了这点\that even Mill's utilitarianism can take account of it,为了全面检视对穆勒的这两点\fully to examine these two worries or objections,质疑或担忧\to Mill's defense我们需要更进一步\we need to push further.我们要问就更高的或更有价值的快乐而言\And we need to ask in the case of higher or worthier pleasures是否存在"良善生活"的理论\are there theories of the good life that能为快乐的价值\can provide independent moral standards提供独立的道德标准\for the worth of pleasures?如果存在那会是怎样的理论\If so, what do they look like?这是一个问题\That's one question.就公正和权利而言如果我们怀疑\In the case of justice and rights, if we suspect that穆勒其实也隐约靠向了个人尊严\Mill is implicitly leaning on notions of human dignity或尊重个人的观点\or respect for person而严格说来这不属于功利主义范畴\that are not strictly speaking utilitarian,我们就需要看看\we need to look to see有没有更强有力的权利理论\whether there are some stronger theories of rights能解释穆勒的这点隐约的直觉\that can explain the intuition which even Mill shares,即尊重个人不利用个人的理由\the intuition that the reason for respecting individuals and not using them甚至胜过了长远看来的功利\goes beyond even utility in the long run.今天我们讨论其中一项强有力的权利理论\Today, we turn to one of those strong theories of rights.这些强有力的权利理论认为\Strong theories of rights say个人很重要不仅仅是用来\individuals matter not just as instruments实现更高社会目标的工具\to be used for a larger social purpose或为了实现功利最大化的工具\or for the sake of maximizing utility,个人是独立的存在\individuals are separate beings有独立的生命值得尊重\with separate lives worthy of respect.这些强有力的权利理论认为\And so it's a mistake,下列看法是错误的\according to strong theories of rights, it's a mistake不该只以偏好和价值的加总\to think about justice or law来考虑公正或法律\by just adding up preferences and values.我们今天要讨论的权利理论是自由主义\The strong rights theory we turn to today is libertarianism.自由主义非常重视个人权利\Libertarianism takes individual rights seriously.它被称为自由主义\It's called libertarianism是因为它宣称个人的基本权利是自由权\because it says the fundamental individual right is the right to liberty就因为我们都是独立存在的个体\Precisely because we are separate individual beings,我们不能被利用\we're not available to any use去满足社会可能的需求\that the society might desire or devise就因为我们是独立存在的个体\Precisely because we are individual separate human beings,我们享有自由的基本权利\we have a fundamental right to liberty,即我们有权自由选择\and that means a right to choose freely,过自己喜欢的生活\to live our lives as we please只要尊重他人同等的权利\provided we respect other people's rights to do the same.这是它的基本理念\That's the fundamental idea.罗伯特·诺齐克\Robert Nozick,本课涉及到的一位自由主义哲学家\one of the libertarian philosophers we read是这样说的\for this course, puts it this way:个人有权利\Individuals have rights.这些权利如此强大如此深远\So strong and far reaching are these rights以至引发一个问题如果有的话政府可以做什么\that they raise the question of what, if anything, the state may do.自由主义对于政府或国家的角色\So what does libertarianism say有什么看法呢\about the role of government or of the state?大部分当代政府所做的三种事\Well, there are three things that most modern states do在自由主义理论看来是不合法\that on the libertarian theory of rights不公正的\are illegitimate or unjust.第一家长式的立法\One of them is paternalist legislation.即制定保护人们免受自身行为伤害的法律\That's passing laws that protect people from themselves,诸如系安全带骑摩托车带头盔的法规\seatbelt laws, for example, or motorcycle helmet laws.自由主义者说系安全带也许是件好事\The libertarian says it may be a good thing if people wear seatbelts但这应由人们自己作主\but that should be up to them政府没有资格\and the state, the government, has no business用法律来强迫人们系安全带\coercing them, us, to wear seatbelts by law.这是强迫\It's coercion,所以第一点不应有家长式的立法\so no paternalist legislation, number one.第二点不应有道德式的立法\Number two, no morals legislation.很多法律试图提高公民的品德\Many laws try to promote the virtue of citizens或者试图树立\or try to give expression to the moral values整个社会的道德标准\of the society as a whole.自由主义者说这也违反了个人的自由权\Libertarian say that's also a violation of the right to liberty.举一个经典的例子\Take the example of, well, a classic example以弘扬传统道德之名立法\of legislation authored in the name of promoting morality 历来都有法律\traditionally have been laws禁止同性恋性行为\that prevent sexual intimacy between gays and lesbians.自由主义者认为\The libertarian says其他人没有因此受到伤害也没被侵权\nobody else is harmed,nobody else's rights are violated,所以政府不应该插手此事\so the state should get out of the business entirely of不该试图立法弘扬道德\trying to promote virtue or to enact morals legislation.第三种不被自由主义认同的\And the third kind of law or policy法律或政策是\that is ruled out on the libertarian philosophy任何为了劫富济贫进行收入或财富再分配\is any taxation or other policy that serves the purpose而制定的税收或其他政策\of redistributing income or wealth from the rich to the poor.仔细想想再分配这个概念\Redistribution is a - if you think about it,按自由主义者的话来说就是强迫\says the libertarian is a kind of coercion.它相当于政府施行的盗窃\What it amounts to is theft by the state若是民主政府的话则是大多数人施行的盗窃\or by the majority, if we're talking about a democracy,其对象是工作出色而赚得大钱的人\from people who happen to do very well and earn a lot of money.诺齐克和其他自由主义者认为\Now, Nozick and other libertarians allow that可以有这样一种"小政府"\there can be a minimal state它的税收只用来提供所有人都需要的服务\that taxes people for the sake of what everybody needs,包括国防治安\the national defense, police force,强制履约和保护产权的司法系统\judicial system to enforce contracts and property rights,不过仅此而已\but that's it.我想听听你们对\Now, I want to get your reactions自由主义第三种观点的态度\to this third feature of the libertarian view.看看你们当中谁赞同\I want to see who among you agree with that idea谁不赞同以及为什么\and who disagree and why.但为了更形象看看问题何在\But just to make it concrete and to see what's at stake,以美国的财富分配状况为例\consider the distribution of wealth in the United States.在所有发达的民主国家之中\United States is among the most inegalitarian society as far as{\an8}{\fn方正黑体简体\fs18\b1\bord1\shad1\3c&H2F2F2F&}该图为美国10%的人口占据了70%的社会财富美国财富分配不均的问题最为严重\the distribution of wealth of all the advanced democracies.这样是否公平\Now, is this just or unjust?自由主义者们怎么说\Well, what does the libertarian say?他们说你不能仅从这个事实来判断\Libertarian says you can't know just from the facts I've just given you.你无法判断财富分配是否公平\You can't know whether that distribution is just or unjust.你不能仅凭分配格局\You can't know just by looking at a pattern或分配结果\or a distribution or result来判断其是否公平\whether it's just or unjust.你得知道它是怎么来的\You have to know how it came to be.不能只关注最终结果\Y ou can't just look at the end stage or the result.{\an8}诺齐克收入分配怎样才公平你得考虑两个原则\You have to look at two principles.第一个原则诺齐克称之最初占有的公正原则\The first he calls justice in acquisition or in initial holdings.这很简单就是说\And what that means simply is人们是否公平地获得生产资料\did people get the things they used to make their money fairly?我们需要了解最初的占有是否来得公平\So we need to know was there justice in the initial holdings?让他们赚到钱的土地工厂或者商品\Did they steal the land or the factory or the goods这些生产资料是不是偷来的\that enabled them to make all that money?如果不是偷来的如果他们\If not, if they were entitled to whatever it was有权享有那些生产资料\that enabled them to gather the wealth,那就算符合第一条原则\the first principle is matched.第二条原则财富的分配是否\The second principle is did the distribution arise基于自由达成的交易\from the operation of free consent,基于自由市场的买卖\people buying and trading on the market?可以看出自由主义眼中的公平\As you can see, the libertarian idea of justice相当于自由市场理念下的公平\corresponds to a free market conception of justice 只要生产资料的获取是公平的\provided people got what they used fairly,不是偷来的\didn't steal it,只要分配的结果是出自\and provided the distribution results自由市场上个体的自由选择\from the free choice of individual's buying and selling things,这样的分配就是公平的\the distribution is just.反之则不公平\And if not, it's unjust.为了进一步限定讨论的话题\So let's, in order to fix ideas for this discussion,我们举个真实的例子\take an actual example.美国最有钱的人是谁\Who's the wealthiest person in the United States -全世界最有钱的人是谁比尔·盖茨\wealthiest person in the world? Bill Gates.的确是没错这就是他\It is. That's right. Here he is.要是你你也会很开心的\You'd be happy, too.他的净资产有多少有人知道吗\Now, what's his net worth? Anybody have any idea?{\an8}{\fn方正黑体简体\fs18\b1\bord1\shad1\3c&H2F2F2F&}净资产400亿美元[《福布斯》2009年数据]数字非常巨大\That's a big number.克林顿当政期间\During the Clinton years,有个竞价捐款记得吧\remember there was a controversy donors?参与的大手笔捐款人都被邀请\Big campaign contributors were invited to在白宫的林肯卧室留宿一晚\stay overnight in the Lincoln bedroom at the White House?你要是捐到2.5万美元以上也可以啊\I think if you've contributed twenty five thousand dollars or above.有人算出来\Someone figured out at按能受邀在林肯卧室留宿一夜\the median contribution that got you invited所需捐款额的中位数计算\to stay a night in the Lincoln bedroom,比尔·盖茨完全付得起在林肯卧室\Bill Gates could afford to stay in the Lincoln bedroom every night住上6万6千年\for the next sixty six thousand years.还有人算出了\Somebody else figured out,他一个小时能挣多少钱\how much does he get paid on an hourly basis?他们算出自从他创立了微软\And so they figured out, since he began Microsoft,假设他每天工作14个小时合理的猜测\I suppose he worked, what 14 hours per day, reasonable guess,然后你算算他的净资产\and you calculate this net wealth,结果算出他的工资率在150美元以上\it turns out that his rate of pay is over 150 dollars,不是每小时也不是每分钟\not per hour, not per minute而是每秒钟150美元以上\150 dollars, more than 150 dollars per second这意味着如果盖茨在上班路上\which means that if on his way to the office,就算看到地上有一张百元大钞\Gates noticed a hundred dollar bill on the street,都不值得他停下来去捡\it wouldn't be worth his time to stop and pick it up.你们很多人会说\Now, most of you will say这么有钱的人我们当然可以向他收税\someone that wealthy surely we can tax them以满足那些得不到教育\to meet the pressing needs of people who lack in education 缺乏食物或者无家可归者的迫切需求\or lack enough to eat or lack decent housing.他们比他更需要这些钱\They need it more than he does.如果你是个功利主义者你会怎么办\And if you were a utilitarian, what would you do?你会制定怎样的税收政策\What tax policy would you have?你会马上进行再分配对吧\You'd redistribute in a flash, wouldn't you?因为作为一个优秀的功利主义者你知道\Because you would know being a good utilitarian that收走一些对他们来说根本无关痛痒的钱\taking some, a small amount, he'd scarcely going to notice it,却能大大改善社会底层那些人的生活\but it will make a huge improvement in the lives增加他们的福利\and in the welfare of those at the bottom.但是记住\But remember,自由主义理论说\the libertarian theory says我们不能那样\we can't just add up简单加总偏好和满足\an aggregate preferences and satisfactions that way.我们必须要尊重个人\We have to respect persons and如果他公平地赚到钱\if he earned that money fairly没有侵犯到他人权利\without violating anybody else's rights完全遵守了那两条公正原则\in accordance with the two principles最初占有公正原则和转让公正原则\of justice in acquisition and in justice in transfer,那么向他多征税就是错的\then it would be wrong,这无异于强取豪夺\it would be a form of coercion to take it away.迈克尔·乔丹没有比尔·盖茨那么富有\Michael Jordan is not as wealthy as Bill Gates但他也自有一番成就\but he did pretty well for himself.想看迈克尔·乔丹这就是他\You wanna see Michael Jordan. There he is.他一年的收入有3100万\His income alone in one year was 31 million dollars另外他为耐克和其他公司代言\and then he made another 47 million dollars又能赚4700万\in endorsements for a Nike and other companies.所以他一年的总收入有7800万\So his income was, in one year, $78 million.假设让他拿出三分之一的收入\To require him to pay, let's say, a third of his earnings交给政府来支持公益事业\to the government to support good causes为穷人提供食物医疗保障住房和教育\like food and health care and housing and education for the poor,这就是强迫是不公平的\that's coercion, that's unjust.侵犯了他的权利\That violates his rights.正因如此再分配是错误的\And that's why redistribution is wrong.有多少人同意自由主义者的这一驳论\Now, how many agree with that argument,认为为了帮助穷人\agree with the libertarian argument that redistribution进行财富再分配不对\for the sake of trying to help the poor is wrong?有多少人不同意这个观点\And how many disagree with that argument?好我们先从那些不同意的人开始\All right, let's begin with those who disagree.自由主义者反对再分配怎么不对了\What's wrong with the libertarian case against redistribution?请说\Yes.received在社会中工作\we're talking about working within a society他们从社会中得到的更多\and they received larger gift from the society因此他们该承担更大的责任\and they have a larger obligation通过财富再分配来回报社会\in return to give that through redistribution, you know,你可以说乔丹也许和那些\you can say that Michael Jordan may work just as hard as some who works,一天洗12甚至14小时衣服的人一样辛苦\you know, doing laundry 12 hours, 14 hours a day,但他得到的更多\but he's receiving more.如果说这都是靠他自己辛苦挣来的\I don't think it's fair to say that, you know, it's all on him,是他天赋所赐我觉得这不算公平\on his, you know, inherent, you know, hard work.好我们来听听自由主义者的辩护\All right, let's hear from defenders oflibertarianism.为何向富人征税救济穷人在原则上是错的\Why would it be wrong in principle to tax the rich to help the poor?说吧\Go ahead.我名叫乔我收集滑板\My name is Joe and I collect skateboards.我已经买了100个滑板了\I've since bought a hundred skateboards.我居住的社区有一百人\I live in a society of a hundred people.我是唯一有滑板的人\I'm the only one with skateboards.突然大家都想要滑板了\Suddenly, everyone decides they want a skateboard.他们跑到我家来\They come to my house,拿走了我的99个滑板\they take my they take 99 of my skateboards.我觉得这是不公平的\I think that is unjust.我认为在某些情况下\Now, I think in certain circumstances我们需要忽视这种不公平\it becomes necessary to overlook that unjustness,容忍这种不正义\perhaps condone that injustice例如在救生艇里被当作食物的男孩\as in the case of the cabin boy being killed for food.如果人们在死亡边缘挣扎\If people are on the verge of dying,也许忽视这样的不公平是必要的\perhaps it is necessary to overlook that injustice,但我认为即使这样我们依然要铭记\but I think it's important to keep in mind 我们的行为不公正\that we're still committing injustice这是在占有他人的财物或资产\by taking people's belongings or assets.你是说按33%a 33 percent tax rate来支持公益事业解决温饱是盗窃行为吗\for good causes to feed the hungry is theft?我觉得这不公正\I think it's unjust.我确实认为这是盗窃\Y es, I do believe it's theft但也许我们有必要容忍它\but perhaps it is necessary to condone that theft.但它依然是盗窃\But it's theft.是的\Yes.为什么是盗窃呢乔\Why is it theft, Joe?因为\Because --为什么这和你收集滑板有相同之处呢\Why is it like your collection of skateboards?这是盗窃是因为至少在我看来\It's theft because, or at least, in my opinion在自由主义者的观点看来\and by the libertarian opinion他公平地取得收入这些收入都是属于他的\he earned that money fairly and it belongs to him.拿走他的收入毫无疑问就是盗窃\So to take it from him is by definition theft.有人想反驳乔吗你请说\Who wants to reply to Joe? Yes, go ahead.我觉得他的例子不恰当\I don't think this is necessarily a case不是你有99个滑板而政府...\in which you have 99 skateboards and the government...或你有100个滑板\or you have a hundred skateboards而政府收走99个\and the government is taking 99 of them.恰当的例子是你的滑板多到\It's like you have more skateboards每天用一个都不重样\than there are days in a year.你的滑板多到\You have more skateboards一辈子也用不完\than you're going to be able to use in your entire lifetime而政府只是拿走其中的一些\and the government is taking part of those.如果你生活在一个那样的社会\And I think that if you are operating in a society in which这个社会中\the government's not,政府不进行财富再分配\in which the government doesn't redistribute wealth,就等于允许一些人无限累积过多的财富\then that allows for people to amass so much wealth以至于那些不在同一起跑线的人\that people who haven't started from this very the equal footing当然这只是假设\in our hypothetical situation,现实中是不存在的\that doesn't exist in our real society他们将余生都将没有机会翻身\get undercut for the rest of their lives.所以你担心\So you're worried that如果没有一定程度的再分配\if there isn't some degree of redistribution of some 照顾社会底层\or left at the bottom,就不会有名副其实的机会均等\there will be no genuine equality of opportunity.很好关于税收是盗窃这个观点\All right, the idea that taxation is theft,诺齐克要更进一步\Nozick takes that point one step further.他同意这是盗窃而且比乔苛刻\He agrees that it's theft. He's more demanding than Joe.乔说这是盗窃但在极端情况下也许可以原谅\Joe says it is theft, maybe in an extreme case it's justified,例如为了养活饥饿的家人\maybe a parent is justified in stealing a loaf of bread而去偷面包的家长\to feed his or her hungry family.乔你会如何称呼自己\So Joe I would say, what would you call yourself,慈悲的自由主义者吗\a compassionate quasi-libertarian?仔细想想\Nozick says, if you think about it,诺齐克说征税相当于强占收入\taxation amounts to the taking of earnings.换句话说就是\In other words, it means强占劳动果实\taking the fruits of my labor.但如果政府有权\But if the state has the right强占我的收入或劳动果实\to take my earning or the fruits of my labor,这在道义上不就等同于\isn't that morally the same政府有权\as according to the state the right让我做部分义务劳动吗\to claim a portion of my labor?所以实际上\So taxation actually征税道义上等同于强迫劳动\is morally equivalent to forced labor因为强迫劳动会强占我的\because forced labor involves the taking of闲暇时间还有努力\my leisure, my time, my efforts,就像征税会强占我的劳动所得一样\just as taxation takes the earnings that I make with my labor.所以对诺齐克和其他自由主义者来说\And so, for Nozick and for the libertarians,再分配的税收是盗窃就像乔说的\taxation for redistribution is theft, as Joe says,但不仅如此\but not only theft is morally equivalent盗窃还在道义上等同于\to laying claim to certain hours强占生命和劳动的时间\of a person's life and labor,因此等于强迫劳动\so it's morally equivalent to forced labor.如果政府有权强占我的劳动果实\If the state has a right to claim the fruits of my labor,这就说明它确实有权强迫我劳动\that implies that it really has an entitlement to my labor itself.什么是强迫劳动\And what is forced labor?诺齐克指出强迫劳动就是奴役\Forced labor, Nozick points out, is what, is slavery,因为如果我连对自己劳动的独占权都没有\because if I don't have the right, the sole right to my own labor,这就说明\then that's really to say政府或是政治共同体\that the government or the political community是我的部分主人\is a part owner in me.政府是我的部分主人又意味着什么呢\And what does it mean for the state to be a part owner in me?仔细想想这就意味着我是一个奴隶\If you think about it, it means that I'm a slave,我不是自己的主人\that I don't own myself.这些推理把我们带回到\So what this line of reasoning brings us to自由主义的权利主张\is the fundamental principle所隐含的基本原则\that underlies the libertarian case for rights.那是什么原则呢\What is that principle?我是我自己主人的原则\It's the idea that I own myself.是尊重人权\It's the idea of self possession自然会接受自我拥有的原则\if you want to take right seriously.如果你不想只把人看成是各种偏好的集合\If you don't want to just regard people as collections of preferences,那你必将走向\the fundamental moral idea这一基本道德理念\to which you will be lead is the idea我们是自己的主人\that we are the owners or the propietors of our own person,功利主义的问题就出在这里\and that's why utilitarianism goes wrong.这也是为什么\And that's why it's wrong摘取健康人的器官是错的\to yank the organs from that healthy patient.你这么做仿佛他的器官属于你或这个社会\Y ou're acting as if that patient belongs to you or to the community.但我们只属于我们自己\But we belong to ourselves.也正是出于这一理由\And that's the same reason才不该制定法律保护我们免受自己伤害\that it's wrong to make laws to protect us from ourselves或告诉我们该如何生活\or to tell us how to live,该秉持怎样的道德规范\to tell us what virtues we should be governed by,rich to help the poor就算是为了公益事业\even for good causes,就算是为了帮助卡特里娜飓风的灾民\even to help those who are displaced by the Hurricane Katrina.请富人去搞慈善活动吧\Ask them to give charity.但如果对他们征税就成了强迫劳动\But if you tax them, it's like forcing them to labor.你能强迫乔丹放弃下周的比赛\Could you tell Michael Jordan he has to skip the next week's games必须下灾区去帮助卡特里娜飓风的灾民吗\and go down to help the people displaced by Hurricane Katrina?道义上说两者是一样\Morally, it's the same.因此利害关系很分明的\So the stakes are very high.我们现在已经听了一些对自由主义的反对声\So far we've heard some objections to the libertarian argument.。

多恩布什宏观经济学第十版课后习题答案09

多恩布什宏观经济学第十版课后习题答案09CHAPTER 9INCOME AND SPENDINGSolutions to the Problems in the Textbook:Conceptual Problems:1. In the Keynesian model, the price level is assumed to be fixed, that is, the AS-curve is horizontal andthe level of output is determined solely by aggregate demand. The classical model, on the other hand, assumes that prices always fully adjust to maintain a full-employment level of output, that is, the AS-curve is vertical. Since the model of income determination in this chapter assumes that the price level is fixed, it is a Keynesian model.2. An autonomous variable’s value is determined outside ofa given model. In this chapter the followingcomponents of aggregate demand have been specified as being autonomous: autonomous consumption (C*) autonomous investment (I o), government purchases (G o), lump sum taxes (TA o), transfer payments (TR o), and net exports (NX o).3.Since it often takes a long time for policy makers to agree on a specific fiscal policy measure, it isquite possible that economic conditions may drastically change before a fiscal policy measure is implemented. In these circumstances a policy measure can actually be destabilizing. Maybe the economy has already begun to move out of a recession before policy makers have agreed to implement a tax cut. If the tax cut is enacted at a time when the economy is already beginning to experience strong growth, inflationary pressure can be created.While such internal lags are absent with automatic stabilizers (income taxes, unemployment benefits, welfare), these automatic stabilizers are not sufficient to replace active fiscal policy when the economy enters a deep recession.4. Income taxes, unemployment benefits, and the welfare system are often called automatic stabilizerssince they automatically reduce the amount by which output changes as a result of a change in aggregate demand. These stabilizers are a part of the economic mechanism and therefore work without any case-by-case government intervention. For example, when output declines and unemployment increases, there may be an increase in the number of people who fall below the poverty line. If we had no welfare system or unemployment benefits, then consumption would drop significantly. But since unemployed workers get unemployment compensation and people living in poverty are eligible for welfare payments, consumption will not decrease as much. Therefore, aggregate demand may not be reduced by as much as it would have without these automatic stabilizers.5. The full-employment budget surplus is the budget surplus that would exist if the economy were at thefull-employment level of output, given the current spending or tax structure. Since the size of the full-employment budget surplus does not depend on the position in the business cycle and only changes when the government implements a fiscal policy change, the full-employment budget surplus can be used as a measure of fiscal policy. Other names for the full-employment budget surplus are the structural budget surplus, the cyclically adjusted surplus, the high-employment surplus, and the standardized employment surplus. These names may bepreferable, since they do not suggest that there is a specific full-employment level of output that we were unable to maintain.Technical Problems:1.a. AD = C + I = 100 + (0.8)Y + 50 = 150 + (0.8)YThe equilibrium condition is Y = AD ==>12 Y = 150 + (0.8)Y ==> (0.2)Y = 150 ==> Y = 5*150 = 750.1.b. Since TA = TR = 0, it follows that S = YD - C = Y - C. ThereforeS = Y - [100 + (0.8)Y] = - 100 + (0.2)Y ==> S = - 100 +(0.2)750 = - 100 + 150 = 50.1.c. If the level of output is Y = 800, then AD = 150 + (0.8)800 = 150 + 640 = 790.Therefore the amount of involuntary inventory accumulation is UI = Y - AD = 800 - 790 = 10.1.d. AD' = C + I' = 100 + (0.8)Y + 100 = 200 + (0.8)YFrom Y = AD' ==> Y = 200 + (0.8)Y ==> (0.2)Y = 200 ==> Y = 5*200 = 1,000Note: This result can also be achieved by using the multiplier formula:Y = (multiplier)(?Sp) = (multiplier)(?I) ==> ?Y = 5*50 = 250, that is, output increases from Y o = 750 to Y 1 = 1,000.1.e. From 1.a. and 1.d. we can see that the multiplier is 5.1.f. Sp2001500 750 1,000 Y2.a. Since the mpc has increased from 0.8 to 0.9, the size of the multiplier is now larger and we shouldtherefore expect a higher equilibrium income level than in 1.a.AD = C + I = 100 + (0.9)Y + 50 = 150 + (0.9)Y ==>Y = AD ==> Y = 150 + (0.9)Y ==> (0.1)Y = 150 ==> Y = 10*150 = 1,500.2.b. From ?Y = (multiplier)(?I) = 10*50 = 500 ==> Y 1 = Y o + ?Y = 1,500 + 500 = 2,000.2.c. Since the size of the multiplier has doubled from 5 to 10, the change in output (Y) that results from achange in investment (I) now has also doubled from 250 to 500.2001503.a. AD = C + I + G + NX = 50 + (0.8)YD + 70 + 200 = 320 + (0.8)[Y - (0.2)Y + 100]= 400 + (0.8)(0.8)Y = 400 + (0.64)YFrom Y = AD ==> Y = 400 + (0.64)Y ==> (0.36)Y = 400==> Y = (1/0.36)400 = (2.78)400 = 1,111.11The size of the multiplier is (1/0.36) = 2.78.3.b. BS = tY - TR - G = (0.2)(1,111.11) - 100 - 200 = 222.22 - 300 = - 77.783.c. AD' = 320 + (0.8)[Y - (0.25)Y + 100] = 400 + (0.8)(0.75)Y = 400 + (0.6)YFrom Y = AD' ==> Y = 400 + (0.6)Y ==> (0.4)Y = 400 ==> Y = (2.5)400 = 1,000The size of the multiplier is now reduced to 2.5.3.d. BS' = (0.25)(1,000) - 100 - 200 = - 50BS' - BS = - 50 - (-77.78) = + 27.78The size of the multiplier and equilibrium output will both increase with an increase in the marginal propensity to consume.Therefore income tax revenue will also go up and the budget surplus should increase.3.e. If the income tax rate is t = 1, then all income is taxed. There is no induced spending and equilibriumincome only increases by the change in autonomous spending, that is, the size of the multiplier is 1.From Y = C + I + G ==> Y = C o + c(Y - 1Y + TR o) + I o + G o==> Y = C o + cTR o + I o + G o = A o4. In Problem 3.d. we had a situation where the following was given:Y = 1,000, t = 0.25, G = 200 and BS = - 50.Assume now that t = 0.3 and G = 250 ==>AD' = 50 + (0.8)[Y - (0.3)Y + 100] + 70 + 250 = 370 + (0.8)(0.7)Y + 80 = 450 + (0.56)Y.From Y = AD' ==> Y = 450 + (0.56)Y ==> (0.44)Y = 450==> Y = (1/0.44)450 = 1,022.73BS' = (0.3)(1,022.73) - 100 - 250 = 306.82 - 350 = - 43.18BS' - BS = -43.18 - (-50) = + 6.82The budget surplus has increased, since the increase in tax revenue is larger than the increase in government purchases.5.a. While an increase in government purchases by ?G = 10 will change intended spending by ?Sp = 10,a decrease in government transfers by ?TR = -10 will change intended spending by a smaller amount,that is, by only ?Sp = c(?TR) = c(-10). The change in intended spending equals ?Sp = (1 - c)(10) and equilibrium income should therefore increase byY = (multiplier)(1 - c)10.5.b. If c = 0.8 and t = 0.25, then the size of the multiplier isα = 1/[1 - c(1 - t)] = 1/[1 - (0.8)(1 - 0.25)] = 1/[1 - (0.6)] = 1/(0.4) = 2.5.The change in equilibrium income isY = α(?A o) = α[?G + c(?TR)] = (2.5)[10 + (0.8)(-10)] = (2.5)2 = 55.c. ?BS = t(?Y) - ?TR - ?G = (0.25)(5) - (-10) - 10 = 1.25Additional Problems:1. "An increase in the marginal propensity to save increases the impact of one additional dollar inincome on consumption." Comment on this statement. In your answer discuss the effect of sucha change in the mps on the size of the expenditure multiplier.The fact that the marginal propensity to save (1 - c) has risen implies that the marginal propensity to consume (c) has fallen. This means that now one extra dollar in income earned will affect consumption by3less than before the reduction in the mpc. When the mpc is high, one extra dollar in income raises consumption by more than when the mpc is low. If the mps is larger, then the expenditure multiplier will be larger, since the expenditure multiplier is defined as 1/(1-c).2. Using a simple model of the expenditure sector without any government involvement, explainthe paradox of thrift that asserts that a desire to save may not lead to an increase in actual saving.The paradox of thrift occurs because the desire to increase saving leads to a lower consumption level. But a lower level of spending sends the economy into a recession and we get a new equilibrium at a lower level of output. In the end, the increase inautonomous saving is exactly offset by the decrease in induced saving due to the lower income level. In other words, the economy is in equilibrium when S = I o. Since the level of autonomous investment (I o) has not changed, the level of saving at the new equilibrium income level must also equal I o.This can also be derived mathematically. Since an increase in desired saving is equivalent to a decrease in desired consumption, that is, ?C o = -?S o, the effect on equilibrium income is ?Y = [1/(1 - c)](?C o) = [1/(1 - c)](-?S o).Therefore the overall effect on total saving isS = s(?Y) + ?S o = [s/(1 - c)](-?S o) + ?S o = 0, since s = 1 - c.3. "When aggregate demand falls below the current output level, an unintended inventoryaccumulation occurs and the economy is no longer in an equilibrium." Comment on this statement.If aggregate demand falls below the equilibrium output level, production exceeds desired spending. When firms see an unwanted accumulation in their inventories, they respond by reducing production. The level of output falls and eventually reaches a level at which total output equals desired spending. In other words, the economy eventually reaches a new equilibrium at a lower value of output.4. For a simple model of the expenditure sector without any government involvement, derive themultiplier in terms of the marginal propensity to save (s) rather than the marginal propensity to consume (c). Does this formula still hold when the government enters the picture and levies an income tax?In the text, the expenditure multiplier for a model without any government involvement was derived as α = 1/(1 - c).But since the marginal propensity to save is s = 1 - c, the multiplier now becomes α = 1/s = 1/(1-c).In the text, we have also seen that if the government enters the picture and levies an income tax, then the simple expenditure multiplier changes toα = 1/[1 - c(1 - t)] = 1/(1 - c').By substituting s = 1 - c, this equation can be easily manipulated, to getα’ = 1/[1 - c + ct] = 1/[s + (1 - s)t] = 1/s'.Just as s = 1 – c, we can say that s' = 1 - c', sinces' = 1 - c' = 1 - c(1 - t) = 1 - c + ct = s + (1 - s)t.This can also be derived in another way:S = YD - C = YD - (C* + cYD) = - C* + (1 - c)YD = - C* + sYD If we assume for simplicity that TR = 0 and NX = 0, thenS + TA = I + G ==> - C* + sYD + TA = I* + G* ==>s(Y - tY - TA*) + tY + TA* = C* + I* + G* ==>4[s + (1 - s)t]Y = C* + I* + G* - (1 - s)TA* = A* ==>Y = (1/[s + (1 - s)t])A* = (1/s')A*.5. The balanced budget theorem states that the government can stimulate the economy withoutincreasing the budget deficit if an increase in government purchases (G) is financed by an equivalent increase in taxes (TA). Show that this is true for a simple model of the expenditure sector without any income taxes.If taxes and government purchases are increased by the same amount, then the change in the budget surplus can be calculated asBS = ?TA o - ?G = 0, since ?TA o = ?G.The resulting change in national income isY = ?C + ?G = c(?YD) + ?G = c(?Y - ?TA o) + ?G= c(?Y) - c(?TA o) + ?G = c(?Y) + (1 - c)(?G) since ?TA o = ?G.==> (1 - c)(?Y) = (1 - c)(?G) ==> ?Y = ?GIn this case, the increase in output (Y) is exactly of the same magnitude as the increase in government purchases (G). This occurs since the decrease in the level of consumption due to the higher lump sum tax has exactly been offset by the increase in the level of consumption caused by the increase in income.6. Assume a model without income taxes and in which the only two components of aggregatedemand are consumption and investment. Show that, in this case, the two equilibrium conditions Y = C + I and S = I are equivalent.We can derive the equilibrium value of output by setting actual income equal to intended spending, that is, Y = C + I ==> Y = C* + cY + I* ==> (1 - c)Y = C* + I* ==> Y = [1/(1 - c)](C* + I*) = [1/(1 - c)]A*.But since S = YD - C = Y - [C* + cY] = - C* + (1 - c)Y,we can derive the same result fromS = I* ==> S = - C* + (1 - c)Y = I*==> (1 - c)Y = C* + I* ==> Y = [1/(1 - c)](C* + I*) = [1/(1 -c)]A* .7. In an effort to stimulate the economy in 1976, President Ford asked Congress for a $20 billiontax cut in combination with a $20 billion cut in government purchases. Do you consider this a good policy proposal? Why or why not?This is not a good policy proposal. According to the balanced budget theorem, equal decreases in government purchases and taxes will decrease rather than increase income. Therefore theintended result would not be achieved.8. Assume the following model of the expenditure sector:Sp = C + I + G + NX C = 420 + (4/5)YD YD = Y - TA + TR TA = (1/6)YTR o = 180 I o = 160 G o = 100 NX o = - 40(a) Assume the government would like to increase the equilibrium level of income (Y) to thefull-employment level Y*= 2,700. By how much should government purchases (G) be changed?(b) Assume we want to reach Y*= 2,700 by changing government transfer payments (TR)instead. By how much should TR be changed?(c) Assume you increase both government purchases (G) and taxes (TA) by the same lump sumof ?G = ?TA o= + 300. Would this change in fiscal policy be sufficient to reach the full-employment level of output at Y* = 2,700? Why or why not?(d) Briefly explain how a decrease in the marginal propensity to save would affect the size of theexpenditure multiplier.5a. Sp = C + I + G + NX = 420 + (4/5)[Y - (1/6)Y + 100] + 160 + 180 - 40= 720 + (4/5)(5/6)Y + 80 = 800 + (2/3)YFrom Y = Sp ==> Y = 800 + (2/3)Y ==> (1/3)Y = 800 ==>Y = 3*800 = 2,400==> the expenditure multiplier is α = 3From ?Y = α(?A o) ==> 300 = 3(?A o) ==> (?A o) = 100Thus government purchases should be changed by ?G = ?A o = 100.b. Since ?A o = 100 and ?A o = c(?TR o) ==>100 = (4/5)(?TR o) ==> ?TR o = 125.c. This is a model with income taxes, so the balanced budget theorem does not apply in its strictest form,which states that an increase in government purchases and taxes by a certain amount increases national income by that same amount, leaving the budget surplus unchanged. Here total tax revenue actually increases by more than 100, since taxes are initially increased by a lump sum of 100, but then income taxes also change due to the change in income. Thus income does not increase by ?Y = 300, as we can see below.Y = α(?G) + α[(-c)(αTA o) = 3*300 + 3*[-(4/5)300] = 900 - 720 = 180This change in fiscal policy will increase income by only ?Y = 180, from Y0 = 2,400 to Y1 = 2,580, and we will be unable to reach Y* = 2,700.d. If the marginal propensity to save decreases, people spend a larger portion of their additionaldisposable income, that is, the mpc and the slope of the [C+I+G+NX]-line increase. This will lead to an increase in the expenditure multiplier and equilibrium income.9. Assume a model with income taxes similar to the model in Problem 9 above. This time, however,you have only limited information about the model, that is, you only know that the marginal propensity to consume out of disposable income is c = 0.75, and that total autonomous spending is A o = 900, such that Sp = A o + c'Y = 900 + c'Y. You also know that you can reach the full-employment level of output at Y* = 3,150 by increasing government transfers by a lump sum of ?TR = 200.(a) What is your current equilibrium level?(b) Is it possible to determine the size of the expenditure multiplier with the information youhave?(c)Assume you want to change the income tax rate (t) in order to reach the full-employmentlevel of income Y* = 3,150. How would this change in the income tax rate affect the size of the expenditure multiplier?a. Since ?A = c(?TR) = (0.75)200 = 150,the new [C+I+G+NX]-line is of the form Sp1 = 1,050 + c1Y.For each model of the expenditure sector we can derive the equilibrium level of income by using the following equation: Y* = αA o = 1/(1-c’) ==> 3,150 = α1,050 ==> the expend iture multiplier is α = 3.If we now change autonomous spending by ?A = 150, then income will have to change by ?Y = α(?A) ==> ?Y = 3*150 = 450.Therefore the old equilibrium level of income must have been Y = 3,150 - 450 = 2,700.b. From our work above we can see that the size of the multiplier is α = 3.c. The new [C+I+G+NX]-line is of the form Sp2 = 900 + c2Y. This new intended spending line intersectsthe 45-degree line at Y = 3,150. Thus the slope of the new intended spending line can be derived as c2= (3,150 - 900)/(3,150) = 5/7.From Y = Sp2 ==> Y = 900 + (5/7)Y ==> (2/7)Y = 900 ==> Y = (7/2)900 = (3.5)900 = 3,150.6The new value of the multiplier is 3.5Sp3,1509002,700 3,150 Y10. Assume you have the following model of the expenditure sector:Sp = C + I + G + NX C = 400 + (0.8)YD I o = 200 G o = 300 + (0.1)(Y* - Y) YD = Y - TA + TR NX o = - 40 TA = (0.25)Y TR o =50(a) What is the size of the output gap if potential output is at Y* = 3,000?(b) By how much would investment (I) have to change to reach equilibrium at Y* = 3,000, andhow does this change affect the budget surplus?(c) From the model above you can see that government purchases (G) are counter-cyclical, thatis they are increased as national income decreases. If youcompare this specification of G with a constant level of G, how is the value of the expenditure multiplier affected?(d) Assume the equation for net exports is changes such that NX o = - 40 is now NX1 = - 40 - mY,with 0 < m < 1. How would this affect expenditure multiplier?a. Sp = 400 + (0.8)YD + 200 + 300 + (0.1)(3,000 - Y) - 40= 1,160 + (0.8)(Y - (0.25)Y + 50) - (0.1)Y = 1,200 + [(0.8)(0.75) - (0.1)]Y = 1,200 + (0.5)Y Y = Sp ==> Y = 1,200 + (0.5)Y ==> (0.5)Y = 1,200 ==>Y = 2*1,200 = 2,400The output gap is Y* - Y = 3,000 - 2,400 = 600.b. From ?Y = (mult.)(?A) ==> 600 = 2(?I) ==> ?I = 300BuS = TA - TR - G = (0.25)(2,400) - 50 - [300 + (0.1)(600)] = 600 - 50 - 300 - 60 = 190BuS* = (0.25)(3,000) - 50 - 300 = 400, so the budget surplus increases by ?BuS = 210.c. If government purchases are used as a stabilization tool, the size of the multiplier should be lower thanif the level of government spending is fixed. In the model of the expenditure sector above, the slope of the [C+I+G+NX]-line is c' = 0.5 compared to c" = 0.6, when government purchases were defined as G = 300.d.With this change, net exports decrease as national income increases. This additional leakage impliesthat the size of the multiplier will decrease. In the model above, the slope of the [C+I+G+NX]-line decreases from c' = (0.5) to c" = (0.5) - m. Therefore the expenditure multiplier will decrease from 1/[1 - (0.5)] to 1/[1 - (0.5) + m].711. Assume you have the following model of the expenditure sector:Sp = C + I + G + NX C = C o + cYD YD = Y - TA + TR TA = TA oTR = TR o I = I o G = G o NX = NX o(a) If a decrease in income (Y) by 800 leads to a decrease in savings (S) by 160, what is the sizeof the expenditure multiplier?(b) If a decrease in taxes (TA) by 400 leads to an increase in income (Y) by 1,200, how large isthe marginal propensity to save?(c) If an increase in imports by 200 (?NX = - 200) leads to a decrease in consumption (C) by800, what is the size of the expenditure multiplier?Recall that the expenditure multiplier for such a simple model can be calculated as:α = 1/(1 - c)a. (?S)/(?Y) = 1 - c = (-160)/(-800) = .2 ==> 1/(1 - c) = 1/(.2) = 5 ==> the multiplier is α = 5.b. From (?Y) = α[-c(?TA o)] ==> (?Y)/(?TA o) = (-c)α = (-c)/(1 - c) ==>(1,200)/(-400) = - 3 = (-c)/(1 - c) ==> -3(1 - c) = -c ==> c = 3/4==> mps = 1 - c = 1/4 = 0.25.c. ?Y = ?C + ?NX = -800 + (-200) = - 1,000==> c = (?C)/(?Y) = (-800)/(-1,000) = .8 ==> multiplier = α = 1/(1 - c) = 1/(.2) = 512. Explain why income taxation, the Social Security system, and unemployment insurance areconsidered automatic stabilizers.Income taxes, unemployment benefits, and the Social Security system are often called automatic stabilizers becausethey reduce the amount by which output changes as a result of a change in aggregate demand. These stabilizers are a part of the structure of the economy and therefore work without any actual government intervention. For example, when output declines and unemployment increases. If we had no unemployment insurance, people out of work would not receive any disposable income and then consumption would drop significantly. But since unemployed workers get unemployment compensation, consumption will not decrease as much. Therefore, aggregate demand may not be reduced by as much as it would have without these automatic stabilizers.13. Assume a simple model of the expenditure sector with a positive income tax rate (t). Showmathematically how an increase in lump sum taxes (TA o ) would affect the budget surplus. From BS = TA - G - TR = tY + TA o - G – TR==> ?BS = t(?Y) + ?TA o = t(mult.)(-c)(?TA o) + ?TA o= t[1/(1 - c + ct)](-c)(?TA o) + ?TA o = ([ -(ct) + 1 - c + (ct)]/[1 - c + (ct)])(?TA o)= (1 - c) /[1 - c + (ct)])(?TA o) > 0, since c < 1In other words, a lump sum tax increase would increase the budget surplus.14. True or false? Why?"A tax cut will increase national income and will therefore always increase the budget surplus." False. Although a tax cut raises national income, not all of the increase in income is spent, nor is it completely taxed away. Income tax revenues fall and the budget deficit rises. Assume the following model of the expenditure sector:Sp = C + I + G + NX I = I oC = C o + cYD G = G oYD = Y - TA +TR NX = NX oTA = TA o + tY BS = TA - G - TRTR = TR o8From Y = Sp ==> Y = C o + c(Y - TA o - tY + TR o) + I o + G o + NX o ==>Y = C o - cTA o + cTR o + I o + G o + NX o + c(1 - t)Y = A o + c'Y ==>Y = [1/(1 - c')]A o with c' = c (1- t)Thus ?Y = [1/(1 - c')][(-c)(?TA o)]and ?BS = t(?Y) + (?TA o) = {[t(-c)]/(1 - c') + 1}(?TA o) ==> = {[-(ct) + 1 - c + (ct)]/[1 - c + (ct)]}(?TA o) = {(1 - c)/[1 - c + (ct)]}(?TA o) > 0 if ?TA > 0. Therefore, if taxes fall, that is, if ?TA < 0, the budget surplus decreases.15. Assume a simple model of the expenditure sector with a positive income tax rate (t). Showmathematically how a decrease in autonomous investment (I o ) would affect the budget surplus.A decrease in autonomous investment (I o) will have a multiplier effect and will therefore decrease national income and tax revenue. The budget surplus will decrease as shown below: ?BS = t(?Y) = tα(?I o) < 016. "An increase in government purchases will always pay for itself, as it raises national income andhence the government's tax revenues." Comment on this statement.An increase in government purchases will increase the budget deficit. If we assume a model of the expenditure sectorwith income taxes, then the multiplier equals [1/(1 - c')] with c' = c (1- t). The change in the budget surplus that arises from a change in government purchases can be calculated as ?BS = t(?Y) - ?G = t[1/(1 - c')](?G) - ?G = {[t - 1 + c - (ct)]/[1 - c + (ct)]}(?G) = - {[(1 - c)(1 - t)]/(1 - c + (ct)]}(?G) < 0, sine ?G > 0.Therefore, if government purchases are increased, the budget surplus will decrease.17. Is the size of the actual budget surplus always a good measure for determining fiscal policy?What about the size of the full-employment budget surplus?The actual budget surplus has a cyclical and a structural component. The cyclical component of the budget surplus changes with changes in the level of income whether or not any fiscal policy measure has been implemented. This implies that the actual budget surplus also changes with changes in income and is therefore not a very good measure for assessing fiscal policy. The structural (full-employment) budget surplus is calculated under the assumption that the economy is at full-employment. It therefore changes only with a change in fiscal policy and is a much better measure for fiscal policy than the actual budget surplus. One should keep in mind, however, that the balanced budget theorem implies that the government can stimulate national income by an equivalent and simultaneous increase in taxes and government purchases, thereby affecting the actual or the full-employment budget surplus.18. Assume a model of the expenditure sector with income taxes, in which people who pay taxes,have a higher marginal propensity to consume than people who receive government transfers, and the consumption function is of the following form: C = C o + c(Y - TA) + dTR, withc < d.(a) What will happen to the equilibrium level of income and the budget surplus if governmentpurchases are reduced by the same lump sum amount as taxes?(b) W hat will happen to the equilibrium level of income and the budget surplus if governmenttransfers are reduced by the same lump sum amount as taxes?a. Assume that ?TA o = ?G = - 100 ==>Y = [(-c)/(1 - c')(?TA o) + [1/(1 - c')](?G) = [(1 - c)/(1 - c')](-100) < 0 c' = c(1 - t)National income would decrease.BS = t(?Y) + ?TA o - ?G = t(?Y) < 09The budget surplus would decrease by the loss in income tax revenue.b. Assume that ?TA o = ?TR o = - 100 ==>Y = [(-c)/(1 - c')](?TA o) + [d/(1 - c')](?TR o) = [(d - c)/(1 - c')](-100) < 0 c' = c(1 - t)National income would increase.BS = t(?Y) + ?TA o - ?TR o = t(?Y) < 0The budget surplus would decrease.19. True or false? Why?"The higher the marginal propensity to import, the lower the size of the multiplier."True. Imports represent a leakage out of the income flow. An increase in autonomous spending will raise income and we will see the usual multiplier effect. However, if imports are positively related to income, this effect is reduced since higher imports reduce the level of domestic demand.Closed Economy Model Open Economy ModelSp = C + I + G Sp = C + I + G + NXC = C o + cY C = C o + cYG = G o G = G oI = I o I = I oNX = NX o - mY with m > 0From Y = Sp ==>Y = (C o + I o + G o) + cY Y = (C o + I o + G o + NX o) + (c - m)YY = A o + cY Y = A o + (c - m)YY = [1/(1 - c)]A o Y = [1/(1 - c + m)]A oTherefore the multiplier is defined as[1/(1 - c)] [1/(1 - c + m)]Clearly the open economy multiplier falls short of the closed economy multiplier. This is because leakages reduce demand. If income taxes were included in these models, they too would reduce the multipliers, as income taxes represent another leakage from the income flow.10。

后香农时代的十大数学问题

后香农时代的十大数学问题Throughout history, mathematicians have always been fascinated by unsolved problems that challenge the limits of human knowledge and ingenuity. The ten mathematical problems proposed by the Clay Mathematics Institute in 2000, known as the Millennium Prize Problems, are some of the most difficult and significant challenges in the field of mathematics today. These problems, which include the Riemann Hypothesis, the P vs NP Problem, and the Birch and Swinnerton-Dyer Conjecture, represent the cutting edge of mathematical research and have the potential to revolutionize our understanding of the natural world.在数学史上，数学家们总是被未解之谜所吸引，这些问题挑战着人类知识和智慧的极限。

克莱数学研究所于2000年提出的十大数学问题，被称为千禧年大奖问题，它们是当今数学领域中最困难和重要的挑战之一。

这些问题包括黎曼假设、P vs NP问题和伯奇-斯温顿-戴尔猜想，代表着数学研究的前沿，并有可能彻底改变我们对自然界的理解。

One of the most famous of these problems is the Riemann Hypothesis, which concerns the distribution of prime numbers andhas baffled mathematicians for over 150 years. The hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part equal to 1/2. If proven true, the Riemann Hypothesis would have far-reaching implications for number theory, cryptography, and computer science. Many mathematicians have tried and failed to prove the hypothesis, making it one of the most tantalizing unsolved problems in mathematics.其中最著名的问题之一是黎曼假设，它涉及素数的分布，困扰了数学家150多年。

2022年考研考博-考博英语-西南政法大学考试全真模拟易错、难点剖析AB卷(带答案)试题号：27

2022年考研考博-考博英语-西南政法大学考试全真模拟易错、难点剖析AB卷（带答案）一.综合题(共15题)1.单选题The high cost for getting there was（）by the fact that I found a very cheap place to live. 问题1选项A.offsetB.participatedC.elaboratedD.reclaimed【答案】C【解析】动词辨析题。

A选项offset“抵消，弥补”；B选项participate“参与，分享”；C选项elaborate“详细描述，变复杂”；D选项reclaim“开拓，抗议”。

句意：到这里的成本高是因为事情变得复杂，我找到了一个非常便宜的地方住。

选项C更符合语境。

2.单选题The most successful way to solve the language problem while a foreign play is being performed is（）translation.问题1选项A.homogeneousB.simultaneousC.instantaneousD.spontaneous【答案】B【解析】形容词辨析题。

A选项homogeneous“均匀的，同质的”；B选项simultaneous“同时的”；C选项instantaneous“瞬间的”；D选项spontaneous“自发的，无意识的”。

句意：解决外交活动中的语言问题最成功的方法是同声传译。

选项B符合句意。

3.单选题Most people in the north wanted to（）slavery while the slave owners of the south held that slavery should be allowed.问题1选项A.extinguishB.vanishC.abolishD.withdraw【答案】C【解析】动词辨析题。

2023-2024学年安徽六安人教版高考专题英语高考复习习题及解析

2023-2024学年安徽六安人教版高考专题英语高考复习1.填空题第1题.根据句意以及所给提示词，完成句子（1）The volunteers tried to calm the ________（alarm）children down.（2）________ (attract) by the sweet voice, l walked into the old house.（3）Nobody likes ________ (make) fun of.（4）目击者称他在昨天晚上八点看到那个失踪的小男孩正在快速地穿过这条马路。

(see sb. doing sth.)The witness says he ________ at eight o'clock yesterday night.（5）我们是时候要下定决心改掉熬夜的习惯。

(It's time…) ________ to break the habit of staying up late.（6）The key is to know to prepare yourself for the event and how to keep yourself ________ (focus).（7）lt is the most instructive lecture that l ________ (attend) since l came to this school.（8）I'd ________ if you could confirm your involvement at your early convenience.如果你能在方便的时候确认你的参与，我将不胜感激。

（9）他是如此聪明的一个孩子以至于老师们都很喜欢他。

(so/such....that)①________ that teachers like him very much.②________ that teachers like him very much.【答案】'（1）alarmed（2）Attracted（3）being made（4）saw the boy going across the road quickly'"（5）It's time for us"'（6）focused（7）have attended（8）appreciate it（9）He is so clever a boy; He is such a clever boy'【解答】（1）alarmed 考查形容词。

多恩布什宏观经济学第十版课后习题复习资料08

•CHAPTER 8Solutions to the Problems in the Textbook:Conceptual Problems:1. The first question you should ask yourself as a policy maker is whether a disturbance is transitory orpersistent. You should then ask yourself how long it would take to put a suggested policy measure into effect and how long it will take for the policy to have the desired effect on the economy. In addition, you need to know how reliable the estimates of your advisors are about the effects of the policy. If a disturbance is small and probably transitory, you may be best advised to do nothing, because any measure you take is likely to have its effect after the economy has recovered. Therefore your action might only further aggravate the problem.2.a. The inside lag is the time it takes after an economic disturbance has occurred to recognize andimplement a policy action that will address the disturbance.2.b. The inside lag is divided into three parts. First, there is the recognition lag, that is, the time it takes forpolicy makers to realize that a disturbance has occurred and that a policy response is warranted.Second, there is the decision lag, that is, the time it takes to decide on the most desirable policy response after a disturbance is recognized. Finally, there is the action lag, that is, the time it takes to actually implement the policy measure.2.c. Inside lags are shorter for monetary policy than for fiscal policy since the FOMC meets on a regularbasis to discuss and implement monetary policy. Fiscal policy, on the other hand, has to be initiated and passed by both houses of the U.S. Congress and this can be a lengthy process. The exceptions are the so-called automatic stabilizers; however, they only work well for small and transitory disturbances2.d Automatic stabilizers have no inside lag; they are endogenous and function without specificgovernment intervention. Examples are the income tax system, the welfare system, unemployment insurance, and the Social Security system. They all reduce the amount by which output changes in response to an economic disturbance.3.a. The outside lag is the time it takes for a policy action, once implemented, to have its full effect on theeconomy.3.b. Generally, the outside lag is a distributed lag with a small immediate effect and a larger overall effectover a longer time period. The effect is spread over time, since aggregate demand responds to any1 / 1policy change only slowly and with a lag.3.c. Outside lags are longer for monetary policy since monetary policy actions affect short-term interestrates most directly, while aggregate demand depends heavily on lagged values of income, interest rates, and other economic variables. A change in government spending, however, immediately affects aggregate demand.4. Fiscal policy has smaller outside lags, but significant inside lags. Monetary policy, on the other handhas smaller inside lags and longer outside lags. Therefore large open market operations should be undertaken to get an immediate effect, but they should be partially reversed over time to avoid a large long-run effect. If the shock is sufficiently transitory and small, policy makers may be best advised not to undertake any policy change at all.5.a. An econometric model is a statistical description of all or part of the economy. It consists of a set ofequations that are based on past economic behavior.5.b. Econometric models are generally used to forecast the behavior of the economy and the effects ofalternative policy measures.5.c. There is considerable uncertainty about how well econometric models actually represent the workingsof the economy. There is also great uncertainty about the expectations of firms and consumers and their reactions to policy changes. Any policy is bound to fail if the information on which it was based is poor.6.The answer to this question is student specific. The main difficulties of stabilization policy arise fromthree sources. First, policy always works with lags. Second, the outcome of any policy depends on the way the private sector forms expectations and how those expectations affect the public's behavior.Third, there is considerable uncertainty about the structure of the economy and the shocks that hit it.It can be argued that a monetary policy rule would greatly reduce uncertainty about the Fed's policy responses. If the government behaved in a consistent way, then the private sector would also behave more consistently and economic fluctuations could be greatly reduced. A monetary growth rule would also reduce any political pressure the administration might exert on the Fed. It is often initially unclear whether a disturbance is temporary or persistent and a monetary policy rule would prevent policy mistakes in cases where the disturbance is, in fact, temporary. If active monetary policy is applied to a temporary disturbance, then the lags involved will guarantee that the economy will1 / 1actually be destabilized.On the other hand, the workings of the economy are not completely understood and events cannot always be predicted. Thus it is difficult to argue for a fixed policy rule. Unanticipated large disturbances warrant an activist policy, especially if they appear to be persistent. It is also possible to construct a more activist monetary growth rule. For example, Equation (8) suggests that the annual monetary growth rate should be increased by two percent for every one percent that unemployment increases above its natural rate. Such a rule is based on the quantity theory of money equation (which relates money supply growth to the growth of nominal GDP) and on Okun's law (which relates the unemployment rate to economic growth). Obviously, because of the long lags for monetary policy, any monetary growth rule will work much better in the long run than in the short run.Fiscal policy rules may make more sense than monetary policy rules, since fiscal policy has long inside lags but shorter outside lags. In a way, built-in stabilizers, although generally not considered "rules", already provide some stability without any inside lag. Many of the arguments against monetary policy rules are also valid for fiscal policy rules and many economists oppose them. The frequently proposed constitutional amendment requiring an annually balanced budget is an example of a fiscal policy rule. There are significant problems associated with such an amendment, since it would greatly limit the government's ability to undertake active fiscal stabilization policy.7. The arguments for a constant growth rate rule for money are based on the quantity theory of moneyequation, that is,MV = PY.From this equation we can derive%∆P = %∆M - %∆Y + %∆V.If the long-run trend rate of real output (Y) and the long-run trend of velocity (V) are assumed to be fairly stable, and if wages and prices are sufficiently flexible, then a constant monetary growth rate (M) would insure a constant rate of inflation, that is, a constant rate of change in the price level (P).Also, since monetary policy has long outside lags, active monetary policy can actually be more destabilizing than stabilizing. In addition, since we do not know exactly how the economy works or may react to specific policies, it is best to follow a rule rather than undertake actions that have uncertain outcomes. However, rules are not without problems, as they would not allow flexibility in responding to major disturbances.1 / 18. Dynamic inconsistency occurs if, after having committed themselves to a specific policy actiondesigned to achieve a long-run objective, policy makers find themselves in a situation where it seems advantageous to abandon their original policy, in order to achieve a short-run goal. Such action will impede the long-run objective.9. Real GDP targeting is the best option if the primary policy goal of monetary policy is to achieve fullemployment. If policy makers forecast potential GDP correctly, then full employment combined with low inflation can be achieved. However, real GDP targeting bears the greater risk that the secondary goal of achieving a low inflation rate will be missed. If the rate at which potential GDP grows is overestimated, then policy makers may stimulate the economy too much. In this case, they will not be successful in achieving price stability. By targeting nominal GDP, the central bank creates a policy tradeoff between inflation and unemployment. If the rate at which potential GDP grows is overestimated and policy makers stimulate the economy too much, we will get less growth but also less inflation than under real GDP targeting. Which targeting approach should be chosen depends greatly on how steep or flat the Phillips curve is perceived to be.Technical Problems:1. If actual GDP is expected to be $40 billion below the full-employment level and the size of thegovernment spending multiplier is 2, then government spending should be increased by $20 billion over its current level. For the next period, when actual GDP is expected to be $20 billion below potential, government spending should be cut by $10 billion from its new level, that is, to $10 billion over its original level. In period three, when actual GDP is expected to be at its full-employment level, the level of government spending should again be cut by $10 billion from the last period's level to bring it back to the original level of Period 0.2.a. If there is a one-period outside lag for government spending, then nothing can be done to close thecurrent GDP-gap. The government should decide to spend $10 billion more for the next period and reduce spending again to its original level after that.2.b. Graph I below shows the path of GDP for Problem 1 with no outside lag and Graph II shows the pathof GDP for problem 2.a. with a one-period outside lag. In each of the graphs the path of actual GDP is shown, first assuming that no policy action takes place and then assuming that the policies proposed in Problems 1 and 2.a. are undertaken.1 / 1Graph IGDP GDPpotential GDP potential GDP0 time 0 timeGDP with fiscal policy GDP without fiscal policyGraph IIGDP GDPpotential GDP potential GDP0 time 0 timeGDP with fiscal policy GDP without fiscal policy3.a. Since the government multiplier for the first period is 1, the level of government spending must beincreased by G = $40 billion to close the GDP-gap of $40 billion. But since the government multiplier in the next period for the amount spent in this period is 1.5, the effect of an increase in government spending in the first period by $40 billion would be an increase in GDP by $60 billion in the second period.3.b. For the second period a GDP-gap of $20 billion is expected. However, as we saw in 3.a., GDP willincrease by $60 billion in the second period if the government increases spending by $40 billion in the first period. Therefore, the government has to reduce spending in the second period by $40 billion from its new level (back to its original level), since the multiplier for a spending change in the same period is 1.3.c. In this problem, fiscal policy has an outside lag. This means that the effect of an increase ingovernment spending is felt both in the period in which the spending increase takes place and (to an even larger degree) in the following period. The increase in government spending needed to close the1 / 1GDP-gap in the first period is guaranteed to overshoot the desired goal in the next period. Thus the government will be forced to reverse its increase in spending to the original level in the second period to offset the destabilizing effect. In a case like this, the government has to be much more active in its fiscal policy than in a situation where no distributed lag exists.4. If there is uncertainty about the size of the multiplier, then fiscal policy becomes much morecomplicated. If the multiplier is 1, then an increase in government spending by $40 billion will close the GDP-gap in the first period. If the multiplier is 2.5, we will overshoot potential GDP by $60 billion. An increase in spending by 40/2.5 = $16 billion is optimal if the multiplier is 2.5. Thus a cautious government will probably increase spending by no more than $16 billion in the first period, and then reduce the level of spending by $8 billion in the next period ($8 billion above the original level). Such a policy action is designed to close the GDP-gap to some degree over the first two periods while never overshooting potential GDP. In Period 3 we will again be back at the full-employment level. The extent to which a less cautious government might exceed these suggested spending increases depends largely on that government's level of concern about unemployment versus inflation.5. To follow an established rule for its policy, the Fed needs to know the source of each disturbance. If adisturbance comes from the goods sector, it is better to have a monetary growth target; if the disturbance comes from the money sector, it is better to have an interest rate target.a. Assume a disturbance comes from the money sector. If an increase in money demand increases theinterest rate, the Fed should try to maintain a constant interest rate by increasing the supply of money.This will re-establish the old equilibrium values of the interest rate and output and effectively offset the disturbance.b. Assume a disturbance comes from the goods sector. If an increase in autonomous investmentincreases the interest rate, then it is not advisable to maintain a constant interest rate. Trying to lower the interest rate again by increasing the money supply would aggravate the disturbance. On the other hand, maintaining a constant money supply, while not offsetting the disturbance, will at least not make things worse.6.a. Students will have to check the Federal Reserve Bulletin in early 2000 and compare the forecasts ofthe Federal Reserve Board with the actual performance of the economy in 1999.6.b. Regardless of how detailed it is, no econometric model can accurately represent the economy, sincewe do not completely understand the way the economy works. Therefore, we can never expect perfect1 / 1forecasts. It is impossible to incorporate all the relevant information on which individuals and firms base their expectations about the future and to determine how these expectations affect actions in any given situation. Forecasts are generally based on the information available at the time, which may be flawed or outdated. In addition, any unexpected change, such as a supply shock, an unanticipated international change, or an unanticipated domestic policy change, can render the initial predictions wrong.1 / 1。

Goldbach's Rule

a rX iv:mat h /5191v1[mat h.GM ]19May2GOLDBACH‘S RULE by Metin Aktay with thanks to Cliﬀord H.Taubes for considering the pre-posterous Content: A.Overview B.List of Deﬁnitions and Variables C.The Proof of Goldbach‘s Conjecture D.Evaluation of Observations and Further Thought Appendix A A.OVERVIEW Goldbach‘s Conjecture,”every even number greater than 2can be expressed as the sum of two primes”is renamed Goldbach‘s Rule for it can not be other-wise.The conjecture is proven by showing that the existence of prime pairs adding to any even number greater than 2is a natural by-product of the existence of the prime sequence less than that even number.First it is shown that the remain-der of cancellations process which identiﬁes primes less than an even number also remainders prime pairs adding to that even number as a natural part of the process.Then a minimum limit for the number of remaindered prime pairs adding to an even number is expressed in terms of that even number and shown to exist for every even number greater than 2.Furthermore,the reasonings and formulations used in the proof are demonstrated to hold against observations.B.LIST OF DEFINITIONS AND VARIABLES Let E be any even number >2.Let N be any positive integer <E.Let i,n be counters,each of integers in natural order beginning with 1.Let a number couple be a couple where the order of two integers matters.Let a number pair be a pair where the order of two integers does not matter.Let N symmetric integers be two integers having identical absolute diﬀerence with N .Let P (i )be any prime ≤(E −1).Let P (i )-prime be indivisibility by P (i )where P (i )is not indivisible and 1is indivisible.Let P (i )-composite be divisibility by P (i )where P (i )is divisible.Let G 1be the number E/2symmetric N which are P (i )-prime for all P (i ).Let G 2be the number of E/2symmetric primes adding to E.Let GP be the number of prime pairs adding to E .Let r.f.P (i )be the E/2symmetric P (i )-prime remaindering frequency for E of1any P(i)divisor.Let stsp.m()be the step truncated series product with steps from i=1to i=m.Let P E be the largest prime<E.Let NP E be the number of primes≤P E.Let GR be Goldbach Ratio,(GP/NP E).C.THE PROOF OF GOLDBACH‘S CONJECTUREC.1The”Remainder”Nature Of The Prime SequencePrimeness of an integer is divisibility by no other than unity and itself.The prime sequence is identiﬁed by cancelling divisibilities and retaining indivisibil-ities.The integers not cancelled as divisible constitute the prime sequence.It is crucial to note that this identiﬁcation process is an indirect process rather than direct,that primes are remainders,not direct creations but remnants after cancellations.C.2Method Of Identiﬁcation Of The Prime Sequence Less Than EThe Sieve of Eratosthenes identiﬁes the prime sequence up to any integer by cancelling divisibilities by prime divisors less than the square root of that integer and thus”remainder”ing indivisibilities.This suﬃces because a prime larger than that square root is multiplied by a prime less than that square root to produce any composite less than that integer.Therefore the sequence of primes less than E are those N which are not divisible by the divisors P(i) dividing with frequency P(i),where P(i)were deﬁned to be primes≤The above may be visualised as matched integers on two integer lines matched head to tail,(E−1)to1and1to(E−1).These matched integers constitute couples adding to E.If both members of such couples are prime then they are E/2symmetric primes,if both are composites then they are E/2symmetric composites.Couples with one prime and one composite members contain E/2 asymmetric primes and E/2asymmetric composites.C.4Concepts of E/2Symmetricity And Asymmetricity Of P(i)DivisorsAny P(i)divisor dividing N divides either symmetrically or asymmetrically with respect to E/2.E/2symmetric P(i)divisors divide E/2and(E/2)+nP(i) and(E/2)−nP(i).E/2asymmetric P(i)divisors never divide E/2symmetric N.E/2symmetricity of any P(i)divisor depends on divisibility of E/2by that P(i).If E/2is divisible by a P(i)then that P(i)divides and”remainder”s symmetrically with respect to E/2.If E/2is not divisible by a P(i)then that P(i)divides asymmetrically with respect to E/2.This dependence applies to all P(i)except P(1)=2which is E/2symmetric independent of E/2divisibility, with either two divisibilities or two indivisibilities bracketing E/2.If all P(i)divisors were E/2symmetric,then both composite and prime N would be E/2symmetric,and thus all primes<E would be members of prime couples adding to E,except for P(i)themselves which are symmetric with com-posites divided by themselves.If all P(i)divisors were to be E/2asymmetric, except for P(1)=2which can not be so,then there would be minimal E/2 symmetric composites and primes.C.5Concept Of P(i)-Primes And The Primes As Inter-secting Sets Of P(i)-primesIf being P(i)-prime is indivisibility by P(i),where1is deﬁned P(i)-prime and P(i)itself is deﬁned divisible,there would be1P(i)-composite and(P(i)−1) P(i)-primes every P(i)consecutive N for any P(i)divisor.The N which are P(i)-prime for all P(i)can not be but prime except for1.Therefore,the primes <E are the N which are P(i)-prime for all P(i),with the addition of the P(i) themselves the deduction of1.3C.6Existence And Frequency Of E/2Symmetric P(i)-primes For Any P(i)Any E/2symmetric P(i)divisor will divide1E/2symmetric P(i)-composite and remainder(P(i)−1)E/2symmetric P(i)-primes every P(i)consecutive N.Any E/2asymmetric P(i)divisor will divide1E/2asymmetric P(i)-composite, and will remainder1E/2asymmetric P(i)-prime and(P(i)−2)E/2symmetric P(i)-primes for every P(i)consecutive N.The above may be visualised with the two lines of integers matched head-to-tail.Any P(i)divisor cancelling P(i)-composites and remaindering P(i)-primes on each line begins dividing from opposite ends,thus may or may not meet at the midpoint since that P(i)may or may not be E/2symmetric.Let the two lines be counted by n counting from one end only,beginning with n=1,which counts one line forward and the other line backwards.If the cancellations by any P(i)divisor on each line is accounted for with respect to n,then the frequency with respect to n of divisibility by that P(i)of both lines is1/P(i)since both lines move by1for every unit change in n.However,for an E/2asymmetric P(i)divisor,the divisibility of each line by that P(i)may lead or lag the other line with respect to n.If a P(i)divisor is E/2symmetric it will divide one matched integer couple and remainder((P(i)−1)matched integer couples as P(i)-primes every P(i)n.If a P(i)divisor is E/2asymmetric it will divide1 P(i)-composite on theﬁrst line matched with a P(i)-prime on the second,and 1P(i)-composite on the second line matched with a P(i)-prime on theﬁrst,and will remainder(P(i)−2)couples with P(i)-primes on each line every P(i)n.P(1)=2divisor can not be E/2asymmetric.P(1)=2would render all 2-Primes E/2asymmetric if it could be E/2asymmetric.P(1)=2divisor will always remainder1E/2symmetric2-Prime for every2consecutive N.E/2asymmetric P(2)=3divisor will remainder1E/2symmetric3-prime for every3consecutive N and E/2symmetric P(2)=3will remainder2.P(i) divisors larger than3will remainder more than1E/2symmetric P(i)-primes for every P(i)consecutive N,i.e.E/2asymmetric P(3)=5divisor will re-mainder3E/2symmetric5-primes every5consecutive N and E/2symmetric P(3)=5will remainder4.Therefore E/2symmetric P(i)-primes exist for each P(i)divisor of E.4C.7Lower Limit For The Number of E/2symmetric P(i)-Primes for all P(i)Toﬁnd the number of E/2symmetric P(i)-primes for a P(i)divisor,the number of N,which is E−1,is multiplied with the E/2symmetric P(i)-prime remaindering frequency of that P(i).Thereforenumber of E/2symmetric P(i)−primes=(E−1)∗r.f.P(i) This may result in a fractional result which may need truncation or rounding up.Thereforenumber of E/2symmetric P(i)−primes≥truncate((E−1)∗r.f.P(i))Toﬁnd a lower limit for G1,the number of N which are E/2symmetric P(i)-prime for all P(i),E−1is multiplied consecutively by r.f.P(i)with trun-cations at each step.This reduction with consecutive remaindering frequencies is valid because P(i)are indivisible by each other.ThereforeG1≥stsp.m((E−1)∗r.f.P(i))Expectation of G1will be minimum where all r.f.P(i)are minimum. Thereforemin.G1≥stsp.m((E−1)∗sp(min.r.f.P(i))).All r.f.P(i)will be minimum where all P(i)divisors are E/2asymmetric. Thereforemin.r.f.P(i)=((P(i)−2)/P(i)),except for r.f.P(1)=1/2.If any product((a)∗(b/c)),where a,b,c are positive integers,and where c≤a,is truncated,then the result can not be less than b.With this logic,lower limits can be found for the successive truncations of the step truncated series product by comparing the denominator of each multiplier with the numerator of the multiplied.Let stsp.m((E−1)∗sp(min.r.f.P(i)))begin from min.r.f.P(m)and work backwards,and let theﬁrst truncation result be T(1)and the last truncation result be T(m),where T(m)≤min.G1.ThereforeT(1)=truncate((E−1)∗min.r.f.(P(m))).Given that except for P(1)=2min.r.f.P(i)=((P(i)−2)/P(i))ThenT(1)=truncate((E−1)∗((P(m)−2)/P(m)))Given that(E−1)≥(P(m))2ThenT(1)≥(P(m)∗(P(m)−2)).5Given thatT(2)=truncate(T(1)∗min.r.f.(P(m−1)))andmin.r.f.P(m−1)=((P(m−1)−2)/P(m−1))ThenT(2)=truncate(T(1)∗(P(m−1)−2)/P(m−1))ThenT(2)≥truncate(P(m)∗(P(m)−2)∗(P(m−1)−2)/P(m−1)) Given thatP(i)≤(P(i+1)−2)except for P(1)=2ThenP(m−1)≤(P(m)−2)ThereforeT(2)≥(P(m)∗(P(m−1)−2)).If lower limits for T(n)are found consecutively as aboveThenT(n)≥(P(m)∗(P(m−n+1)−2)).ThereforeT(m−1)≥((P(m))∗(P(2)−2)).Since P(2)=3thenT(m−1)≥((P(m))Given that r.f.P(1)=1/2,thenT(m)≥truncate((P(m))∗1/2)Thereforemin.G1≥truncate(P(m)/2).C.8Lower Limit For The Number Of Prime Pairs Adding To ESince1is deﬁned P(i)-prime for all P(i),G1,which is the number of N E/2 symmetric P(i)-prime for all P(i),may count1if the E/2symmetric counter-part of1is also P(i)prime for all P(i).Given that G2is the number of E/2 symmetric primes adding to E,and given that G1may count1and(E−1), and given that G1excludes P(i),and given that min.G1is a truncated result, thenG2≥min.G1−(2or0)+2∗(number of P(i)adding to E with other primes) Thereforemin.G2≥(min.G1−2).Thereforemin.G2≥(truncate(P(m)/2)−2)GP,the number of prime pairs adding to E,is half of G2since every E/2sym-metric prime is a member of a pair of primes adding to E.If the halving of G2 is fractional,then it is rounded up since an odd numbered G2indicates that E/2itself is prime and is counted once.G2is even numbered where E/2itself6is not prime.Thereforemin.GP≥rounded−up(1/2∗min.G2).Thenmin.GP≥rounded−up(1/2∗(truncate(P(m)/2)−2)). Thereforemin.GP≥1for E≥50where P(m)≥7.The logic of the above proof also proves that there is at least one prime pair adding to E with primes>P(m)for every E≥50.Given the existence of at least one prime pair adding to E for every E<50, thenmin.GP≥1for every E.ThereforeGP≥1for every E.Magnitude of min.GP∼=√E for large E.C.9Conclusion Of The Proof of Goldbach‘s ConjectureThus it is proven that the remaindering of the prime sequence<E by divi-sors P(i)≤E/4for large E.Therefore there will always be E/2symmetric prime pairs adding to E for any E since the remaindering process for the prime sequence<E can not avoid remaindering E/2symmetric primes.Therefore there will be at least one pair of E/2symmetric primes adding to E for any E.Therefore every even number greater than2can be expressed as the sum of two primes.7D.EVALUATION OF OBSERVATIONS AND FURTHER THOUGHTD.1Table Of Observations And CalculationsBehaviour with respect to E of the number of prime pairs adding to E may be deduced from Appendix A,Observations and Calculations,where the formu-lations utilised above and actual counts are tabulated for sample E evenly and saliently spread up to10,000.D.2Validation Of Assumed RelationshipsGP may be approximated as(E/2)∗sp(r.f.P(i)),which would undercount since it fails to check the P(i)excluded by r.f.P(i),and would undercount since it omits rounding up where needed,and would overcount since it omits trunca-tion where needed,and would overcount since it may count1.In Appendix A, GP∼=(E/2)∗sp(r.f.P(i))is calculated as is without the reduction rationalisa-tions used in the proof.It is observed that the proportion of error in this raw term decreases with larger E.More important than decreasing error is that itis observed to trackﬂawlessly the volatility with respect to E of actual GP.D.3Observations and Evaluations Of Goldbach RatioAppendix A shows that GR varies as expected.GR is low where E/2is indivisible by smaller P(i)and high where E/2is divisible by smaller P(i), since divisibility of E/2by smaller P(i)increases E/2symmetricity of primes substantially.E with E/2prime or divisible only by P(1)=2have low GR.All E withE/2divisible by3have high GR because avoidance of an E/2asymmetric3 divisor doubles symmetricity of primes.For example,E=210has a high GR, 41%,since E/2=105is divisible by3,5,7,which thus remainder primes sym-metrically with respect to E/2.A GR value of41%is considered high since GR is deﬁned by utilising pairs of primes adding to E,which means that the maximum possible GR is50%,where every prime is a member of a pair of primes adding to E.D.4Deductions In Relation To The Goldbach CometE/2symmetricity of P(i)divisors,which is divisibility of E/2by P(i),ex-plains the dense cluster bands which form when GP is plotted against E,a plot called the Goldbach Comet on account of these cluster bands.The densest asymptotic cluster band is formed by E with E/2asymmetric smaller prime divi-sors.E/2symmetricity of smaller prime divisors explain other bands.The next8dense band are the E with E/2symmetric P(2)=3divisor,at(2/3)/(1/3)=2 times the heights of the asymptotic lowest band.Then the next dense band are the E with E/2symmetric P(3)=5divisor,at(4/5)/(3/5)=4/3times the heights of the asymptotic lowest band.Joint E/2symmetricity of the smaller primes also form distinct dense bands,such as the E with E/2symmetric P(2)=3and P(3)=5divisors,at(2/3∗4/5)/(1/3∗3/5)=8/3times the heights of the asymptotic lowest band.D.5Further ThoughtThe number of prime pairs adding to E increases with E but with high volatility.It could be that this volatility isﬂuctuation around a fundamental relationship.Assuming the prime sequence to be a discrete wave function,and assuming the estimate N/lnN of J.S.Hadamard to be it‘s frequency for N,this fundamental relationship is likely to be GP=E/(2∗(lnE)2).Metin AktayIhsan Aksoy sok.EVA apt.No:7/2,Camlik,Etiler,Besiktas,Istanbul80630, TurkeyPhone:+902122651016Fax:+902122577374Mobile:+905322741771E-mail:maktay@,or maktay@9APPENDIX A:OBSER V ATIONS AND CALCULATIONSE NPE E/2observed calcul.≤P(m)calcul.GP 1271121033199132,3,5,741-1121113P23-5549919P16-3350919211-967723323-1499731P11-111,02131213-271,49937514-92,01743P9-42,039432882,477473,526-103,01953P10-123,51159710-24,00361P9-124,0936129-114,679675,715-135,00367P9-115,591732,3,5,11,1727-65,98773P826,577792,11,13,2310-37,01983P8-37,309832,3,23,5318-97,99389P8-38,191892758,609892,3,5,7,4126-29,01389P9-89,497973,521-49,97397P8-3 Sample calculations for GP calculated as((E−1)/2)∗sp(r.f.P(i))GP for E=2490:(2,489/2)∗(1/2∗2/3∗4/5∗5/7∗9/11∗11/13∗15/17∗17/19∗21/23∗27/29∗29/31∗35/37∗39/41∗41/43∗45/47)GP for E=3022:(3021/2)∗(1/2∗1/3∗3/5∗5/7∗9/11∗11/13∗15/17∗17/19∗21/23∗27/29∗29/31∗35/37∗39/41∗41/43∗45/47∗51/53)10。

庞氏与传销

预处理语句作者：德布拉答：情人节，总法律顾问美国联邦贸易委员会关于发表于国际货币基金研讨会试论影响我国中央银行目前的法律问题华盛顿特区1998年5月13日我想感谢发言的机会对于日益恶化的国际问题，您传销。

是什么样的计划，引人注目的是，虽然他们是非常古老的欺诈形式，现代技术大大损害我们的公民为乘以他们的潜力。

在几乎没有时间上网一个多车道的高速公路到世界各地的新兵金字塔建造者特别优惠。

介绍首先，让我告诉你有关委员会，联邦贸易。

（1）该委员会是一个独立的政府机构，国会1914年成立的研究。

我们执行了政府的核心功能- 确保自由市场的工作。

这就要求生产者和消费者手中的准确信息的竞争，以便以最低的价格来产生最好的产品，促进效率和创新，加强经济。

蓬勃发展的竞争，消费者必须对现有产品和服务的认识。

我们的消费者保护局确保消费者在市场信息不欺骗或误导。

自由市场也意味着消费者有竞争力的价格之间的产品和服务的选择。

我们的竞争局确保市场是反竞争的合并和其他不公平商业惯例的自由，例如操纵价格或将零售价格楼层。

随着航空旅行和保险像少数地区外，该委员会已超过几乎所有经济领域的广泛的执法权威。

不幸的是，我们现在看到传销侵入我们的部门监督很多。

什么是金字塔计划和什么是合法营销？传销现在是在如此多的形式，他们可能难以立即承认。

然而，他们都有一个共同的首要特征。

消费者或投资者，他们承诺巨额利润主要招募其他人加入这个计划，而不是从任何实际利润为基础的投资或物品的真实出售给公众的。

有些计划可能意在销售产品，但他们往往简单地使用他们的产品以隐藏金字塔结构。

有两个说，故事中，一个产品，只是被用来掩盖一传销：装载清单和零售缺少的迹象。

当库存载入公司的激励计划部队新兵购买更多的产品比以往任何时候都可以高价卖出往往。

如果发生这种情况在整个公司的分配制度，在金字塔顶端的获利甚丰的人，尽管很少或根本没有移动产品推向市场。

在底层的人们为他们的存货，只是在地下室积累过多付款。

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2
theorem is also valid for q = q1 q2 . Consider Zq as Zq1 ⊕ Zq2 . And deﬁne functions g1 , g2 , g3 over Z∗ q1 by 1 fi ((a, b)). gi (a) = φ(q2 ) b∈Z∗
q2
1 2 3
p1 ,p2 ,p3 prime −3 −2 where S(n) = p∤n (1+(p − 1) ) p|n (1 − (p − 1) ) and A is a positive constant. Nowadays Vinogradov’s theorem has become a classical result in additive number theory. Later, using a similar method, van der Corput [2] proved that the primes contain inﬁnitely many non-trivial 3-term arithmetic progressions (3AP). On the other hand, another classical result due to Roth [8] asserts that a set A of integers contains inﬁnitely many non-trivial 3APs provided that d(A) > 0, where |A ∩ [1, x]| . d(A) = lim sup x x→∞ Roth’s theorem is a special case of the well-known Szemer´ edi theorem [9], which states that any integers set A with d(A) > 0 contains arbitrarily long arithmetic progressions. For a set X of positive integers and its subset A, deﬁne the upper density and lower density of A relative to X by |A ∩ [1, x]| |A ∩ [1, x]| and dX (A) = lim inf . dX (A) = lim sup x→∞ |X ∩ [1, x]| x→∞ |X ∩ [1, x]|
2000 Mathematics Subject Classiﬁcation. Primary 11P32; Secondary 11B05, 11AP70. This work was supported by the National Natural Science Foundation of China (Grant No. 10471090).
b∈Zq2 b∈Z∗ q2
ห้องสมุดไป่ตู้1 (g1 (a) + g2 (a) + g3 (a)) = K, φ(q1 ) a∈Z∗
q1
φ(q2 )K.
This concludes the proof of our induction. Thus we only need to prove Theorem 1.1 when q is the power of a prime. Assume that q = p where p 5 is a prime. Let Si = a=0 fi (a) for i = 1, 2, 3. Clearly S1 + S2 + S3 = (p − 1)K . Assume on the contrary that there exists some n ∈ Zp such that for any x, y, z ∈ Z∗ p with x + y + z = n, f1 (x) + f2 (y ) + f3 (z ) < K. We ﬁrstly consider the case n = 0. Observe that f1 (x) =
q
We shall make an induction on the number of prime divisors of q . First, assuming that Theorem 1.2 holds for two co-prime integers q1 and q2 , we claim that this
1
Let P denote the set of all primes. In [4], Green obtained a Roth-type generalization of van der Corput’s result. Green showed that if P0 is a subset of P with dP (P0 ) > 0 then P0 contains inﬁnitely many 3APs. One major ingredient in Green’s proof is a transference principle, which transfers a subset of primes with relative positive density to a subset of ZN = Z/N Z (where N is a large prime) with positive density. Subsequently, this principle was greatly improved (in a diﬀerent way) in the proof of Green and Tao’s celebrated theorem [5] that the primes contains arbitrarily long arithmetic progressions. The Hardy-Littlewood circle method [11] is commonly applied in Vinogradov’s, van der Corput’s, Roth’s and Green’s proofs. So in this paper, we shall use Green’s idea to extend the Vinogradov theorem as follows. Theorem 1.1. Suppose that P1 , P2 , P3 are three subsets of P with dP (P1 ) + dP (P2 ) + dP (P3 ) > 2. Then for suﬃciently large odd integer n, there exist p1 ∈ P1 , p2 ∈ P2 and p3 ∈ P3 such that n = p1 + p2 + p3 . Notice that the result of Theorem 1.1 is the best possible in the following sense: Letting P1 = P2 = {p ∈ P : p ≡ 1 (mod 3)} and P3 = P \ {3}, then dP (P1 ) = dP (P2 ) = 1/2 and dP (P3 ) = 1, but 6k + 5 ∈ P1 + P2 + P3 for any integer k . For a positive integer q , let Zq = Z/q Z and Z∗ q = {b ∈ Zq : (b, q ) = 1}. The key of our proof is an addition theorem: Theorem 1.2. Let q be a positive integer with (q, 6) = 1. Let f1 , f2 , f3 be three ∗ real-valued functions over Z∗ q . Then for any n ∈ Zq , there exist x, y, z ∈ Zq such that n = x + y + z and f1 (x) + f2 (y ) + f3 (z ) 1 (f1 (a) + f2 (a) + f3 (a)), φ(q ) a∈Z∗
arXiv:math/0701240v2 [math.NT] 12 Feb 2007
TERNARY GOLDBACH PROBLEM FOR THE SUBSETS OF PRIMES WITH POSITIVE RELATIVE DENSITIES
HONGZE LI AND HAO PAN
Abstract. Let P denote the set of all primes. Suppose that P1 , P2 , P3 are three subsets of P with dP (P1 ) + dP (P2 ) + dP (P3 ) > 2, where dP (Pi ) is the lower density of Pi relative to P . We prove that for suﬃciently large odd integer n, there exist pi ∈ Pi such that n = p1 + p2 + p3 .
Thus for any n = (n1 , n2 ) ∈ Zq1 ⊕ Zq2 , by the induction hypothesis, there exist x1 , y1 , z1 ∈ Z∗ q1 such that n1 = x1 + y1 + z1 and g1 (x1 ) + g2 (x1 ) + g3 (x1 ) i.e., (f1 ((x1 , b)) + f2 ((y1 , b)) + f3 ((z1 , b))) Deﬁne functions h1 , h2 , h3 over Z∗ q2 by h1 (b) = f1 ((x1 , b)), h2 (b) = f2 ((y1 , b)), and h3 (b) = f3 ((z1 , b)). Then applying the induction hypothesis again, there exist x2 , y2 , z2 ∈ Z∗ q2 such that n2 = x2 + y2 + z2 and 1 h1 (x2 ) + h2 (y2 ) + h3 (z2 ) (h1 (b) + h2 (b) + h3 (b)) K. φ(q2 ) ∗