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HW13_solutions

# HW13_solutions - Intermediate Microeconomics 2008 Problem...

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Intermediate Microeconomics, 2008 Problem Set No 13: Solutions Q1) For the following utility functions and total endowments: Find the market clearing price(s) and the competitive allocation. Hint: You can set one price equal to one. We will use the following strategy to find market clearing prices: First, without loss of generality, we will set p 2 = 1 and for ease of notation we will set p 1 = p . These assumptions are convenient because they allow us to refer to p both as the price of good 1 and as the price ratio ( p 1 p 2 = p ). Given these prices we will write down the agents’ wealth functions y A ( p ) and y B ( p ) and calculate demand functions. These will all be functions of the price. Then, we will impose the condition that the markets must clear. We can do this either by setting q A 1 + q B 1 = e 1 or q A 2 + q B 2 = e 2 . By Walras’ Law, we only need to impose one of these two market clearing conditions. The price that clears the market is then the competitive equilibrium price. (a) U A ( q 1 , q 2 ) = U B ( q 1 , q 2 ) = q 1 q 2 . e A = (20 , 20), e B = (100 , 60). Agent A ’s wealth, y A ( p ) is p 1 e A 1 + p 2 e A 2 = 20 p +20. So A ’s demand for good 1 is q A 1 = 1 2 20 p +20 p and A ’s demand for good 2 is q A 2 = 1 2 (20 p + 20). Likewise, agent B ’s wealth, y B ( p ) is 100 p +60. So B ’s demand for good 1 is q B 1 = 1 2 100 p +60 p . Now that we have the demand functions for one of the goods we can impose the market clearing condition. There are 120 total units of good 1 in the economy, so the total demand for good 1 at the market clearing price must be 120: q A 1 + q B 1 = e 1 1 2 20 p + 20 p + 1 2 100 p + 60 p = 120 60 + 40 p = 120 40 p = 60 p = 2 3 Now that we know the price, we can plug into agent A ’s demand functions to find A ’s allocation: q A 1 = 25 and q A 2 = 50 3 . By the market clearing condition we know that the remainder of the two goods must go to agent B : q B 1 = e 1 - q A 1 = 120 - 25 = 95 and q B 2 = e 2 - q A 2 = 80 - 50 3 = 190 3 . (b) U A ( q 1 , q 2 ) = U B ( q 1 , q 2 ) = q 1 2 1 + q 2 . e A = (20 , 20), e B = (100 , 60). Since the endowments are the same as in part (a), so must be y A ( p ) and y B ( p ). 1

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To find the demand functions, we set MRS = p 1 p 2 for each agent. For agent A this implies that 1 2 1 q A 1 = p q A 1 = 1 4 p 2 Plugging this demand function into agent A ’s budget constraint, we find y A ( p ) = pq A 1 + q A 2 20 p + 20 = 1 4 p + q A 2 q A 2 = 20 p + 20 - 1 4 p Similarly, q B 1 = 1 4 p 2 . Now impose that the market for good 1 must clear: q A 1 + q B 1 = e 1 1 4 p 2 + 1 4 p 2 = 120 p 2 = 1 240 p = 1 4 15 0 . 0645 At this price A consumes q A 1 = 60 and q A 2 = 5 15 + 20 - 15 17 . 418. And so, q B 1 = 60 and q B 2 = 80 - 5 15 + 20 - 15 62 . 582. (c) U A ( q 1 , q 2 ) = q 1 + q 2 , U B ( q 1 , q 2 ) = 1 2 q 1 2 1 + 1 2 q 1 2 2 . e A = (20 , 20), e B = (20 , 60). First note that MRS A = 1. So, if p < 1 then agent A would demand all of the good 1 in the economy and no good 2. Therefore, the market for good 1 could not possibly clear because agent B will also demand some of good 1. Likewise, if p > 1 then agent A would demand all of the good 2 and the market for good 2 could not possibly clear. Therefore it must be the case that p = 1.
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HW13_solutions - Intermediate Microeconomics 2008 Problem...

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