90 MICROECON

1. Consider an individual faced with two job choices: she can either accept a position with a xed annual salary of I 3: I] which requires Lz units of labor per annum, or she can start her own business producing widgets. The technology for producing widgets is given by y = fEL,T] = Ly'lT, 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 good1 the price will be p3, and if it is bad, the price will be p; -::2 3:3. The probability that the economy will be good is 'JT, and the probability that it will be bad is {1 it}. The individual must determine T and L before knowing which state occurs. Assume she has the {von Neumann - Morgenstern] utility function um, L] = NEIL L], where m. is money, L is her labor, and L is her endowment of time [L :2:- LI}. 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 choose1 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 = l, t = l, and (11.51} = {5, 11E} Show that ifshe was certain the economy would be good and if 313 = 11], then she would choose to start her own rm, whereas if she was certain the economy would be bad and ifpb = S, then she would rather accept the offer of [1,LI}. For the remainder of the problem1 suppose the widget technologyr is changed to y = LlTl-'u. 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 to. Again, she must decide on all input amounts bore 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 xed salary I and labor requirement LI, 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 prot 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 to and supplement this using her own lab-or. Set up the new decision problem and discuss how this would affect the likelihood of her starting her own business. 2. Consider an economy with two goods, two consumers, and one rm. The finn produces fEL] = min{5L,15} units of food when it uses L E I] unite of labor as input. It also can dispose of either good without cost. Consumer I initially owns 2 units of time l[for labor or leisure] and gets utility cf from consuming c E [l' units of food and E 2 [1 units of leisure. Consumer 2 owns the rm and, initially1 no goods. Consumer '2 cares only about food and wants as much ofit as possible. a. Graph the production Function of the rm. Describe its returns to scale. b. Is there a Pareto efcient {Pareto optimal} allocation in which consumer I 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 efcient? Are there any other competitive equilibrium allocations? Suppose now that there is technical progress. The rm has a new production function g{L} = 313T for L 33 I]. The rest of the economy remains the same as before. e. Find a competitive equilibrium and compare the allocation and the welfare of the con sumers to what they were in the equilibrium of part d. f. Is the allocation in part e Pareto efcient 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. 3. A. monopoly rm can hire from a population of qualied workers Half of the workers are of type 1 and half of type '2. Type I? workers [5' = 1,2} produce Bil + t} units of output when they are assigned a job of difculty type t 2 U. A. worker of type 3 who is paid to units of output gets utility or c} when working at a job of type t, where c is twice continuously differentiable. Denoting partial derivatives by subscripts, efng?) :a U, can, I9] {I 1, cm, I9] = D1c[,}= Cl , ci[2,2} 3: 2, for all t :2:- I] and H E I]. The monopoly can offer workers jobs of any type t 13 I] and wages depending on the job type. It seeks to maximize its expected prot1 where the prot 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 maro'mize their utility, getting reservation utility of I] 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 recognise every worker's type 9 before it makes any contract offers. Formulate the monopoly's optimization problem. Find an optimal job difculty t for each type I? [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 specic as possible. c. Suppose now that the monopoly cannot recognize anyr particular worker's type, but knows all the information given above part a. Formulate the monopoly's optimization problem when it can oEer a menu of contracts {job diiculty types and oorresponding wages}. Characterize the optimal contracts that workers aooept 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 specic as possible with the given infomation. Compare the expected prot of the monopoly to what it is in part b. What fraction of the workers in the population does the monopoly want to hire'lI d. What can be said about the efciency of the outcomes in parts b and c? If an outoome is Pareto inefcient, 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