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Question 1 (20 points) Consider the Mortensen-Pissarides model in continuous time. Labor force is nor- malized to 1. Unemployed workers, with measure u - 1, and firms with one vacancy each and total measure v search for each other, and v is determined endogenously by free entry. A CRS matching function, m(u, v), brings together unemployed workers and vacant firms; m is increasing in both arguments. As is standard, let 0 = v/u denote the market tightness and q(0) = m/v the arrival rate of unemployed workers to the typical firm. What is different here compared to the baseline model is that a "match" and a "productive job" are not equivalent by default. When a worker and a vacant firm meet, the firm must train the worker before she can start producing. A formed match turns into a productive job at a stochastic rate, a E (0, +co), so that 1/a can be thought of as the average time necessary for the training to be completed. Assume that the firm and the worker determine the wage level when they first meet (i.e., even before training starts), through Nash bargaining, with 3 6 (0, 1) representing the worker's power. However, the wage upon which they have agreed will only be paid to the worker when she starts producing. To close the model, we will make a few more standard assumptions. The output of a productive job is p > 0 per unit of time, and while a firm is searching for a worker it has to pay a search (or recruiting) cost, pe > 0, per unit of time. Firms that are training their workers do not pay this cost (they are done recruiting). Productive jobs are exogenously destroyed at rate > > 0 (only productive jobs are subject to this shock; matches at the training stage cannot be terminated). All agents discount future at the rate / > 0, and unemployed workers enjoy a benefit z > 0 per unit of time. While at the training stage the worker does not receive an unemployment benefit (a trainee is not unemployed). a) Define the value function of the typical firm for all the possible states of the world it may find itself in. b) Do the same for the typical worker. c) Combine the free entry condition with the expressions you provided in part (a) in order to derive the job creation (JC) curve of this economy. d) Using the same methodology as in the lectures (adjusted to accommodate the differences in the new environment), derive the wage curve (WC) for this economy. Hence, two parties who met at time, say, t are negotiating over an object that will be paid in the future (at time t + 1/a, in expected terms). But, as is always the case, the Nash Bargaining problem is to split the generated surplus as of time t. 2 e) Provide a restriction on parameter values such that a steady state equilibrium pair (w, 0) exists. Is it unique? (No need for a lengthy discussion.) f) What is the effect of a decrease in a on the equilibrium w and ? Explain analytically and intuitively (but shortly). g) Describe the Beveridge curve (BC) of this economy by looking at the flows of workers in and out of the various states. What effect will the decrease in a (discussed in the previous part) have on equilibrium unemployment?Question 4 (20 points) This question considers a TFP shock in the New Keynesian model. The representative household's utility function is Nite (4) In linearized form, the equilibrium conditions for this model are as follows. The house- hold's Euler equation and labor supply conditions are: EMHI - 4 = - ( 1 - ElAHI) (5) (6) The linearized equilibrium conditions for firms are: mic, = thy - de The resource constraint is D = C Monetary policy follows a simple Taylor Rule: (11) (Linearized) TFP follows an AR(1) process (12) er B i.id. In percentage deviations from steady state: me, is real marginal cost, & is con- sumption, i, is the real wage, n is hours worked, y, is output and a, is Total Factor Productivity. In deviations from steady state: i, is the nominal interest rate, #, is inflation. A is a function of model parameters, including the degree of price stickiness." Assume that o, > 1, 0 0 Output is sold at the nominal price of P, dollars per unit of output. Firms pay labor in advance of production - they borrow the wage bill from financial intermediaries. Hence the cost of labor inputs is: Cost = R,W h, where R, denotes the (gross) nominal interest rate and W, is the nominal wage. Financial intermediaries receive funds from two sources: Households make deposits (/, ) at the beginning of the period before the current state of the world is known and the current monetary transfer (which determines the current state) is received from the government. The financial intermediary inelastically provides the funds in the form of loans to businesses to finance their wage bill. That is, financial intermediaries make no profits by assumption: I, +8,M. = Why The income from the loans is distributed entirely to the households. Note this implies that the law of motion for the money supply is: M, =(1+8, )M, Households make deposit, consumption, and labor decisions in order to maximize lifetime expected utility. Preferences are given by: . Ep [Inc, + A(1 - hill As mentioned above, households allocate part of their nominal wealth to the banking sector before knowing the current realization of the monetary growth rate. Then, after observing g, , they make consumption and labor decisions. It is assumed that the consumption is subject to a cash-in-advance constraint. Note that, as implied by the discussion above, current labor income can be used to finance current consumption. The only source of uncertainty in the economy is due to the monetary growth rate; this random variable is assumed to be independently and identically distributed with E(g, )> 0. Given this environment, do the following: a) Set up the firm's and household's maximization problem. Derive and interpret the associated necessary condition. In your answer, demonstrate how this model is distinguished from a typical cash-in-advance model b) Define a stationary monetary equilibrium in this economy. c) Prove that the liquidity effect is present in this economy. d) Prove that consumption and real wages are procyclical. e) What is the correlation of savings (i.e. funds placed in the banking sector) and nominal interest rates? 2. In the analysis of monetary policy, it is currently common practice to use a monetary environment in which the behavior of the monetary aggregate is often ignored. (The next question presents an example of this approach.) In contrast, monetary theory typically imposes strong restrictions on the behavior of the money supply, nominal interest rates, and income, both nominal and real. Is this neglect of the monetary aggregate justified? In your answer, be specific in discussing the pros and cons of this approach. 3. Consider the following basic New Keynesian model x, = E,xHI -(-E,x.)+", (IS) 1, = BE,,. + Kx, +e, (Inf) 1, =Yx, +6 y, (PR) where x, is the output gap; it is the inflation rate; i, is the nominal interest rate controlled by the central bank; and u, and e, are two uncorrelated and white noise disturbances. Answer the following questions concisely: a) Comment on the micro-foundational origins of expression (/S) and briefly discuss common alternative specifications. What consequences do these variations have on the optimal choices of y and o? b ) Comment on the micro-foundational origins of expression (Inf) and briefly discuss common alternative specifications. What consequences do these variations have on the optimal choices of yand of c) Solve the model by expressing (E,M.4. E,y...) as a function of (x, y, )' and the shocks only. d) Discuss how it would affect the choices of yand 6 if the monetary authority's only goal were output stability. e) If e, were a persistent (but not necessarily unit root) autoregressive process, discuss what effect this would have on the optimal choices of y and 6. () Suppose the inflation equation becomes IT, = Bite, instead. Does the new trade-off between inflation and output variability change your answer to (d)?1. a. Write down a stochastic monetary model in which the Fisher relationship holds and one in which it does not. b. The U.S. now issues inflation-indexed bonds. It has been stated that by comparing the yields on inflation indexed bonds and equivalent maturity nominally denominated bonds one can uncover investors' forecast of inflation. Do you agree? Use your answer in part (a) to support your argument. 2. Consider the following variant of Stockman's cash-in-advance model. The primary distinction is that asset and goods markets are assumed to meet sequentially with the asset market preceding the goods market. The assets in the economy are: capital (k, ) , one- period nominal bonds B,, one-period real bonds (b, ), and money (M, ) . Nominal bonds costs IS in period t and return S/, in the following period while real bonds cost one unit of consumption in period ? and return R, units of consumption in period +1. In the asset market, agents receive the returns from capital (i.e. the revenue from the sale of output in last period's goods market), the returns from bonds, money not spent in last period's goods market, and the lump-sum monetary transfer and use this to buy new bonds, money, and purchase investment (investment is not subject to the CIA constraint). Next agents visit the goods market where money is used to finance consumption (i.e. consumption is subject to the cash-in-advance constraint). The aggregate money stock grows at the constant rate # > 0. (Note that, as in Stockman, agents' use capital to produce output via the production function f(k, ) . This output is sold in the goods market. The law of motion for capital is standard with depreciation of 6 E (0,/).) Households have standard time separable preferences as in Stockman. (a) Set up the agent's maximization problem as a dynamic programming problem. Identify the state and control variables. (b) Prove that in this economy, the Lagrange multiplier associated with the budget constraint is equal to agent's marginal utility of consumption. Explain. (c) Define a steady-state equilibrium. Is money superneutral in this economy? (d) Show that the Fisher relationship holds. Monetary Preliminary Examination, July 2004. 3. Suppose the economy is described by the following new-Keynesian model y, = E,y.I -G"(i, - E,x,.)+8, *, = BE,*.., + ky, + v, 1, = $. ( 1 , - 7') +0,y, where y is the output gap, > is the inflation rate and &, and v are i.i.d. random disturbances. We also assume 1>B > 0, k > 0, G > 0. (a) Briefly comment on the micro-foundations behind the new-Keynesian model just presented. (b) If #, = 0, explain intuitively why @, > 1 is needed to ensure that the equilibrium will be unique. (c) If both or and p, are nonnegative, the condition *(0 -1)+(1-B)d, >0 (1) implies that the economy can still have a unique, stable equilibrium even when k Explain intuitively why some values of o, 0.Question 6 (20 points) Consider the following OLG economy. Time is discrete and goes from { = ( to infinity. Agents live for two periods, and there is a constant measure 1 of individuals in each generation. Young agents born in t have preferences over consumption streams of a single good that are ordered by a(c) tu(d+,) where u(c) = _, and where c, is the consumption of an agent born at i in time f. Assume that o > 0 and that when o = 1, u(c) = Inc. Each young agent born at t 2 0 has identical preferences and endowment pattern (w1, w2), where wi is the endowment when young and us is the endowment when old. Assume () 0 and with probability 1 - # the price will be zero. What happens after the price is zero? How does the price p change with a? e) Assume o = 1 and my = 0. Show that there are equilibria in which in some periods agents are optimistic (so that #, is high) and the price of money is high, and some periods in which they are pessimistic (so that #, is low) and the price of money is low. Argue that this model produces non-fundamental fluctuation in asset prices