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# Answer 2 question 1 1 in an economy with three agents

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2 Question 1-1. In an economy with three agents (A, B, and C) and two goods (X and Y), define what is meant by a Pareto-efficient allocation between A, B, and C. Use no more than 50 words. ANSWER: PART 2: TWO SHORT-ANSWER QUESTIONS [20 marks each] Question 2-1 (20 marks) Consider an exchange economy with only 2 agents, Arnold and Brigitte, and 2 goods, X and Y. Let Arnold have utility function 3 / 1 ) , ( A A A A A Y X Y X U = and Brigitte B B B B B Y X Y X U = ) , ( . Here ) , ( A A Y X and ) , ( B B Y X represent the good bundles of Arnold and Brigitte, respectively. Let Arnold have an initial endowment of ) 0 , 3 8 ( ) , ( = E A E A Y X and Brigitte ) 5 , 3 1 ( ) , ( = E B E B Y X . (a) Draw the initial endowment, as well as a few indifference curves of Arnold and Brigitte in an Edgeworth box. Prove graphically (i.e. using the Edgeworth box) whether or not the initial endowment is Pareto-efficient . (b) To avoid confusion, draw up a new Edgeworth box and indicate clearly the contract curve , the set of individually rational trades between Arnold and Brigitte, and the core of this economy. (c) Let the price of X be 3 and the price of Y be 1. Derive Arnold’s optimum bundle graphically —do this again in a fresh Edgeworth box— and mathematically. (d) Prove mathematically that the prices given in (c) constitute competitive equilibrium prices, and find the competitive equilibrium allocation. ANSWERS:

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4 Question 2-2 (20 marks) Consider a firm with production function , ) , ( 2 / 1 KL L K F = where K represents the production factor capital and L labour. Let the price of capital be 2 = r and the price of labour be 2 = w . (a) Give an expression of an isocost line for this firm using the factor prices given above. Graph all input combinations that lead to a cost level of \$100,000. (b) Give a mathematical representation of the cost minimization problem that this particular firm solves. Assume that the firm intends to produce q units of output. (c) Derive the optimum amounts of capital and labour for this firm as a function of q. (d) Give the cost function for this firm. (e) Print the correct version of the following line in your answer booklet: “The firm’s average cost decreases / increases / stays constant because the production technology exhibits constant / decreasing / increasing returns to scale.” ANSWERS: <end of quiz>
Economics 203 Intermediate microeconomics ANSWERS QUIZ 2 Question 1-1 Sketch: Draw an isoquant (nice and convex, and downward sloping) and an isocost curve (downward-sloping line) in a L-K diagram. Mention that production function is F ( K, L ) and that in the SR K = K. Then fi rst fi nd the combination of L and K with minimum costs, i.e. the point where the isocost line just touches the isoquant. This is the LR optimum bundle. This isocost line characterizes the long-run minimum cost level. Second, mark the point that crosses the isoquant and with K = K . This is the SR optimum bundle.

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