Households receive labor income and profits from firms. They pay taxes on their labor income, keeping the fraction S,, and receive government goods. Individual households, being atomistic, take G, as given. Households save by purchasing risk-free bonds. Let B, denote the quantity of bonds bought at time t, measured in terms of output. 1 unit of bonds delivers (1 + r,) units of output at time { + 1. Households face the usual initial, non-negativity and No-Ponzi-Game conditions. (a) Solve the consumer's and the producer's problems. (b) Note that there is no capital in this economy, so that bonds represent trades between consumers. Imposing equilibrium, find the resource constraint and the labor allocation condition. Use this result to express output and consumption in terms of the after-tax rate S,, and then solve for the equilibrium value of (1 + r,). (c) Do higher labor tax rates (lower values of S, ) increase or decrease output? Briefly explain in terms of income and substitution effects. (d) Let lower-case letters with carats " " denote deviations of logged variables around their steady state values. Show that the log-linearized expressions for labor hours and output are: 71 = 08, 020. Find the log of average labor productivity, apt, = 1 - 6. (e) Will labor productivity to be pro- or counter-cyclical in this model? Briefly ex- plain. Is the cyclicality of labor productivity found in the model consistent with that in the data? If not, how might the model be modified?1. Consider an economy composed of heterogeneous agents who live for two periods. Agents derive utility from consumption c according to utility function u(c), and discount utility of the second period at a discount rate of 8. Agents enter the economy with zero assets. In the first period, they receive an endowment w = w; and in the second period, they have a probability of 0.5 to receive endowment us = (1 + 6)w, and a probability of 0.5 to receive endowment wa = (1 - 5)w, where 0 0, n 0, is the optimal level of asset carried to the second period equal to 0 or greater than 0? Provide a proof and an intuitive explanation for your answer. (e) Under the assumptions (1 + r)3 = 1 and u(c) = cl-5/(1 - o), where o > 0, suppose now agents face greater uncertainty about future income, i.e. have a probability of 0.5 to receive endowment as = (1 + 6 )w, and a probability of 0.5 to receive endowment un = (1- 8 )w, where 0 ; where ; is agent i's marginal utility of income) and explain why, in general, the planner can increase the social welfare. (c) Suppose the I agents have expected, discounted quadratic utilities of the form Compute the equilibrium prices as a function of the aggregate output w = >w' and of the parameters o, (a')(=1, of the preferences. What is unusual in this example? (d) Using the result of question (b), show that the equilibrium of an economy with quadratic utilities as in (c) is CPO. If you were asked to do a proof that generically the equilibrium of a T-period economy with T > 2 is not CPO, how would you parameterize the family of economies that you consider?Question 2 (20 points) Consider the following extension of the Mortensen-Pissarides model in continuous time. Labor force is normalized to 1, but there are two types of workers, Type 1, with measure I E (0, 1), and Type 2, with measure 1 - #. The two types of workers have different productivities: when a type I worker is matched with a firm, she can produce p > Ounits of the numeraire good per unit of time, but when a type 2 worker is matched with a firm, she cannot produce anything (the type 2 is a total lemon). Although firms would clearly prefer to match with Type 1 workers, they can only observe the worker's type after they have matched. This implies that the matching process is "unbiased", i.e., when a firm matches with a worker, the probability that this worker is of Type 1, depends only on the relative measure of Type 1 workers in the pool of unemployed.' On the flip side, this assumption means that the arrival rate of jobs to a worker does not depend on her type (since firms cannot discriminate, even though they would like to). Once a match has been formed, the worker's type is immediately revealed. If the worker is of type 1, the two parties negotiate over the wage as in the baseline model (with E (0, 1) denoting the worker bargaining power) and production starts right away. If the worker is type 2, clearly, there is no need for any negotiation, since there is no production and no surplus to split. In this case, by law, the firm must pay the worker a fixed wage um per unit of time (think of it as the minimum wage), until it can prove that the worker is a lemon. The firm will eventually be able to prove this in a court of law, but the court decision takes a random amount of time. Specifically, the decision of the court arrives at a Poisson rate a > 0. When the decision is made, the firm can (finally) fire the unproductive worker and stop paying her the amount wm- Let the measure of unemployed workers of Type i be u,, and let the total measure of unemployed workers be u = uj + u2. There is a very large measure of (identical) firms that can enter the market and search for workers. A firm can enter the labor market with exactly one vacancy, and the total measure of vacancies v will be determined endogenously by free entry. A CRS matching function brings together unemployed workers and vacant firms, and, due to the "unbiased" matching technology assumed here, the total number of matches depends only on v, u, i.e., m = m(u, v), and it is increasing in both arguments. As is standard, let @ = v/u denote the market tightness. To close the model, we will make a few more standard assumptions. While a firm is searching for a worker it has to pay a search (or recruiting) cost, pc > 0, per unit of time. Productive jobs are exogenously destroyed at Poisson tate A 2 0, and, as already explained, unproductive jobs are terminated (through the legal process) at the rate a > 0. To avoid weird equilibria, assume that a > A. All agents discount future at the rate r > 0, and all unemployed workers enjoy a benefit z > 0 per unit of time. We will impose p > wm > z. a) Describe the Beveridge curve (the relationship between unemployment, u,, and market tightness, #) for each type of worker.' 1 Suppose there are a unemployed workers out there looking for jobs, and 75% of them are of Type 1. Then, conditional on meeting a worker, the probability that this worker is a Type 1 is 75%. 'Hint: This economy will have two Beveridge curves, one for each type. To find them equate the inflows and outflows out of the pool of unemployment for each type. 3 b) For i = 1, 2, use your findings in part (a) to define the fraction of Type i workers who are unemployed (i.e., the unemployment rate within the Type i population). Denoting this term by y, show that 71