@micro homework.

[{1} Show that the solution in part {c} also satises the additional constraints, IRE, ICI, in the original problem GP in {b}. {e} Argue that based on part {d}? the optimal contracts found in {c} are? in fact? also optimal for the original problem GP in {b}. [f] Now suppose the fraction of type 2 buyers goes up to 31.34 and the fraction of type 1 buyers goes down to 1J4. What does the optimal menu of contracts look like now for the seller? Give an economic explanation of the comparison. 1. Consider an island economy with a single agent who is endowed with one unit of time all or part of which can be supplied as labor {L} to produce food [3;] or consumed as leisure [if]. Food is produced according to the technology y = L}. However. production also generates pollution as a byproduct which adversely aects the welfare of the agent. Overall. the agent's utility is given by (y. E] = yf' y]. where cfyl denotes the welfare cost of producing 33;. a. 'What is an allocation in this economy? What quantities must be determined? Identify the feasible allocations. Be precise. b. Set up the choice problem facing the agent. assuming that the agent can choose any feasible input and output combination. For the remainder of the problem. suppose f[L} = L and cfy} 2 g. c. Solve the agents choice problem in this case. d. Argue that the solution in part c is Pareto ecient. e. Explain the meaning of the Second Welfare Theorem in this context. Does the theorem "hold" here? That is. does the conclusion follow under these circumstances? Next. suppose instead there are two identical individuals each with the tastes and en dowment as described above. Denote agent is quantities as L... E. and 33;. for i = 1. 2. Then total output is y = f{ L1 + L3] = y] + \"9'2. total {pollution} costs are y]. and \"dye fat} 2 Earl-l: Ely}- f. Set up the Social Planner's Problem for determining the Pareto efcient allocations. g. If each individual were to contribute the same amount of labor and receive the same amount of food as in part c. would the outcome be Pareto efficient? Explain. h. Instead. suppose that each agent decides how to allocate its time separately. taking the other agent's decision as given. and assuming it will consume what its labor produces. Argue that in this case. the outcome would be the same as in part c and explain why. i. Discuss the proposal to partition the island into two halves and have each of the individuals engage in production separately [incurring only the welfare cost associated with their own production]. rather than produce jointly. Evaluate this proposal. 2. IConsider two duopolists? 1 and 23. who produce a homogeneous product. Their cost functions are given1 respectively? by cyl] = y? and cyg] = y; + km, for some I: 2? U. The rms act as lCournot competitors facing uncertain {future} market demand described by the inverse demand function p[y?o}? where y is aggregate output and or is a random variable distributed on the interval [g E] with known density f {or}. The rms must choose their outputs prior to the realization of market demand (or). a. In light of the above? set up the decision problem facing rm 1' assuming each rm seeks to maximize its expected prot taking the other rms output as given. [Do not solve the problem.} b. For the special case in which p[y?or) = y + orJ solve for the lCournot equilibrium as a function of the expected value of :15. denoted E. c. Argue that if a 3} 3k? then rm 1's revenue will be greater and it's costs less than rrn Ts in Cournot equilibrium1 and hence 1's prots will be greater. d. In light of part c... suppose that the government decides to subsidise rm 2 in the event of a loss in order to ensure that there remain at least two competitors in the industry. This works as follows. As before? the rms rst engage in Cournot competition, choosing their output levels to maximize their expected prots. Market demand is then realized. If the market price is below rm 2's breakeven priceJ denoted ag), the government would raise the price for rm 2 [only] so that it would be able to break even. Assuming rm 2 knows it will be subsidized in the event of a loss1 explain how the government intervention would affect its decision problem. e. Discuss the effects of E and R: on the likelihood of the need for government subsidisa tion. f. Nextj suppose that even when subsidies are not necessary [i.e.} when rm 2 would be protable otherwiseL the government decides to provide a price subsidy to rm 2 since rm 1 has a cost advantage. In this case, rm 1 would receive the market price yio}, but rm 2 would receive pfy, or) + 5 per unit. Prove that this would always have the desired effect of increasing rm Ts market share relative to rm 1's. 3. Apple's market research shows that there are two types of consumers for its iMac computer and bMac computer {baby iMac}. The bMac is physically the same as the iMac but has some functions disabled. Both iMac and bMac cost 300 dollars to produce per unit. Both consumer typesJ h and I, get utility or if they have m dollars and no computer: ah[[l,m] = mf,m] = m. High type consumers with m dollars [after paying for a computer} get utility ah (if? m} = 1500 +11\": and al}? m) : 8DD+my wheres denotes having one iMac and I] having one bMac. Low types get utility 31H? m] = 600 + m and