Advanced Microeconomics Aggregate Demand Ronald Wendner Department of Economics University of Graz, Austria Course # 320.911
Advanced Microeconomics
Aggregate Demand
Ronald Wendner
Department of EconomicsUniversity of Graz, Austria
Course # 320.911
Aggregate Demand and a Representative Household
� Aggregate demand and aggregate wealth
� Consistency of aggregate demand with the WARP
� Aggregate demand and the existence of a representative household
R. Wendner (U Graz, Austria) Microeconomics 2 / 19
Aggregate wealth
Aggregate Demand and Aggregate Wealth
� Consumers i, i = 1, ..., I
preferences, wealth, demand are consumer-specific
I %i
I wi
I xli (generic notation)
Aggregate demand and aggregate wealth
x(p,w1,w2, ...,wI ) =I∑
i=1xi(p,wi)
w =I∑
i=1wi
R. Wendner (U Graz, Austria) Microeconomics 3 / 19
Aggregate wealth
When are we allowed to write aggregate demand as a function of aggregatewealth?
x(p,w1,w2, ...,wI ) = x(p,w) ⇔I∑
i=1xi(p,wi) = x
(p,
I∑i=1
wi
)
I Aggregate demand depends on aggregate wealth, not on the distribution ofwealth
I∑i=1
∂ xli(p,wi)∂ wi
dwi = 0 , l = 1, ...,L ,I∑
i=1dwi = 0
for any wealth redistribution
I∑i=1
wi =I∑
i=1w′
i ⇔I∑
i=1dwi = 0
R. Wendner (U Graz, Austria) Microeconomics 4 / 19
Aggregate wealth
Necessary and sufficient condition for the above
∂ xli(p,wi)∂ wi
= ∂ xlj(p,wj)∂ wj
I straight and parallel income expansion paths
R. Wendner (U Graz, Austria) Microeconomics 5 / 19
Aggregate wealth
PropositionA necessary and sufficient condition for income (wealth) expansion paths acrossconsumers to be straight and parallel is that %i admit indirect utility functions ofthe Gorman form:
vi(p,wi) = ai(p) + b(p)wi
I Examples: homothetic %; quasilinear %
I Under Gorman %, for any wealth distribution:
x(p,w1,w2, ...,wI ) = x(p,w)
R. Wendner (U Graz, Austria) Microeconomics 6 / 19
Aggregate wealth
� A less restrictive condition: wi = wi(p,w)
I p: affect the value of one’s endowmentsI w: government may base individual’s taxes (final wealth position wi) on an
individuals wage income and the aggregate wealth
� Wealth distribution rule (WDR)
Family of functions (p,w) 7→ (w1(p,w), ...,wi(p,w)) with∑
i wi(p,w) = w
Example: wi = αi w, αi ≥ 0,∑
i αi = 1
(wi indpt of p)
R. Wendner (U Graz, Austria) Microeconomics 7 / 19
Aggregate wealth
PropositionIf wi are generated by a WDR, then:
x(p,w) =∑
ixi(p,wi(p,w))
I x(p,w1,w2, ...,wI ) = x(p,w)
if % admit the Gorman form
or wi generated by a WDR
R. Wendner (U Graz, Austria) Microeconomics 8 / 19
Weak Axiom of Revealed Preference
Aggregate Demand and the WARP
We assume simple WDR from here on: wi = αiw
To which extend do properties of xi(p,wi) carry over to x(p,w1, ...,wI )?
Properties that carry over:
I continuityI HD0 in (p, w)I Walras law
Weak Axiom may not carry over, even if we assume a WDR
Definitionx(p,w) satisfies WARP if p · x(p′,w′) ≤ w and x(p,w) 6= x ′(p,w) implyp′ · x(p,w) > w′ for any (p,w) and (p′,w′).
R. Wendner (U Graz, Austria) Microeconomics 9 / 19
Weak Axiom of Revealed Preference
Example: I = L = 2; WDR: αi = 1/2; (p,w), (p′,w)
xi(p,w), xi(p′,w), i = 1, 2 satisfy WARP
x(p,w), x(p′,w) does not satisfy WARP, where
12p · x(p′,w/2) < w
2 ,12p′ · x(p,w/2) < w
2
R. Wendner (U Graz, Austria) Microeconomics 10 / 19
Weak Axiom of Revealed Preference
� Problem: wealth effects
WARP ⇔ compensated law of demand (CLD)
(p′ − p) · [x(p′,w′)− x(p,w)] ≤ 0 where w′ = p′ · x(p,w)
Problem: if (p,w)-change is compensated in aggregate, it needs not becompensated individually⇒ wealth effects (in addition to substitution effects)
[w′ = p′ · x(p,w)] 6=⇒ [αiw′ = p′ · xi(p, αiw)] ∀ i
I wealth effects may dominate substitution effects ⇒
CDLi does not hold for some i ⇒ CDL may not hold in aggregate ⇐⇒WARP may not hold
R. Wendner (U Graz, Austria) Microeconomics 11 / 19
Weak Axiom of Revealed Preference
� Under which condition does x(p,w) satisfy WARP?
Definitionxi(p,wi) satisfies die uncompensated law of demand (ULD) if:
(p′ − p) · [xi(p′,wi)− xi(p,wi)] ≤ 0 for all p, p′,wi .
The analogous definition applies to x(p,w).
PropositionIf every xi(p,wi) satisfies the ULD then x(p,w) satisfies ULD. As a consequence,x(p,w) satisfies the WARP.
I wealth effects are restricted
R. Wendner (U Graz, Austria) Microeconomics 12 / 19
Weak Axiom of Revealed Preference
PropositionIf %i are homothetic, then xi(p,wi) satisfies the ULD.
CorollaryIf %i are homothetic, then x(p,w) satisfies the WARP.
R. Wendner (U Graz, Austria) Microeconomics 13 / 19
Representative household
Aggregate Demand and a Representative Household
� Under which conditions can aggregate x(p,w) be the result of utilitymaximization of a representative household, whose preferences can be used tocompute meaningful aggregate (social) welfare measures?
Assumptions used in the following:
– WDR: w 7→ (w1(p,w), ...,wI (p,w))
– wi(p,w) cont., HD1
I x(p,w) exists and satisfies cont., HD0, Walras law
R. Wendner (U Graz, Austria) Microeconomics 14 / 19
Representative household
DefinitionA positive representative consumer exists if there exists a rational % on RL
+ suchthat x(p,w) is the Walrasian demand generated by this %.That is x(p,w) � x whenever x 6= x(p,w) and p · x ≤ w.
PropositionA positive representative consumer exists if:x(p,w) satisfies Walras law, HD0 and negative semidefiniteness of the Slutskymatrix.Then there exists a utility function u(x) generating x(p,w), where u(x) is derivedfrom a rational, continuous and monotone preference relation.
I ULD ⇒WARP ⇒ S(p,w) is negative semidefinite ⇒ positive representativeconsumer exists (e.g., % are homothetic)
R. Wendner (U Graz, Austria) Microeconomics 15 / 19
Representative household
� A normative representative consumer: stronger restrictions
∃ positive- but @ normative representative consumer is possible
DefinitionA (Bergson-Samuelson) social welfare function is a function W that assigns toevery utility distribution (u1, ..., uI ) ∈ RI a utility value. That is, W : RI → R.
Example 1: W =∑
i αi ln ui , αi > 0,∑
i αi = 1 (inequality aversion)
Example 2: W =∑
i ui (utilitarian)
R. Wendner (U Graz, Austria) Microeconomics 16 / 19
Representative household
� % of normative representative consumer depend on specific W
V (p,w) = maxw1,...,wI
W (v1(p,w1), ..., vI (p,wI )) , s.t.∑
iwi ≤ w (P1)
PropositionSuppose the WDR (w1(p,w), ...,wI (p,w)) solves problem (P1). Then the valuefunction V (p,w) is an indirect utility function of a positive representativeconsumer for the aggregate x(p,w).
I Optimal WDR (w1(p,w), ...,wI (p,w)) depends on specific W ⇒
V (p,w) depends on W ⇒ % depend on W
R. Wendner (U Graz, Austria) Microeconomics 17 / 19
Representative household
DefinitionThe positive representative consumer % for x(p,w) is a normative representativeconsumer relative to W if the WDR solves problem (P1) and therefore V (p,w) isan indirect utility function for %.
I In addition to requirements for positive representative consumer, requirementon WDR (on w-distribution)
I Additional requirement depends on W
R. Wendner (U Graz, Austria) Microeconomics 18 / 19
Representative household
Example I.
Consider W from Example 1. Then the optimal WDR is wi(p,w) = αiw. If%i are homothetic ⇒ ULD holds and S(p,w) is nsd ⇒ x(p,w) exists andV (p,w) is an indirect utility function for % of the normative representativeconsumer (relative to W ).
Example II.
% admit vi(p,wi) = ai(p) + b(p)wi (Gorman) and W =∑
i ui (utilitarian).Then any WDR solves (P1) and V (p,w) =
∑i ai(p) + b(p)w is the indirect
utility function of the normative representative consumer (relative to W ).
R. Wendner (U Graz, Austria) Microeconomics 19 / 19