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**(pi) and x*x*(pi) for all pI< p. If preferences are continuous then where x*Nx* (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**(O)and 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