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. (10) Consider the discrete time monetary-search model we saw in class. As in the baseline model, in the day time trade takes place in a decentralized market characterized by anonymity and bilateral meetings (call it the DM), and at night trade takes place in a Walrasian or centralized market (call it the CM). There are two types of agents, buyers and sellers, and the measure of both is normalized to the unit. The per period utility is u(q) +U(X) H, for buyers, and q +U(X) H, for sellers; q is consumption of the DM good, X is consumption of the CM good (the numeraire), and H is hours worked in the CM. In the CM, one hour of work delivers one unit of the numeraire. The functions u; U satisfy standard propertiesGoods are non storable, but there exits a storable and recognizable object, called at money, that can serve as a means of payment. The supply of money, controlled by the monetary authority, follows the process Mt+1 = (1 + )Mt , and new money is introduced via lump-sum transfers to buyers in the CM. What is dierent here is that only a fraction of buyers turn out to have a desire to consume the DM good in the current period; let us refer to these buyers as C-types (for consumption) and to the remaining 1 buyers as N-types (for no-consumption). The shock that determines each buyers type in every period is iid. A buyer learns her type after all CM trade has concluded but before the DM opens. To make things interesting we will assume that between the CM and the DM there is a third market, where C-types and N-types can meet and trade liquidity", i.e., money. Let us refer to this market as the loan market (LM).4 The LM is a bilateral market for loans, where N-types, who may carry some money that they do not need, meet C-types, who may need additional liquidity. A CRS matching function f(; 1 ) brings the two types together. Importantly, the LM is not anonymous, so that agents can make credible (and enforceable) promises. Hence, when an N-type and a C-type meet, they mutually benet from a contract specifying that the N-type will give l units of money to the C-type right away, and the C-type will repay d (for debt) units of the numeraire good in the forthcoming CM. After the LM trades have concluded (for the agents who matched with someone), C-types proceed to the DM, where they use money to purchase goods from sellers. Assume that all C-type buyers match with a seller. Notice that I have not said anything about the splitting of the various surpluses (i.e., bargaining), because this information will not be necessary for what I am asking here. Let W(:) be the CM value function of a buyer, and V (:) the DM value function of a C-type buyer (since only these buyers visit the DM). Also, let i (:) be the LM value function of a type-i buyer, i 2 fC; Ng. Your task in this question is to describe these value functions. I am not asking you to analyze them. I recommend that you draw a graph summarizing the timing of the model. (a) Describe the function W(:), and show that it is linear in all its arguments/state variables (what these arguments are, however, is for you to determine). (b) Let (q; p) be the quantity of good and the units of money exchanged in a typical DM meeting. Let (l; d) be the size of the loan (in dollars) and the promised repayment (in terms of the numeraire) specied in a typical LM meeting. What variables do the terms q; p; l; d depend on? Hint: Provide quick answers of the form q is a function of the money holdings of the (C-type) buyer". (c) Describe the function V (:), where, again, determining the state variables is your task. (d) Describe the functions i (m), i 2 fC; Ng, for a buyer who enters the LM with m units of money. Hint: Recall that some buyers (of either type) will match in the LM and some will not, and the outcome of the matching process will critically aect a buyers continuation value. Make sure that this is reected in the expression you provide. 4 Since it is very important to understand the model setup

(20) Consider the standard growth model in discrete time. There is a large number of identical households normalized to 1. Each household wants to maximize life-time discounted utility U(fctg 1 t=0) = X1 t=0 tu(ct); 2 (0; 1): Each household has an initial capital k0 at time 0, and one unit of productive time in each period that can be devoted to work. Final output is produced using capital and labor, according to a CRS production function F. This technology is owned by rms (whose measure does not really matter because of the CRS assumption). Output can be consumed (ct) or invested (it). Households own the capital (so they make the investment decision), and they rent it out to rms. Let 2 (0; 1) denote the depreciation rate of capital. Households own the rms, i.e. they are claimants to the rmsprots, but these prots will be zero in equilibrium. The function u is twice continuously dierentiable and bounded, with u 0 (c) > 0, u 00(c) < 0, u 0 (0) = 1, and u 0 (1) = 0. Regarding the production technology, we will introduce the useful function f(x) F(x; 1)+ (1)x, 8x 2 R+. The function f is twice continuously dierentiable with f 0 (x) > 0, f 00(x) < 0, f(0) = 0, f 0 (0) = 1, and f 0 (1) = 1 . In this model the government taxes householdsinvestment at the constant rate 2 [0; 1]. The government returns all the tax revenues, T, to the households in the form of lump-sum transfers. Throughout this question focus on recursive competitive equilibrium (RCE). (a) Write down the problem of the household recursively.3 Carefully distinguish between aggregate and individual state variables. Then, dene a RCE. Hint: Writing down the budget constraint correctly is essential for this question, so think carefully: the household can choose to allocate its wealth between consumption and investment in any way it likes, but for any unit of resources allocated into investment, a fraction of that amount will be subtracted from the households budget (and it will be returned to them in the form of a lump-sum transfer). (b) Write down the dynamic equation that the aggregate capital stock follows in this economy. Hint: Obtain the Euler equation for the typical household and impose the RCE conditions. (c) Now focus on steady-states. Describe the steady-state equilibrium value of the aggregate capital stock in this economy, and denote it by K ( ). If the formula you arrived at involves the function F, I recommend that you replace it with the function f in order to answer the next parts. (d) Describe the value of K when = 0 and when = 1. (e) In class, we studied the RCE steady state level of capital in an economy where the government taxed the income from renting capital (as opposed to investment, which is the case here). In that model, we saw that for = 1 the equilibrium capital stock reached zero. Based on your answer to part (d), does this also happen here? Provide an intuitive explanation of why (or why not) (f) Let F(K; N) = KaN1a , a 2 (0; 1). Provide a closed-form solution for K ( ). (g) Now focus on the special case where F(K; N) = K 1 2 N 1 2 . Calculate the governments total tax revenue, T, and plot it as a function of the tax rate (the so-called Laer curve). Which value of maximizes tax revenu