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. Monetary policy in the overlapping generations model with ex ante heterogeneity Time: discrete, innite horizon, t = 1. 2,3... Demography: A mass 2N of newborns enter in every period. Everyone lives for 2 periods except for the rst generation of old people who live for 1 period. 1iui'lz'ithin the population there are two types of household A and B who di'er according to their endowments {see below}. The population is split exactly in half between the groups. Preferences: for the generations born in and after period 1; rtfcie- Elie\") = lnfiel + 31nfit+1l 1-: 24,3 where cg, is consumption in period t and stage 9 of life for type i = A. B individuals. For the initial old generation {11:23]} = Int-:31} for i = A. B Endowments: Except for the initial old, in the rst period of life type A people receive 1 unit of the consumption good and type B people receive 2 units. No one gets any endowment in their second period of life. In period 1 the rst generation of old are endowed with Hg units of money spread equally among them which can be stored but provides no utility in consumption. The money supply grows each period so that the aggregate money supply in period t is Hn + G'Jt. The new money transfers occur by helicopter drop Il:i.e. lump sum} to each old person at the beginning of the period in which they are old. Information: There is complete information with perfect foresight. Solution concept: Competitive equilibrium. Each period there are markets for the con sumption good and money. Let, pt, be the price for goods in terms of money in period t which is taken as given by every participant. Solution concept: Competitive equilibrium. Each period there are markets for the con (a) smnption good and money. Let, pt, be the price for goods in terms of money in period t which is taken as given by every participant. 'Write out and solve the problem faced by the members of each type in each generation t. Use iwf'l to represent the nominal money demand of each type i = 31,3 individual born in period t. (Hint: If Pt+l drops out of your rst order condition equation don't 1worry about it-) 'Write down the market clearing conditions and dene a competitive equilibrium- 'We will focus on a steady state monetary equilibrium- Obtain an expression for the transfers made to each household- Solve for money demand in each period in terms of the current price of goods, 10:, and the transfer amount. You do not have to solve for pi. Solve for the amount of consumption in each period of life for each generation also in terms of 01 and the transfer amount By comparing consumption of type A individuals with that of type B, comment on the extent to which money growth, a 21?- I} has a redistributive effect. Briey interpret the result- Solution concept: Competitive equilibrium. Each period there are markets for the con (a) smnption good and money. Let, pt, be the price for goods in terms of money in period t which is taken as given by every participant. 'Write out and solve the problem faced by the members of each type in each generation t. Use iwf'l to represent the nominal money demand of each type i = 31,3 individual born in period t. (Hint: If Pt+l drops out of your rst order condition equation don't 1worry about it-) 'Write down the market clearing conditions and dene a competitive equilibrium- 'We will focus on a steady state monetary equilibrium- Obtain an expression for the transfers made to each household- Solve for money demand in each period in terms of the current price of goods, 10:, and the transfer amount. You do not have to solve for pi. Solve for the amount of consumption in each period of life for each generation also in terms of 01 and the transfer amount By comparing consumption of type A individuals with that of type B, comment on the extent to which money growth, a 21?- I} has a redistributive effect. Briey interpret the result- 4. The three equations below are the fundamental equations from the New Keynesian model. DC' 1 :2 L: = E: Zn 343 [NEH -|- M2 ,] {Loss} 56 = Em\" 51 {5: Ett+1l + u: {Enlerl TH = .5E5Ft+l + 53?: {Fi} it = z: + dim + 533: {TR} where E; denotes the expectation conditional on information available at time t, TI} denotes %, where .Pg is price at time t, a: is the output gap, dened as y, g, where 3; denotes output, the superscript 3' denotes fullemployment output, and ination at time t, dened as hats denote percent deviations from the nonstochastic steady state, it is the nominal interest 1 rate, ct is the interest rate target, 5 cf. 1 is the discount factor, CF is the intertemporal elasticity of substitution, A and n: are parameters which depend on the underlying structural parameters of the model, and m, = E: \"5+1 : is a stochastic error. All level variables should be understood as proportionate deviations from their exible price equilibrium levels, while all rate deviations are differences in the rate. {a} Explain the meaning of each equation and how the rst three are derived from a model of monopolistic competition with optimization. {b} Assume that it is the policy instrument and solve for the optimal value for 3:. [Hint Substitute equation {TR} into equation {Euler}.][s there a tradeoff between stabilizing the output gap and stabilizing ination? Explain intuitively assuming that the monetary authority implements the optimal response you derive. {c} In the Taylor Rule, set st at its optimal value and set 5? = 5,, = 1']. Substitute the interest rate into the demand equation to derive a system in two variables and their expectations. {d} 'What are the requirements on the roots of the dynamic system for there to be a unique [d] 'What are the requirements on the roots of the dynamic system for there to be a unique equilibrium? {e} Explain, without solving, how you would choose parameters in the Taylor Rule, rim and y to assure a unique equilibrium