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NOTE: Please only include Student ID number (no name or unikey) on your submitted answers. To receive full marks, show your workings for algebraic manipulations.

1. Consider an economy where consumers maximize their utility, without uncertainty:

subject to the following budget constraint:

ct is the consumption in period t,

bt represents one-period nominal bonds,

it−1 is the nominal interest rate determined in period t − 1,

β is the discount factor,

wt is the real wage rate,

Nt is labor supply,

kt is capital,

rt−1 is the return on capital, determined at t − 1

πt is inflation

(a) Does bt represent a real or a nominal amount?

(b) Show that the households’ optimality conditions imply the Fisher equation and interpret it.

(c) Assume that the economy is subject to a shock to β - in particular, it increases tem porarily. Also, assume that the production structure is such that there are nominal price rigidities. Explain why a sudden increase in β acts both as a shock that affects aggregate demand and aggregate supply. (this is an essay question, no math is required)

(d) Propose a modification to the supply side of the model that can help you ensure that a temporary increase in β leads to a decline in inflation.

2. (Gali 2015, Chapter 4) Consider the following version of the three-equation New-Keynesian model with a classical Taylor rule.

πt and yt represent respectively inflation and the output gap, and both values are zero in steady state. rt is the real interest and rt n is the natural interest rate., which is exogenous and time-varying. All the parameters are positive.

(d) Find a solution to the system above

(e) The system above is similar, yet different, from the representation we had in lecture 8 (Dynare). Using the fact that the eigenvalues of the inverse matrix are equal to the inverse of the eigenvalues of the original matrix, re-state the Blanchard and Khan conditions with the system above in terms of the eigenvalues of matrix A.

(f) For which ranges of ϕπ the model admits a unique solution around the steady state? To answer this question, you will have to use the following fact: define f(λ) = λ 2+a1λ+a0. The function f(λ) has its two roots within the unit circle provided that (i) |a0| < 1 and (ii) |a1| < 1 + a0.

(g) Interpret this result.