高级微观经济学 Part 2_Consumption Theory I

5.WA and demand law
• Walrasian demand function x(p,w) satisfied
WA if for any
See the fig
we have:
.
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• Changing in price will change wealth too. But
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2.The Байду номын сангаасule of choice
• We can only observe what a man choice under
a certain constrain. • Is the man’s preference rational when we observed several choice of him? • Or if his preference was rational, what rules will his choice take?
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• Weak Axiom of Revealed Preference (WARP
or WA) : if for certain and for and must • Revealed Preference “ ”: ,
, then
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• Wealth effects:
– Engel function: given price over wealth – Wealth expansion path:
– Wealth effects:
, function
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3.Competive budget
• Consumption set X and it’s price
with
the wealth w>0. • Definition: Walras (competive) budget set
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Consumption set
rice Rice of Hangzhou
4
4
bread
Bread of Shanghai
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expansion path
x2
x1
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• Propersition1: if
– – – If
is rational, then: is irreflexive and transitive is reflexive and transitive and symmetric
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• Slutsky matrix (substitution matrix)
• Substitution effects • is n.s.d • Giffen good is necessary inferior good.
• Proposition4: if X is concave, then is
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4.Demand function
• For any p and w are strictly positive, the
corresponding demand set x(p,w) are nonempty, if x(p,w) is single point, we call it the Walrasian( Marshallian) demand function. • x(p,w) is homogenous of degree zero and satisfied Walras’ law. That is px=w for all
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1.Preference
• An (acceptable) alternatives set (or a consumption
set
) – A (weak) preference relation“ ”: a binary relationship on X, for any , means” x is at least as good as y”. – A strict preference “ ”:
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• Wealth effects:
– Commodity l is normal goods if – Commodity l is inferior goods if – The demand is normal if
See the fig
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• Choice construct: a choice under certain
budget: – Budget set: and , is a budget family, represented a limited list of all possible choice under certain system, nature, technology and other social environments that the decision-maker must take. – Choice rule: , maybe contain more than one elements.
how can we tell the demand changing by price changing from wealth changing? • Given a changing from , and people will not get worse that is here wealth changing (compensation ) was called “Slutsky wealth compensation” and “(Slutsky) compensated price changing”.
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• Price effects:
– Supply curve: – Price effects:
– Giffen good:
See the fig.
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• Cournot Aggregation: if Walrasian demand
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• Choice set • Proposition2: if are rational, then choice
construct satisfied WA. • Prop.2 indicate that if the decision-maker have rational preference, than his choice satisfied WA. Is his preference rational if his choice is observed satisfy WA?
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• Proposition5: x(p,w) satisfied WA if and only
if:
,
and when
• Prop.5 indicates,
, or that’s called “demand law ”, or “compensation demand law ”.
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• Definition: for
, we say “rational preference rationalization • Propersition3: if satisfied WA and contain all subsets of X that is ternary and less than ternary. – Example: X={x,y,z}, B ={{x,y},{y,z},{x,z}}, C({x,y})={x},C({y,z})={y},C({x,z})={z},(B , C(.)) satisfied WARP, but there is no rationalization preference relation over it.
Advanced Microeconomics
(Consumption Theory I)
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Preference and Choice
• Content:
– Preference and rationality – The rule of choice – Budget – Demand function – WA and demanded law
function x(p,w) is HD0 and satisfied Walras Law , then:
• Engel Aggregation: if Walrasian demand
function x(p,w) satisfied Walras Law, then:
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– An indifference relation “ ”:
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• Definition: preference on X is rational if:
– It’s complete: – It’s reflexive: – It’s transitive:
Wealth effects
x2
x2
x2
x1
x1 luxury goods
x1
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Supply curve:
x2
x1
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Giffen good
x2
x1
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WA and demand law
x2 x2 x2
x1
x1
x1
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WA and demand law
x2
x2
x1
x1
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Consumption set
x2
x2
3 2 1
x1
x1
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Consumption set
Leisure
24
Leisure
24
8
x1
x1
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