Unformatted text preview: Econ 100A – UC Berkeley Fall 2011 Prof. Santesteban Midterm 2 Review Problems II These are optional review problems. You are not required to do any of them. Multi‐Part Problems 1. Suppose that you are given a cost function c(w,r,q)=2w1/2r1/2q3/2 where w is the wage rate for labor, r is the rental rate of capital and q is the output level. a. Does the production process that gives rise to this cost function have increasing, decreasing or constant returns to scale? (Make sure to support your answer.) b. Derive the marginal and average cost functions. c. Calculate the supply function for the firm. How does output by the firm change as input and output prices change? d. If the cost function had been c(w,r,q)=2w1/2r1/2q1/2 instead, how would your answer to (c) change? Does that make sense? 2. Suppose there are different ways of producing computer chips. If you hire one worker (for the day) for each machine that you rent (for the day), you can produce 10 chips per day with each worker/machine pair for the first 60 machine/worker pairs. For the next 60 worker/machine pairs (assuming still that you hire them as pairs for the day), you are able to produce 20 chips per day with each of the additional pairs. Once you have 120 worker/machine pairs, you can only get 5 additional chips per day for each additional pair. But hiring 1 worker for each machine is not the only way to produce computer chips. Suppose you are starting from a production plan where you are using exactly as many workers as machines to produce a given level of chips. The technology is such that, starting at the production plan where you are using the same number of workers as machines, you can replace 1 or more workers with two machines (for each worker) and get just as many chips produced. Alternatively (and again starting at the production plan where you use exactly as many workers as machines), you can replace 1 or more machines with 2 workers (for each machine) and get just as many chips produced. a. On the template below, illustrate all the different ways that 600 chips can be produced per day. (Hint: The isoquant you should draw is composed of two line segments.) b. On the same graph, illustrate the different ways of producing 400 chips and the different ways of producing 1,800 chips. (Label each isoquant with the relevant output quantity). c. Is this production technology homothetic? d. If machines cost $100 per day, for what range of daily wages will you decide to use exactly as many workers as machines? e. Suppose both machines and workers cost $100 per day. Illustrate the long run cost curve for this firm. f. Illustrate the long‐run marginal and average cost curves. 3. Same as Problem 2 but now suppose the daily wage and rental rate are both equal to $100 and the firm currently has 120 units of capital. a. Illustrate the short run production function (assuming labor is variable in the short run but capital is not). (Label the intercept as well as any kink points.) b. Derive the short run cost function. (Label the intercept as well as any kink points.) c. Derive the short run marginal and average cost functions. d. How low can price fall in the short run before a firm shuts down? e. What does the average expenditure – i.e. the curve that includes all short run costs but also expenditures that are not costs in the short run – look like? Explain how this curve relates to the long run average cost curve. f. We have said that the long run supply responses to output price changes are larger than short run supply responses. In what sense is this true for the firm you have analyzed here? 4. Suppose GE produces 1 million light bulbs per month While labor is variable both in the short run and the long run, capital is fixed in the short run. Labor is sold at a rate w and capital is rented at a rate r. a. On a graph with labor on the horizontal axis, illustrate the current isocost and isoquant for GE. Carefully label the slope of the isocost. b. For the rest of the problem, suppose a new tax on capital is implemented but GE intends to continue to produce 1 million light bulbs per year. What will GE do differently in the short run and the long run? Explain using your graph from part (a). c. Using your answer to part (b), explain what happens to the short run cost curve in the short run. What happens to this short run curve in the long run? Do costs rise more or less in the long run than they do in the short run? d. Do total costs rise more or less in the long run than total expenditures do in the short run? Explain. 5. Suppose all firms in an industry have a production technology described by the production function . The cost of labor is 2 and the cost of capital is 4, and each firm faces a recurring fixed cost of 300. a. Derive the long run cost and average cost functions for each firm. (Hint: Given the shapes of the isoquants implied by the production function, you should be able to do this without solving a calculus problem.) b. What is the long run equilibrium output price? c. How much does each firm produce in long run equilibrium? d. Suppose market demand is given by long run equilibrium? . How many firms are in the industry in e. Suppose the industry is currently in long run equilibrium. Derive the short run cost function for each firm (assuming labor is variable but capital is fixed in the short run). f. Now suppose that demand falls to . What happens to output price in the short run? What happens to price and the number of firms in the long run? 6. Suppose that the market demand curve is a. Calculate the equilibrium price and output level. and the market supply curve is . b. Suppose a price ceiling of 6 is imposed. What is the new equilibrium quantity transacted in the market? c. How does the price consumers pay (including any marginal effort costs) compare to the price firms receive? d. What is the total cost of the additional effort exerted by consumers? 7. Which of the following is definitely true for a per‐unit tax in the goods market where neither demand nor supply is perfectly inelastic: a. The more price inelastic demand is, the lower deadweight loss will be. b. The more price inelastic supply is, the higher deadweight loss will be.
c. A demand become more price inelastic, the after tax price for consumers rises. d. Both (a) and (b) e. Both (b) and (c) f. Both (a) and (c) g. All of the above h. None of the above ...
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