ecplain using graphs

1. Let (p', y') for t = 1,.... N be a set of observed choices that satisfy WAPM, let YI and YO be the inner and outer bounds to the true production set Y". Let a (p), a(p), and a (p) be profit functions associated with YO, Y, and Y I correspondingly. (a) Show that for all p. a (p) > *(p) 2 #-(p). (b) If for all p, a(p) = "(p) = * (p), what you can say about YO, Y, and Y/? Provide formal arguments. (c) For (p'. y' ) = ([1, 1]. [-3, 4]), and (p3, y') = ([2, 1], [-1, 2]) construct YI and YO (graphically). What can you say about returns to scale in the technology these observations are coming from? Hint: think y = (-c,y). 2. Given the production function f(21, 12, 13) = 17 min (x2, 23)", (a) Calculate profit maximizing supply and demand functions, and the profit function. What re- striction you have to impose on a? (b) Fix y. Calculate conditional demands and the cost function c(w1, w2. (). (c) Solve the problem py - c(un, w2, y) - maxy, do you obtain the same solution as in 20? Explain your findings. 3. Given the production function f(21, 12) = 21 + 29, where b > 0, calculate the cost function c(1, 1, y). How would costs respond to the changes in wi, wa, and y? How would factor demands respond? 4. Consider a firm with conditional factor demand functions of the form (output has been set equal to 1 for convenience): 12 = 1+ dufw5. What are the values of the parameters a, b, c, and d and why? 5. The cost function is c(w1, w2, y) = wjugg. (a) What do we know about a and b? (b) What are the conditional factor demands? What is the production function? (c) What can you tell about returns to scale? 6. Let e(w1, u2. 7) = 2(g) be the isocost and y = f(21, 12) = 7 be the isoquant corresponding to a fixed output level 7 = 1. (a) What are the slopes of these lines? (b) Draw the isocost and isoquant for Cobb-Douglas technology ? = 27 2; corresponding to J = 1. (c) Suppose c(w1, w2, 1) = aw + buy. Draw the isocost and the corresponding isoquant (use the slopes to obtain the shape of the isoquant). (d) Repeat for c(w1, w2, 1) = minfawn, buy}. (e) Draw conditional demand z, as a function of " for 7c and 7d. Hint: If you have trouble in 7c - 7e, think what technology these costs came from.1. Explain why the "optimal" level of COz emissions is not zero? 2. The government is trying to choose between an emissions tax system and a cap and trade system. (read the relevant sections of the Stern Review) a. Assume that the government is certain about the shapes and location of the marginal abatement cost and marginal damage cost functions associated with some pollutant. How would you decide whether to use an emissions tax system or a cap and trade system to internalize the pollution externality? b. Assume that the government is uncertain about the shapes and location of the marginal abatement costs and marginal damage cost functions associated with some pollutant. How would you now decide whether to use an emissions tax system or a cap and trade system to internalize the pollution externality. 3. You are doing a cost-benefit analysis to determine whether, how much, and when CO2 emissions should be reduced from "Business as Usual Levels" over the next 100 years. (read the relevant sections of the Stern Review) a. Explain why the discount rate chosen to calculate the present value of mitigation costs and damages of climate change can have a large effect on the results? b. What factors should go into choosing the correct discount rate.1. Explain why the "optimal" level of COz emissions is not zero? 2. The government is trying to choose between an emissions tax system and a cap and trade system. (read the relevant sections of the Stern Review) a. Assume that the government is certain about the shapes and location of the marginal abatement cost and marginal damage cost functions associated with some pollutant. How would you decide whether to use an emissions tax system or a cap and trade system to internalize the pollution externality? b. Assume that the government is uncertain about the shapes and location of the marginal abatement costs and marginal damage cost functions associated with some pollutant. How would you now decide whether to use an emissions tax system or a cap and trade system to internalize the pollution externality. 3. You are doing a cost-benefit analysis to determine whether, how much, and when CO2 emissions should be reduced from "Business as Usual Levels" over the next 100 years. (read the relevant sections of the Stern Review) a. Explain why the discount rate chosen to calculate the present value of mitigation costs and damages of climate change can have a large effect on the results? b. What factors should go into choosing the correct discount rate.1. Consider a very simple "two node" model in which natural gas is produced in region A and is transported by pipeline to region B. The price of natural gas in Region B is $6 per thousand cubic feet (Mef) and the price of transportation service from Region A to Region B is $1/Mef. a. If the market for producing gas in region A is perfectly competitive and there is a perfectly elastic supply of transportation service what will be the equilibrium price of natural gas in Region A? b. What will happen if the government places a cap of $4/Mof on price that producers in Region A can charge for the natural gas they produce? Discuss how your assumptions about the shape of the competitive supply curve in Region A affects your answer. 2. Now consider a more complex natural gas supply and demand system such as the one we have in North America. There are multiple gas production areas and many consuming areas that are remote from production areas and rely on pipelines to transport gas to them. a. Assume that the price of natural gas at Henry Hub (Louisiana --- a gas producing area) is $5/Mof and assume that the price of natural gas in Los Angeles (a gas consuming area) is $4/Mef. Explain how such a price pattern can emerge? b. What incentives are there to expand pipeline capacity from producing areas in the Western U.S. to consuming areas in the Eastern U.S. 3. The United States maintains a Strategic Petroleum Reserve (SPR) that now contains 700 million barrels of crude oil. a. What factors would you take into account to design a policy to determine when and how much oil is released from the SPR? b. How would the expected supply behavior of OPEC affect your policy design?1. Suppose that you are trying to determine whether OPEC is an effective cartel. Suppose there is an exogenous and unanticipated shock to world oil supplies --- c.g. political unrest that causes oil exports from Venezuela to fall to zero. Discuss how you would go about distinguishing between the following models of imperfect competition if you could observe market prices, output levels for individual countries, and the short run marginal oil production costs of individual countries before and after the demand shock. a. Textbook cartel b. Dominant firm/fringe firm c. Cournot (quantity) competition d. Perfect competition 2. If oil and natural gas are close substitutes discuss the effect of a negative shock to world oil supplies as in (1) on field prices, import prices, quantities, and market clearing conditions for natural gas sold in the U.S. under the following conditions: a. Field prices for natural gas produced in the U.S. and prices for imported gas are unregulated. Pipelines pass through the cost of the gas they purchase in the field in the prices they charge to local distribution companies. b. The field price of natural gas produced in the U.S. is subject to a price ceiling equal to the pre-shock domestic price of natural gas while imported gas prices are unregulated. Pipelines are required to charge the average prices they pay for domestic and imported gas to local distribution companies. c. The pipeline network in the U.S. was operating at full capacity prior to the supply shock.3. Consider the following infinitely repeated game. . There is an Incumbent and a (potential) Entrant. . Each date t has two stages: - At the first stage, the Entrant decides whether to Enter or Stay Out. - If the Entrant decides to Enter, it bears an entry cost of k = 0.1. The two players then play a standard "Cournot" game: * Each of the two firms simultaneously decide on a non-negative output qe and qr- * The demand curve is given by p = 1-Q, where Q = qe + qr, so the price of output is given by p = 1 -Q. (Notice that we allow negative prices here; but you should be able to ignore that for the sake of this problem.) * The marginal cost of production is zero, so the profits are given by: qIP = qEP - k. - If the Entrant Stays Out, it gets a payoff of 0, and the Incumbent decides on a production level or to produce, facing the same demand curve. . The game is infinitely repeated, with a discount factor 6 e (0, 1). . Notice that the entrant bears the cost & in each round in which she enters. (a) (10 points) Find the subgame perfect equilibrium of the non-repeated version of this game (the game that takes place at any t). (b) (10 points) Prove that the following strategy profile is a subgame perfect equilib rium for sufficiently high o, and find the minimal o for which it is: . The Entrant Enters every round. . Every producer produces q = 1, so long as no producer has ever produced a quantity other than 3- . If any producer has ever produced a quantity other than }, "trigger" to play- ing the subgame perfect equilibrium from part (a) in every period. (c) (10 points) For o = .9, is there a subgame perfect equilibrium in which entry is always deterred (i.e., in which the Entrant never Enters)? If so, find one. If not, explain why not. (d) (5 points) Describe strategies that can implement the same outcome as part (b) for some 6