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Part III Exercise #1. [10 pts.] A monopolist has an inverse demand curve given by p(y) = 12-y and a cost curve given by c(y) = y". Describe how to determine its profit-maximizing level of output, and compute it. Show your work: to receive full credit you should show how you compute the optimal quantity of output. Exercise #2. [15 pts.] You are the manager of a new amusement park that has monopoly power over the service it offers: roller coaster rides. You have figured that the park will attract 1,000 people per day, and each person will take a = 50 - 50p rides, where p is the price of a ride. The marginal cost of a ride is essentially zero. (a) If admission to the park were free and you had to choose the profit maximizing price of a ride p', which value would you set? (b) Suppose now that you can use a two-part tariff, i.e., you can set a price for admission to the park, and another one for each roller coaster ride. Which admission fee and price of a ride would you choose? Show your work: to receive full credit you should show how you compute the optimal prices. Exercise #3. [20 pts.] Your company has monopoly over product H. It sells it in the UK and in the US. The cost function for your firm is c(q) = 10q. Your marketing department has identified the UK and the US demand curves for H to be qV8 = 50,000 - 2,000pus qUK = 10,000 - 500PUK; where qus and qui represent the quantities demanded by US and UK consumers respectively, and pus and pux the (US dollar) prices charged for the product in the US and in the UK, respectively. (a) If you were to charge the same price in both countries, how many units of H should you sell, and what price should you charge in order to maximize your company's profits? (b) If you could charge different prices in the US and the UK what prices would you choose and how many units would you sell in the US and how many in the UK? Exercise #4. [20 pts.] A company sells product A in a competitive market. Its long-run cost function is given by "(y) = y + 10 for y >0 c(y) = 0 for y = 0, where y represents the quantity of good A. (a) What is the lowest price at which this company will supply a positive amount of product A in the long-run? (b) Suppose that the market price for A is p = 2v10, and that 100 firms are operating in this market in the long-run. What is the equilibrium market demand for the product? Part II Exercise #5. [17 pts.] The rental price of machinery K (measured in machine-hours) is $10 per hour, while the hourly wage rate for labor, L (measured in labor-hours), is $6. Find the cost function associated with the following technology: y = 10K + L, where K represents machinery (measured in machine-hours) and L labor (measured in labor-hours). Part I Exercise #6. [18 pts.] John likes books and restaurant meals. His utility function is u(It, I'm) = (1)02 (1m)0.4 where r; represents the quantity of books that John "consumes" in a month, and Im the amount of restaurant meals that he consumes. Suppose that both books and restaurant meals are perfectly divisible goods. The relative price of books in terms of restaurant meals is 0.4, i.e., buying a book costs 40% of the price of a restaurant meal. The nominal price of a restaurant meal is $40. John's income is $1,000 per month. Compute the quantity of books and restaurant meals that John consumes in a month.Exercise 10.4 (Moderate) For the purposes of this exercise, assume that you have data on annual national job creation ", and job destruction D, for NV years, so f = 1. .. N. Show that if annual national job reallocation , and net job creation NET, have a negative covariance, then the variance of job destruction must be greater than the variance of job creation. Recall the definition of variance of a random variable X for which you have A observations, fa. ] var(.1)= = (x -17. where I is the mean of X. Similarly, recall the definition of the covariance of two variables X and Y'. If there are / observations each, fromlin, then: cow(X,Y)= = >(x, - = -D). These definitions and the definitions of NET, and /, provide all the information necessary to answer this exercise. 94 Unemployment Exercise 10.5 (Easy) Consider the employment statistics in chart below. Compute each of the following five measures: (1) the economy-wide rate of job creation a; (1) the economy-wide rate of job destruction do; (ill) the net rate of job creation note; (iv) the upper bound on the number of workers who had to change employment status as a result of the gross job changes, and (v) the lower bound on the number of workers who had to change employment status as a result of the gross job changes for each each of the years 1991, 1992, 1993, 1994, and 1995. Year net.UB LB 1990 1000 0 500 800 100 1992 1200 200 700 1993 1000 600 1994 800 800 500 1995 400 1200 600 1996 200 1400 600 1997 0 2000 500 Exercise 10.6 (Moderate) For each of the following statements, determine if it is true, false or uncertain and why. If possible, back your assertions with specific statistical evidence from DHS. 1. Foreign competition is destroying American manufacturing jobs. 2. Robots and other capital improvements are replacing workers in factories. 3. Most job creation occurs at plants that grow about 10% and most job destruction occurs at plants that shrink about 10% 4. Diversified plants are better able to withstand cyclical downturns. 5. Every year, high-wage manufacturing jobs are replaced by low-wage manufacturing jobs. O O C.Exercise 19.4 (Moderate) For this exercise, we will consider what happens when the government and the private sector repeatedly interact. Unemployment in period tut, inflation at and inflationary ex- Exercises 241 pectations * ? are related by the simple Phillips curve: " =u; + 7("; - m), for all t = 0, 1, ... , 00. The parameter y is fixed over time. The government knows about the Phillips curve, but the private sector does not. The government has preferences over unemployment and inflation in period t of: V(71, 1;) = -12 -x , for allt = 0, 1, .. .; 00. The private sector sets inflationary expectations based on last period's inflation. This is known as adaptive expectations. As a result, a; is given by: of = *1-1, for all t = 1, 2, . .. , DO. Assume that ; = 0, that is, the private sector begins by believing that inflation will be zero. Answer the following questions: 1. Assume that the government takes as given expectations in a period , and picks the inflation rate at which gives it the highest payoff in period t. Find the government's choice rule *; (;). 2. If the government sets inflation n = " (";), how do expectations evolve over time? Thus right down a law of motion for inflation, ,("-1). 3. What do the trajectories of inflation and unemployment look like over time? Are they rising or falling? Do they settle down? If so, where? 4. How would your answer have been different if, instead of the initial expected infla- tion being zero, it had been some very large number instead? 5. Now assume that the Phillips curve is augmented with a mean zero shock term, E, so: Assume that the government knows the value of a, and reacts appropriately. Now what happens? Exercise 19.5 (Easy) To answer this exercise, you need to answer Exercise 194 above. Imagine that the pri- vate sector has adaptive expectations about the government's inflationary policy over time, but that part of expected inflation is the government's announced inflation target. This an- nounced inflation target is merely an announcement and has nothing to do with reality. If " is the announced target for period-t inflation, expectations satisfy: of = 6-1 + (1 -8)m/, for all t = 1, 2, ... , DO. Here 0 0 given, (1) a = [5( in) - 8 - p-ga- b(p+ m)k, (2) and the condition that for any fixed pair (v, to), where to 2 0 and v S to. lim dare- fio(((-))-5+mydo = 0. (3) Notation: K, = Ki/(T,L,) and & = Ci/(T,L.) = c/It, where K, and C are aggregate capital and aggregate consumption, respectively, L, is population = labor supply, and 7, is the technology level, all at time t. Finally, f is a production function on intensive form, satisfying f (0) = 0, S' > 0, f" To- After this shock everybody rightly expects T to grow forever at the same rate, g, as before. d) Illustrate by the phase diagram (or a new one) what happens to & and on impact, i.e., immediately after the shock, and in the long run. e) What happens to the rate of return on impact and in the long run? f) Why is the sign of the impact effect on the real wage ambiguous (at the theoretical level) as long as f is not specified further? g) What happens to the real wage in the long run? V.3 Fiscal sustainability. Consider the government budget in a small open economy (SOE) with perfect mobility of financial capital, but no mo- bility of labor. The real rate of interest at the world financial market is a positive constant r. Time is continuous. Let Y = GDP at time t. Ge = government spending on goods and services at time f, T, = net tax revenue (gross tax revenue - transfer payments) at time f, Be = public debt at time t. All variables are in real terms (i.e., measured with the output good as nu- meraire). Taxes and transfers are lump-sum. Assume there is no uncertainty and that the budget deficit is exclusively financed by debt issue (no money financing). a) Write down an equation describing how the budget deficit and the increase per time unit in public debt are linked.VI.I A carbon tar and Tobin's q We consider a small open economy (henceforth called SOE) with perfect mobility of financial capital but no mo- bility of labor. The SOE faces a constant real interest rate r > 0, given from the world market for financial capital. The technology of the representative firm is given by a neoclassical production function with constant returns to scale, Y, = F(Ki, Ly, Me), F. > 0, F. 0, ifj. The firm faces strictly convex capital installation costs and the installation cost function is homogeneous of degree one: J = G(1, K;) = Kig(1/ K.). 4(0) = '(0) = 0, g" > 0. In national accounting what is called Gross Domestic Output (GDP) is aggregate gross value added, Le., GDP, = Y - J- PMML (1) where pur > 0 is the exogenous real price of oil which we treat as a shift parameter. The labor force of the SOE is a constant L. There is perfect competition in all markets. There is a tax, + > 0, on use of fossil energy (a "carbon tax"). In equilibrium with full employment the following holds: M. = M(K., (1 + 7)PM). My > 0, Mp. 0 given, (4) where m(1) = 0, m' = 1/g", and & is the capital depreciation rate whereas g is the shadow price of installed capital along the optimal path, satisfying the differential equation i = (r+5) - MPK(K, (1 + +)PM) + 9(m(q:)) - m(g.)(q -1). (5) Moreover, a necessary transversality condition is lim Kage " = 0. c) Briefly interpret (4) and (5): what is the economic "story" behind these equations? O C