Midterm Preparation, Microeconomics
The midterm exam will consist of one question from the first question cluster and:
(a) Let f(x1, x2) = 2x 0.3 1 x
0.2 2 be a production function. Find the short run average cost curve for each of x2 = 1, 1.5, 2, 2.5, 3 and for generic prices w1, w2. Find the marginal cost curve as well. Find the long run average cost curve for specific prices of w1 = 0.5 and w2 = 1. Graph your results. Be sure to include (and label) in your graph the levels of output for which the long run average cost curve would coincide with each of the short run curves above (for the specific prices). Find the long run marginal cost curve at each of these levels.
(b) Let c(y1, y2) = y1 + y2 + (y1y2) 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 scope exist if for a positive set of y1 and y2, c(y1, y2) < c(y1, 0) + c(0, y2). [Hint: Be very careful to handle the case of y2 = 0 separately.]
(c) Let y = Axα. But assume we do not what A and α are. Here’s what we do know. Suppose that we see when p = 2 and w = 1 then x = 3. Also when p = 1 and w = 1.5, then x = 4. Can we identify the parameters of the production function from these two observations? Graph what an economist who didn’t know the functional form of the production function would conclude about the production set if in the first observation y = 8 and in the second y = 10. [Hint: In the first part of this question you can now no longer use just the production function and instead need to work with the maximization condition which is loosely w = pTRS. For the second, you should work with isoprofit curves and a revealed profitability argument.]
(d) Suppose a Cobb Douglass production function with two inputs and exponents inside the production function y = xα11 x α2 2 that are less than one. Derive the profit maximizing choices of x1, x2, and y for arbitrary prices. How does this simply if α1 and α2 sum up to one?
(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, i.e. the amount of output where production is cheaper if it is concentrated in one firm. Derive the long run, industry, average cost for this function if production is undertaken by the most cost efficient number of firms for each level of output.
(f) Assume a market demand function of D(P ) = 50 − P and a per firm cost function of C(q) or 20q. Find the competitive equilib- rium. Assume that in order to boost wages, the local gov’t applies a minimum price on output of 40 and establishes a monopoly to serve it. Assume that a monopolist will fill all of the demand at the regulated price floor. What is the effect of establishing a monopoly on output?
Suppose there is an innovation which can lower the cost of providing the good to 10q but only one firm has access to it. What is the value of this innovation, in a market without any price control: to consumers, producers and society as a whole (total surplus)? [Note you are looking at the change in these types of surpluses after innovation not the level.] You should assume that the industry is a monopoly as long as the price is at or below the previous competitive equilibrium price. What is the value of this innovation under a market with a price floor and a monopoly? Assuming that innovation is the result of costly investments by producers, how does this type of price control affect incentives to innovate? That is, in which market structure will the firm with access to the innovation be willing to pay the most for it?
(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”? How would the above result change if a firm faced a cost of (q1 − q2)2 if it changed output from q1 to q2 after a price change.
(h) Assume an isoquant for a fixed level of output equal to ȳ =1/2x 1 2 1 x 1 3 2 Fix w1 = 1. Show how the cost minimizing bundle changes as w2 moves from 1 to 2. You can graph x1 and x2 on separate graphs as a function of w.
(i) Describe a hypothetical situation in which periods of low inflation can be used to estimate the effects of eliminating usury laws (a price ceiling on interest rates) for similar values of real interest rates without low inflation. Be as specific as possible in terms of the estimation you would run in terms of the counterfactual implicit in such an estimation. Explain graphically how such a comparison is consistent with economic theory. 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 . You own a fixed amount of non-resaleable reserves k̄. 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? [A tax profile is a set of time specific marginal tax rates.] Suppose instead the production 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) Derive the isoquant for a firm that has two inputs which are perfect compliments. Use this to solve for the firm’s 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. You should be able to due this for any preferences within these classes and your answer should have a specific, though not-necessarily, polynomial analytical form.
(l) Given our discussion in class about eyeglasses and correcting for any typos in the slides, we can speculate about another similar historical event. In 1977, the Supreme Court ruled that indi- vidual states could not ban lawyers from advertising either their availability or their prices. Unlike with eyeglasses, all states had previously enacted a ban. What do you think was the effect of this decision on (long run) legal fees?