Journal of International Money and Finance (1992), 11, 3-16 Realistic cross-country consumption correlations in a two-country, equilibrium, business cycle model MICHAEL B. DEVEREUX,ALLAN W. GREGORY, AND GREGOR W. SMITH* Queen’s University, Kingston, Ontario, Canada K7L3N6 A well-known feature of one-good, mtilti-agent, Arrow-Debreu economies with identical, additively-separable, homothetic preferences is that the consumptions of all agents are perfectly correlated. Such economies are widely used in interpreting business cycles but seem to be inconsistent with observed cross-country correlations of aggregate consumption. This paper provides an example of a two-country real business cycle model in which preferences are not separable between consumption and labor supply. The model has a simple closed-form solution, and allows for fluctuations in labor supply in equilibrium. Moreover, it generates correlations between national consumption rates which are close to some of those observed in historical data. (JEL F41). A well-known property of Arrow-Debreu economies with one good and stationary, additively-separable preferences is that the consumption of each agent is a deterministic, increasing function of aggregate consumption. Moreover, if preferences are identical and homothetic then the consumptions of two agents are perfectly correlated (see, for example, Townsend, 1987; Brennan and Solnik, 1989; Stulz, 198 1; and Wilson, 1968).’ While preferences with these features are adopted widely, this implication of optimal risk-sharing is not evident empirically. For example, the correlation between US and Canadian quarterly private consumption from 197 I : 1- 1988 :4 (source: OECD Department of Economics and Statistics Quarter/y National Accounts) in deviations from linear trends is 0.564. Alternative transformations such as first differences produce even lower consumption correlations. Moreover, the correlation for the USA and Canada appears to be higher than that for many other pairs of countries. Evidence on cross-country correlations is given by Backus et al. ( 1989; Table 2), Tesar ( 1989; Table 1 ), and Baxter and Crucini ( 1990; Table A-6), and in Table 1 below. This paper alters the preferences commonly used in real business cycle studies in a way which may resolve the discrepancy between data and theory with respect to the cross-country consumption correlations. We do not address other shortcomings of business cycle models (see McCallum, 1989, for a survey).2 We construct a simple, *David Backus, seminar participants at Concordia and Queen’s, and two referees provided helpful comments. We thank Campbell Harvey and Tony Wirjanto for data and acknowledge the financial support of the Social Sciences and Humanities Research Council of Canada. 0261P5606/92/01 /0003- 14 ;<B 1992 Butterworth-Heinemann Ltd
Transcript
Journal of International Money and Finance (1992), 11, 3-16
Realistic cross-country consumption correlations in a two-country, equilibrium, business cycle model
MICHAEL B. DEVEREUX, ALLAN W. GREGORY, AND GREGOR W. SMITH*
A well-known feature of one-good, mtilti-agent, Arrow-Debreu economies with identical, additively-separable, homothetic preferences is that the consumptions of all agents are perfectly correlated. Such economies are widely used in interpreting business cycles but seem to be inconsistent with observed cross-country correlations of aggregate consumption. This paper provides an example of a two-country real business cycle model in which preferences are not separable between consumption and labor supply. The model has a simple closed-form solution, and allows for fluctuations in labor supply in equilibrium. Moreover, it generates correlations between national consumption rates which are close to some of those observed in historical data. (JEL F41).
A well-known property of Arrow-Debreu economies with one good and stationary, additively-separable preferences is that the consumption of each agent is a deterministic, increasing function of aggregate consumption. Moreover, if preferences are identical and homothetic then the consumptions of two agents are perfectly correlated (see, for example, Townsend, 1987; Brennan and Solnik, 1989; Stulz, 198 1; and Wilson, 1968).’ While preferences with these features are adopted widely, this implication of optimal risk-sharing is not evident empirically. For example, the correlation between US and Canadian quarterly private consumption from 197 I : 1 - 1988 :4 (source: OECD Department of Economics and Statistics Quarter/y
National Accounts) in deviations from linear trends is 0.564. Alternative transformations such as first differences produce even lower consumption correlations. Moreover, the correlation for the USA and Canada appears to be higher than that for many other pairs of countries. Evidence on cross-country correlations is given by Backus et al.
( 1989; Table 2), Tesar ( 1989; Table 1 ), and Baxter and Crucini ( 1990; Table A-6), and in Table 1 below.
This paper alters the preferences commonly used in real business cycle studies in a way which may resolve the discrepancy between data and theory with respect to the cross-country consumption correlations. We do not address other shortcomings of business cycle models (see McCallum, 1989, for a survey).2 We construct a simple,
*David Backus, seminar participants at Concordia and Queen’s, and two referees provided helpful comments. We thank Campbell Harvey and Tony Wirjanto for data and acknowledge the financial support
of the Social Sciences and Humanities Research Council of Canada.
Canada 0.565 France 0.701 0.773 Germany 0.276 0.743 0.760 Japan 0.556 0.561 0.782 0.665 Sweden 0.300 0.767 0.494 0.458 0.299 UK -0.331 0.003 -0.338 - 0.080 0.009 0.298
USA -0.136 0.441 0.069 0.315 0.195 0.417 0.517
Data are quarterly, seasonally adjusted, total private consumption expenditures in constant prices of 1985, for 1971:l to 1988:4. The first observation is lost in calculating differences. Source: OECD Quarter/y National Accounts.
MICHAEL B. DEVEREUX et al. 5
two-country, model economy in which preferences exhibit a particular non- separability between consumption and labor supply. To make the argument as transparent as possible we study a model which has a closed-form, analytical solution as in Long and Plosser (1983), Dellas (1986), and Cantor and Mark (1988). In contrast to those papers the model allows for fluctuations in labor supply in equilibrium. In contrast to Backus et al. (1989), the form of the non-separability between consumption and labor supply generates correlations between national consumption rates which are close to some of those observed in data even with little curvature in the period utility function.
Our results depend on the assumption that labor is internationally immobile, as opposed to the perfect and costless mobility of goods across borders. The immobility of labor seems to us a more realistic assumption than the opposite case. Alternative frictions in the environment also could account for observed, low, consumption correlations. For example, proportional shipping costs drive a wedge between consumption rates in Dumas’s (1988) one-good, two-country economy. Other explanations might involve capital controls, incomplete international asset markets (as studied by Kollman, 199 1, for example), measurement error in consumption rates, or non-traded goods (see Backus and Smith, 1991).
Section I of the paper describes the model economy. We focus on a technology with 100 per cent depreciation; this allows analytical solutions but (as Appendix B shows) is not necessary for the results on consumption comovements across countries. Section II discusses the empirical implications of the model and considers a simple method for gauging the closeness of the model’s consumption correlation to that in data. While the model has some obvious deficiencies, its consumption correlation is close to some of those observed (quarterly) for OECD countries from 1971 to 1988. Section III considers some extensions. Section IV concludes the paper.
I. A two-country economy
We develop a two-country model of a world economy in which there is a common good produced in each country. The countries follow identical production techniques but each national technique is subjected to independent, country-specific, productivity shocks. Let the countries be called ‘home’ and ‘foreign’ for concreteness, and denote all foreign variables with an asterisk. The model is characterized through the following series of assumptions:
Assumption I. Preferences of the home and foreign countries are given by
(1)
(2)
Et 1 B’u(c,, n,); u = log(c, - ynf); t=o
m
-6 c P’u*(c:,$); u* = log(cF - yn?“); r=o
where p ~(0,1), p > 1, c, is home consumption, and n, is home employment. The stochastic environment will be described below. The special feature of these preferences is that the income elasticity of leisure is zero. Note in particular that the marginal utility of consumption is not independent of labor supply. A similar representation of preferences is adopted by Greenwood et al. (1988).
6 Cross-country consumption correlations
Assumption 2. Technologies are Cobb-Douglas:
(3) y, = Q,K;n’l-a’ f 3
(4) y: = (j:jq+-$"-"',
where CI E (0, 1 ), yr, and y: are output levels for each country, K, and K: are capital stocks, and d1 and 0: are productivity coefficients, identically and independently distributed across both time and countries. Each shock has a mean of unity, a constant variance g2, and takes only positive values with positive probability.
Assumption 3. The rate of physical depreciation in the capital stock of each country is unity. Thus I, + 1 = K, + I and IT+ r = K:+ 1. This assumption is required for exact, closed-form analytical solutions for policy functions in the program outlined below. A similar assumption is made, for identical reasons, by Long and Plosser ( 1983) and Cantor and Mark (1988). While this assumption is useful for illustrative purposes, it is in no way required for the main point of the paper, as Appendix B shows.
It is assumed that there is a unified world capital market whereby households from either country can trade in goods and assets, but that labor is immobile across national boundaries. One way to approach the problem would be to define and derive a recursive competitive equilibrium for the world economy. However, since a world competitive equilibrium with complete markets is obviously Pareto efficient, we can exploit the equivalence between a competitive equilibrium and a social planning optimum by solving instead a social planning problem in which a weighted sum of national utilities is maximized subject to technologies and aggregate resource constraints. By the results of Negishi (1960) and Mantel (1971), there exist weights such that the allocations that solve the social planning problem for these weights are identical to those of a competitive equilibrium for a given set of initial endowments. This approach avoids the detailed specification of trading institutions involved in solving directly for a competitive equilibrium. It is the exact analogue for the two-country model of the social planning problem solved by Long and Plosser ( 1983) for a closed, linear logarithmic economy.
The state vector for the planning problem is s, = (K,, K:, O,, O:), which is Markov with transition probability density function ,f(s,+ 1 Is,). For a random variable x, adapted to the information generated by {s,: z = 0, 1,2,. , t} let E,x,+ 1 denote
Sx,+l(St+l)f(Sl+llSt)dst+l. The social planner then faces the following problem:
(Pl) Choose (c,, c:, n,, PI;“, K,,,, K,*,,j to maximize
E, i: FCu(c,, n,) + u*(c:, $)I, t=o
subject to
c, + c: + K,,, + K,*,, = O,K:n:-” + 8~Kj%j+-“,
4 _ i.i.d.( 1, (r2), 19: - i.i.d.( 1, a’),
K, = K,* given.
Since the model is entirely symmetric and the initial capital stocks of each country are assumed equal, the weights in the social planner’s objective function are equal.
Appendix A derives the solution for optimal consumption, hours worked and capital stocks. Given Assumption 1 the marginal rate of substitution between consumption and labor supply must be independent of consumption. Equality between (the negative of) this marginal rate of substitution and the marginal product of labor in each country must then imply that hours are determined by the conditions:
These give
(6)
(7)
n, = [~tl,K~]l’lm,
n* = (l - a) &$Kjw l’lJJ I
[ PLY 1 ’ where w = p - (1 - N) > 0. Conditional on a given capital stock, hours worked for each country respond positively to current domestic productivity shocks, with an elasticity of l/o, but do not depend on productivity shocks in the other country. Thus Assumption 1 allows for intratemporal response of labor supply to productivity shocks, but no intertemporal response, in the sense that real interest rate movements will have no affect on hours worked. Thus the model does not imply (as one with time-separable preferences would) that a shock which leads to postponement of leisure also acts to reduce consumption (see Barro and King, 1984). Note that as ,D becomes very large the elasticity of labor supply tends to zero. In that case, by equations (6) and (7) the level of labor supplied tends to unity, and the model reduces to the simple log linear growth model with a fixed labor supply.
It is clear given equation ( 5) that an optimal program will entail marginal utilities of consumption being equalized across countries. This implies that (c, - Yrzf) = (CT - yn:“) must hold in equilibrium. However, given y # 0, consumption is not equalized across countries.
Substituting labor supply from equations (6) and (7) into the home and foreign production functions, we derive the output levels for each country,
(8) y, = v(&K;1)P’“,
(9) y: = v(Bj”K,*“)P’“,
where v = [( 1 - a)/,u~y] (l -a)io Given p/o > 1, the elasticity of output with respect . to an innovation in the current productivity coefficient is greater than one. The reason is that hours worked can respond immediately to a productivity shock. Because labor supply is determined by the atemporal conditions (6) and (7), however, a productivity shock in one country in a given period has no effect within that period on production in the other country. This effect can occur only through time as investment responds to the productivity disturbance.
Given these solutions for output levels, it is easy to manipulate the first-order conditions (see Appendix A) to arrive at the following ‘portfolio-type’ characterization
8 Cross-country consumption correlutions
of the optimal choice of capital stocks across countries:
(10) E, i d&+, K~+l)~!“‘;(Kt+l[\‘(OrK;I+,)~i’” + \(O:K,*,“,)“‘“- K,+z - K1*f2]);
Since 0, and 0: are i.i.d., it is clear that K,,, = K,*, 1 solves this expression.3 Manipulating further gives the optimal policy rule for the capital stock
which gives the familiar Brock Mirman law of motion as ,LL tends to infinity. The capital stock in each country depends on the sum of a convex function of the two productivity coefficients. Productivity shocks in either country contribute equally to investment in each country for the next period. Since pcx/tu < 1, it is clear that the capital stock will converge to a stationary distribution.
Now using the solution for the capital stock, ( II), the fact that the effective consumption index I’, - yn; is equated across countries, and the solution for hours worked in each country, (6) and (7), we may obtain the consumptions of the home and foreign countries:
The first term is common to each of these equations. It is the value of effective consumption that optimal risk-sharing equates across countries. It depends upon the SUWI of a function of each of the productivity outcomes 8, and 0:. The second term, however, is country-specific, and captures the fact that domestic hours worked and so domestic consumption will respond directly to the domestic productivity shock. The presence of this second term implies that consumption rates are not perfectly correlated across countries.
The conditional (given K,) correlation coefficient between period t consumption rates, based on period t ~ 1 information, is
(14) &1((‘,,(‘,*) = l/(1 + ry2!(2(11 + ;)i)),
where < = $( w//L)( 1 - (px~;‘to)) and ‘1 = ;,W’ -“‘. Using the definition of V, it can be seen that lim (p + 1~ )rl = 0, so that for large values of /L the conditional correlation is near unity. In general, however, it is always below one. In the numerical analysis below, we compute the unconditional correlation and its sampling variability by simulating the model with sequences of shocks.
II. Empirical implications
Given parameter values and sequences of shocks, the model generates time paths for domestic and foreign output, consumption, labor supply, savings, investment, and the trade balance. Thus the predictions of the model can be compared with historical evidence on such facts as the cyclicality of the trade balance or the correlation between domestic savings and investment. Since our interest is in the consumption correlation implied by this model, it seems natural to focus on this moment by first parameterizing the model and then determining both the population consumption correlation and
MICHAEL B. DEVEREUX et al. 9
the sampling distribution for the corresponding sample moment. The latter provides useful information in measuring the match between data and model in this dimension. We set parameter values as follows: the discount factor, p = 0.95; y = 1.00; p = 1.58; the share of capital, CY = 0.30; so that o = 0.88. For these parameter values the population conditional correlation coefficient is pt 1 (c,, cr) = 0.445, from equation
(14). These numbers are similar to those of other studies. For example, the intertemporal
elasticity of substitution in labor supply is approximately (p - l)- ‘. If p = 1.58 then this elasticity is 1.7, which is the value used by Greenwood et al. (1988) in a closed-economy business cycle model. We could estimate some of the parameters using a simulation estimator as studied by Gregory and Smith (1990) and Lee and Ingram ( 199 1).
The shocks are drawn independently from a common density such that 0 = exp(x) with x - n.i.d.( -0.0002, 0.022); thus 8 is log normally distributed with mean unity. The shocks thus have the same distribution and moments as that of the unconditional shock density in Kydland and Prescott (1982) and Backus et al. (1989), based on US Solow residuals.4 Initial conditions for the two capital stocks are set arbitrarily and then 50 initial observations are discarded to clear the initial conditions5
We generate 1000 replications of 71 observations (the same number as in the historical data summarized in Table 1) of the model. For each replication we calculate the sample correlation between consumptions. Table 2 gives the population moment (obtained by letting the sample size become arbitrarily large until convergence is achieved) and the approximate 95 per cent confidence interval given by the quantiles of the empirical density function. Figure 1 shows the histogram of the simulated correlations, along with a normal density on the same scale.
Table 2 also presents the observed sample correlation between US and Canadian private consumption expenditures, quarterly from 1971 to 1988 and thus also based on 71 observations. We quote sample correlations found after using several different detrending methods. In each case, a p-value gives the estimated probability of finding a correlation less than or equal to the corresponding sample correlation. The p-value is simply the proportion of replications in which the simulated correlation coefficient is less than the historical value. Thus we treat these sample moments as critical values in testing the business-cycle model, using as a metric the variability in the model itself. This method of testing is outlined in greater detail by Gregory and Smith ( 1991). The second column in Table 2 gives the average consumption correlation (for each detrending method) for all pairs ofcountries from the following list: Australia, Canada, France, Germany, Japan, Sweden, UK, and USA. Then the third and fourth columns give the maximum and minimum correlations. The complete correlation matrices are given in Table 1.
Table 2 and Figure 1 show that the population correlation coefficient between consumptions in the model is well below one. They also show that the sampling variability (based on that of Solow residuals, as in other business cycle models) of this moment is sufficient to reconcile the model with some observed, historical correlations such as that between detrended US and Canadian consumption. However, the average consumption correlation for the eight countries lies outside the model’s 95 per cent confidence interval. This result does not depend on the method of detrending, although other methods also could be examined (as in Canova and Dellas, 1990). We refer to the generated correlations as realistic because the model also cannot generate correlations as large as some of those (shown in the third column
10 Cross-country consumption correlations
TABLE 2. Consumption correlations-f
Theory:
Historical data:
Population moment (95 per cent); (min, max) 0.586 (0.39, 0.70); (0.260, 0.791) Sample moments (p-value)
t The population unconditional correlation is calculated from one replication with 5000 observations. The
confidence interval, maximum and minimum, and p-values are based on 1000 replications. Correlations
use real quarterly private consumption: 1985 = 100, s.a., 1971:l to 1988:4. OECD Department of Economics and Statistics Quarterly National Accounts; comparative tables. We lose the first observation in
differencing, leaving calculations for 1971:2 to 1988:4, i.e., 71 observations. With R reolications, standard
errors for p-values are given by Jp(l--p)lR.
1.0
0.9
0.8
.- 0.7 5
6 0.4 ._ r E e 03 P.
0.2
0.1
80
'; 60 2
u
0.22 0.38 0.54 0.70
Correlation
FIGURE 1. Sample correlations.
0.86
MICHAEL B. DEVEREUX et al. 11
of Table 2) in the data and because adding realistic depreciation rates and persistence in shocks would lower the generated correlations (see below).
The sensitivity of the conditional correlations to the parameter settings can be studied using equation ( 14). Both conditional and unconditional correlations are positively related to the value of p; the lower is p the more responsive is labor supply to a country-specific productivity shock and hence (because of non-separability) the larger is the country-specific effect on consumption and the lower the correlation. As one varies the intertemporal elasticity of labor supply from 0.2 to 2.5 (a range suggested by empirical studies) p takes on values from 6 to 1.4 and the population unconditional correlation ranges from 0.953 to 0.543. The results (conditional and unconditional) are not sensitive to the values of 0, a, y, and rr. While the population correlation is inversely related to cr (the standard deviation of the shock) varying 0 across a very wide range of values affects the unconditional correlation only in the second decimal place and likewise has little effect upon the sampling variability.
The population moment of 0.586 can be compared to the value of 0.78 found by Backus et al. (1989), who extend the equilibrium business cycle model of Kydland and Prescott (1982) to a two-country setting. They use power utility of the form u(c,, n,) = [cy( 1 - n,)‘-“lK/~ with an exponent K = - 1 (a coefficient of relative risk aversion of 1 - IC = 2). Measuring (1 - n,) as a distributed lag in leisure has little effect on the correlation, while introducing a one-quarter time to build lag in the technology lowers it to 0.52. They do not report the sampling variability of these moments. For IC = 0 the consumption correlation is one, since then preferences are additive in logs, and a planner will actually equate consumption rates across countries. The correlation is less than one if the curvature parameter differs from zero. For example, they find that raising IC to 5 lowers the consumption correlation to 0.56; but this is perhaps an unreasonable degree of curvature. In contrast to that of Backus et al., our model implies that even with little curvature (an exponent of zero in our case), the theoretical consumption correlation may be quite low. Given that an optimal program equalizes marginal utilities of consumption across countries, the preference structure adopted by those authors (for reasonable curvature values) comes very close to equalizing consumption rates. In our model, as we have shown above, there is no such implication.
Finally, we examine the effects on the correlations for Canada and the USA of measuring consumption without seasonal adjustment, of excluding expenditure on durables, and of measuring consumption in per capita terms. For Canada the non-seasonally-adjusted data are quarterly personal expenditure in 198 1 prices (D10131), quarterly personal expenditure on durables in 1981 prices (DlOl32), and population (Dl), where the brackets contain numbers from the CANSIM base of Statistics Canada. The corresponding non-seasonally adjusted US data originate in the Bureau of Economic Analysis of the Department of Commerce and are those
used in Ferson and Harvey ( 1987). These measure nominal consumption expenditures on durables, nondurables, and services, and are deflated by the appropriate CPI.
With the same four detrending methods, the correlations are as follows: total consumption expenditures, not seasonally adjusted (0.939, 0.657, 0.929, 0.601); consumption expenditures excluding durables, not seasonally adjusted (0.946, 0.744, 0.940, 0.662); total consumption expenditures, not seasonally adjusted, per capita (0.938,0.694,0.929,0.654). Excluding durables and deflating by population have little effect on the correlations. Using unadjusted data has a large effect since the seasonal patterns are correlated across countries. In theory the model’s predictions apply to
12 Cross-country consumption correlations
risk-sharing at all frequencies but the business-cycle frequencies seem most appropriate (and also stringent) for testing since we do not have a complete seasonal model.
III. Extensions
The practical advantage of studying a business cycle model with the preferences in Assumption 1 is that closed-form solutions can be found. One disadvantage is that the model cannot be used to study risk-sharing on steady-state growth paths. For the analysis of steady-state growth the intertemporal elasticity of substitution in consumption must be independent of the scale of consumption, and income and substitution effects of productivity growth on labor supply must be offsetting, since hours cannot grow. King et ul. (1988a) describe the restrictions on preferences necessary for the study of steady-state growth. The preferences described in Assumption 1 do not satisfy these restrictions. Hercowitz and Sampson (1991) suggest modifying the period utility function in equation ( 1) to
(15) u = log(c, - yh,nf),
in which h, is an index of knowledge which adds to the productivity of home or leisure activities. This amendment removes the income effect of wage increases on labor supply. A positive productivity shock then raises the real wage but there can be a parallel increase in the productivity of leisure time. Thus steady-state growth is possible if the index h, grows, even though hours are bounded. Hercowitz and Sampson examine some of the empirical predictions of a closed-economy growth model with the preferences in equation (15). In another model which is consistent with steady-state growth, Leme (1984) has proposed some utility-based measures of capital market integration which likewise involve both secular and cyclical components of aggregate, national consumptions.
Our aim has been to demonstrate the point concerning consumption correlations rather than to construct a complete numerical business-cycle model. For this reason we have not studied the correspondence with data for the other moments of the model (for instance, the autocorrelation coefficient ofconsumption or of labor supply). However, the same non-separable preferences have been used by Greenwood et al.
( 1988) in a numerical one-country business cycle model. They allow for realistic depreciation and persistence in shocks, as well as adopting a different production technology. The model has some empirical success in reproducing the relative variabilities and autocorrelations in output, consumption, and investment of the postwar US economy. As a check on the accuracy of our model under more general specifications, in Appendix B we solve the model under the assumption of less than 100 per cent depreciation rates. For a depreciation rate of 10 per cent the unconditional correlation coefficient for consumption across countries is calculated to be 0.552, which again is comparable to the historical values in Table I.
What effect would the introduction of realistic persistence and cross-country correlation in the shocks have on the results? Backus et al. estimate the parameters of a bivariate first-order process for Solow residuals, using the USA and a group of other countries. Their estimates indicate a very high degree of persistence in the shocks within a country, but a low cross-country correlation of the shocks. Introducing these persistence and cross-country correlation effects in our model would reduce the implied consumption correlation. With persistence and low international correlation of shocks, a shock in the home country, for example, would lead not only to a rise
MICHAEL B. DEVEREUX et al. 13
in labor supply in the home country, but also to a shift of world investment away from the foreign country towards the home country. This would magnify the effects of the shock on the country-specific component of consumption, reducing the theoretical correlation coefficient.
IV. Conclusion
Accounting for observed, cross-country correlations in aggregate consumption can be problematic for standard Arrow-Debreu models with additively separable period utility. We study a simple, two-country example with non-separability between consumption and leisure of a form which has been used with some success in modelling business cycles and growth. The example can be solved analytically. Moreover, it generates correlations between national consumption rates which are comparable to some of those observed in historical data for OECD countries.
Appendix A: Solution of the social planning problem
Given the solutions (6) and (7) of the text, the first-order conditions for the dynamic programming problem (5) can be obtained by substituting the aggregate resource constraint into the expression u(c,, n,) (alternatively, (6) and (7) come out of the first-order conditions of the full problem Pl). This leaves three control variables to be chosen at any date: c:, K,, and K:. First-order conditions are given by:
The full solutions are then obtained by substituting for n, and n:. (Al ) simply gives the condition of equal marginal utilities of consumption in an optimal program. (A2) and (A3), when combined with (A4) and (A5 ) updated by one period, give condition ( 10) ofthe text.
Appendix B: Realistic depreciation
In this Appendix we construct a version of the model which does not rely upon the artificial construct of 100 per cent depreciation rates per period. We show that the main argument of the paper-that the predicted consumption correlation coefficient can be realistically low-remains unchanged with this alteration.
The mode1 with less than 100 per cent depreciation cannot be solved analytically. To obtain solutions we linearize the first-order conditions of the social planning problem around the deterministic steady state as in King et al. ( 1988b). Population moments can easily be computed from the resulting linear stochastic difference equations.
Maintain assumptions I and 2, but instead of assumption 3 let investment be given by
(Bl) I - K,+, *+1 - -(l-6)K,andIf+,=K:+,-(l-6)Kf,
with 6 l (0,l). The appropriate resource constraint is then
= E, { d( F+ 1 KY, Pi K++ , - (1 - h)/{[L’(fl,K:+,)‘~” + v(o:K::,)‘:“‘- I,,, - I;c+,]}).
For the same reasons as in the specification in the text of the paper, the solution implies identical capital stocks across countries. Then taking the Euler equation for optimal consumption in either country and linearizing around the deterministic steady state, we may derive the following solution difference equation, where r?, denotes the deviation of the time t variable X from its steady-state level:
where i. is the stable root of the second-order difference equation (B4). Using the same approach as in the text, it is then easy to demonstrate that consumption in each country may be written (in terms of deviations from the deterministic steady state) as
(B6) c*, = aI?, + bRt+, + CO, + do:,
<Bl) e,* = aI?, + bR,+, + CO: + do,,
where a, b, c, and d are functions of the parameters. Using the solution (B5) the population correlation coefficient for consumption across countries can be computed readily. Given the assumption that the technology shocks are drawn from the same distribution across countries this does not depend upon the moments of the 0 process. With the parameter settings in the text and the additional assumption that 6 = 0.1, (10 per cent depreciation) the unconditional population correlation coefficient, p(c,, CT), is computed to be 0.552.
Notes
1. Strictly speaking one consumption is a monotone increasing and deterministic function of the other. Since this function need not be linear the correlation coefficient need not be one. For identical, homothetic preferences it is exactly one and for simplicity we refer to it as taking that value.
2. Other quantitative studies of two-country, real, business cycle models include those of Mendoza ( 1989), Backus et al. ( 1989), and Baxter and Crucini ( 1990), which focus on such issues as income correlations, comparisons with autarky, current account dynamics, and correlations between savings and investment. Canova and Dellas (1990) extend the log-linear model of Dellas (1986) to include two goods and specialized production. They study trade flows and their relation to comparative output dynamics across countries.
3. We could have assumed more generally, as in Cantor and Mark (1988) that the 0, and 0: processes are only i.i.d. across time, and not across the two countries. Then it is easy to show that condition (10) gives an implicit equation in the share of world investment carried out by the home and foreign countries, where the special case we have chosen
MICHAEL B. DEVEREUX et al. 15
gives a share of one half. However, in the more general case there is no exact solution for the share, and we require the exact solution below.
4. One could also adopt the moments of the country-specific shocks identified by Costello (1989).
5. Simulations are done in RATS386TM. All data and simulation codes used in the paper are available from the authors.
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