HW10-REV - Economics 11, Winter 2006: Homework #10...

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Unformatted text preview: Economics 11, Winter 2006: Homework #10 (Corrected) Not to be turned in The first 3 problems treat a model of the economies of two countries of equal size. In each country, labor can be turned into Butter B (think of this as consumer goods) or Guns G (think of this as national defense). Country A is more "pacifistic" that is, the people care more about consumer goods and less about national defense than Country B. Country A: goods: L = labor, B = butter, G = guns 1 consumer (or many identical consumers) endowment = 1 unit of labor L (each) utility function u(B, G) = B 4/5 G1/5 consumer does not care about own labor industry production functions B = 5L, G = 5L Country B: goods: = labor, B = butter, G = guns 1 consumer (or many identical consumers) endowment = 1 unit of labor (each) utility function u(B, G) = B 3/5 G2/5 consumer does not care about own labor industry production functions B = 10L, G = 2L Notice that the countries have different tastes and different technologies. 1. Consider Country A by itself. (a) What variables are to be determined at equilibrium? 1 (b) What equations must hold at equilibrium? (c) Solve for the equilibrium. Express all prices relative to the the price of butter pB 2. Consider Country B by itself. (a) What variables are to be determined at equilibrium? (b) What equations must hold at equilibrium? (c) Solve for the equilibrium. Express all prices relative to the the price of butter pB 3. Consider the economy in which butter and guns can be traded but labor cannot move; that is, A labor must be used in A industries, B labor must be used in B industries. (a) Which industries will operate at equilibrium? (b) What variables are to be determined at equilibrium? (c) What equations must hold at equilibrium? (d) Solve for the equilibrium. Express all prices relative to the price of butter pB (e) Compare equilibrium consumptions and utilities with equilibrium consumptions and utilities from the previous 2 problems. Explain briefly. The next 2 problems will give you a little experience with non-linear production functions. Consider an economy with two kinds of consumers. Type A consumers are endowed with 1 unit of unskilled labor K and care about their own labor K and sculpture S; their utility function is uA (K, S) = K 2/3 S 1/3 . Type B consumers are endowed with 1 unit of skilled labor L and care about their own labor and sculpture S; their utility function is uB (L, S) = L1/3 S 2/3 . [Caution: Unskilled labor and skilled labor are really different goods are are NOT interchangeable.] There is one industry which turns capital and labor into sculpture according to the production function S = K 1/2 L1/2 . 2 4. Suppose there is 1 consumer of type A and 1 consumer of type B. (a) What variables are to be determined at equilibrium? (b) What equations must hold at equilibrium? (c) Solve for the equilibrium. Express all prices relative to the the price of sculpture pS . 5. Suppose there are M consumers of type A and N consumers of type B. (a) What variables are to be determined at equilibrium? (b) What equations must hold at equilibrium? (c) Solve for the equilibrium. Express all prices relative to the the price of sculpture pS . (d) What happens to the utility of type A consumers when M and N stays fixed? (e) What happens to the utility of type B consumers when M and N stays fixed? (f) What happens to the utility of type A consumers when N and M stays fixed? (g) What happens to the utility of type B consumers when N and M stays fixed? The last three problems are about non-competitive firms (monopolies). 6. TrendyTees produces T-shirts for pre-school kids in Manhattan Beach. Its production cost is C(Q) = 5Q and it faces the market demand function Q= 1000 - 40p 0 if if p 25 p > 25 (Prices in dollars.) If TrendyTees wants to maximize its profit, how much should it charge? How many T-shirts will it sell? What will its profit be? 3 7. Amazon is planning to begin selling personalized book wrappers. Amazon estimates that it can produce wrappers at a cost of $0.50 each, and that if it charges the price p per wrapper, it will be able to sell Q = (5, 000, 000)p-3/2 wrappers. If Amazon wants to maximize its profit, how much should it charge? How many wrappers will it sell? What will its profit be? (The answer may be a little messy.) 8. The exponent -3/2 in the previous problem was carefully chosen. Show that if the market demand function is Q = pa with 0 > a > -1 and the cost function is C(Q) = Q, then the profit function does not have a maximum. 4 ...
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This note was uploaded on 04/01/2010 for the course ECON Econ 11 taught by Professor Mcdevitt during the Fall '07 term at UCLA.

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