For practice, Economics 2010d not to be turned in Spring 2014

Problem Set 6 1. In one version of the New Keynesian model with Calvo or Rotemberg price rigidity, the time

paths of output and the price level can be determined from the following four equations: t t ty m p (1) t ty (2) 1 1 t t t t t tp p E (3) 1 t t tm m (4)

y is output, m is the money supply, p is the price level, is the inflation rate, and is the markup of price over marginal cost. Where relevant, assume that variables are in log deviations from steady state. The steady-state inflation rate is zero. , , and are all positive parameters. In addition, assume that (0,1) and (0,1). is a mean-zero, iid error term.

a) Consumption is one major component of output, and consumption growth depends on the real interest rate (via the consumers Euler equation). So how can this model solve for output without any explicit solution for the real interest rate? Explain the economics at work.

b) Suppose there is a 1 percent shock to at time t = 0. What are the predicted time paths of

output, the price level, and the inflation rate? If you want to solve this question analytically, it will be helpful if you assume 1. Note: Answering this question requires you to solve a second-order difference equation. Feel free to do so analytically, or numerically in Dynare for reasonable parameter values. In any case, give a tight verbal explanation.

c) Nekarda and Ramey (2010) argue that in US data one often cannot reject the hypothesis that

= 0. Suppose it is indeed true that = 0. How does this change your solution to part (b)? d) Nekarda and Ramey (2010) suggest that the lack of evidence proving > 0 is a major

challenge to New Keynesian Economics (NKE). Suppose they had instead found that is definitely much bigger than zero. Which result would be worse for NKE? Why?

For practice, Economics 2010d not to be turned in Spring 2014 2. Consider the following Dynamic New Keynesian (DNK) model.

Consumers face the problem:

1 1

0

11

Max 1 1

s.t.

1

t j t jjt

j

t t tt t t t

t t t

C HE a

B W BC H iP P P

with standard notation. Ct is a Dixit-Stiglitz aggregate of a continuum of consumption varieties Cit with elasticity of substitution > 1. are profits rebated lump-sum to consumers. B represents private (inside) debt, and is zero in equilibrium. Assume an ad-hoc money demand function:

.t tt

M YP

A firm producing output of type i has the production function: 1it itY ZH

. There is no investment, so .t tY C Due to Rotemberg frictions in changing prices, inflation follows the NKPC: 1 t t t tE , where is marginal cost. Assume 1, 0 1, 0, 0, 0, and 0.a Z

A. Derive the consumers static and dynamic first-order conditions for utility maximization.

Using these conditions, express the real wage as a function of C, H, and parameters. Log-linearize this expression to solve for / .t tW P

B. Assuming initially that = 0, solve for the real marginal cost of production for each firm.

Use your result from Part A and equilibrium conditions to express t as a function of .tY C. Assume that 0, 0, and 0. Suppose there is a 1% permanent increase in M.

Explain why this shock has real effects on output in the short run but has no real effect in the long run in this model.

For practice, Economics 2010d not to be turned in Spring 2014 D. Now allow for > 0. Consider the same 1% permanent increase in M. Is there any value of

that would make output, Y, rise by 1% both on impact and permanently, given the admissible ranges for the other parameters? That is, would some value of make money non-neutral in the long run in this model? Be sure to comment on how the values of and affect your answer. Is there any possible value of if 1? (Hint: Assume the hypothesized resultthat money has a one-for-one, permanent effect on outputand see if you can derive a contradiction.)

E. What is the relationship between your answer to part D, above, and your answers to question

2, parts c and d? Comment on the implications for big-picture issues in macroeconomics.