Figure 5.3: Hodrick-Prescott Þltered cyclical observations from the model. Investment has been shifted down by 0.10 for visual clarity.

This coarse overview of the one sector real business cycle model shows that there are some aspects of the data that the model does not explain. This is not surprising. Perhaps it is more surprising is how well it actually does in generating ‘realistic’ time series dynamics of the data. In any event, this perfect markets—no nominal rigidities ArrowDebreu model serves as a useful benchmark against which reÞnements can be judged.

5.2Calibrating a Two-Country Model

We now add a second country. This two-country model is a special case of Backus et. al. [5]. Each county produces the same good so we will not be able to study terms of trade or real exchange rate issues. The presence of country-speciÞc idiosyncratic shocks give an incentive to individuals in the two countries to trade as a means to insure each

Notes: ext is US net exports divided by GDP. Foreign country aggregates data from France, Germany, Italy, and the UK. All variables are real per capita from 1973.1 to 1996.4 and have been passed through the Hodrick—Prescott Þlter with λ = 1600.

the same capital shares and Cobb-Douglas form of the production function, and identical utility. Let the social planner attach a weight of ω to the domestic agent and a weight of 1 −ω to the foreign agent. In terms of e ciency units, the social planner’s problem is now to maximize

In the market economy interpretation, we can view ω to indicate the size of the home country in the world economy. (5.28) and (5.29) are the

152 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES

Cobb—Douglas production functions for the home and foreign counties, with normalized labor input N = N = 1. (5.30) and (5.31) are the domestic and foreign capital accumulation equations, and (5.31) is the new form of the resource constraint. Both countries have the same technology but are subject to heterogeneous transient shocks to total productivity according to

2.40e−4, and Σ12 = Σ21 = 6.17e−5. The contemporaneous correlation of the innovations is 0.26.

Apart from the objective function, the main di erence between the two-county and one-country models is the resource constraint (5.32). World output can either be consumed or saved but a country’s net sav-

cate the dependence of consumption on the state by ct = g(λt), and

For future reference, the derivatives of g and h are,

g1 = g2 = −γ,

g3 = fk(A, k) + (1 − δ), g4 = fk(A , k ) + (1 − δ),

g5 = f(A, k)/A, g6 = f(A , k )/A ,

g7 = −1,

h1 = h2 = · · · = h6 = 0, h7 = 1.

Next, transform the constrained optimization problem into an unconstrained problem by substituting (5.34) and (5.35) into (5.27). The problem is now to maximize

At date t, the choice variables available to the planner are kt+1, kt+1, and ct . Di erentiating (5.36) with respect to these variables and rearranging results in the Euler equations

(5.39) is the Pareto—Optimal risk sharing rule which sets home marginal utility proportional to foreign marginal utility. Under log utility, home and foreign per capita consumption are perfectly correlated, ct = [ω/(1 − ω)]ct .

The Two-Country Steady State

From (5.37) and (5.38) we obtain y/k = y /k = (γ/β+δ−1)/α. We’ve already determined that c = [ω/(1 − ω)]c = ωcw where cw = c + c is world consumption. From the production functions (5.28)—(5.29) we get k = (y/k)1/(α−1) and k = (y /k )1/(α−1). From (5.30)—(5.31) we get i = i = (γ + δ − 1)k. It follows that c = ωcw = ω[y + y −(i + i )] = 2ω[y − i].

Thus y − c − i = (1 − 2ω)(y − i) and unless ω = 1/2, the current account will not be balanced in the steady state. If ω > 1/2 the home country spends in excess of GDP and runs a current account deÞcit. How can this be? In the market (competitive equilibrium) interpretation, the excess absorption is Þnanced by interest income earned on past lending to the foreign country. Foreigners need to produce in excess of their consumption and investment to service the debt. In a sense, they have ‘over-invested’ in physical capital.

In the planning problem, the social planner simply takes away some of the foreign output and gives it to domestic agents. Due to the

You can write second—order stochastic di erence equation (5.51) as

(1 −m1z −m2z ) = (1 −ω1L)(1 −ω2L) satisfy m1 = ω1 + ω2 and m2 = −ω1ω2. Under the parameter values used to calibrate the model and us-

the unit circle, and the unstable root |z2| < 1 lies inside the unit circle. From the law of motion governing the technology shocks (5.33), you

have

these forecasting formulae to solve the unstable root forwards and the stable root backwards. The solution for the world capital stock is

Now you need to recover the domestic and foreign components of

The date t+1 world capital stock is predetermined at date t. How that

capital is allocated between the home and foreign country depends on

˜ ˜ the realization of the idiosyncratic shocks At+1 and At+1.

˜ ˜

Given kt, and kt , it follows from the production functions that the