Microeconomic Analysis

Overview

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1. (Supply function) The cost function of a firm is C(y) = 13 y3 − 2y2 + 4y + 1. Suppose the firm is a price taker.
   1. (a) What is the lowest price for the firm to keep operating in the short term? Find its short-term supply function.
   2. (b) What is the lowest price for the firm to keep operating in the long term? Find its long-term supply function.
2. (Short-term competitive equilibrium) The market demand function is: X(p) = 21−5p. There are 5 identical firms with firm cost function C(y) = y2 + 1.2. (a) Solve for the short-term competitive equilibrium price in the market. (b) What is the profit for each firm?
3. (Long-term competitive equilibrium) In the above question, suppose there is no barrier for entry and exit. (a) What will happen when other firms see the result you get in the above question.
   (b) Solve for the long-term equilibrium price, output, and number of firms in the market. (c) Compute the consumer surplus, producer surplus, and total surplus.
4. (Welfare) In the above question, suppose government levies a quantity sales tax of t = 0.3.
   1. (a) Find the new long-term equilibrium price, output, and number of firms.
   2. (b) Compute the consumer surplus, producer surplus, and total surplus.
   3. (c) How does government tax affect social welfare (total surplus)? Show your idea with a clearly defined graph.
5. (General Equilibrium) Sarah and Jane are two representative individuals living in an economy that produces two goods, X and Y. Sarah’s and Jane’s utility functions are given as: Sarah : US = 100X0.3Y 0.7 Jane : UJ = 50X0.5Y 0.5 The market determined prices of X and Y are $10 and $20, respectively. Current outputs are 58 units of X per time period and 36 units of Y. Jane’s current income is $600 per time period, while Sarah’s income is $700 per time period. (a) Write expressions for Sarah and Jane’s marginal rates of substitution. 1

1. (b) Determine the quantities of X and Y that Sarah and Jane should consume given the prices and income.
2. (c) Do the values calculated in part (b) satisfy the conditions for equilibrium in exchange? Explain using numbers.
3. (d) Examine your answers in parts (b) and (c). If equilibrium has not been achieved, what would be necessary to reach equilibrium? If equilibrium has been achieved, comment on the process by which equilibrium was reached.

6. (General Equilibrium) Consider an exchange economy with two consumers (1 and 2) and two goods (A and B). The preference of both consumers are given as u1(x1A,x1B) = x1/4x3/4,

1A 1B u2(x2A,x2B) = x3/4x1/4. The initial endowments for consumer 1 is w1 = (w1A,w1B) = (3,5)

2A 2B and the initial endowments for consumer 2 is w2 = (w2A,w2B) = (5,3). Both consumers are

both price-takers. They meet in the market and trade with each other. Suppose the price for good A is normalized to be always 1, pA ≡ 1. (Note: w1 = (w1A,w1B) = (3,5) means that the mighty God gives consumer 1 with 3 units of good A and 5 units of good B before any economic activity. w2 is similarly defined.)

1. (a) Draw the Edgeworth Box for this economy. Draw four indifference curves for each consumer. Highlight the Pareto efficient set.
2. (b) (Compute the general equilibrium) What is the competitive equilibrium price(pB)? (Hint: use the definition of equilibrium. What is the budget constraint in this case? Suppose the price is p = (pA,pB), then the budget constraint is pAx1A+pBx1B = pAw1A+ pBw1B for consumer 1. Similar for consumer 2.)
3. (c) Is this competitive equilibrium Pareto efficient?