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Question 2 (20 points) This question studies the co-existence of money and credit. Time is discrete with an infinite horizon. Each period consists of two subperiods. In the day, trade is bilateral and partially anonymous as in Kiyotaki and Wright (1991) (call this the KW market). At night trade takes place in a Walrasian or centralized market (call this the CM). There are two types of agents, buyers and sellers, and the measure of both is normalized to 1. The per period utility for buyers is u(q) + U(X) - H, and for sellers it is -q + U(X) - H, where q is the quantity of the day good produced by the seller and consumed by the buyer, X is consumption of the night good (the numeraire), and H is hours worked in the CM. In the CM, all agents can turn one unit of labor into a unit of good. The functions u, U satisfy the usual assumptions; I will only spell out the most crucial ones: There exists X* E (0, co) such that U'(X*) = 1, and we define the first-best quantity traded in the KW market as q' = (q : u'(q') = 1}. Here we will assume that there are two types of sellers. Type-1 sellers, with measure 1 - o, never accept unsecured credit. Hence, in any meeting with this type of seller, the buyer must pay on the spot (quid pro quo) with money. In contrast, type-2 sellers, with measure o, accept money but they also accept unsecured credit, in the form of a promise by the buyer to repay the seller in the forthcoming CM with numeraire good. However, there is a credit limit: the buyer can credibly promise to repay only an amount up to C 8 - 1. New money is introduced, or withdrawn if a 0. Is this intuition (always) correct in this model and why?1. In this question, we will use the framework of the real business cycle model to study con- sumption in the face of anticipated productivity shocks. Welfare of the representative agent is given by FEo [InGi + , In(1-4)1. 0 b. Whenever employed, the wage (in the same job) is subject to change with probability y. New wages are also drawn from F(.). Note: (i) the worker does not get a choice to stay on the old wage but can quit to unemployment, (ii) this is not a one-time wage adjustment - as long as the worker remains employed at the firm he is subject to further wage adjustments. Also when employed with probability & the job is destroyed and the worker moves to un- employment. Assume that the events of getting a wage adjustment and job destruction are mutually exclusive (i.e. not independent) and that ) + 9 0 (U) where Ci represents consumption, G is government spending, and 1-4 is leisure, with unity as the time endowment and & as labor supply. Output is produced only with labor according to: (Y) The government's budget constraint is given by G1 =TY (G) and the agent's budget constraint by But = (1 + r) B, + Y (1-7) - C (FBC) where B, are bonds, 7, is the output tax rate. (a) Derive the first order condition on B+1 and solve for the Euler equation between current and future consumption. Derive the first order condition on labor and use it to solve for the equilibrium relationship between labor and consumption. (b) Aggregate equation (FBC) across agents imposing the equlibrium values for B, and Bit1. (Note that there are no government bonds since the government's budget constraint is always balanced). Use the resulting equation, together with equation (Y), to solve for f, as a function of parameters and possibly 7. Then substitute into equation (Y) to solve for output as a function of parameters and possibly 7. Explain how an increase in the tax rate affects labor supply and output. (c) Use your reduced-form solutions , and Y, the Euler equation, the equilibrium condition for B, and government and agent budget constraints (aggregated across agents) to solve for steady state values for r. f, G. Y, B. and G as functions of parameters and the steady- state value of r, denoted by 7. Denote steady-state values by dropping time subscripts. (d) Substitute steady-state values into the utility function (equation U) and solve for the utility-maximizing value for 7. Is the steady-state utility maximizing tax rate positive, negative, or zero? Explain. Substitute the optimum value for 7 into steady-state solu- tions for r, {. G, Y. B. and C which contain 7. (e) Log-linearize your reduced-form solutions for Y, and , as well as equations (G). (FBC) and your solution for the Euler equation around steady-state values with the tax rate set at the optimum. Denote linearized values with hats. Collect the linearized equations of the model at the end of your answer. (Do not worry that equations with hatted variables do not obey adding up constraints - this is due to the linearization, not to your errors!) (f) Analyze the complete dynamic effects of a transitory increase in fo. (The tax rate rises in period 0 and then returns to its steady-state value.) Explain your answer. (g) Now, analyze the complete dynamic effects of a permanent increase in +, > 0 vt. (The tax rate rises above its steady-state value and stays there forever.) (h) How does an increase in the tax rate affect welfare (utility of the representative agent). Explain (do not derive). Suggest a modification of the problem that would reverse your conclusion. 3. One-sided search with temporary layoff. Time: Discrete, infinite horizon. Demography: Single worker who lives forever. Preferences: The worker is risk-neutral (1.e. u(r) = r). He discounts the future at the rate F. Endowments: Endowments are contingent on which of the 3 possible employment states the worker is in. When unemployed, the worker receives income b per period. Also with probability a he gets an offer of employment at a wage w ~ F(.) on [0, @] where w > b. When employed, the worker faces two possibilities. With probability o the job is destroyed and he moves to unemployment. Or, with probability A, he is moved to "temporary lay-off" (in that case he is not occupied in the job but the job still exists). When on temporary lay-off, the worker receives beach period and with probability a draws a wage offer from F(.). The difference from unemployment is that he can also get recalled back to his old job at his old wage. This happens with probability p. It is also possible, with probability 6, that while he is on temporary lay-off his former job gets destroyed in which case he looses any connection to the firm and becomes unemployed. If a temporary lay-off worker gets hired by someone other than his former employer the connection with the former employer is lost. Note: in each state the events that can happen to him are assumed to be mutually exclusive. This requires that a to + A+ p b. Whenever employed, the wage (in the same job) is subject to change with probability y. New wages are also drawn from F(.). Note: (i) the worker does not get a choice to stay on the old wage but can quit to unemployment, (ii) this is not a one-time wage adjustment - as long as the worker remains employed at the firm he is subject to further wage adjustments. Also when employed with probability ) the job is destroyed and the worker moves to un- employment. Assume that the events of getting a wage adjustment and job destruction are mutually exclusive (i.c. not independent) and that A + 7 a? Briefly interpret this result. (f) If this person were one out of a unit mass (continuum) of similar workers, obtain an expression for u, the steady state share of the population in unemployment, in terms of parameters and F(w' ). 3. Consider the following version of a Lucas tree model. Welfare of the representative agent is given by Fo ( [Bu(C) ), 00. " (9) 20, "(9) 50 where c, denotes consumption. There are two types of trees in this economy. The first is a safe tree which produces d dividends each period. The second is a risky tree which produces 2d dividends with probability 0.5 and no dividends with probability 0.5. The dividends from both trees are not storable. The economy starts off with each household owning one such tree. Let pf be the price at time t of a title to all future dividends from a safe tree and of the price of all future dividends from a risky tree. Let RT = Ry (di) be the time-t price of a risk-free discount bond that pays one unit of consumption at time t + 1. Finally, let r, denote the consumer's financial resources, which she allocates between risk-free bonds (by), safe stocks (s/), risky stocks (s ) and consumption (C). (a) Write down the consumer's problem in recursive form (Bellman equation) and find the first order conditions. Write expressions for the bond Euler equation and for the stock Euler equation. (b) What is the equilibrium value for consumption in periods when risky dividends are high? In periods when risky dividends are low? Label each with superscripts h and I. (c) Use the values for equilibrium consumption in your first order conditions to take the expectations writing out the first order conditions in terms of probabilities of each state and the marginal utility of consumption in those states. Label the future price of each type of stock with an additional superscript, h, / to denote its value when the price is high versus low. Rearrange your first order conditions to express prices of both stocks and the price of bonds (Ry!) on the left-hand- side. (d) Will the interest rate (not the price of bonds) be higher or lower if the current period is one with high dividends? Explain. (e) Now, recognize that the current price of stocks is state-dependent and is not explicitly dependent on time. Therefore, p," = pit and so forth. Drop the time subscripts and solve for the state dependent prices of each stock price, where the state is high or low dividends. (f) Compare your solutions for the prices of the risky and safe asset for the high-income state and determine whether the risky asset or the safe asset has the higher price. Explain