In need of 3 your help..

1. The Laffer 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 Laffer Curve for an income tax. Utility for the representative agent is given by (U) 1=0 where Of represents consumption, G, is government spending, and 1 -4, is leisure, with unity as the time endowment and & as labor supply. Output is produced with capital (Kr) and labor according to: (Y) The government's budget constraint is given by GI = TY (G) 7: is the income tax rate. The agent's budget constraint is Kit1 = Ki (1 - 8) + (nek, + me) (1 - 74) - C. (ABC) where o is the rate of depreciation of capital. (Recall that Y, = nik, + wyf since production is constant returns to scale.) (a) Derive the first 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 profit-maximizing problem for a representative firm using capital and labor to produce output. Take first order conditions with respect to K, and &. (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 affect / and Y? [Hint: Begin by solving for _ and . Then express the labor-leisure choice as a non-linear function of / and parameters. Since this is non-linear in f, you do not have to actually solve the equation for 2. Solve for levels of Y, C, and K as a function of { and parameters including 7.] (f) Tax revenue is given by TY. Explain how tax revenue varies as 7 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 Laffer Curve? If so, derive the tax-revenue maximizing value of the tax rate.Question 2 (30 points} This question studies the cur-existence of diecreut currencies. Time is discrete with an innite horizon. Each period consists oftwo subperiods. In the day, trade is bilateral and anonymous as in Kiyotalci and Wright [1939} (call this the KW 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 1. The per period utility for buyers is 115(9) + UIXJ H, and for sellers it is n'; + UUCI H, 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 mluleraire], and H is hours worked in the CM. 111 the CM, all agents have access 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, so) such that U'(X') = 1, and we dene the Erstvbest quantity traded in the K'W market as q' E {q : 15(q'} = l}. \"is will assume that there are two types of money, 1111 and mail There are also two types of sellers. For reasons that we will leave out of the model, Typevl sellers, with measure er E [1}, 1], do not recognize 1112, thus, they accept only the local currency m1. Type-2 sellers, with measure 1 or, recognize and, hence, accept 111%, as well as 1111. Hence, local currency has a liquidity advantage over the foreign one, since it is recognized by all sellers. All buyers meet a seller in the KW market, so that or is the proballity with which a buyer meets a type-l seller, and l a" is the probability with which she meets a type-2 seller. In any type of meeting, buyers have all the bargaining pm. The rest is standard. Goods are non storable. rT'he supply of each money is conv trolled by an individual authority, and evolves according to Mu\" = [1 + ijiy. New money [of both types) is introduced, or withdrawn if p, r: 1'}, via lump-sum transfers to buyers in the CM. Throughout this question focus on steady states. a] Describe the value function of a buyer and a seller who enter the Walrasian market with arbitrary money holdings (mhmgl. b] DBSlibE the terms of trade in each type of K\"ir meeting. c] Describe the objective function of the typical buyer, J(m'1, 11133]. d) For any given [p51 , [13], p..- 2 ,8 1, for all 2', describe the steadyvstate equilibrium, summarized by the variables {qhqmahzg}, where q,- s the amount of special good traded in a KW meeting with a seller of type i = {1, 2}, and 2,- denotes the equilibrium ' One possible interpretation is that this is a model of a Latin Arlierloan mommy, and ml is the local veilrreuuc-p.r {sq}, the peso} while we; is the US dollar. Of course, this is a very simple model, so true should not take this suggestion too literally. real balances of money i = 1,2. (Hint: For now, all you need to do is provide 4 equations, the solution to which yields the equilibrium values for our 4 variables.) e} I now ask you to characterize the equilibrium in more detail. To that end, let us assume that the KW utility function is quadratic, i.e., ch) = [l + yjq 5: which 2 3 implies q' = 1r. For the various possible combinations of (p1,pg],p..- 2 f3 1, provide closed-form solutions for (91,4;3). Does currency 2 circulate in this economy (i.e., is 2); > U] in every equilibrium? Ifnot, can you provide a condition on policy parameters that would guarantee :2 2' ll? Question 1 (20 points) Consider the Mortensen-Pissarides model in discrete time. The labor force is normalized at 1. Let u denote the unemployment rate. There is a large number of firms who can enter the market and search for a worker. Firms who engage in search have to pay a fixed cost & per period until they find a worker. If in any given period a measure v of vacant firms search for workers, then the total number of matches created in the economy is given by m(u, v) = = Each vacant firm has one job opening. Within each match, the firm and the worker bargain (a la Nash) for the wage, w, with 7 denoting the bargaining power of the worker. If they agree, they move on to production, which will deliver output equal to p per period. All agents discount future at rate B E (0, 1). At the end of every period (after production has taken place), existing matches get destroyed with probability 6.' So far this is just the standard model (in discrete time). We now make two as- sumptions that depart from the baseline model. First, the unemployment benefit, z, does not represent utility of leisure or value of home production, as we conveniently assumed in class. Here, z is a payment made by the government and, naturally, this payment needs to be funded somehow. We assume that the government raises these funds by imposing a lump-sum (flat) tax (per period) on every matched firm. Thus, the government chooses both s and 7, and must do so in a way so that the budget constraint is satisfied at any t. The second assumption concerns the duration of unemployment benefits. In partic- ular, we will assume that workers are eligible for unemployment benefits only for one period." (This assumption would be quite realistic for the US, if we were to assume that a period of the model corresponds to 6 months.) a) Describe the Beveridge curve (BC) of this economy in steady state, i.e., express u as a function of the market tightness 0 = v/u. b) This model predicts that a certain level of unemployment will persist even in the steady state. What is perhaps a little more subtle is that workers who are currently A worker who was part of a match that got hit by the destruction shock at the end of period t, will be unemployed for sure in t + 1 and will try to find a job again in (the beginning of ) period t+ 2. Consider again the worker described in footnote 1, ie., a worker who was part of a match that got hit by the destruction shock at the end of period t. This worker will be unemployed for sure in f + 1 and will receive z. Starting in period t + 2, she will try to find a new job. If she is successful, she will move (immediately) into production. If she is unsuccessful, she will remain unemployed for another period, and, importantly, during that period she will not be eligible for unemployment benefits. This process will continue until the worker finds employment. 2 in the pool of unemployment have been unemployed for different periods of time. This is especially relevant in our question, where unemployment benefit eligibility depends on the duration of unemployment. Describe the measure of workers who have been unemployed for i periods, i = {1, 2,3,...}.3 Verify that your result is correct by adding up the various unemployment durations. (They should add up to the steady state u!) c) Describe the value function for a vacant firm (V) and a firm that has filled its vacancy (J). d) Describe the value function of a typical worker in the various states. e) Exploiting the usual free entry argument, derive the job creation (JC) condition. f) Describe the wage curve (WC) in this economy. g) What is the relationship between + and z, u so that the government's budget constraint is satisfied in every period? Use this condition in order to get rid of 7 in the WC and JC expressions you derived earlier. h) Plot the JC curve in the (w,) space. Does it have the standard shape? i) Plot the WC in the (w, 0) space. Does it have the standard shape? j) Shortly discuss the existence and uniqueness of a steady state equilibrium.Question 4 (20 points) This question considers a distortionary labor income tax in the New Keynesian model. The representative household's utility function is: 1-0 Nite 1 - o It (6) The household's budget contraint is: Q Biti + P.C. - Be + (1 - n)mPM, + 1, + PT. (7) C is consumption, N is hours worked, w is the real wage, 7 is the distortionary labor income tax rate, II are firm profits distributed lump sum, T are hump-sum taxes. B are bonds that are in zero net supply. P is the aggregate price level. Q, is the bond price. In linearized form, the household's Euler equation and labor supply conditions are: (8) in = of, + win+ t (9) The linearized equilibrium conditions for firms are: (10) (11) A = BE (1+1) + Amic, (12) The resource constraint is: 1 = & (13) Monetary policy follows a simple Taylor Rule: (14) The (linearized) labor income tax rate follows an AR(1) process 71 = PTI-ITE , (15) er i i.i.d. and tax revenues are redistributed lump-sum to households. In percentage deviations from steady state: mic, is real marginal cost, & is con- sumption, why is the real wage, i, is hours worked, y, is output. In deviations from steady state: i, is the nominal interest rate, it, is inflation and f, is the income tax rate.? A is a function of model parameters, including the degree of price stickiness." Assume that or > 1, 0