Answers to Microeconomics Prelim of August 7, 0. Consider an individual faced with two job choices: she can either accept a position with a fixed annual salary of x > 0 which requires L x units of labor per annum, or she can start her own business producing widgets. The technology for producing widgets is given by y = f(l, T ) = L T, where L denotes labor (again per annum) and T denotes the quantity of another input. If she were to start her own business, she would devote her own labor time to production rather than hiring outside labor. Both the market for T and the market for widgets are perfectly competitive. However, while the price of T is known to be t, the price of widgets depends upon the state of the economy: if the economy is good, the price will be p g, and if it is bad, the price will be p b < p g. The probability that the economy will be good is π, and the probability that it will be bad is ( π). The individual must determine T and L before knowing which state occurs. Assume she has the (von Neumann - Morgenstern) utility function u(m, L) = m( L L), where m is money, L is her labor, and L is her endowment of time ( L > L x ). Her only source of money is from the job she chooses. a. Explain precisely how the individual would choose between the two jobs. b. If the individual were to start her own business, determine the optimal quantities of y, T, and L she would choose, as functions of the parameters. c. Discuss how changes in the parameters affect her decision whether to start her own business. Increases in which parameters increase the likelihood that she starts her own business? d. Next, suppose L =, t =, and (x, L x ) = (6, ) Show that if she was certain the economy would be good and if p g = 0, then she would choose to start her own firm; whereas if she was certain the economy would be bad and if p b = 8, then she would rather accept the offer of (x, L x ). For the remainder of the problem, suppose the widget technology is changed to y = L / T /4. e. Now, assume that if she starts her own business, instead of devoting her own labor to production, she hires outside labor at the wage w. Again, she must decide on all input amounts before learning the state of the economy. In this case, her business is her only source of income and she does not work otherwise. Alternatively, she can accept the job with the fixed salary x and labor requirement L x, as before. Set up her decision problem. f. Compare the individual s optimal decisions in part e to what she would choose if she were to maximize expected profit instead of expected utility as the owner of her own business. g. Finally, suppose that if she starts her own business she can both hire outside labor at wage w and supplement this using her own labor. Set up the new decision problem and discuss how this would affect the likelihood of her starting her own business.. Consider an economy with two goods, two consumers, and one firm. The firm produces f(l) = min{5l, 5} units of food when it uses L 0 units of labor as input. It also can dispose of either good without cost. Consumer initially owns units of time (for labor or leisure) and gets utility cl from consuming c 0 units of food and l 0 units of leisure. Consumer owns the firm and, initially, no goods. Consumer cares only about food and wants as much of it as possible. a. Graph the production function of the firm. Describe its returns to scale. b. Is there a Pareto efficient (Pareto optimal) allocation in which consumer consumes no food and no leisure? Explain. c. Find all Pareto efficient allocations in this economy. How do they differ from one another?
d. Find a competitive equilibrium for this economy. Is the allocation Pareto efficient? Are there any other competitive equilibrium allocations? Suppose now that there is technical progress. The firm has a new production function g(l) = 8 L for L 0. The rest of the economy remains the same as before. e. Find a competitive equilibrium and compare the allocation and the welfare of the consumers to what they were in the equilibrium of part d. f. Is the allocation in part e Pareto efficient for the economy after there is technical progress? g. Does the technical progress lead to Pareto improvement if the economy is competitive? Explain why or why not. Answers: a. The graph of the production function f(l) has a kink at (3, 5). To the left of this point, the graph is linear with slope 5, joining (3, 5) to (0, 0). To the right of (3, 5), the graph is horizontal. There are constant returns to scale over the input range L [0, 3) and decreasing returns for increases in input from any L > 0 to any L > max{l, 3} since tl = L implies tf(l ) = 5tL > 5 = f(tl ). b. and c. A Pareto efficient allocation must solve max c l s.t. c +c f(l), l +l +L, c i 0, l i 0, i =,, and c c for some c. For feasible L, f(l) = 5L. If, at a solution, c l > 0, then l = 0 and the first two constraints and last constraint bind. Otherwise c l can be raised. At such a solution, the problem can be rewritten as max c l s.t. c = 5( l ) c, which is solved at c = (0 c )/, l = (0 c )/0, L = + ( c /0), f(l) = 5 + ( c /), c = c, l = 0. There are multiple solutions with c l = 0 in the optimization problem above, but for the allocation to be Pareto efficient, c must be maximized subject to feasibility, which implies c = l = l = 0, L =, f(l) = 0 = c. The different efficient allocations differ according to how much food consumer gets. The more food consumer gets, the more consumer works and the less it consumes. d. Let p and w be equilibrium prices of food and time. If p 0, then the consumers have no optimal consumption, so p > 0 in equilibrium. The firm s profit is 5pL wl if L 3, so if w < 5p, the firm demands L = 3, which is not feasible. If w > 5p, then the firm demands L = 0, but consumer demands a positive amount of food, so in equilibrium, w = 5p > 0 and the profit is 0. Consumer has Cobb-Douglas demands c = w/(p) = 5, l = w/(w) =. Consumer has no wealth, so no consumption. The firm demands L = l = and produces f(l) = 5. These conclusions follow from the definition of equilibrium, so there is no other competitive allocation. The allocation is Pareto efficient from part b or, alternatively, from the first welfare theorem since the preferences are locally nonsatiated. e. As in part d, c = w/p, l =. Consumer demands no leisure, so L = l = maximizes the firm s profit 8p L wl. This implies 4p = w, c = 4, g(l) = 8, c = 4, where c can be found using feasibility or the budget constraint of consumer : pc = 8p wl = 8p w. The wage rate in terms of food falls compared to the equilibrium in part d. Consumer works and gets as much leisure as in part d, but consumes less food. The firm produces more and gets more profit and consumer gets more food than in part d. f. The allocation is Pareto efficient by the first welfare theorem. g. The technical progress does not lead to Pareto improvement. The marginal product of consumer at the initial equilibrium labor supply falls as a result of the progress, since the productivity of smaller amounts of labor rises. As a result, the wage rate falls in terms of food and consumer, whose income comes only from labor, is hurt.
3. A monopoly firm can hire from a population of qualified workers. Half of the workers are of type and half of type. Type θ workers (θ =, ) produce θ( + t) units of output when they are assigned a job of difficulty type t 0. A worker of type θ who is paid w units of output gets utility w c(t, θ) when working at a job of type t, where c is twice continuously differentiable. Denoting partial derivatives by subscripts, c tt (t, θ) > 0, c tθ (t, θ) <, c(0, θ) = 0, c t (0, θ) = 0, c t (, ) >, for all t > 0 and θ 0. The monopoly can offer workers jobs of any type t 0 and wages depending on the job type. It seeks to maximize its expected profit, where the profit from any particular worker is the value of the worker s output minus the wage the monopoly pays. Workers are free to accept or reject any contracts offered to them. They maximize their utility, getting reservation utility of 0 if they reject all offers. a. Interpret the assumptions about c and its derivatives and describe how the types of workers differ from each other. b. Suppose the monopoly can recognize every worker s type θ before it makes any contract offers. Formulate the monopoly s optimization problem. Find an optimal job difficulty t for each type θ (optimal for the monopoly). Find corresponding wages that are optimal for the monopoly for each of these job types. Show how these optimal contracts can look in a graph. Compare the contracts the two worker types receive, being as specific as possible. c. Suppose now that the monopoly cannot recognize any particular worker s type, but knows all the information given above part a. Formulate the monopoly s optimization problem when it can offer a menu of contracts (job difficulty types and corresponding wages). Characterize the optimal contracts that workers accept and show how they can look in a graph. The graph can be the same one you used in part b. Compare the contracts received by the two types of workers to each other and to those they receive in part b. Be as specific as possible with the given information. Compare the expected profit of the monopoly to what it is in part b. What fraction of the workers in the population does the monopoly want to hire? d. What can be said about the efficiency of the outcomes in parts b and c? If an outcome is Pareto inefficient, could a government that has the same information as the monopoly obtain a Pareto improvement by restricting the set of contracts the monopoly can offer? Explain. Answers: 3a. More difficult jobs are more costly for both types of workers. The marginal cost (cost of a small amount of additional difficulty per unit of difficulty) is near 0 for the more productive worker (type ) when the difficulty is near 0 and the cost and marginal cost are less for the more productive worker than for the other type worker. The marginal cost rises for both worker types as the job difficulty increases and is above for the more productive worker when the difficulty is above. b. The monopoly offers type θ a wage w and job type t to maximize θ( + t) w s.t. w c(t, θ) 0. The optimal t maximizes θ( + t) c(t, θ) and is the unique solution to θ c (t, θ), with equality if t > 0. Denote this t by ˆt θ. The corresponding wage is ŵ θ = c θ (ˆt θ ). Since c (0, ) = 0 and c (, ) >, we have 0 < ˆt <. If ˆt > 0, then implicit differentiation of the first order condition yields dˆt θ /dθ > 0, so the higher productivity worker is assigned a more difficult job. The same conclusion holds if ˆt = 0 since ˆt > 0. As in the graphs, the wage can be higher for type or for type, but type strictly prefers the contract received by type to its own contract if type is paid a positive wage (since the graph of c(t, ) lies above that of c(t, ), except possibly at t = 0, and these graphs are indifference curves of the two types at utility level 0.) Each type gets utility 0 and a wage below its productivity. c. The monopoly cannot do better than to offer two contracts (t θ, w θ ), θ =, to maximize its expected profit ( + t w + + t w )/ from a given worker subject to (Pθ) w θ c(t θ, θ) (participation constraints) and 3
4 (Iθ) w θ c(t θ, θ) w θ c(t θ, θ), θ, θ =, (incentive constraints). In the usual way, it is possible to show that maximizing the objective function subject to (P) and (I) alone yields a solution that satisfies the other two constraints and satisfies (P) and (I) with equality. The first order conditions for the monopoly optimization imply = c (t, ) and = c (t, ) c (t, ) > c (t, ), hence t = ˆt and t < ˆt (unless ˆt = 0). Also w = c(t, ) and w satisfies I with equality. It follows that w < ŵ unless ˆt = 0 and that t > t, w > w, w c(t, ) > 0, and w > ŵ. The monopoly expected profit is less than in part b, but it is still positive, so the monopoly wants to hire all the qualified workers. d. If workers of type θ get (t θ, w θ ), then w θ = θ( + t θ ). If w θ is higher, then the firm makes an expected loss on that contract and gets a higher payoff by withdrawing it. If wθ is lower, then a firm whose contract is rejected by type θ can raise its payoff by offering a slightly higher wage and the same job difficulty, an offer that type θ accepts. The same argument shows that (θ, w) maximizes w c(t, ) s.t. w = + t, so t = ˆt. Let ( t, w ) satisfy w = + t and w c(t, ) = w c( t, ) so that type is indifferent between ( t, w ) and (t, w). Since type prefers (t, w) to (t, w) if t < t, type must get its most preferred contract satisfying t t and w = + t. If ˆt < t, then (t, w) = ( t, w ). Otherwise, (t, w) = (ˆt, + ˆt ). In any case, t > t and w > w. Type gets the same job difficulty as in part b (more than in part c unless ˆt = 0) and a higher wage than in both b and c. If ˆt t, then type gets the same job difficulty as in b and c and a higher wage. If ˆt < t, then type gets a more difficult job than in b and c and gets a higher wage., 6 F B N /3 /3 A E 5, 6 D C D C 0, 3 9, 0 0, 3 9, 5 4. Nature (N) and players and play the game represented by the tree above. N plays left with probability /3 and right with probability /3. The payoff of player is listed first at each terminal node. a. List all the pure strategies of player and of player. b. Represent the extensive form game above as a strategic form game in which the payoffs corresponding to strategies of players and are the expected payoffs. c. For the game in part b, find all the Nash equilibria (NE) in which players and play pure strategies. d. Returning to the extensive form game, consider a behavioral strategy profile in which player plays A with probability a, plays B with probability b, and plays C with
5 probability c. Let α be the belief probability that player attaches to the node following A. Find all the sequential equilibria in which players and play pure strategies (i.e., where each of a, b, c takes the value of either 0 or ). Compare your results to those in part c. e. There is a sequential equilibrium in which b = but 0 < a < and 0 < c <. Find it and describe it in words. Answers: 4a. Player : AB, AF, EB, EF. Player : C, D. \ C D AB 9, 0/3 0, 3 b. AF 0/3, 6/3 /3, 4 c. The pure NE are {AB, C} and {EF, D}. EB 9/3, 4 0/3, 5 EF 4, 6 4, 6 d. If a and b are both 0, then α can be any number in [0, ]. To show this, we need to construct sequences of totally mixed strategies with probabilities a k and b k of A and B converging to 0 with conditional probabilities of the node after A converging to α. If α (0, ), then the sequences a k = α/k and b k = ( α)/k have this property. If α = 0, then a k = /k and b k = /k work and if α =, then a k = /k and b k = /k do. (/3)a If a and b are not both 0, then α = (/3)a+(/3)b +(/)(b/a) if α > 3/5 if c > 5/9 Player s best response: c = [0, ] if α = 3/5 Player right: a = [0, ] if c = 5/9 0 if α < 3/5 0 if c < 5/9 Player left: b = if c > /9 [0, ] if c = /9 0 if c < /9 Then c = {a =, b = } α = 3 c =. Also c = 0 {a = 0, b = 0} α [0, ]. In order to generate c = 0, α 3. So pure strategy 5 sequential equilibria are: {a =, b =, c = } with α = /3 and {a = 0, b = 0, c = 0} with α 3/5. e. From the answer to part d, if 0 < a, c < in SE, we must have α = 3/5 and c = 5/9. If b =, then 3/5 = α = implies a = 3/4. Thus, {a = 3/4, b =, c = 5/9} +(/)(b/a) with α = 3/5 is a sequential equilibrium. In it, the left type of player chooses to interact with player for sure and the right type of player chooses to interact with player with probability 3/4. If player interacts with player, player believes that it is the right type with probability 3/5 and cooperates (accommodates by choosing C) with probability 5/9.,