my economics assign

Consider the following economy. There are two periods, 0 and 1, and a single consumption good. The economy is populated by two types of agents: nanciers (F) and entrepreneurs (E). There is a mass one of each type. All agents in the economy have preferences given by U = C1 + C2 , where Ct denotes consumption in period t and is a parameter between 0 and 1. Financiers have deep pockets (i.e., they have a large endowment every period). Entrepreneurs have access to a variable scale project: investing I units in period 0 yields RI units of the consumption good in period 1 in case of success and 0 in case of failure. If the entrepreneur puts eort, then the probability of success is pH. If entrepreneur i shirks, then the probability of success is pL < pH but she gets a private benet BiI. Assume that Bi is observable by Es and Fs. Entrepreneurs dier in their private benet of shirking Bi, which is distributed according to the cumulative distribution function G(Bi) in the compact set [B;B], with E[Bi] = B. E's have limited liability at both dates (C1 0;C2 0) and access to initial funds Ni, distributed in the population according to cumulative distribution function F(Ni) in the compact set [N;N] with E[Ni] = N. Assume that the distribution of private benet Bi is independent of the distribution of net worth Ni. Denote the interest rate of the economy by r. Assume that pHR 1 + r > 1 > pLR + B 1 + r a) State the conditions an optimal contract must satisfy. Let i R Bi pHpL . Show that iI is the maximum amount an entrepreneur of type i can commit to repaying at t = 1 (for this reason we will call i as \pledgeability"). b) State the entrepreneurs' problem. Show that Es invest up to the maximum possible scale, i.e., I = Ni 1 pHi 1+r : How does the project scale depend on Bi? Explain. Hint: Since Fs have a large endowment, the Es keep all the surplus from the project. c) Let I R B B R N N I(Ni;Bi)dF(Ni)dG(Bi) be the aggregate investment in this economy. Argue that if the nanciers' endowment is suciently large, then 1 + r = 1 . Let e0 denote the Fs' endowment in period 0. What is the minimum value of e0 such that 1 + r = 1 in equilibrium? d) Start from a situation in which there is no heterogeneity so that B = B = B and N = N = N. Does an increase in heterogeneity in Ni (but keeping the average 6 constant) increase, decrease or not change aggregate investment I? How about an increase in heterogeneity in Bi (keeping the average constant)? Hint: If X is a random variable and h(x) is a convex function of x, then E[h(X)] > h(E[X]). Now suppose that entrepreneurs can hire a monitoring service (at a cost), which reduces the private return of shirking to biI where bi = Bi with 2 (0; 1). We will study whether entrepreneurs have an incentive to hire the monitoring service. e) Suppose that the cost of monitoring is cI and an entrepreneur i hires the service. How does the contract with Fs change? Show that the scale of the project is larger with monitoring if and only if c (1 )pH R i 1 + r Hint: Carefully state the conditions a contract must satisfy. f) What is the maximum cost such that E chooses monitoring

Consider the Mortensen-Pissarides model in continuous time. Labor force is normalized to 1 and there are 2 types of workers. A worker of type i = fL;Hg enjoys a benet zi while unemployed, and zL < zH. Let 2 (0; 1) denote the measure of type-H workers. With random matching the probability with which a rm matches with a certain type of worker depends only on the relative measure of this type in the pool of unemployed. The rest of the model is standard. Unemployed workers and rms (with one vacancy each) search for each other. Let the measure of unemployed workers be denoted by u = uL + uH, where ui is the measure of unemployed workers of type i. Also, let the measure of vacant rms be denoted by , which will be determined endogenously by free entry. A CRS and increasing (in both arguments) matching function, m(u; ), brings together unemployed workers and vacant rms. It will be useful to dene the market tightness as =u. Once a match has been formed, the wage is determined through Nash bargaining, with 2 (0; 1) representing the worker's power. The output of all jobs is p > 0 per unit of time, i.e., p does not depend on the worker's type and p > zi, for all i. Also, while a rm is searching for a worker it has to pay a search (or recruiting) cost, pc > 0, per unit of time. All jobs are exogenously destroyed at rate > 0. All agents discount future at the rate r > 0. Throughout this question focus on steady state equilibria. a)Write down the value functions for a rm and a worker of each type in all possible states. b) Exploiting the free entry condition of rms, derive the analogue of the job creation (JC) curve for this economy. c) Using the same methodology as in the lectures (adjusted to accommodate the dierences in the new environment), derive the wage curve (WC) for this economy. d) Combine the JC curve and the WC curve determined in the previous parts in order to provide an equation that (implicitly) determines the equilibrium . Compare with the analogous equilibrium condition from the lectures. For the remainder of this question, we will make an important change in the environment. We will assume that the total production of a lled job is given by px, where x is the match-specic (idiosyncratic) productivity. (But we will maintain the assumption that there are two types of workers and they enjoy a dierent unemployment benet.) As in the theory of \endogenous job destruction" (Chapter 2 of Pissarides' book), we will assume that existing matches get hit by a productivity shock at rate ; when that happens the random variable x attains a new value drawn from the cdf G(x). This distribution is iid over time and has support in the interval [0; 1]. 2 New jobs are always created at x = 1. Assume that every time a match obtains a new value x, the rm and the worker decide whether it is worth keeping the match alive and, if yes, they renegotiate over the wage. To answer the following questions make a conjecture about the equilibrium form, similar to the one we made in class in the endogenous job destruction theory. e) In this new environment what is the unemployment rate of workers of type i = L and H? Is it the same? f) Write down the value functions for a rm in all possible states. g) Write down the value functions for a worker of each type in all possible states. h) Without characterizing equilibrium (just using the economic intuition you have developed by studying these types of models) answer the following question: In the model with exogenous job destruction L-type workers are worse o compared to H-type workers because they get paid a lower wage. How is the well being of L-types compared to that of H-types aected as we switch from the model of exogenous to the model of endogenous job destruction? (Hint: Compare the wage the two types receive but also the length of time for which they get to keep this wage, since now job destruction is endogenous.) 3

This question considers the macroeconomic eects of a time-varying sales tax in the New Keynesian model. There are a continuum of identical households. The representative household makes consumption (C) and labor supply (N) decisions to maximize lifetime expected utility: E0 X1 t=0 t lnCt N1+ t 1 + ! (1) subject to their budget constraint: (1 + s t )Ct + Bt = wtNt + (1 + it1) Pt1 Pt Bt1 + Dt + Tt (2) where wt is the real wage, Nt is hours worked, Bt are real bond holdings at the end of period t, it1 is the nominal interest rate paid between t 1 and t, Pt is the price of the nal consumption good and Dt are real prots from rms that are distributed lump sum. Tt are real lump sum transfers from the government. As usual, 0 < < 1 and > 0. s t is a sales tax charged on the purchases of consumption goods. The production side of the model is the standard New Keynesian environment. Monopolistically competitive intermediate goods rms produce an intermediate good using labor. Intermediate goods rms face a probability that they cannot adjust their price each period (the Calvo pricing mechanism). Intermediate goods are aggregated into a nal (homogenous) consumption good by nal goods rms. The production side of the economy, when aggregated and linearized, can be described by the following set of linearized equilibrium conditions (the production function, the optimal hiring condition for labor and the dynamic evolution of prices): ^yt = ^nt (3) w^t = m^ct (4) ^t = Et(^t+1) + m^ct (5) The resource constraint is: ^yt = ^ct (6) Monetary policy follows a simple Taylor Rule: ^it = ^t (7) The (linearized) sales tax rate follows an AR(1) process: ^ s t = ^ s t1 + et (8) et is i.i.d. and tax revenues are redistributed lump-sum to households. In percentage deviations from steady state: m^ct is real marginal cost, c^t is consumption, w^t is the 4 real wage, ^nt is hours worked, ^yt is output. In deviations from steady state: ^it is the nominal interest rate, ^t is in ation and ^ s t is the sales tax rate. is a function of model parameters, including the degree of price stickiness.1 Assume that > 1, 0 < < 1. a) First consider the representative household's problem. Write down the household's problem in recursive form and derive the household's rst order conditions. b) Show that the linearized rst order condition for labor supply from part (a) is: ^ wt = ^ct + ^nt + ^ s t (9) and that the exible price natural rate of output (in linearized form) is given by: ^yn t = 1 1 + ^ s t (10) Hints: you will need to use equations (3), (4), (6) and (9). To simplify the algebra, dene ^ s t = ln 1+s t 1+s ss and assume a zero steady state tax rate, s ss = 0.2 c) This model can be reduced to two equations: Et~yt+1 ~yt = (^t Et^t+1) + 1 + (1 )^ s t (11) ^t = Et(^t+1) + ~yt (12) plus the stochastic process for the sales tax. ~yt = ^yt^yn t is the output gap. = (1+ ). Using the method of undetermined coecients, nd the response of the output gap and in ation to an exogenous cut in sales taxes when prices are sticky and monetary policy follows the Taylor Rule above. To do this, guess that the solution for each variable is a linear function of the tax shock ^ s t . d) Discuss how, and why, a sales tax cut aects the natural rate of output, the output gap and in ation in this model. e) Now suppose the monetary policymaker attempts to target the natural real interest rate. Is this policy optimal from a welfare perspective in this model? Explain. You do not need to derive anything, answer using your knowledge of this model. (Hints: To answer this question, think about what the rst best allocation, ^ye t , would look like and whether the policymaker can close the welfare relevant output gap ^xt = ^yt ^ye t . Assume the steady state is ecient). 1 = (1)(1) where is the probability that a rm cannot adjust its price. 2If the tax rate is suciently small, this implies ^ s t = s t (approximately). Another way to think about this is that the policy choice variable is (1+ s t ), so ^ s t = (1+s t )(1+s ss) (1+s ss) but with a steady state tax rate of zero s ss = 0. 5

Consider the following economy. There are two periods, 0 and 1, and a single consumption good. The economy is populated by two types of agents: nanciers (F) and entrepreneurs (E). There is a mass one of each type. All agents in the economy have preferences given by U = C1 + C2 , where Ct denotes consumption in period t and is a parameter between 0 and 1. Financiers have deep pockets (i.e., they have a large endowment every period). Entrepreneurs have access to a variable scale project: investing I units in period 0 yields RI units of the consumption good in period 1 in case of success and 0 in case of failure. If the entrepreneur puts eort, then the probability of success is pH. If entrepreneur i shirks, then the probability of success is pL < pH but she gets a private benet BiI. Assume that Bi is observable by Es and Fs. Entrepreneurs dier in their private benet of shirking Bi, which is distributed according to the cumulative distribution function G(Bi) in the compact set [B;B], with E[Bi] = B. E's have limited liability at both dates (C1 0;C2 0) and access to initial funds Ni, distributed in the population according to cumulative distribution function F(Ni) in the compact set [N;N] with E[Ni] = N. Assume that the distribution of private benet Bi is independent of the distribution of net worth Ni. Denote the interest rate of the economy by r. Assume that pHR 1 + r > 1 > pLR + B 1 + r a) State the conditions an optimal contract must satisfy. Let i R Bi pHpL . Show that iI is the maximum amount an entrepreneur of type i can commit to repaying at t = 1 (for this reason we will call i as \pledgeability"). b) State the entrepreneurs' problem. Show that Es invest up to the maximum possible scale, i.e., I = Ni 1 pHi 1+r : How does the project scale depend on Bi? Explain. Hint: Since Fs have a large endowment, the Es keep all the surplus from the project. c) Let I R B B R N N I(Ni;Bi)dF(Ni)dG(Bi) be the aggregate investment in this economy. Argue that if the nanciers' endowment is suciently large, then 1 + r = 1 . Let e0 denote the Fs' endowment in period 0. What is the minimum value of e0 such that 1 + r = 1 in equilibrium? d) Start from a situation in which there is no heterogeneity so that B = B = B and N = N = N. Does an increase in heterogeneity in Ni (but keeping the average 6 constant) increase, decrease or not change aggregate investment I? How about an increase in heterogeneity in Bi (keeping the average constant)? Hint: If X is a random variable and h(x) is a convex function of x, then E[h(X)] > h(E[X]). Now suppose that entrepreneurs can hire a monitoring service (at a cost), which reduces the private return of shirking to biI where bi = Bi with 2 (0; 1). We will study whether entrepreneurs have an incentive to hire the monitoring service. e) Suppose that the cost of monitoring is cI and an entrepreneur i hires the service. How does the contract with Fs change? Show that the scale of the project is larger with monitoring if and only if c (1 )pH R i 1 + r Hint: Carefully state the conditions a contract must satisfy. f) What is the maximum cost such that E chooses monitoring

Consider the following economy. There are two periods, 0 and 1, and a single consumption good. The economy is populated by two types of agents: nanciers (F) and entrepreneurs (E). There is a mass one of each type. All agents in the economy have preferences given by U = C1 + C2 , where Ct denotes consumption in period t and is a parameter between 0 and 1. Financiers have deep pockets (i.e., they have a large endowment every period). Entrepreneurs have access to a variable scale project: investing I units in period 0 yields RI units of the consumption good in period 1 in case of success and 0 in case of failure. If the entrepreneur puts eort, then the probability of success is pH. If entrepreneur i shirks, then the probability of success is pL < pH but she gets a private benet BiI. Assume that Bi is observable by Es and Fs. Entrepreneurs dier in their private benet of shirking Bi, which is distributed according to the cumulative distribution function G(Bi) in the compact set [B;B], with E[Bi] = B. E's have limited liability at both dates (C1 0;C2 0) and access to initial funds Ni, distributed in the population according to cumulative distribution function F(Ni) in the compact set [N;N] with E[Ni] = N. Assume that the distribution of private benet Bi is independent of the distribution of net worth Ni. Denote the interest rate of the economy by r. Assume that pHR 1 + r > 1 > pLR + B 1 + r a) State the conditions an optimal contract must satisfy. Let i R Bi pHpL . Show that iI is the maximum amount an entrepreneur of type i can commit to repaying at t = 1 (for this reason we will call i as \pledgeability"). b) State the entrepreneurs' problem. Show that Es invest up to the maximum possible scale, i.e., I = Ni 1 pHi 1+r : How does the project scale depend on Bi? Explain. Hint: Since Fs have a large endowment, the Es keep all the surplus from the project. c) Let I R B B R N N I(Ni;Bi)dF(Ni)dG(Bi) be the aggregate investment in this economy. Argue that if the nanciers' endowment is suciently large, then 1 + r = 1 . Let e0 denote the Fs' endowment in period 0. What is the minimum value of e0 such that 1 + r = 1 in equilibrium? d) Start from a situation in which there is no heterogeneity so that B = B = B and N = N = N. Does an increase in heterogeneity in Ni (but keeping the average 6 constant) increase, decrease or not change aggregate investment I? How about an increase in heterogeneity in Bi (keeping the average constant)? Hint: If X is a random variable and h(x) is a convex function of x, then E[h(X)] > h(E[X]). Now suppose that entrepreneurs can hire a monitoring service (at a cost), which reduces the private return of shirking to biI where bi = Bi with 2 (0; 1). We will study whether entrepreneurs have an incentive to hire the monitoring service. e) Suppose that the cost of monitoring is cI and an entrepreneur i hires the service. How does the contract with Fs change? Show that the scale of the project is larger with monitoring if and only if c (1 )pH R i 1 + r Hint: Carefully state the conditions a contract must satisfy. f) What is the maximum cost such that E chooses monitoring