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

Second agent has a utility function minal, (1-a)x21 At price vector(1, 1)-agent 1 consumes a units of the first good and 1-a units on the second good Agent 2 consumes 1-a units of the first good and a units of the second good At price vector(1+0, 1-0)the first agent consumes a units of the first good and 1-a(1+6) units of the second good Agent 2 consumes exactly units of 1+-2a6 the first good and units of the 1+6-2a6 second good

Second agent has a utility function minx1,1  x2 . At price vector (1, 1)– agent 1 consumes  units of the first good and 1   units on the second good. Agent 2 consumes 1   units of the first good and  units of the second good. At price vector 1  , 1   the first agent consumes  units of the first good and 11 1 units of the second good. Agent 2 consumes exactly 1 12 units of the first good and  12 units of the second good

Question 1: Is the old aggregate bundle affordable at the new prices? Certainly- the old aggregate consumption was 1 and 1 which is still doable Question #2: Is the new bundle affordable at the old price This also is a possibility- aggegate consumption at old prices comes to 1-2a0+c which can be rewritten 2-4ae-2a62+4a2e (1+6-206(1-0) For this to be affordable at the old prices it

Question # 1: Is the old aggregate bundle affordable at the new prices? Certainly– the old aggregate consumption was 1 and 1 which is still doable. Question # 2: Is the new bundle affordable at the old price: This also is a possibility– aggegate consumption at old prices comes to 12 1  1 12 , which can be rewritten: 2  4  22  422 1    21   For this to be affordable at the old prices it

must be that 1 >1-2a8-204+2a-0 (1+-2a)(1 or 1-02-2aO+2ae2>1-2a0-a02+2a2 or0>(1-2a)(1-a) This calculation suggests that as long as a is greater than. 5, the weak axiom fails So lets assume that a=75 and let 0=1/3 In this case, the switch in prices pushed consumer one from consuming(.75,. 25) to(.75,0) The switch in prices pushed consumed two from consuming(.25. 75)to consuming (3,9

must be that 1  122222 121 or 1  2  2  22  1  2  2  222 or 0  1  21   This calculation suggests that as long as  is greater than .5, the weak axiom fails. So let’s assume that  . 75 and let   1/3. In this case, the switch in prices pushed consumer one from consuming (.75, .25) to (.75, 0). The switch in prices pushed consumed two from consuming (.25, .75) to consuming (.3, .9)

Aggregate consumption has changed from (1,1)to(1.05,9) Which was, after all. affordable at the old prIce. How did I rig this example? The key is that the individual who receives a negative income shock by the change in prices, needs to respond very sharply to this income shock(which means cutting back spending on the cheaper good)

Aggregate consumption has changed from (1, 1) to (1.05,.9). Which was, after all, affordable at the old price. How did I rig this example? The key is that the individual who receives a negative income shock by the change in prices, needs to respond very sharply to this income shock (which means cutting back spending on the cheaper good)

The individual who receives a positive income needs to respond slowly, which means increasing spending on the more expensive good Endowing the big consumer of the good that is getting cheaper with leontief preferences, ensures that there is almost no substitution into the cheap good Endowing the big consumer of the good that is getting more expensive with very elastic preferences for the other good ensured that the income effect(really a budget set effect)would dominate in that case The key to Wa failure lies in the income effects of price changes

The individual who receives a positive income needs to respond slowly, which means increasing spending on the more expensive good. Endowing the big consumer of the good that is getting cheaper with leontief preferences, ensures that there is almost no substitution into the cheap good. Endowing the big consumer of the good that is getting more expensive with very elastic preferences for the other good ensured that the income effect (really a budget set effect) would dominate in that case. The key to WA failure lies in the income effects of price changes