answer the followimng

Consider a Cobb-Douglas production function Y = AK?L1??with ? = 0.6 and A = 1.5. Assume that the savings rate is s = 0.04, the population growth rate is n = 0.02 and the depreciation rate is ? = 0.05. 1. Using the Solow-Swan model, compute the steady state values of k, y, and of consumption per person c. 2. Find the level of s that maximizes c, and the corresponding steady state value of k.

3. Does the model allow for long-term growth? 4. Suppose the World Bank wants to help this country by either (i) directly increasing k, or (ii) by doubling A. How would each of these actions affect the steady state values of y and c and the transition dynamics of k?

A firm has inventory of second-generation chips for $100,000 based on their price a year ago. Since then, third-generation chips have been introduced in the market and the price of the old chips has been reduced by 10%. What is the opportunity cost of holding the inventory of the old chips today? Explain.

Company ABC uses two inputs, labour and capital, in its production process. Prices for these inputs are $11 per hour and $33 per unit for labour and capital, respectively. The marginal products of the two inputs are 24 for labour and 48 for capital, given the current amount of inputs. Is the firm using the cost-minimizing combination of labor and capital? If not, should ABC hire more or less labour?

Explain why we say that the marginal rate of substitution for perfect complements is zero.

If the price elasticity of demand for lamps is -1.2, how should the LampShop Co. change its price in order to raise its revenue? Explain.

Instructions. Each question is 33 points. Good Luck! 1. Let P be the set of lotteries over {a, b, c} x {L. M. R). In which of the following pairs of games the players' preferences over P are the same? (a) L M R L M R 2,-2 1,1 -3.7 a 12,-1 5.0 -3,2 1,10 0.4 0.4 5,3 3,1 3,1 -2,1 1,7 -1,-5 -1,0 5,2 1,-2 (b) L M R L M R 1.2 7,0 4,-1 1,5 7,1 4-1 6.1 2.2 8.4 6.3 2,4 8.8 3,-1 9,2 5.0 9,5 5,1 2. Let P be the set of all lotteries p = (Pr, py; p.) on a set C = {r, y, z } of consequences. Below, you are given pairs of indifference sets on P. For each pair, check whether the indifference sets belong to a preference relation that has a Von-Neumann and Morgenstern representation (ie. expected utility representation). If the answer is Yes, provide a Von-Neumann and Morgenstern utility function; otherwise show which Von-Neumann and Morgenstern axiom is violated. (In the figures below, setting p. = 1 - Pr - Py; we describe P as a subset of IR?.) (a) = (plex = 2py + 1} and 12 = {plPr = 4py + 1} (b) h = (plps = 2py + 1} and 12 = {plpr = 2p,} (c) h = (plp, 1/2} (d) h = {ply = (pz)" + 1/2} and 12 = {plpy = (px)}} 3. On a given set of lotteries, find a discontinuous preference relation > that satisfies the independence axiom.1. Compute a sequential equilibrium of the following game. 1 X D (2, 0 ) I 2 R 1 a b (7) (9) (3) (4) (4 ) 2. Consider the following centipede game. There are 27 dates t = 1, 2. ..., 27. At each odd date t = 1, 3,.... player 1 gets to choose between exit, which ends the game, and stay, after which the game proceeds to * + 1. At each even date t = 2, 4, ..., 27, player 2 chooses between exit and stay. At 27, the game ends even after stay. Player 1 has two types, namely, rational and irrational, with probabilities 1 - & and &, respectively for some & E (0, 1/2), and player 2 has only one type. The irrational type gets -1 if he exits and 0 otherwise. For all the other types, if player i exits at t, player i gets t + 1 and the other player gets -1. At t = 27, after stay, rational player 1 gets 27 + 2 and player 2 gets 2T. (a) Compute the sequential equilibrium. (You do not need to show that it is unique.) (b) For every T > 2, find the smallest & under which the rational type of player 1 stays with probability 1 at t = 1. Briefly discuss your finding. (c) (Bonus) Prove or disprove the following statement. There exists an E > 0 such that for every & E (0, z), the unique Nash equilibrium outcome is that either (the rational) player 1 exits at t = 1 or player 2 exits at t = 2 (if player 1 happens to be irrational). 3. Fix a finite extensive-form game G* and consider a family of extensive-form games G' in which everything is as in G except for the probabilities assigned by the nature at the histories the nature moves. Assume that for any history h at which nature moves and for any available action a E A (h), the probability a" (a|h) nature assigns to a at h in game G" converges to the probability a" (ajh) nature assigns to a at h in game G". Show that for any sequence of assessments (o", ("), if (o, " ) is a sequential equilibrium of G'" for each m and (o", /") - (o", "), then (@", /" ) is a sequential equilibrium of G*.1. Compute a sequential equilibrium of the following game. 1 X D (2, 0 ) I 2 R 1 a b (7) (9) (3) (4) (4 ) 2. Consider the following centipede game. There are 27 dates t = 1, 2. ..., 27. At each odd date t = 1, 3,.... player 1 gets to choose between exit, which ends the game, and stay, after which the game proceeds to * + 1. At each even date t = 2, 4, ..., 27, player 2 chooses between exit and stay. At 27, the game ends even after stay. Player 1 has two types, namely, rational and irrational, with probabilities 1 - & and &, respectively for some & E (0, 1/2), and player 2 has only one type. The irrational type gets -1 if he exits and 0 otherwise. For all the other types, if player i exits at t, player i gets t + 1 and the other player gets -1. At t = 27, after stay, rational player 1 gets 27 + 2 and player 2 gets 2T. (a) Compute the sequential equilibrium. (You do not need to show that it is unique.) (b) For every T > 2, find the smallest & under which the rational type of player 1 stays with probability 1 at t = 1. Briefly discuss your finding. (c) (Bonus) Prove or disprove the following statement. There exists an E > 0 such that for every & E (0, z), the unique Nash equilibrium outcome is that either (the rational) player 1 exits at t = 1 or player 2 exits at t = 2 (if player 1 happens to be irrational). 3. Fix a finite extensive-form game G* and consider a family of extensive-form games G' in which everything is as in G except for the probabilities assigned by the nature at the histories the nature moves. Assume that for any history h at which nature moves and for any available action a E A (h), the probability a" (a|h) nature assigns to a at h in game G" converges to the probability a" (ajh) nature assigns to a at h in game G". Show that for any sequence of assessments (o", ("), if (o, " ) is a sequential equilibrium of G'" for each m and (o", /") - (o", "), then (@", /" ) is a sequential equilibrium of G*