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Solution_Winter2007

Solution_Winter2007 - Econ 51 Final Exam Suggested...

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Econ 51: Final Exam - Suggested Solutions 1. General Equilibrium (27 points) a. i. (3 points) These are log utilities, so all we need to do is to equate MRSs. This gives us b i a i = b j a j for each i; j . In addition, to be feasible, we must have the sum of the bananas equal to 5 N and the sum of the apples equal to 10 N . These two conditions must mean that the set of all Pareto E°cient allocations is the set of allocation that satisfy b i a i = 1 2 for all i . ii. (3 points) Normalize p a = 1 and denote p b = p . We will clear the apple market. Excess demand by each individual (they are all identical) is given by z i a ( p ) = 1 2 10+5 p 1 ° 10 = 2 : 5 p ° 5. Total excess demand in the market is simply z a ( p ) = N (2 : 5 p ° 5). To clear the market, we must have p = 2. Allocation for each agent is then given by (10 ; 5), that is the initial endowment. The solution does not depend on N . This is because, essentially, the general equilibrium already assumes very many identical agents of each type, so having N types that are all identical is exactly like an economy with a single individual. b. i. (3 points) Now the MRSs must also be equal the MRT, which is 1. Thus, we must have b i a i = 1 for all i . To achieve this, the production plan for the ±rm must make total number of apples equal total number of bananas. This is achieved by a production plan of ( y a ; y b ) = ( ° 2 : 5 N; 2 : 5 N ). ii. (3 points) We know that the ±rm will be in business (due to the shortage in bananas), so prices must be p a = p b = 1 (we normalized the price of apples as before). At these prices, allocations to each individual are (7 : 5 ; 7 : 5). The ±rm’s production plan is ( y a ; y b ) = ( ° 2 : 5 N; 2 : 5 N ). For the same reason as in part a.ii, the answer doesn’t depend on N . iii. (2 points) It won’t change any of the answers. The overall production plan in the industry will remain the same (for both parts), and it can be arbitrarily split between the two ±rms, as they make zero pro±ts anyway. That is, production plans must satisfy ( y A a ; y A b ) = ( ° x; x ) and ( y B a ; y B b ) = ( ° w; w ) such that x; w ± 0 and x + w = 2 : 5 N . iv. (2 points) It won’t change any of the answers. The overall production plan in the industry will remain the same (for both parts), and it can be arbitrarily split between the two ±rms, as they make zero pro±ts anyway. That is, production plans must satisfy ( y A a ; y A b ) = ( ° x; x ) and ( y B a ; y B b ) = ( w; ° w ) such that x; w ± 0 and x ° w = 2 : 5 N . c. i. (4 points) Normalize p a = 1 and denote p b = p . We will clear the apple market.
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