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1. The Laer Curve states that as the tax rate rises from a low level, tax revenue initially rises, reaches a peak, and then falls. This question asks you to derive a steady-state Laer Curve for an income tax. Utility for the representative agent is given by U = X1 t=0 t " ln Ct + ln Gt + (1 `t) 1 1 # ; 0 < < 1; ; > 0 (U) where Ct represents consumption, Gt is government spending, and 1`t is leisure, with unity as the time endowment and `t as labor supply. Output is produced with capital (Kt) and labor according to: Yt = K t ` 1 t (Y) The governments budget constraint is given by Gt = tYt (G) t is the income tax rate. The agents budget constraint is Kt+1 = Kt (1 ) + (rtKt + wt`t) (1 t) Ct ; (ABC) where is the rate of depreciation of capital. (Recall that Yt = rtKt + wt`t since production is constant returns to scale.) (a) Derive the rst order conditions and write them as an Euler equation, labeled (EE) and a labor leisure choice labeled (LL). (b) Write the equation for the resource constraint in steady state. (c) Write the prot-maximizing problem for a representative rm using capital and labor to produce output. Take rst order conditions with respect to Kt and `t : (d) List the equations required to solve for the endogenous variables in equilibrium. (e) Denote steady-state values by dropping the time subscripts and solve for equilibrium quantities. How does the tax rate aect ` and Y ? [Hint: Begin by solving for Y K and C K : Then express the labor-leisure choice as a non-linear function of ` and parameters. Since this is non-linear in `, you do not have to actually solve the equation for `. Solve for levels of Y; C; and K as a function of ` and parameters including :] (f) Tax revenue is given by Y: Explain how tax revenue varies as rises from zero to something close to unity. Explain how capital and labor react to the tax rate generating the response of income to the tax rate. Do we have a Laer Curve? If so, derive the tax-revenue maximizing value of the

Endowment model with "cash" and "credit" goods (inspired by Lucas and Stokey [1987]) Consider the following economy: Time: discrete, innite horizon, t = 1; 2; ::: Demography: A single representative household (takes prices as given) Preferences: The household likes to consume two goods. Good 1 is a cash good which requires cash in advance of the period in which the purchase is made. Good 2 is a "credit" good which does not require cash for its acquisition. The household gets utility u(c1;t; c2;t) from consumption of ci;t units of good i = 1; 2 at time t: Assume that u is strictly increasing in both arguments and strictly concave with lim c1;t!0 u1(c1;t; c2;t) = lim c2;t!0 u2(c1;t; c2;t) = 1 The households discount factor is < 1: Endowments: In period t; the household receives yt units of generic perishable consumption good and can choose freely how much of it to take to the market as a cash good, y1;t and how much to take to the market as a credit good, y2;t (y1;t + y2;t = yt). The usual assumption that you cannot consume your own output applies so you have to bring the goods to market. Note: This assumption means that both goods will always trade at the same price, pt : (If not, people would only bring the higher priced good to market and our assumptions on preferences rule out such a corner solution.) Institutions: There is a government that issues currency so that the stock at time t is Ht : the time t growth rate of the money supply is t so that Ht+1 = (1 + t)Ht : The money is distributed (or collected if t < 0) lump sum so that households period t transfer (tax) is t : The household can only acquire the cash good using money which has to have been acquired in the previous period. The credit good can be acquired on credit in that the household can use the current periods proceeds from sales to obtain it. This means they can eectively swap their own goods for credit goods and do not need to go through cash. (a) Write down the problem faced by the representative household. (Note: there is no capital in this model so the households only state variable will be Mt their nominal money holdings.) (b) Obtain the rst-order, complementary slackness and envelope conditions. (c) Obtain an intertemporal (Euler type) condition that relates the marginal utility of consuming good 2 today and good 1 tomorrow. Explain this expression. (d) Write down the market-clearing conditions and the government budget constraint. Dene a competitive equilibrium. (e) Now suppose t = and yt = y for all t and that we are in a steady state so that c1;t = c1 and c2;t = c2 for all t. What does this mean for the real return on money, pt=pt+1 and the optimality of the Friedman rule? Explain your answer.

Consider a two-period world with uncertainty. Utility for the representative agent is given by U = u (C1) + u (C2) < 1 Production in each period t = 1; 2 is given by Yt = AtK t L 1 t ; where we assume A1; K1 are given L1 = L2 = 1 A2 = (1 + g) A1 where g is normally distributed with E (g) > 0 Agents believe that they can save and dissave using capital and risk-free bonds, denoted by B (view B as an asset). Risk-free bonds earn an endogenous interest rate of r: Assume that capital depreciates at rate and that the agent begins period 1 with no risk-free bonds. (a) Write equations for the agents period 1 and period 2 budget constraints, where the agent knows there is no period 3. Assume that the agent chooses K2 and B2 to maximize expected utility and write Euler equations, one using bonds, and one using capital. (b) Consider four alternative specications of utility: Linear: u (C) = aC Quadratic: u (C) = aC b 2 C 2 for C < a b Log: u (C) = ln C CES: u (C) = C 1 1 for > 0 Dene certainty equivalence. Which utility functions exhibit certainty equivalence? Which bond Euler equations have certainty equivalence? (c) Linearize the marginal utility of consumption in the CES case about C2 = C1: Evaluate whether the linearized bond Euler equation has certainty equivalence. What are the implications of your answer for whether or not the linearized RBC model has certainty equivalence? (d) Dene risk aversion and evaluate which utility functions are characterized by risk aversion. What characteristic of the utility function characterizes risk aversion? (e) Dene precautionary saving and evaluate which utility functions are characterized by precautionary savings. How does precautionary savings dier from risk aversion? (f) Write an expression for the risk-premium on capital (relative to the risk-free interest rate) using the bond and capital Euler equations with generic utility (U (C)). Dene the gross return on capital as 1 + R = 1 + Y2 K2 ; and the risk premium as ER r: Use your expression to explain the sign of the risk premium. Do all of the utility functions above have a positive risk pr

Diamond-Mortensen-Pissarides with two types of worker Time: Discrete, innite horizon Demography: A mass of 1 of workers with innite lives. A proportion of the workers are high, h; productivity the rest are low, l; productivity workers. A workers type, i = h; l is observable by the rm when they meet and it never changes. There is a large mass of rms who create individual and identical vacancies. The number of vacancies is controlled by free-entry. Preferences: Workers and rms are risk neutral (i.e. u(x) = x). The common discount rate is r: To simplify the algebra the value of leisure for workers (usually referred to as b) is set to zero. The cost of holding a vacancy for rms is a utils per period. Productive Technology: A rm matched to a worker of productivity i produces pi > 0; i = h; l units of the consumption good per period ( ph > pl). With probability each period, jobs (lled or vacant) experience a catastrophic productivity shock and the job is destroyed. Matching Technology: With probability m() each period unemployed workers encounter vacancies. Here = v=u; v is the mass of vacancies, u = uh + ul is the total mass of unemployed workers and ui is the mass of type i = h; l unemployed workers. The function m(:) is increasing concave and m() < 1 for all : Also lim!0 m0 () = 1; lim!1 m0 () = 0; and m() > m0 (): The rate at which vacancies encounter unemployed workers is then m()= which is decreasing in : (Assume that job destruction and matching are mutually exclusive so m() + < 1:) Note: as everyone matches and loses their jobs at the same rate the proportion of unemployed workers who are high productivity will be the same as in the whole population, i.e. uh = u and ul = (1 )u: Institutions: The terms of trade are determined by symmetric Nash bargaining. That is, both rms and workers having bargaining power 1 2 : (a) Let V u ; and V e ; be respectively the present discounted expected values of a type i unemployed worker and a type i employed worker. Let V f be the value to a rm of having hired a type i worker and let Vv be the value to a rm of holding a vacancy (freeentry will drive this to zero.) Write down the ow value or Bellman type equations for workers and rms based on a type i = h; l worker receiving wage wi : Hint: the vacancies will meet with workers according their population shares. Assume all meetings lead to matches. (b) Symmetric Nash bargaining means that the surplus is split evenly between workers and rms so that (V i f Vv) = 1 2 (V i f Vv) + (V e V i u ) : By reference to this bargaining equation and the value equations dene a free-entry search equilibrium. (c) Solve for an expression that characterizes equilibrium in terms of the market tightness : (d) How does equilibrium market tightness vary with the proportion of the population who are high productivity, ? What does this mean for the value to being an unemployed low productivity person. Explain the result and how this might inuence education poli

For 3 of the following 6 statements, state whether the statement is true, false, or uncertain, and give a complete and convincing explanation of your answer. Note: Such explanations typically appeal to specic macroeconomic models. 1. The government benets paid to unemployed U.S. workers have recently become less generous. Although this cut will reduce the number of unemployed workers, it will not aect people already employed. 2. Increasing the tax rate on capital income will have very little eect on workers, as most workers own very few assets. 3. Ination is exclusively a monetary phenomenon 4. As cash-in-advance induces an indirect utility function over money, it is simply a special case of money in the utility function. 5. During the Great Recession 2007-2009, unemployment rose, because the natural rate of unemployment rose. The search and matching approach to labor markets explains why the natural rate rose. 6. Real business cycle theory provides an explanation of how a malfunction of debt markets can reduce national income and product.

Consider the following variant of the Lucas tree model. The preferences of the representative consumer are E0 X1 t=0 t ln(ct) ; 0 < < 1; where ct denotes consumption, and Et () denotes expectations conditional on time-t information. Output is produced by an innite-lived tree: each period, the tree produces dt units of non-storable output. Output is a non-negative, exogenous random variable governed by a Markov process with the stationary transition density f (d 0 ; d). The economy starts o with each consumer owning one tree apiece. Let pt = p (dt) be the price at time t of all future prots from a tree, which we will call a unit of ideal stock. Let q (d 0 ; dt) be the kernel used to price the one-stepahead contingent claim that delivers one unit of the consumption good when dt+1 = d 0 . Finally, let xt denote the consumers nancial resources, which she allocates between ideal stock, contingent claims, and consumption. (a) Write down the consumers problem in recursive form and nd the rst order conditions. (b) Show that the equilibrium price for ideal stock, p(dt), can be written as pdt . (c) Now suppose that consumers have access to common stock, which allows them to act as residual claimants. Each period a unit of common stock provides a dividend, gt , as follows: gt = g(dt) = dt d0; if dt d0 0; if dt < d0 : Let ot = o (dt) be the price at time t to all future dividends from a unit of common stock. i. If an individual buys a unit of common stock in period t and sells it one period later, his income is the sum of the dividend gt+1 and the resale price ot+1. Combine this fact with the contingent claim price q (d 0 ; dt) to derive an expression for o (dt). ii. Impose the equilibrium allocation to derive an Euler equation for common stock. (d) It is useful to rewrite the common stock dividend as gt = g t (dt)dt ; g t (dt) maxfdt d0; 0g dt : Show that the equilibrium price for common stock, o(dt), can be written as tpt , with t = (dt). Intuitively, what does (dt) repre