Economics 303Advanced Macroeconomics I Fall 2006

Problem Set 1: Due in class Tuesday, September 12, 2006

1. Consider an AR(1) process.

xt = φxt−1 + εt.

where εt+1 ∼ i.i.dN(0, σ2ε ).

(a) Represent this process as a MA(∞).

(b) Suppose the following shocks are observed:

t 0 1 2 3 4 5 6εt 0 0 1 0 0 0 0

Let x0 = 0. Compute the observed values for xt for t = 1, 2, ..., 6.

2. Consider the following ARMA(2,1) process:

yt+1 = φ1yt + φ2yt−1 + εt+1 + θ1εt

where εt+1 ∼ i.i.dN(0, σ2ε ).

(a) Map this process into a vector AR(1) representation of the form:

xt+1 = Axt + Cwt+1

with Ewt+1w′t+1 = I.

(b) Does this process have the Markov property? Explain briefly.

(c) Let φ1 = 0.5, φ2 = 0.25, θ1 = .2 and σε = 1.

Suppose the following shocks are observed:

t 0 1 2 3 4 5 6 7εt 0 0 0 1 0 0 0 0

If y0 = y1 = 0, compute the observed values for yt for t = 2, 3, 4, 5, 6, 7.

I suggest you have a computer do the work for you. If so, please attach your computercode to your write up. If you want the response of yt and not the whole state vector,multiple by [1 0 0 ...0] to pull off yt, the first element of the state vector.

3. Ljungqvist-Sargent, page 74, Exercise 2.2.

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Economics 303Advanced Macroeconomics I Fall 2006

4. Consider a country, Victimus, that is involved in a war. The enemy has a ‘neutron’ bombthat destroys people but not capital. As a result of the war, Victimus’ population is reducedbut its stock of physical capital is left unchanged. The growth rate of the population and thesaving rate are unaffected. Use the Solow growth model to answer the following questions.

(a) What is the immediate impact on total output and on output-per-person?

(b) What will the effect of the war be on the steady-state growth rate of capital, thesteady-state growth rate of output, and the steady-state growth rate of productivity?

(c) What will the effect of the war be on the growth rate of the capital-to-labor ratio, thegrowth rate of output-per-person, and the growth rate of total factor productivity onthe way to the post-war steady-state?

(d) What effect will the war have on the Golden Rule level of k?

5. Consider a world with two countries, A and B. Both countries have identical productionfunctions relating output per worker to the capital-to-labor ratio: yi,t = f(ki,t) = Akα

i,t.Both countries have the same constant savings rate, s, and depreciation rate, δ. Populationgrowth in both countries is zero. So in the absence of trade, kit, would evolve according thedifference equation:

ki,t+1 = (1− δ)ki,t + sAkαi,t, i = A, B

(a) Use the diagram with kt on the x-axis and kt+1 on the y-axis to describe the evolutionof the capital-to-labor ratio that would occur in either country in the absence of anytrade between the two. Explain briefly.

(b) Continue to assume no trade, and assume the initial capital -to-labor ratios for eachcountry are such that kA0 > kB0. On a single diagram, plot both countries’ capital-to-labor ratios kit as functions of time, t, in a way that is consistent with your answerto (a). Explain briefly.

(c) Now assume free trade in capital goods, so that new investment (but not old capital)flows to the country where its return is the highest (with profit repatriated to theinvestors later on). On the same diagram used in (b) [redraw if you need a clean one],plot the time path of the capital-to-labor ratios for both countries as a function oftime, t. Explain briefly.

(d) Let country A (the one with high capital-to-labor ratio) represent North America, andlet country B represent Sub-Saharan Africa. Does the Solow model correctly predictwhere new investment is occurring? Why or why not?

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