Provide

2. A risk-averse investor with W, to invest has two assets available. One is risk free and pays a return p, while the other is risky and pays a random return of r. Assume that the investor chooses a , the proportion of W, to invest in the risky asset, so as to maximize the expected utility of W = (ar+ (1-a).p).W. . a. Suppose that r could be either r, orr , each with probability one half and assume that r, > r . Write out the first-order condition for maximization of expected utility. Is the second-order condition satisfied? Explain. Are restrictions required concerning the values off, , r, and p for a to be bounded? b. Suppose that for some parameterization, the optimal value of a is zero. Show that the optimal value of a becomes positive if r, is increased. C. Give the Arrow-Pratt measure of relative risk aversion. d. Show that if the investor's measure of relative risk aversion is decreasing, then the optimal a increases with W. 3. Suppose that there are two goods (X and Y). The representative consumer's preferences over these two goods are given by the utility function U = VX . Y , and income is given by I. 1. Let the prices of the two goods be represented by Pr and Py. What are the demand functions for these two goods? Derive the expenditure function. b. Suppose that the government gives the consumer a voucher that can be used to purchase X, but not Y. The value of the voucher is V. Show how this changes the consumer's decision problem. C. Suppose that instead of a voucher that can only be used to purchase X, the government gives the consumer a voucher that can be used to buy either X or Y. Again, the value of the voucher is V. Keeping the value of I the same as in part (b), are there values of / for which the consumer would be indifferent between the two programs? Are there values of / for which the consumer would prefer one program to the other? d. Instead of providing a voucher of value , the government can reduce the price of X. By what percent would the government have to reduce this price in order to have this policy be equivalent to a voucher?Question 2 (20 points) Consider the Mortensen-Pissarides model in continuous time. Labor force is normal- ized to 1, but here there are N types of workers (this will be the only difference in comparison to the baseline model seen in class). A worker of type i enjoys a benefit equal to & while unemployed, and z 0 per unit of time, i.e., p does not depend on the worker's type and p > z, for all i. Also, while a firm is searching for a worker it has to pay a search (or recruiting) cost, pe > 0, per unit of time. All jobs are exogenously destroyed at rate A > 0 (again, independently of the worker's type). All agents discount future at the rater > Q Throughout this question focus on steady state equilibria. a) Based only on your knowledge of the environment (i.e., without analyzing the model), explain whether the following statement is true or false: "In this economy, all workers will be paid the same wage, since they are equally productive". b) Write down the value function of a firm with an unfilled vacancy (V) and the value function of a firm matched with a worker of type i (J;). c) Write down the value function of a worker of type i while unemployed (U.) and while employed (W.). d) Combine the free entry condition (i.e., V =0) with the expressions for J, provided earlier in order to derive the job creation (JC) curve for this economy. e) Using the same methodoly as in the lectures (adjusted to accommodate the differ- ences in the new environemnt), derive the wage curve (WC) for this economy. f) Combine the JC curve and the WC curve determined in the previous parts in order to provide an equation that (implicitly) determines the equilibrium e. g) Describe the equilibrium wage for a worker of type i (w;) as a fuction of the model's parameters and the equilibrium # (which was implicitly determined in part f). h) Is w, increasing or decreasing in z,? What happens to the distribution of equilibrium wages as the distribution of workers' types becomes more dispersed? Hint: The right-hand side (RHS) of the JC curve should look identical to the one seen in class. On the left-hand side (LHS). instead of just w, you should have an expression that represents the average wage that a firm expects to pay. *Hint: There will be two differences compared to the WC seen in class: 1) On the LHS, instead of just w. you should have the average wage that a firm expects to pay (the same as in part d); 2) On the RHS, instead of just z, you should have the average unemployment benefit, i.e. . = = 2, ME.Question 3 {21'} points] This mieson studies the err-existence of money and credit. Time is discrete with an innite horizon. Each period mnsists of two subperiods. In the day, trade is partially bilateral and anonymous as in Kiyotaki and 'Wright [1939] {call this the K'W 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 l. The per period utility or buyers is as] + nix; H, and or sellers it is q+ our] s, 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 Chi. In the CM, all agents have accem to a technology that turns one unit of work 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, no] such that U'tX'] -- l, and we dene the rstvbest quantity traded in the KW market as q' E {q : u'fq'} = l}. The only difference cnmapared tn the baseline model is that there are two types of sellers. Type- sellers, with measure a E [I], 1], accept credit. More precisely, in meetings with a type- seller [type-t} meetings], no medium of exchange {MOE} is necessary, and the buyer can purchase day good by promising to repay the seller in the forthcoming GM with muneraire good [this arrangement is called an IOU). The buyer can premise to repay any amount [no credit limit], and her promise is credible [buyers never default}. Type-l sellers, with measure 1 a, never accept credit, hence, any purchase of the day good must be paid for on the spot {quid pro qua) with money. All buyers meet a seller in the KW market, so that or is the probability with which a buyer meets a tyPB-l} seller, and l o" is the probability with which she meets a type-l seller. The rest is standard. rGoods are non storable, but there exits a storable and recog- nizable object, at money, that can scrve as a MOE in type-l meetings. Money sup- ply is controlled by a monetary authority, and we consider simple policies of the form Mei-1 = [1 +pLJM1, pl. 3:- ,B 1. New money is introduced, or withdrawn if p. a: l}, via lumpvsum transfers to buyers in the CM. Let it denote the unit price of money [in terms of the mimeraire]. In any type of meeting, buyers have all the bargaining power. a} Describe the CM value function, W011, d}, of a typical buyer, where m denotes her money holdings as she enters the CM, and d [for 'debt'] denotes the amount of numeraire good she may owe to a (type-{l} seller that she met earlier in the KW market. Show that W[m,dj is linear in both arguments. Now let 113mm] and Wmfm) denote the CM value functions for sellers of type t} and 1, respectively (d stands for the amount of numeraire good to be paid to the type- seller in the CM, and m stands for the amount of money that the type-l seller received in the KW market). To answer the next question you can take as given that Wsolidj = its+ri, and WSIIImJ = A31 + dim {treat the A terms as constants]. b} Let 9, denote the quantity of day good traded in a type-j = II}, 1 meeting. Let d denote the amount of numeraire the buyer promises to repay in the forthcoming CM in exchange for no, and let 3: denote the amount of money the buyer pays in exchange for in. Describe the bargaining solution in a typical type'j = , 1 meeting [recall that, under our assumptions, sellers will not carry any money as they enter the KW marketj.'1 c} Describe the objective function of the typical buyer, Jfr], where the hat denotes next period's choice.5 d] Describe the equilibrium variabIEs gmql as functions of the model's parameters, including the nominal interest rate, 2'. How is ql related to the real balances z E ism?\" _. _.__ _._ _ _ . .. .. .2 Question 4 (20 points) Consider the planner's problem for a real business cycle model with inelastic labor supply (essentially the stochastic growth model) and no trend growth. Preferences are given by: In Ct (1) Output is produced using capital K Y = AKa (2) where K, is the capital stock at the start of period , and A, is a TFP shock and is governed by a discrete state Markov chain. There are adjustment costs to capital, which evolves according to the following production function: (3) When 6 = 0, this becomes the simple model we saw in class with full depreciation (i.e. where Kit1 = I,). The resource constraint is a) Write down the recursive formulation of planner's problem. Use two constraints: the typical resource constraint and the capital production function. Denote the Lagrange multiplier on the resource constraint as ), and the one on the capital production constraint as Aqt- bj Derive the first order conditions. Provide an intuitive explanation of q- c) Using guess and verify, solve the model and find the policy functions for Kit1, It and C, (Hint: start by guessing that investment and consumption are a constant share of output). d) Explain why we can interpret o as a parameter that affects the degree of capital adjustment costs in this model. Briefly discuss how, and why, the responses of consump- tion and investment to TFP shocks vary with 6. Why do some macroeconomic modelers prefer to include capital adjustment costs in their models