微观经济学讲义(黄有光)4

Advanced Microeconomics Topic 3: Consumer DemandPrimary Readings: DL – Chapter 5; JR - Chapter 3; Varian, Chapters 7-9.3.1 Marshallian Demand FunctionsLet X be the consumer's consumption set and assume that the X = R m +. For a given price vector p of commodities and the level of income y , the consumer tries to solve the following problem:max u (x )subject to p ⋅x = y x ∈ X∙ The function x (p , y ) that solves the above problem is called the consumer's demand function .∙ It is also referred as the Marshallian demand function . Other commonly known namesinclude Walrasian demand correspondence/function , ordinary demand functions , market demand functions , and money income demands .∙ The binding property of the budget constraint at the optimal solution, i.e., p ⋅x = y , is theWalras’ Law .∙ It is easy to see that x (p , y ) is homogeneous of degree 0 in p and y .Examples:(1) Cobb-Douglas Utility Function:.,...,1,0 ,)(1m i x x u i mi i i =>=∏=ααFrom the example in the last lecture, the Marshallian demand functions are:.ii i p yx αα=where ∑==mi i 1αα.(2) CES Utility Functions: )10( )(),(/12121<≠+=ρρρρx x x x uThen the Marshallian demands are:,),( ;),(2112221111rr r r r r p p yp y x p p y p y x +=+=--p p where r = ρ/(ρ -1). And the corresponding indirect utility function is given byrr r p p y y v /121)(),(-+=pLet us derive these results. Note that the indirect utility function is the result of the utility maximization problem:yx p x p x x x x =++2211/121, subject to )(max 21ρρρDefine the Lagrangian function:)()(),,(2211/12121y x p x p x x x x L -+-+=λλρρρThe FOCs are:00)(0)(22112121)/1(2121111)/1(211=-+=∂∂=-+=∂∂=-+=∂∂----y x p x p Lp x x x x L p x x x x L λλλρρρρρρρρ Eliminating λ, we get⎪⎩⎪⎨⎧+=⎪⎪⎭⎫⎝⎛=-2211)1/(12121x p x p y p p x x ρ So the Marshallian demand functions are:rrr rrr p p yp y x x p p yp y x x 211222211111),(),(+==+==--p pwith r = ρ/(ρ-1). So the corresponding indirect utility function is given by:r r r p p y y x y x u y v /12121)()),(),,((),(-+==p p p3.2 Optimality Conditions for Co nsumer’s ProblemFirst-Order ConditionsThe Lagrangian for the utility maximization problem can be written asL = u (x ) - λ( p ⋅x - y ).Then the first-order conditions for an interior solution are:yu i p x u i i =⋅=∇∀=∂∂x p p x x λλ)( i.e. ;)( (1)Rewriting the first set of conditions in (1) leads to,,k j p p MU MU MRS kj kj kj ≠==which is a direct generalization of the tangency condition for two-commodity case.Sufficiency of First-Order ConditionsProposition : Suppose that u (x ) is continuous and quasiconcave on R m +, and that (p , y ) > 0. If u if differentiable at x*, and (x*, λ*) > 0 solves (1), then x* solve the consumer's utility maximization problem at prices p and income y .Proof . We will use the following fact without a proof:∙ For all x , x ' ≥ 0 such that u (x') ≥ u (x ), if u is quasiconcave and differentiable at x , then∇u (x )(x' - x ) ≥ 0.Now suppose that ∇u (x*) exists and (x*, λ*) > 0 solves (1). Then,∇u (x*) = λ*p , p ⋅x* = y .If x* is not utility-maximizing, then must exist some x 0 ≥ 0 such thatu (x 0) > u (x*) and p ⋅x 0 ≤ y .Since u is continuous and y > 0, the above inequalities implies thatu (t x 0) > u (x*) and p ⋅(t x 0) < yfor some t ∈ [0, 1] close enough to one. Letting x' = t x 0, we then have∇u (x*)(x' - x ) = (λ*p )⋅( x' - x ) = λ*( p ⋅x' - p ⋅x ) < λ*(y - y ) = 0,which contradicts to the fact presented at the beginning of the proof since u (x 1) > u (x*).Remark∙ Note that the requirement that (x*, λ*) > 0 means that the result is true only forinterior solutions.Roy's IdentityNote that the indirect utility function is defined as the "value function" of the utility maximization problem. Therefore, we can use the Envelope Theorem to quickly derive the famous Roy's identity.Proposition (Roy's Identity?): If the indirect utility function v (p , y ) is differentiable at (p 0, y 0) and assume that ∂v (p 0, y 0)/ ∂y ≠ 0, then.,...,1 ,),(),(),(000000m i yy v p y v y x ii =∂∂∂∂-=p p pProof . Let x * = x (p , y ) and λ* be the optimal solution associated with the Lagrangian function:L = u (x ) - λ( p ⋅x - y ).First applying the Envelope Theorem, to evaluate ∂v (p 0, y 0)/ ∂p i gives.**)*,(),(*i ii x p L p y v λλ-=∂∂=∂∂x p But it is clear that λ* = ∂v (p , y )/ ∂y , which immediately leads to the Roy's identity.Exercise∙ Verify the Roy's identity for CES utility function.Inverse Demand FunctionsSometimes, it is convenient to express price vector in terms of the quantity demanded, which leads to the so-called inverse demand functions .∙ the inverse demand function may not always exist. But the following conditions willguarantee the existence of p (x ):∙ u is continuous, strictly monotonic and strictly quasiconcave. (In fact, these conditionswill imply that the Marshallian demand functions are uniquely defined.)Exercise (Duality of Indirect and Direct Demand Functions):(1) Show that for y = 1 the inverse demand function p = p (x ) is given by:.,...,1 ,)()()(1m i x x u x u p m j j jii =∂∂∂∂=∑=x x x(Consult JR, pp.79-80.)(2) Show that for y = 1, the (direct) demand function x = x (p, 1) satisfies.,...,1 ,)1,()1,()1,(1m i p p v p v x m j j jii =∂∂∂∂=∑=p p p(Hint: Use Roy’s identity and the homogeneity of degree zero of the indirect uti lityfunction.)3.3 Hicksian Demand FunctionsRecall that the expenditure function e (p , u ) is the minimum-value function of the following optimization problem:,)( s.t. min ),(u u u e m≥⋅=+∈x x p p R x for all p > 0 and all attainable utility levels.It is clear that e (p , u ) is well-defined because for p ∈ R m ++, x ∈ R m +, p ⋅x ≥ 0.If the utility function u is continuous and strictly quasiconcave, then the solution to the above problem is unique, so we can denote the solution as the function x h (p , u ) ≥ 0. By definition, it follows thate (p , u ) = p ⋅x h (p , u ).∙ x h (p , u ) is called the compensated demand functions , also commonly known as Hicksiandemand functions , named after John Hicks when he first discussed this type of demand functions in 1939.Remarks1. The reason that they are called "compensated " demand function is that we mustimpose an artificial income adjustment when the price of one good is changing while the utility level is assumed to be fixed.2. It is important to understand that, in contrast with the Marshallian demands, theHicksian demands are not directly observable.As usual, it should be no longer a surprise that there is a close link between the expenditure function and the Hicksian demands, as summarized in the following result, which is again a direct application of the Envelope Theorem..Proposition (Shephard's Lemma for Consumer): If e (p , u ) is differentiable in p at (p 0, u 0) with p 0 > 0, then,.,...,1 ,),(),(000m i p u e u x ih i=∂∂=p pExample: CES Utility Functions)10( )(),(/12121<≠+=ρρρρx x x x uLet us now derive the Hicksian demands and the corresponding expenditure function.min {p 1x 1 + p 2x 2} subject to)(/121u x x =+ρρρThe Lagrangian function is))((),,(/121221121u x x x p x p x x L -+-+=ρρρλλThen the FOCs are:0)(0)((0)((/121121/12122111/12111=+-=∂∂=+-=∂∂=+-=∂∂----ρρρρρρρρρρρλλλx x u L x x x p x L x x x p x L Eliminating λ, we getρρρρ/121)1/(12121)(x x u p p x x +=⎪⎪⎭⎫ ⎝⎛=- From these, it is easy to derive the Hicksian demand functions given by:121)/1(212111)/1(211)(),()(),(----+=+=r r r r h r r r r h pp p u u x p p p u u x p pwhere r = ρ/(ρ-1). And the expenditure function is.)(),(),(),(/1212211rr r h h p p u u x p u x p u e +=+=p p pAlternatively, since we know that the indirect utility function is given by:,)(),(/121rr r p p y y v -+=p3.4 Recall that (last the indirect utility function v (p (a) e (p , v (b) v (p , eFurthermore, we solutions of both optimization the followinginteresting identities between Marshallian demands and Hicksian demands:x (p , y ) = x h (p , v (p , y )) x h (p , u ) = x (p , e (p , u ))which hold for all values of p , y and u .The second identity leads to a classic differentiation relation between Hicksian demands and Marshallian demands, known as Slutsky equation.Proposition (Slutsky Equation): If the Marshallian and Hicksian demands are all well-defined and continuously differentiable, then for p > 0, x > 0,),,(),(),(),(y x yy x p u x p y x j i j h i j i p p p p ⋅∂∂-∂∂=∂∂where u = v (p , y ).Proof . It follows easily from taking derivative and applying Shephard's Lemma.Substitution and Income Effects∙ The significance of Slutsky equation is that it decomposes the change caused by a pricechange into two effects: a substitution effect and an income effect .∙ The substitution effect is the change in compensated demand due to the change inrelative prices, which is the first item in Slutsky equation.∙ The income effect is the change in demand due to the effective change in incomecaused by the price change, which is the second item in Slutsky equation. ∙ The substitution effect is unobservable, while the income effect is observable.Question: From the above diagram (also know as Hicksian decomposition ), can you see crossing property between a Marshallian demand function and the corresponding Hicksian demand? (Hint: there are two general cases.)Slutsky MatrixThe substitution effect between good i and good j is measured byj i p u x s jh i ij ,,),(∀∂∂=pSo the Slutsky matrix or the substitution matrix is the m ⨯m matrix of the substitution items:⎥⎥⎦⎤⎢⎢⎣⎡∂∂==j h i ij p u x s ),(][p SThe following result summarizes the basic properties of the Slutsky matrix.Proposition (Substitution Properties). The Slutsky matrix S is symmetric and negative semidefinite.Proof . By Shephard’s Lemma (for consumer), we know thatji ihj i j j i j h i ij s p u x p p u e p p u e p u x s =∂∂=∂∂∂=∂∂∂=∂∂=),(),(),(),(22p p p pHence S is symmetric. It is evident that S is the Hessian matrix of the expenditure function e (p , u ).Since we know that e (p , u ) is concave, so its Hessian matrix must be negative semidefinite.Since the second-order own partial derivatives of a concave function are always nonpositive, this implies that s ii ≤ 0, i.e.,i p u x s ih i ii ∀≤∂∂=,0),(pwhich indicates the intuitive property of a demand function: as its own price increases, the quantity demanded will decrease. You are reminded that this is a general property for Hicksian demands.For the Marshallian demands, note that by Slutsky equation,).,(),(),(),(y x yy x p u x p y x i i i h i i i p p p p ⋅∂∂-∂∂=∂∂Then for a small change in p i , we will have the following:.),(),(),(),(i i i i i h i i i i i p y x yy x p p u x p p y x x ∆⋅∂∂-∆∂∂=∆∂∂≈∆p p p pThe first item, capturing the own price effect of the Hicksian demands, is of course nonpositive.The sign of the second item depends on the nature of the good:∙ Normal good : ∂x i (p , y )/ ∂y > 0.∙ This leads to a normal Marshallian demand function: it is decreasing in its ownprice.∙ Inferior good : ∂x i (p , y )/ ∂y < 0.∙ When the substitution effect still dominates the income effect, the resultingMarshallian demand is also decreasing in its own price.∙ When the substitution effect is dominated by the income effect, it will lead to aGiffen good, that is, its demand function is an increasing function of its own price.Because of Slutsky equation, the Slutsky matrix (i.e., the substitution matrix) also has the following form that is in terms of Marshallian demand functions.⎥⎥⎦⎤⎢⎢⎣⎡∂∂+∂∂=⎥⎥⎦⎤⎢⎢⎣⎡∂∂==),(),(),(),(][y x y y x p y x p u x s j i j i j h i ij p p p p SWe will get back to the above Slutsky matrix in the next lecture when we discuss the integrability problem .3.5. The Elasticity Relations for Marshallian Demand FunctionsDefinition . Let x (p , y ) be the consumer’s Marshallian demand functions. Define.),(,),(),(,),(),(yy x p s y x p p y x y x yy y x i i i i jj i ij i i i p p p p p =∂∂=∂∂=εηThen1. ηi is called the income elasticity of demand for good i .2. ε ij is called the price elasticity of the demand for good i with respect to a price change ingood j . ε ii is the own-price elasticity of the demand for good i . For i ≠ j , ε ij is the cross-price elasticity .3. s i is called the income share spent on good i .The following result summarizes some important relationships among the income shares, income elasticities and the price elasticities.Proposition . Let x (p , y ) be the consumer’s Marshallian demand functions. Then1. Engel aggregation :.11=∑=mi ii s η2. Cournot aggregation :.,...,1 ,1m j s s j mi iji =-=∑=εProof . Both identities are derived from the Walras’ Law, namely, the fact that the budget is tight or balanced:y = p ⋅x (p , y ) for all p and y . (A)To prove Engel aggregation, we differentiate both sides of (A) w.r.t. y :∑∑∑====∂∂=∂∂=m i mi i i i i i i mi i i s y x yy y x y y x p y y x p 111,),(),(),(),(1ηp p p pas required.To prove Cournot aggregation, we differentiate both sides of (A) w.r.t. p j :.),(),(),(),(),(01∑∑=≠∂∂=-⇒∂∂++∂∂=mi j i i j jj jj ji j i ip y x p y x p y x p y x p y x p p p p p pMultiplying both sides by p j /y leads to∑∑∑====-⇒∂∂=∂∂=-mi iji j mi i jj i i i mi j j i i j j s s y x p p y x y y x p p p y x y p yy x p 111),(),(),(),(),(εp p p p pas required too.3.6 Hicks ’ Composite Commodity TheoremAny group of goods & services with no change in relative prices between themselves may be treated as a single composite commodity, with the price of any one of the group used as the price of the composite good and the quantity of the composite good defined as the aggregate value of the whole group divided by this price. Important use in applied economic analysis.Additional ReferencesAfriat, S. (1967) "The Construction of Utility Functions from Expenditure Data," International Economic Review, 8, 67-77.Arrow, K. J. (1951, 1963) Social Choice and Individual Values. 1st Ed., Yale University Press, New Haven, 1951; 2nd Ed., John Wiley & Sons, New York, 1963.Becker, G. S. (1962) "Irrational Behavior and Economic Theory," Journal of Political Economy, 70, 1-13.Cook, P. (1972) "A One-line Proof of the Slutsky Equation," American Economic Review, 42, 139. Deaton, A. and J. Muellbauer (1980) Economics and Consumer Behavior. Cambridge University Press, Cambridge.Debreu, G. (1959) Theory of Value. John Wiley & Sons, New York.Debreu, G. (1960) "Topological Methods in Cardinal Utility Theory," in Mathematical Methods in the Social Sciences, ed. K. J. Arrow and M. D. Intriligator, North Holland, Amsterdam. Diewert, W. E. (1982) "Duality Approaches to Microeconomic Theory," Chapter 12 in Handbook of Mathematical Economics, ed. K. J. Arrow and M. D. Intriligator, North Holland, Amsterdam.Gorman, T. (1953) “Community Preference Fields,” Econometrica, 21, 63-80.Hicks, J. (1946) V alue and Capital. Clarendon Press, Oxford, England.Katzner, D.W. (1970) Static Demand Theory. MacMillan, New York.Marshall, A. (1920) Principle of Economics, 8th Ed. MacMillan, London.McKenzie, L. (1957) “Demand Theory Without a Utility Ind ex," Review of Economic Studies, 24, 183-189.Pollak, R. (1969) "Conditional Demand Functions and Consumption Theory," Quarterly Journal of Economics, 83, 60-78.Roy, R. (1942) De l'utilite. Hermann, Paris.Roy, R. (1947) "La distribution de revenu entre les divers biens," Econometrica, 15, 205-225. Samuelson, P. A. (1938) "A Note on the Pure Theory of Consumer's Behavior," Econometrica, 5, 61-71, 353-354.Samuelson, P. (1947) Foundations of Economic Analysis. Harvard University Press, Cambridge, Massachusetts.Sen, (1970) Collective Choice and Social Welfare. Holden Day, San Francisco.Stigler, G. (1950) "Development of Utility Theory," Journal of Political Economy, 59, parts 1 & 2, pp. 307-327, 373-396.Varian, H. R. (1992) Microeconomic Analysis. Third Edition. W.W. Norton & Company, New York. (Chapters 7, 8 and 9)Wold, H. and L. Jureen (1953) Demand Analysis. John Wiley & Sons, New York.11。