Economics Assignment

Q1. The following questions aim to help you understand the impulse responses in Romer Section 5.7, which we also discussed in class. For the following questions, you can use any results that were derived in Section 5.3 and 5.4 for the Baseline Real Business Cycle Model. You can assume that A^~₀ = 1, A^~_t+1 = ρ_AA^~_t, and ρ_A = .95.

(a) Prove that L₀ > L^-.

(b) Prove that L₀ > L₁.

(c) Explain in words two reasons why L₁ decreases from L₀.

(d) Explain why the wage w is still far above its steady state even after the technology shock has decayed substantially.

(e) Building on your previous answers, what are two parameters we could change such that the wage would approach its steady state faster? Be sure to mention whether we would increase or decrease these parameters.

Q2. Consider the IS-LM model from class and Section 6.1 in Romer.

(a) The IS curve (6.8) is a relationship between output and the real interest rate and the LM curve (6.10) is a relationship between output and the nominal interest rate. Under what additional assumption are nominal and real interest rates the same?

(b) Suppose (just for this part) that instead of i = r we had i = π^e + r for π^e > 0. How would this cause the IS or LM curve to shift (if at all) relative to the baseline case (when i = r)? You can assume that no other variables (M_t, P_t, or Y_t+1) are affected.

(c) Suppose (just for this part) that Y_t = C_t + G_t, for G_t > 0. How does this cause the IS or LM curve to shift (if at all), relative to the baseline case (when Y_t = C_t)?

(d) Suppose the parameter θ decreases. What happens to the IS and LM curve (if there is any effect)? What effect does this have on the equilibrium real interest rate?

(e) Suppose the parameter v increases. What happens to the IS and LM curve (if there is any effect)? What effect does this have on the equilibrium real interest rate?

Q3. Models with Phillips Curves

For part (a), (b), and (c), consider the IS-MP, AD-AS model from Section 6.4. As in Q2, sometimes the answer may be that there is no effect. Assume the relevant Phillips Curve is given by equation 6.24.

(a) Suppose the monetary authority sets higher interest rates for all levels of output. How does this affect equilibrium interest rates, output, and inflation? You can assume for now there is no change in expected inflation.

(b) Suppose that in addition to the scenario in part (a), inflation expectations move in the same direction as inflation did in part (a). How does this additional development affect equilibrium interest rates, output, and inflation relative to part (a)?

(c) Show the equilibrium changes relative to the baseline model when θ increases.

(d) What is the most important difference between the micro-founded New Keynesian Phillips Curve derived in Section 7.4 and the traditional Phillips Curve from Section 6.4?

(e) Building on your answer to part (d), explain why this major difference is important for monetary policy.

Q4. Intermediate steps of the Calvo model

(a) On p. 317, Romer notes that the wage W_t is a constant times the price P_t. What is this constant as a function of exogenous parameters in the model?

(b) What simplifying assumption concerning consumer behavior ensures that the ratio P_t/W_t is constant in this model? (There is a straightforward answer and only this answer will be accepted.)

(c) Briefly explain why the solution we derive for p_t is only relevant when inflation is small, β is 1, and the economy is near its flexible price equilibrium. What step of our derivation would break down otherwise?

(d) Write an expression for the following derivative relevant for Section 7.1:

d/dp_{i t=0}Σ^∞q_t(p_i - p*_t)².

(e) Simplify as much as possible the following, given that qk = (1 - α)^k.

_k=0Σ^∞β^kq_k.