Aggregate Demand Advanced Microeconomics - UNIGRAZ · Advanced Microeconomics Aggregate Demand RonaldWendner 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


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


Aggregate wealth

Necessary and sufficient condition for the above

∂ xli(p,wi)∂ wi

= ∂ xlj(p,wj)∂ wj

I straight and parallel income expansion paths


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)


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)


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


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′).



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



� 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



� 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



PropositionIf %i are homothetic, then xi(p,wi) satisfies the ULD.

CorollaryIf %i are homothetic, then x(p,w) satisfies the WARP.


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



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)



� 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)



� % 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



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



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 ).


Documents

Aggregate Demand Advanced Microeconomics - UNIGRAZ · Advanced Microeconomics Aggregate Demand RonaldWendner Department of Economics University of Graz, Austria Course # 320.911