高级微观(汪浩) Micro 05 Aggregate demand

Lecture 4

Chapter 4 Aggregate Demand

To what extent the theory we discussed before can be applied to aggregate demand, a sum of

demands of all consumers? Particularly, we are interested in:

1. When can aggregate demand be expressed as a function of prices and aggregate wealth?

2. When does aggregate demand satisfy the weak axiom?

3. When does aggregate demand have similar welfare significance as what we discussed before?

Aggregate demand and aggregate wealth

Suppose there are I consumers with rational preference i and corresponding demand

function L i i R w p x ∈),(, given prices L R p ∈ and wealth levels R w i ∈.

The aggregate demand is defined as ∑==I

i i

i

I w p x w w p x 1

1),(),...,,(.

The aggregate demand depends not only on the prices but also the distribution of wealth. We

are interested in knowing when we can write the aggregate demand in form of ),(1

∑=I

i i w p x , where

aggregate demand depends only on aggregate wealth.

For this to hold, we need for any ),...,(1I w w and )',...,'(1I w w such that ∑∑===I

i i I

i i w w 1

1

', we

have

∑∑===I

i i

i

I i i

i

w p x w p x 1

1)',(),(. This equation can be rewritten as 0)]',(),([1

=−∑=I

i i

i

i

i

w p x w p x ,

which implies that

0)

,(1

=∂∂∑

=I

i i i

i i dw w w p x with 01

=∑=I

i i

dw

.

This to be true for any ),...,(1I w w and )',...,'(1I w w if and only if for every product l and any

two consumer i and j we have j

j lj i i li w w p x w w p x ∂∂=∂∂)

,(),(.

In words, for any given price and commodity, the wealth effect at the price must be the same whatever consumer we look at and whatever his level of wealth. This condition is equivalent to the

statement that all consumer’ wealth expansion paths are parallel, straight line.

2x

Consumer i

Consumer j

1x

Proposition B1 A necessary and sufficient condition for the set of consumers to exhibit parallel, straight wealth expansion paths at any price vector p is that the preferences admit indirect utility functions of the Gorman form with the coefficient i . That is

i i i i w p b p a w p v )()(),(+=

Corollary Aggregate demand can be written as a function of aggregate wealth if and only if consumers have preferences that admit indirect utility functions of the Gorman form with equal wealth coefficient b (p ).

We may obtain less restrictive conditions if we consider aggregate demand functions that

depend on a wider set of aggregate variables than just the total wealth level.

For example, we can have an aggregate demand function depending only on the statistical distribution of wealth when all consumers possess identical but otherwise arbitrary preferences and differ only in their wealth level.

If individual wealth levels were generated by some underlying process that shapes the wealth

distribution, it may still be possible to write aggregate demand as a function of prices and aggregate wealth.

For example, individual i ’s wealth level is generated by a function of p and w , (,).i w p w We

call 1((,),...,(,))I w p w w p w with (,).i i w p w w ∑= for all (p,w) a wealth distribution rule . In this case we can always wirte aggregate demand as a function (,)(,(,))i i i x p w x p w p w =∑, which depends only on prices and aggregate wealth.