哈佛大学：《中级微观经济学》（英文版）Lecture 5 Aggregating Consumers

Economics 2010a Fa2003 Edward L. Glaeser Lecture 5

Economics 2010a Fall 2003 Edward L. Glaeser Lecture 5

5. Aggregating Consumers Consumer Heterogeneity and a Discrete good b. The Properties of Aggregate Demand The Existence of a representative Consumer d. Externalities e. The Social Multiplier

5. Aggregating Consumers a. Consumer Heterogeneity and a Discrete Good b. The Properties of Aggregate Demand c. The Existence of a Representative Consumer d. Externalities e. The Social Multiplier

f. Equalizing differentials with Heterogeneity g. Welfare Losses with Heterogeneous Consumers h. Application: Price Controls and Aggregation

f. Equalizing Differentials with Heterogeneity g. Welfare Losses with Heterogeneous Consumers h. Application: Price Controls and Aggregation

a general lesson- aggregate outcomes do not always resemble individual outcomes Lets start with a discrete commodity, of which only one unit can be consumed Normalize this to be commodity 1 Proposition Suppose that the preference relation >on X={x1∈{0,1},(x2xL)∈R1},is rational, continuous and locally non-satiated on the commodities other than commodity 1, and if x*={x∈Bo:x≥ y for all y∈Bo}, Where B0={x∈x:x1=0and∑2px≤v and

A general lesson– aggregate outcomes do not always resemble individual outcomes. Let’s start with a discrete commodity, of which only one unit can be consumed. Normalize this to be commodity 1. Proposition: Suppose that the preference relation  on X  x1  0, 1,x2,... xL    L1  , is rational, continuous and locally non-satiated on the commodities other than commodity 1, and if x  x  B0 : x  y for all y  B0 , where B0  x  X : x1  0 and i2 L pixi  w , and

x*(p1)={x∈Bo(p1):x≥ y for all y∈Bo Where B0(D1)= {x∈x:x1=1and∑h2 pixi w-pi Ifx*>(1,0,0.0)then there exists a p,[,w] for all pi>pl,x*x**(p1), at the least x*x*(v)

xp1   x  B0p1 : x  y for all y  B0 where B0p1  x  X : x1  1 and i2 L pixi  w  p1 . If x  1, 0, 0. . . 0 then there exists a  p1 0,w for all p1   p1, x  xp1  and x  xp1  for all p1   p1. If preferences are continuous then where x  x  p1 . Proof: It is obviously true that there exist some values of p1 0,w in which x  xp1  and some values for x  xp1 , at the least x  x0 and x  xw

Furthermore, non-satiation tells us that x*x**(pi) implies that x*>x**(pi) for all pI>p, and thatx*>x**(p1) it must be that p PLB then there exists a continuum of values of pI at which x*≥x*(P1)andx*≤x*(p1) i.e. for which x*N x**(p1 but local non-satiation rules that out. so it must be that pla=pib=p, and by construction for all pi>p, and x*<x**(p1)and

Furthermore, non-satiation tells us that x  x  p1  implies that x  xp1  for all p1   p1, and that x  x  p1  implies that x  xp1  for all p1   p1. Hence, since the set of p1for which x  xp1  is bounded above there exists an upper bound  p1,A. And since the set of p1for which x  xp1  is bounded below there exists a greatest lower bound  p1,B. As it would be impossible to have xp1   x  xp1 , it must be that  p1,A   p1,B. If  p1,A   p1,B then there exists a continuum of values of p1 at which x  xp1  and x  xp1 , i.e. for which x  xp1  but local non-satiation rules that out, so it must be that  p1,A   p1,B   p1 and by construction for all p1   p1, and x  xp1  and

x*>x*(p)for叫lp1<1 Continuity gives you the existence of a price where consumers are indifferent Consider, the sequence x**(p1 -nand the sequence x*, where x**(pI-m)2x* for all n, continuity implies that x**(p12x Likewise consider the sequences x**(pI+n)and x, where x**(PI+D<x* for all n, So x**(p1)<x* which together imply that x**(PI)Nx* Thus each person has a price at which he is indifferent between consuming commodity one and not doing so In this case, individual demand is always downward sloping -no income effects or

x  xp1  for all p1   p1. Continuity gives you the existence of a price where consumers are indifferent. Consider, the sequence xp1  1 n  and the sequence x, where xp1  1 n   x for all n, continuity implies that xp1  x . Likewise consider the sequences xp1  1 n  and x, where xp1  1 n   x for all n, so xp1  x which together imply that xp1   x. Thus each person has a price at which he is indifferent between consuming commodity one and not doing so. In this case, individual demand is always downward sloping– no income effects or

cross-partials to worry about Why? What would happen if we dropped continuity or local non-satiation?

cross-partials to worry about. Why? What would happen if we dropped continuity or local non-satiation?

Assume that there are j consumers each with preferences 2; and wealth wi, facing prices for all goods i>1, p>0, which generates a cutoff price pi for each consumer Let D(D12P,) ithe number of consumers s tpI <pil This function is certainly weakly downward sloping -as pi falls the set of consmers for which pI pi must rise If we want something like continuity, we must assume that a distribution of pi in the population that has positive density for all values pi between zero and some value p which denotes the highest price at which anyone buys anything

Assume that there are J consumers, each with preferences j and wealth wi , facing prices for all goods i1, p  0, which generates a cutoff price  p1 i for each consumer. Let Dp1,p,w   the number of consumers s.t. p1   p1 i This function is certainly weakly downward sloping– as p1 falls the set of consmers for which p1   p1 i must rise. If we want something like continuity, we must assume that a distribution of  p1 i in the population that has positive density for all values  p1 i between zero and some value p which denotes the highest price at which anyone buys anything

If this distribution is denoted f(p), then aggregate demand is D(D1,P,) f1)①D1=1-F(F1) PIpI and aD=f() So aggregate demand is downward sloping, continuous, and will be concave or convex depending on the shape of the density For example if p, is uniformly distributed on the interval p, p ]then demand equals P-e the familiar linear demand curve a demand function with constant price elasticity occurs if reservation prices are distribution so that: f(p)=p and

If this distribution is denoted f  p1 , then aggregate demand is: Dp1,p,w   p1p1 p f  p1d  p1  1  F  p1  and D p1  f  p1. So aggregate demand is downward sloping, continuous, and will be concave or convex depending on the shape of the density. For example if  p1 is uniformly distributed on the interval p,pthen demand equals pp1 pp – the familiar linear demand curve. A demand function with constant price elasticity occurs if reservation prices are distribution so that: f  p1   p1 1 and