6. (10) Consider the following bargaining game between a buyer and a seller. The seller has access to a technology which allows him to produce q units of a good at cost c(q) = q. The buyer's utility from consuming this good is u(q), with u'(q) > 0, u"(q) 0 per unit. What is the economic meaning of ? What is the buyer's optimal choice of m, and how does it depend on i?5. (20) Consider the one-sided job search model that we studied in class. There is a continuum of identical workers whose measure is normalized to the unit. Workers maximize expected discounted utility Eo Atu(BE), 1=0 and, for simplicity, let u(y) = y. The term y takes the value w when the worker is employed at wage w, and = > 0 when the worker is unemployed. Workers who were unemployed in period t - 1 will receive a wage offer in period t. This offer will be drawn from a random distribution with a CDF given by F(w) that has support on the set [0, w]. Assume that = z. (e) Derive an expression for the term dR" /dA. What is the sign of this term? What is the intuition for this sign? (f) Describe the steady state level of unemployment in this economy. How does it depend on A? (full credit requires providing a formula for du/dA)4. (20) Consider the standard growth model in discrete time. There is a large number of identical households (normalized to 1). Each household wants to maximize life-time discounted utility Each household has an initial capital stock ro at time 0, and one unit of productive time in each period that can be devoted to work. Final output is produced using capital and labor services, yt = F(kt, m). where F is a CRS production function. This technology is owned by firms whose number will be determined in equilibrium. Output can be consumed (c) or invested (it). We assume that households own the capital stock (so they make the investment decision) and rent out capital services to the firms. The depreciation rate of the capital stock (r;) is denoted by 6. The capital stock depreciates no matter whether it is rented out to a firm or not. Finally, we assume that households own the firms, i.e. they are claimants to the firms' profits. The functions u and F have the usual nice properties. (You will not explicitly need them, so there is no need to be more precise.) (a) Define an AD equilibrium. How many firms operate in this equilibrium? (b) Write down the problem of the household recursively. Here firms face a static problem. I am not asking you to explicitly spell it out, but it will be critical for a correct definition of the RCE. Be sure to carefully define the state variables and distinguish between aggregate and individual states. Define a recursive competitive equilibrium (RCE). Consider again an Arrow-Debreu setting. c. In this economy, why is it a good idea to describe the AD equilibrium capital stock allocation by solving the (easier) Social Planner's Problem? From now on assume that F(k;, n;) = king -" and 6 = 1. Also, assume that the households' preferences are characterized by "habit persistence". In particular, households wish to maximize s'(lace + ylace-1), 720. d. Fully characterize (i.e. find a closed form solution for) the equilibrium allocation of the capital stock using any method you like. (Hint; You might find it useful to guess that the "policy rule" satisfies the difference equation In(ki+1) = g + a In(k;), where g is a constant to be determined.) e. What happens to the ADE price of labor and capital services as t - co? f. Provide a formula that describes the ADE price of the consumption good in any period t > 0 as a function of the model's parameters. g. Suppose now that the government imposes a lump-sum tax T (i.e. a fixed amount tax) on every agent in the economy. Subsequently, (but in the same period) the government returns the tax earnings to the agents. State whether the following statement is correct or not: "Since the government returns all the tax earnings to the agents, the taxation system under consideration has no effect on the real allocation of the economy. This, in turn, means that instead of studying the competitive equilibrium, one could just solve the Planner's problem, and we know that the equilibrium allocation would coincide with the Planner's solution"
