Due March 10 Economics 2010d Spring 2014

Problem Set 5 1. We have studied flexible-price models without paying attention to the zero lower bound

(ZLB) on nominal interest rates. One often hears that the ZLB is irrelevant if prices are fully flexible. This question asks you to investigate whether that conventional wisdom is correct.

a. Take a basic RBC model where the capital stock is fixed. What sort of shocks may

require the equilibrium (not steady-state!) real interest rate to be negative? b. Suppose that the equilibrium real interest rate is indeed negative, but the nominal interest

rate, i, cannot fall below zero. How is the negative real interest rate to be achieved? (Hint: Remember the Fisher equation: 1.)t t t tr i E

c. Comment on the sense in which the price level has to be fully flexible for your solution in part b to work. Distinguish between prices that can change at an arbitrarily large but finite rate and prices that can change infinitely fast. (While the model is set in discrete time, it may be helpful to think in continuous time, where there is a clear distinction between a variable that can jump and one that cannot.)

2. Take the basic 3-equation DNK model in class or in Walsh (ch. 8), and modify it to include government purchases. Assume that G is financed with lump-sum taxes. As usual, assume that government purchases are pure waste and that G is exogenous. a. Derive the log-linearized equation for fY as a function of Z and .G b. How does the presence of G change the NKIS or NKPC equations? c. Suppose there is an increase in .G How will it change ?fY d. Does your answer to part c depend on the expected persistence of the shock? e. How will the responses of Y and to the shock depend on the monetary policy rule?

Due March 10 Economics 2010d Spring 2014 3. Put the log-linearized 3-equation DNK model (without G) into Dynare. Assume the

following parameters: = 0.99, = 0.01, = 2, = 0.25. Assume that the Taylor rule depends only on inflation. Start with a value of = 2. Assume that Z follows an AR(1) process with persistence parameter 0.95. a. Find the impulse responses of , Y and X to a Z shock (in deviations from steady state). b. Experiment with different values of . What values minimize the output gap? Explain. c. Find the impulse responses to a monetary policy shock, with and without interest rate smoothing. d. Do values of < 1 lead to indeterminacy in this model?

4. Use your answer to question 2 to modify the computer model to allow for government purchases. Assume that the steady-state share of G in Y is 0.20. Assume that G follows an AR(1) process with persistence parameter 0.90. Redo 3a and 3b for shocks to G .