Trade in Capital Goods and International Co-movements of Macroeconomic Variables

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1 Open Econ Rev (2009) 20: DOI /s Trade in Capital Goods and International Co-movements of Macroeconomic Variables Koichi Yoshimine Thomas P. Barbiero Published online: 23 May 2007 Springer Science + Business Media, LLC 2007 Abstract This paper contributes to the ongoing debate on the causes of international co-movements of macroeconomic variables. In particular, existing real business cycle models predict cross-country consumption correlations to be higher than in the actual data, cross-country output correlations to be lower than in the actual data, and cross-country consumption correlations to be relatively higher than the output correlations. We show that cross-country correlations of consumption, investment, employment and output predicted by a standard international real business cycle model are highly sensitive to the share of capital goods in total trade. Our calibrated model shows that when capital goods account for a share of total traded goods greater than 50%, the apparent discrepancy between the data and the simulations is resolved. Keywords International real business cycles Capital goods Cross-country correlations JEL Classification E32 F21 F41 The standard international real business cycle models à la Backus et al. (1995) have difficulty accounting for empirically observed cross-country comovements of macroeconomic aggregates. In such models, predicted crosscountry consumption correlations are higher, and output correlations are lower than what the available data show. Generally, data for the US and other K. Yoshimine (B) T. P. Barbiero Department of Economics, Ryerson University, Toronto, ON M5B 2K3, Canada kyosh@ryerson.ca T. P. Barbiero barbiero@ryerson.ca

2 114 K. Yoshimine, T.P. Barbiero OECD countries show that consumption correlations are lower than output correlations. To date, a number of studies have asked why such a discrepancy arises. One obvious way to tackle this discrepancy is to introduce incomplete asset markets, but studies examining these structures have shown only negligible improvement between the theory and the data. 1 In this paper we show that cross-country co-movements of aggregate variables are highly sensitive to the share of capital goods in total trade, and that these results resolve the difficulty of previous studies to adequately explain the data. Boileau (1999, 2002) has examined the role of capital goods in international business cycles. Boileau (1999) examines the role of capital goods, but the model structure is similar to Backus et al. (1994, 1995) and hence does not explicitly take into account the share of capital goods in total trade. In Boileau (2002), the trade share is explicitly considered, but the main concern of the paper is the role of investment specific technical changes, and the mechanism by which the co-movements of output and consumption are affected is not examined. In both studies, the discrepancies in cross-country co-movements of macroeconomic variables remain unresolved. High consumption correlation can easily arise in the standard model where trade occurs mainly in consumption goods because cross-country differences in consumption are equalized by relatively quick movements of traded consumption goods. However, such a model structure is not appropriate to analyze international transmissions of productivity shocks because more than half of the world s trade consists of capital goods, making consumption smoothing across countries quite difficult. For example, Boileau (2002) argues that equipment accounts for half of total trade in G7 countries. 2 InthecaseoftheUS,he reports 53% of its total trade is made up of capital goods. Note that other production inputs such as chemicals and raw materials are not considered capital goods, even though they account for nearly 10% of trade. Thus, for our purpose it is important that our calibration exercise explicitly take into account the trade share of capital goods that exceeds 50%. Theoretically, the transmission of aggregate fluctuations via traded capital goods is fundamentally different from the transmission via consumption goods because the resulting trade flows also differ. For example, a productivity increase in an economy tends to induce a trade deficit rather than a trade surplus in that nation because the value of imported inputs tends to exceed the value of exports of consumption goods. Specifically, a rise in the share of capital goods in total trade increases transmission of productivity shocks and reduces opportunities for the exports and imports of consumption goods, thus reducing the possibility for consumption smoothing across countries. That is, output correlation must rise and consumption correlation must fall. The purpose of this paper is to analyze how theoretical cross-country co-movements of aggregate variables can approximate the existing US and 1 See, for example, Baxter and Crucini (1995), Heathcote and Perri (2002). 2 See also Baxter (1995).

3 International co-movements of macroeconomic variables 115 OECD data by calibrating for different shares of capital goods in total trade. We find that as the share of capital goods rises above 50% of total trade, our simulation results better fit the existing US and OECD data. This paper is organized as follows. In Section 1, we set out the details of our model. In Section 2, calibration and simulation results are presented and discussed. Section 3 concludes. 1 The model In our model there are two countries, home and foreign, each having a representative firm that uses physical capital k and labour hours n, and we further assume a Cobb Douglas production function in producing output y. The stock of physical capital evolves according to the way commonly assumed in the standard neoclassical model. Using an asterisk to denote foreign variables we have the following: y t = z t k α t n1 α t (1) y t = z t k α t n 1 α t (2) k t+1 = x t + (1 δ)k t (3) k t+1 = x t + (1 δ)k t (4) Total investment x is equivalent to a composite of capital goods, where a share denoted by x 1 is produced domestically (home) and a share denoted by x 2 is produced abroad (foreign). Specifically, x = [ (x 1 ) μ + ω x (x 2 )μ] 1 μ (5) x = [ (x 1 )μ + ω x (x 2 ) μ] 1 μ, (6) where the parameter μ is the elasticity of substitution between x 1 and x 2,and μ<1. The parameter ω x is the relative importance of foreign capital goods compared to home capital goods. The home and foreign productivities evolve according to the following standard vector autoregressive process: z t = Az t 1 + ɛ t (7) where z =[ln(z), ln(z )], A is a matrix of coefficients, and the elements in ɛ =[ε, ε ] are normally distributed with a mean of zero, serially independent, and contemporaneously correlated with each other. In each period, a representative consumer in the home country takes the state of the world s t ={k t, k t, z t, z t } as given and chooses consumption c 1,t and c 2,t, leisure l t, and the next period capital stock k t+1 to maximize the expected value of life-time utility. This optimization problem is written in a recursive form as follows: { v(s) = max u(c, l) + β E[v(s ) s] } (8) c 1,c 2,k,l

4 116 K. Yoshimine, T.P. Barbiero subject to Eqs. 3 through 6 and c 1 + x 1 + p(c 2 + x 2 ) rk + wn (9) where the time subscript is omitted, and the next-period variables are denoted byaprime. 3 The time endowment is normalized to unity so that l = 1 n.the utility function is given by u(c, l) = 1 [ 1 σ c 1 γ l ] γ 1 σ. Total consumption c is a composite of home and foreign final goods: c = [ (c 1 ) κ + ω c (c 2 )κ] 1 κ (10) where κ is the elasticity of substitution between c 1 and c 2,andκ<1. The parameter ω c is the relative importance of foreign final goods. The optimization problems for the home and foreign consumers are entirely symmetric, including the preference parameters. The equilibrium conditions for the home and foreign countries are defined as prices {r, r,w,w, p} and allocations {c, c, x, x, n, n } such that the consumers in both countries solve their respective optimization problem, and both the factor markets and goods markets clear. The market clearing condition for the goods market is given by: y = c 1 + x 1 + c 2 + x 2 (11) y = c 1 + x 1 + c 2 + x 2 (12) The solution to the model yields the optimal allocations 1 κ 1 ω κ 1 c c 1 = (rk + wn x 1 ξ)ψ 1 (13) c 2 = (p/ω c) 1 κ 1 c1 (14) x 1 =[k (1 δ)k]ξ 1 μ (15) x 2 = (p/ω x) 1 μ 1 x1 (16) where ψ = 1 + p κ and ξ = 1 + p μ. Note that the shares of consumption and capital goods in trade crucially depend on the preference parameters ω c and ω x. For example, when trade consists of only consumption goods, ω c is strictly greater than one, but ω x is zero. By contrast, when trade consists of only capital goods, ω c = 0 but ω x > 0. Thus, as the share of capital goods becomes large, ω x tends to increase and ω c tends to decrease. The equations above also imply that the co-movement between home and foreign production is more sensitive to ω x than ω c. To illustrate, suppose there is a transitory productivity increase at home so that the rise in factor prices r and w induces k and n to rise. Equation 15 implies that x 1 certainly rises, 1 μ 1 ω μ 1 x but Eq. 13 implies that c 1 may or may not rise. This in turn implies that c 2 may not rise but x 2 definitely rises. Thus, the productivity shock at home is 3 Here we show our model as a competitive equilibrium rather than a planner s problem because of the insight it can provide.

5 International co-movements of macroeconomic variables 117 more effectively transmitted to the foreign production when capital goods account for a large share of total trade, because ω x takes a relatively large value, compared with the case of no traded capital goods. Note also that the terms of trade becomes more volatile as the value of ω x becomes large. From Eqs. 14 and 16, the terms of trade is defined as p = 1 2 [ ω c ( c 2 c 1 ) κ 1 + ω x ( x 2 x 1 ) ] μ 1. (17) Since investment is more volatile than consumption, p tends to be more volatile as the share of traded capital goods, and hence ω x, becomes large. 2 Calibration and simulation results The basic parameter values for the model have been chosen in accordance with those used in the literature (e.g., Backus et al. 1994, 1995; Boileau 2002). For example, the share of capital in production is α = 0.36, the coefficient of risk aversion is σ = 2.0. The parameters κ and μ are both Given these values, other parameter values have been derived by matching the steady-state conditions of the model to the long-run averages of US aggregate data. 4 Using the quarterly US data, the capital stock per output (k/y) is found to be 11.3, aggregate consumption (c/y) is 0.75, and aggregate investment (x/y) is From the steady state condition x = δk, the depreciation rate is δ = Also, both exports and imports are 10% of total output. We define η = x 2 /(c 2 + x 2 ) = x 2/(c 2 + x 2 ) to be the share of capital goods in trade. As the benchmark case, we assume that there is no trade in capital goods, so that η = 0. Note that based on our previous discussion, we are particularly interested in the case where η>0.5 (We will later examine different values of η in the simulation, but the share of imports c 2 + x 2 = 0.10 is maintained). This benchmark assumption suggests c 2 = 0.10 and hence c 1 = From the first order condition, the optimal allocation of home and foreign consumption goods is given by c 2 = c 1ωc 1/(1 ρ), from which we obtain ω c = The discount factor is obtained as β = 1/(r + 1 δ) = 0.99,wherer is the marginal productivity of the steady state level of k. The parameter γ is obtained from the first order condition n = 1 γ [1 + (r δ)k/w], wherew is the marginal productivity of labour. Using the Bureau of Labor Statistics data, we find the average hours worked to be 31% of the unit endowment of time, thus n = This gives γ = Also, from Eq. 1, the steady-state productivity level is To increase the accuracy of the simulation, this model is solved by the discrete state-space method rather than the approximation method. The possible states of the world s ={k, k, z, z } are defined in grid points, and the 4 We have calibrated our model to the US data, so that it can be compared to other studies such as Boileau (2002).

6 118 K. Yoshimine, T.P. Barbiero equilibrium allocations for each grid point is obtained by iterating on the value function. The simulation results are arrived at by searching for the allocations corresponding to each value of a given productivity shock sequence. Allocations of the decision variables between the grid points and outside the grid points are obtained by interpolation and extrapolation, respectively. To do so, we assume that each of z and z can take the value of z or z, so that there are four possible patterns of z =[ln(z), ln(z )]. We have set the values for z and z at 0.77 and 0.97, respectively, so that the steady state value of z is half the distance between them. The continuous shock process (7) is converted into a four state Markov chain using the method discussed by Tauchen (1986). The stochastic process for productivity is calibrated using the value found in Backus et al. (1994). Productivities z and z have an autocorrelation of and a cross-country feedback of Innovations have a standard deviation of and a cross-correlation of Using these values, we obtain the following Markov transition probability π ij = where each element in the matrix is the probability that the realized productivity at t is z j, when the productivity at t 1 is z i. Using the probability matrix, we arrive at the sequence of productivity shock z, which we used for our simulation. Table 1 shows the simulation results in conjunction with the observed US and OECD data. The first row of the table shows the properties under US bilateral relations obtained by Backus et al. (1995). The second row shows the OECD averages also as reported by Backus et al. (1995) and Ambler et al. (2004). The third row reports the theoretical properties obtained by Backus et al. (1995), and the fourth and fifth rows report Boileau s (2002) resultsin which he looks at the two extreme cases where only final goods are traded and only capital goods are traded. As Table 1 shows, the relative volatility and cyclical properties for consumption, investment, and hours worked are well explained by the Backus et al. (1995) and the Boileau (2002) models. The glaring problem is that Backus et al. (1995) and Boileau (2002) do not replicate the relative magnitude of cross-country correlations. Boileau s (2002) results actually move in the right direction as he changes the trade shares of equipment, but these results are still the opposite of what the actual data reflects. When the trade share changes from final goods to capital goods, the output correlation ρ yy rises while the consumption correlation ρ cc falls, which is what we would expect, but the fact remains that ρ yy is still below ρ cc. 5 In 5 This is likely because he allows the output share of trade to vary depending on the share of capital goods, therefore pure effects of capital goods on transmission of productivity shocks are unclear.

7 International co-movements of macroeconomic variables 119 Table 1 Observed and theoretical properties of international business cycles Relative volatility Cyclical property Cross correlation σc σx σn σp ρcy ρxy ρny ρpy ρzz ρ yy ρ cc ρ xx ρ nn (1) Data (US) a (2) Data (OECD average) b (3) BKK (4) Boileau (final goods) (5) Boileau (capital goods) Simulation results η = η = η = η = η = η = η = η = η = η = η = Note: BKK95 Backus et al. (1995), Boileau Boileau (2002). Results are averages of 5 simulations, each having 100 periods. All variables are log-transformed. η refers to the share of capital goods in trade. Relative volatility is the standard deviation relative to that of output a Obtained from Backus et al. (1995) b Relative volatilities and cyclical properties are averages of 10 OECD countries calculated from Backus et al. (1995). Cross correlations are obtained from Ambler et al. (2004)

8 120 K. Yoshimine, T.P. Barbiero other words, the existing studies do not adequately explain the transmission of productivity shocks, and consumption smoothing across countries. The results derived from our calibration exercise better explain the actual data, as shown by various values of capital goods shares in Table 1. Theshare is indicated by η, whereη = 0 denotes no trade in capital goods, and η = 1 indicates that only capital goods are traded and no consumption goods whatsoever. For each level of η we re-calibrate our model to take into consideration the preferences of foreign consumption goods ω c and foreign capital goods ω x while maintaining the share of trade in output constant. All results are log transformed, but the Hodrick Prescott filter has not been applied because our theoretical model does not generate trends. 6 Moreover, we have assumed that cross-country correlations of productivity is constant at 0.13 in all cases since we want to isolate the effects of changing trade shares of capital goods on the transmission of productivity shocks. Our results show that as η rises, the relative volatilities of consumption σ c, investment σ x, and hours worked σ n remain in the same order of magnitude as the results arrived at by Backus et al. (1995) and Boileau (2002). The relative volatility of the terms of trade σ p rises and becomes similar to Boileau s results as the share of capital goods exceeds the 50% level. The reason why the volatility of the terms of trade rises as η rises is straightforward. With a high trade share of capital goods, each country has more opportunity to obtain investment goods abroad and less opportunity to smooth consumption. Since investment reacts to shocks more rapidly than consumption, the results are higher rates of trade activity and hence a higher terms of trade volatility. In our results the cyclical properties are similar to those of Backus et al. (1995) and Boileau (2002). However, significant differences are seen in crosscountry correlations. As η rises from 0.0 to 1.0, the output correlation more than doubles. Correlations of investment and hours worked rise even more. The consumption correlation rises slightly between η = 0.0 and η = 0.2, but then falls rapidly thereafter. Most importantly, our results show that the consumption correlation goes below the output correlation when η>0.5. The correlations ρ yy and ρ cc at η>0.5converge to the OECD data. Although ρ xx and ρ nn at η = 0.5 are not similar to the OECD data, they are similar to the US data. These results are obtained under fairly reasonable parameter values. Our results show that the higher value of η raises output correlations because presumably productivity shocks are transmitted more efficiently. For example, a rise in home productivity will increase foreign importation of consumption goods, but foreign countries will also increase export of their capital goods to meet the rising home demand for them. Thus, the larger share of capital goods induces higher cross-country correlations of output and lower cross-country correlations of consumption, and vice versa. 6 To verify, we have tested our simulation outputs using the Augmented Dickey Fuller test and found that all the outputs are stationary based on the 1% level of MacKinnon critical value. Note that even when we apply the HP filter, our key findings do not change except that ρ cc falls below ρ yy at η = 0.50 and falls below zero at η 0.7.

9 International co-movements of macroeconomic variables 121 The slight increase in consumption correlation from η = 0.0 to η = 0.1 is explained as follows. When there is no trade in capital goods, a productivity shock in the home country does not affect production in the foreign country. When trade share in capital goods rises, the foreign country can supply more capital goods to the home country and raise its output. But the foreign country increases use of not only capital goods but also consumption goods because its national income rises. Thus, even though a rise in the share of capital goods reduces the trade share of consumption goods, the rise in income allows the foreign country to consume more of its own goods. This positive effect of a rise in income on consumption exceeds the negative effects of the reduced share of consumption good when the share of capital goods is very small. The standard model has difficulty explaining the positive cross correlations of investment as well as that of hours worked. Our results show that both investment and hours worked correlations are indeed very small when the share of capital goods in total trade is small (they are both 8% when η = 0), but both investment and hours worked correlations increase as the share of capital goods increases. In our calibration exercise correlations become very similar to the US data when η is in the range of 0.4 and 0.5. The correlation of investment rises because the rise in the share of traded capital goods allows more investment to flow across countries. Since output correlations rise, the investment correlations must also rise. The correlations of hours worked also rise because investment and hours worked are complements with each other under the assumption of a homogeneous production function. 3 Conclusion We have examined the standard two-country business cycle model to explore the implication of different shares of traded capital goods in the transmissions of productivity shocks across countries. In the standard model, predicted cross-country consumption correlations are higher, and output correlations are lower than what the available data show. We resolve this apparent disparity between the theory and the data by investigating the impact of the different shares of traded capital goods. Our primary finding is that the cross-country transmission of productivities and the cross-country co-movements of aggregate variables are very sensitive to the share of traded capital goods. When we calibrate our model to take into account the share of capital goods in total trade, the disparities between the theory and the existing data disappear. Two limitations of our model are worth noting. First, our model does not include an asset market. Thus, inter-temporal trade is not allowed, and the trade balance is zero in every period. The absence of an asset market prevents us from explaining the statistical properties of net exports, such as cyclical properties ands cross correlations with other variables. Also, our model does not consider non-traded goods. Non-traded goods are often used to explain deviations of exchange rates from parity conditions, but the parity conditions in our model always hold. Thus, one possible direction of future research

10 122 K. Yoshimine, T.P. Barbiero is to combine these structures with the appropriate share of capital goods in total trade. Also, our model is calibrated to the US data, but it is of interest to calibrate the model to other OECD countries, as income shares of international trade tend to be higher in other countries. Acknowledgements We thank Leo Michelis for his comments on an earlier version of this paper. We also thank the anonymous referee for his/her helpful feedback. References Ambler S, Cardia E, Zimmermann C (2004) International real business cycles: what are the facts? J Monet Econ 51: Backus DK, Kehoe PJ, Kydland FE (1994) Dynamics of the trade balance and the terms of trade: The j-curve? Am Econ Rev 84(1): Backus DK, Kehoe PJ, Kydland FE (1995) International business cycles: theory and evidence. In: Cooley TF (ed) Frontiers of business cycle research. Princeton University Press, Princeton, NJ, pp Baxter M (1995) International trade and business cycles. In: Grossman G, Rogoff K (eds) Handbook of International Economics. Elsevier, Amsterdam, The Netherlands Baxter M, Crucini MJ (1995) Business cycles and asset structure of foreign trade. Int Econ Rev 36(4): Boileau M (1999) Trade in capital goods and the volatility of net export and the terms of trade. J Int Econ 48: Boileau M (2002) Trade in capital good and investment-specific technical change. J Econ Dyn Control 26: Heathcote J, Perri F (2002) Financial autarky and international business cycles. Int Monet Econ 49: Tauchen G (1986) Finite state Markov-chain approximations to univariate and vector autoregressions. Econ Lett 20(2):