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Midterm Preparation, Microeconomics March 17, 2017 1. One question from the first question cluster and: (a) Let f(x1, x2) = 2x 0.5 1 x 0.3 2 + 0.0001x 2 1x 2 2 Find the short run average cost curve for x2 = 1, 1.5, 2, 2.5, 3 Find the long run average cost curve. (b) Let c(y1, y2) = y1 + y2 + (y1y 2 ) 1 3 Does this cost function have economies of scale for y1? What about economies of scope for any strictly positive y1 and y2. Hint, economies of scale exist if for a positive set of y1 and y2, c(y1, y2) > c(y1, 0) + c(0, y2) (c) Let y = Axa lpha. Suppose that we see when p = 2,w = 1, x = 4 and y = 8 and when p = 1, w = 0.2, x = 6. Can we identify the production function. Suppose that we know nothing of the functional form and that in the second example y=10. Graph what we know about the production function. (d) Suppose a Cobb Douglass production function with two inputs and exponents inside the production function y = x α1 1 x α2 2 that are less than one. Derive the profit maximizing choices of x1, x2, andy for arbitrary prices. (e) Let AC(y)=1/y+2+0.1y. Find the efficient point of production for this firm. Suppose every firm in the industry has access to the technology, find the region of economies of scope. Derive the long run average cost for this function. (f) Assume a market demand function of D(P)=100-P and a per firm cost function of C(q) or 20q. Find the competitive equilibrium. Assume that in order to boost wages, the local govt applies a minimum price on output of 30. What are the effect of this on output? 1 Suppose instead that the regulation raises the cost of product. Repeat the analysis. Go back to the first part of the analysis with the regulated price but not the additional cost. Suppose there is an innovation which can lower the cost of providing the good to 5q. What is the value of this innovation under both an unregulated market and a regulated market to consumers, producers and society as a whole (total surplus)? (g) Use profit-maximization and revealed choices by firms to show the law of supply. Be sure to define what is meant by the law of supply? (h) Assume an isoquant for a fixed level of output equal to ¯y = 1/2x 1 2 1 x 1 3 2 Fix w1 = 1. Show how the profit bundle changes as w2 moves from 1 to 2. (i) Provide a historical situation in which periods of low inflation can be used to estimate the effect of usury laws (a price ceiling on interest rates) for similar values of real interest rates without low inflation. How do mortgage lenders typically ration credit when price ceilings bind. (j) Lets suppose that a firm owns a proven oil reserve of k units which cannot be transferred and a technology for producing oil from reserves of y = k 1 3 . Suppose the price of oil is constant the interest rate (1+r)=1.03 and the oil expires after 2 periods. Solve for efficient use of oil over two periods. Suppose that an outside policy maker wants to reduce current oil production. Can they do that with a constant tax on oil production. What must be true of any of tax profile which accomplishes the policy makers goal? Suppose instead the cost function equals y = k 1 3 + 1 for any y > 0 but is equal to zero for no production. How do your answers to the above change? (k) Suppose there is an industry with fixed costs equal to 1 and an average variable cost equal to 5 + 0.1y. Suppose that price in this industry equals 10. Find the amount of production and the producer surplus (short run). Be sure to define producer surplus and show it graphically. Find also the short run and long run break even prices. Suppose demand is sufficiently large. What is the long run equilibrium price for this industry? 2 (l) Assume the previous industry is such that there is an incumbent firm which cannot recover the (already paid) fixed cost over any horizon even if it exits the industry but firms that want to enter still must pay it. What is the long run equilibrium price and distribution of profits for this industry? (m) Derive the isoquant for a firm that has two inputs which are perfect compliments. Use this to solve for the firms cost minimizing input bundle and in terms the cost function c(y). Repeat for a firm that has a perfect substitute technology for producing output. (n) Given our discussion in class about eyeglasses and correcting for any typos in the slide, we can speculate about another similar historical event. In 1977, the Supreme Court ruled that individual states could not ban lawyers from advertising either their availability or their prices. Unlike with eyeglasses, all states enacted a ban. What do you think was the effect of this decision on legal fees? (o) Suppose the demand for product X = 10 − 2X + Y where Y is a substitute that is not currently being produced. Assume X has a marginal cost of one dollar. Entry is completely barred and there is a unique monopolist that produces good X. Find the firms price quantity and profit. [Hint: remember that MR(Q)=MC(Q) for the function MR(Q) which is computed by taking the derivative of the demand curve. The market price is taken directly from the demand curve.] i. Next suppose that the incumbent considers introducing good Y . The demand for good Y is Py = 10−2Y −X. Assume that there is a fixed cost of 4 dollars for introducing Y and a constant MC of 1 dollar for each additional unit of Y . Find the optimal values for Px, Py, x, and y. Will this incumbent introduce Y ? Is it in societys best interest for Y to be introduced. Compute T S to show this. ii. Next allow entry to occur but that conditional on entry by one firm into the market for Y , the two firms will split profits evenly. Will the entrant have the incentive to introduce Y ? iii. Finally assume that entry is allowable and that our initial incumbent firm can prevent entry by introducing good Y first 3 (this comes from the property that if the incumbent picks the entire profit maximizing quantity, there arent enough profits left over for the entrants to actually enter. Will the incumbent introduce good Y . iv. Repeat the above exercise with a fixed cost of 6. Is it still optimal from the societal prospective for good Y to be introduced. Look at all the market structures (no entry, entry with cooperation, entry with deterrence). Are there any market rule(s) that will reproduce the TS maximizing rule for the decision to provide Y ? If so, why? If not why not? 4
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