Appendix: Net Exports, Consumption Volatility and International Business Cycle Models.

Appendix: Net Exports, Consumption Volatility and International Business Cycle Models. Andrea Raffo Federal Reserve Bank of Kansas City February 2007 Abstract This Appendix studies the implications of introducing a fully specied home production sector in the BKK environment. It concludes that, without affecting signicantly the business cycle properties of the model, GHH preferences represent an easy yet convenient way to achieve the same objective. Correspondence: Andrea Raffo, Economic Research Department, Federal Reserve Bank of Kansas City, 925 Grand Blvd, Kansas City, MO, 6498. Email: andrea.raffo@kc.frb.org. The views expressed herein are solely those of the author and do not necessarily reect the views of the Federal Reserve Bank of Kansas City or the Federal Reserve System

Appendix. A fully specied BKK model with home production. In this Appendix, I describe a two-country two-goods model à la BKK with home production. For simplicity, I report the ndings assuming that () only shocks to market technology generate business cycles and (2) home technology does not use capital. The rst assumption is not critical, in that what is relevant for the decision rule of agents is the changes of market technology (i.e. real wages) relative to the home technology. This point is also emphasized in Benhabib et al. [99] (BRW henceforth) and McGrattan et al. [997], for example. The second assumption allows to compare the results presented here with the results in the main paper. In fact, if we assume that there is a CES aggregator for home and market goods and that the (linear) home technology uses only labor input, then we can consider the following two scenarios:. unit elasticity of substitution between goods, which delivers a reduced form model with standard isoelastic (log) preferences (as in the BKK model); 2. innite elasticity of substitution between goods, which delivers a reduced form model that is a special case of GHH preferences. This analysis achieves the following two objectives. First, it shows that for sufciently large elasticity of substitution between home and market activities, the model is able to generate large volatility in consumption in the BKK environment. Ultimately, as argued in Raffo [2007], this element is key to explain countercyclical net exports, a very important feature of international data. Second, it conrms that GHH preferences are a convenient way to introduce home production in the BKK model, without affecting its business cycle properties. For a formal derivation of these results, see BRW. For a BKK model with home production that uses capital, see Canova and Ubide [998].

. Preferences and Technology There are two countries (i = ; 2) populated by identical innitively lived agents. There are two intermediate goods (j = A; B) and two nal goods (C i ). Each country specializes in the production of one intermediate good and factors of production are immobile across countries. Households derive utility from consumption and leisure. Preferences are of the standard isoelastic form U(C it ; N it ) = ln (C it ) + ln( N it ) where C i is a composite consumption good, N i is overall labor and the time endowment is normalized to one. The composite consumption good is given by C i = (C " mi + C " hi) =" where C mi is market consumption and C hi is home consumption. Total time spent working is the sum of time devoted to market activities N mi and time devoted to home activities N hi : N i = N mi + N hi Households produce home goods using the following technology C hi = b hi N hi Substituting the home production constraint into the utility function we obtain U(C it ; N it ) = ln(c " mi + (b hi N hi ) " ) =" + ln( N it ) Households produce intermediate goods for international trade using the following production function Y it = e zt K itn where K it is capital and z t is an exogenous technology shock. Feasibility in intermediate goods production requires 2 mit

A t + A 2t = Y t B t + B 2t = Y 2t Final market goods are produced combining intermediates from the two countries according to the Armington aggregator 8 >< G it (A it ; B it ) = >: $i A it + ( $ i )B it i = ( $i )A it + $ i B it i = 2 where $ i > 0:5 determines home bias in the composition of nal goods. The nal market good can be either consumed or invested so that the resource constraint states C mit + X it = G it (A it ; B it ) In each country, capital is subject to a convex adjustment cost. Capital evolves according to the following law of motion K it+ = ( )K it + X it + Xit K it K it where X it is investment and is the depreciation rate. (:) is such that 0 (:) > 0; 00 (:) < 0: In summary the social planner problem can be stated as s:t: PP Max E t t ln(c mit " + (b hit N hit ) " ) =" + ln( N mit N hit ) i t C mt + K t+ + (K 2 t+ K t ) 2 = $ At + ( $ )Bt + ( )K t ( t ) C + K 2t+ + (K 2 2t+ K 2t ) 2 = ( $ 2 )A 2t + $ 2 B2t + ( )K 2t ( 2t ) A t + A 2t = e z t KtN mt ( 3t ) B 2t + B t = e z 2t K2tN ( 4t ) 3

.2 First Order Conditions The rst order conditions associated with the social planner problem are: C mt : C : C " mt [C " mt +(b hn ht ) " ] = t C " [C " +(b 2hN 2ht ) " ] = 2t N mt : ( N mt N ht ) = 3t( )e z t K tn mt N : ( N N 2ht ) = 4t( )e z 2t K 2tN N ht : = b " h N " ht ( N mt N ht ) [C mt " +(b hn ht ) " ] N 2ht : = b " 2h N " 2ht ( N N 2ht ) [C " +(b 2hN 2ht ) " ] A t : A 2t : B t : B 2t : $ A t + ( $ )Bt $ At t = 3t ( $2 )A 2t + $ 2 B2t ( $ 2 )A2t 2t = 3t $ A t + ( $ )Bt ( $ )Bt t = 4t ( $2 )A 2t + $ 2 B2t $ 2 B2t 2t = 4t K t+ : t [ + (K t+ K t )] = E t f t+ [ + (K t+2 K t+ )] + 3t+ e z t+ K t+nt+ K 2t+ : 2t [ + (K 2t+ K 2t )] = E t f 2t+ [ (K 2t+ K 2t+2 )] + 4t+ e z 2t+ K 2t+N2t+ 4

We can rewrite the FOC as follows: [] Combining C m and N m b " h N " ht C " mt n $ = A t + ( $ )Bt $ A t o ( )e z t K tn mt [2] Combining C 2m and N 2m b " 2h N " 2ht C " = n ( $2 )A 2t + $ 2 B2t $ 2 B 2t o ( )e z 2t K 2tN [3] The home production condition in country is ( N mt N ht ) = b " h N " ht [C mt " + (b h N ht ) " ] [4] The home production condition in country 2 is [5] Using A, A 2 ; C and C 2! ( $ 2 ) A2t A t ( N N 2ht ) = b " 2h N " 2ht [C " + (b 2h N 2ht ) " ] $ A t + ( $ )Bt ( $2 )A 2t + $ 2 B2t = C" C mt " [C " mt + (b h N ht ) " ] [C " + (b 2h N 2ht ) " ] [6] Combining A and B (TOT)! Bt = ( N mt N ht )! A t ( N N 2ht ) e z t K tn mt e z 2t K 2t N 5

[7] Euler equation for country is C " mt [C mt " +(b hn ht ) " ] [ + (K t+ K t )] = E t + $ At+ + ( $ )Bt+ $ At+e z t+ K C " mt+ [C " mt+ +(b hn ht+ ) " ] [ (K t+ K t+2 ) t+nmt+] o [8] Euler equation for country is C " [C " +(b 2hN 2ht ) " ] [ + (K 2t+ K 2t )] = E t + ( $ 2 )A 2t+ + $ 2 B2t+ $ 2 B2t+e z 2t+ K C " + [C " + +(b 2hN 2ht+ ) " ] [ (K 2t+ K 2t+2 ) 2t+N+] o And the constraints are [9] C mt + X t = $ A t + ( $ )Bt [0] C + X 2t = ( $ 2 )A2t + $ 2 B2t [] A t + A 2t = Y t [2] B 2t + B t = Y 2t [3] K t+ = ( )K t + X t 2 (K t+ K t ) 2 [4] K 2t+ = ( )K 2t + X 2t 2 (K 2t+ K 2t ) 2 [5] Y t = e z t K tn mt [6] Y 2t = e z 2t K 2tN 6

.3 Additional variables of interest The following variables will be the focus of the analysis in comparing the quantitative predictions of theoretical economies with the data. The terms of trade are dened as the price of import relative to export T OT = G B G A = $ $ A B Net exports over GDP (expressed in terms of nal good units) are dened as NX = A 2 T OT B Y Changes in net exports are determined by changes in quantities (exports and imports of goods) and prices (terms of trade). The variable NXQTY represents the difference between exports and imports when both terms are evaluated at steady state prices, that is.4 Solution and Calibration NXQT Y = A 2 B Y In order to solve the model, I rst compute the non-stochastic steady state by setting the innovations in productivity equal to their unconditional mean values. I then impose that aggregate variables are consistent with long-run ratios observed in the data (namely, consumption-output ratio equal to 0.8, investment-output ratio equal to 0.2 and, for the symmetric case, import-output ratio equal to 0.5). Finally, I log-linearize the system of equations characterizing the solution of the model around the deterministic steady state and solve the resulting system of stochastic difference equations using the method of undetermined coefcients as described in Uhlig (999). Regarding the calibration of parameters, I follow the original BKK work for most of them, hence the reader can refer to their paper for the principles behind the calibration. Table presents these parameter values. [Please insert Table here] 7

In this section, I briey discuss the calibration strategy regarding the new parameters introduced by the home production structure, namely "; ; ; b h : The parameter " governs the elasticity of substitution between home and market consumption. I will consider " to be a free parameter in order to study the implications of this parameter value for my ndings, which is the main objective of this Appendix. Notice, in particular, that if " equals 0 (unit elasticity) the model reduces to the standard BKK model with log utility ( = ). If " equals (perfect substitution), the model has a reduced form representation which is a special case ( = and = ) of the BKK model with GHH preferences proposed in Raffo [2007]. Incidentally, empirical estimates for this elasticity report values between 0:6 and 0:8, i.e. towards a large degree of substitution. The remaining three parameters need to be chosen using the model's steady state relationships. A closer investigation of these equations reveals two important features. First, only two parameters have to be pinned down, that is and the product b " h : Second, different values of " imply different values for the remaining two parameters. I follow BRW and calibrate and b " h by setting the steady state level of hours devoted to market and non-market activities equal to 0:3 and 0:25 respectively..5 Simulation Results Table 2 reports simulation results. The rst line corresponds to the data for U.S. and E.U.. The second line presents the standard BKK model with adjustment cost to investment calibrated to match the volatility of investment relative to the volatility of output. Below, I report moments (standard deviations relative to output standard deviations and correlations with output) for the main variables involved in the analysis. Each row corresponds to a different value for the elasticity of substitution between home and market goods. When " = 0 the model reduces to the BKK case with log utility. In the case of " =, instead, the model is a special case of the GHH preferences in which there is no curvature in labor and the intertemporal elasticity of substitution is one (again, log case). 8

The main message that emerges from the table is twofold. First, high substitution between home and market activities delivers high volatility of market consumption. This result, in turn, represent the core mechanism to generate countercyclical net exports through changes in the net trade in goods. Only with large values for " there is enough response in consumption to productivity shocks so that imports become strongly procyclical. This feature is consistent with the empirical evidence that reports estimates for this elasticity in the order of 0:6 0:8. Second, the business cycle properties produced by GHH preferences are very similar to the fully specied BKK model with home production (and large elasticity). Adopting these preferences, in other words, represents a convenient way to introduce home production into standard models without any loss of generality. Researchers are invited to use GHH preferences if they want to be consistent with countercyclical net exports and high variability in consumption at business cycle frequencies. [Please insert Table 2 here] 9

References [] Backus, D. and P. Kehoe [992], "International Evidence on the Historical Properties of Business Cycles", American Economic Review, September, 82, 864-888. [2] Backus, D., P. Kehoe and F. E. Kydland [994], "Dynamics of the Trade Balance and the Terms of Trade: The J-Curve?", American Economic Review, March, 84, 84-03. [3] Backus, D., P. Kehoe and F. E. Kydland [995], "International Business Cycles: Theory and Evidence", in Frontiers of Business Cycle Research, ed. T. F. Cooley. Princeton University Press. Princeton, pp.33-356. [4] Baxter, M. and U. Jermann [999], "Household Production and The Excess Sensitivity of Consumption to Current Income", American Economic Review, September, 89, 902-920. [5] Benhabib, J., R., Rogerson and R. Wright, [99], "Homework in Macroeconomics: Household Production and Aggregate Fluctuations', Journal of Political Economy, December, 99, 99(6), pp. 67-87. [6] Cooley, T. [995], Frontiers of Business Cycle Research, Princeton University Press. Princeton, New Jersey. [7] Eichenbaum, M. and L.P. Hansen [990], "Estimating Models with Intertemporal Substitution Using Aggregate Time Series data", Journal of Business and Economic Statistics, 8:53-69. [8] Greenwood, J., Z. Hercowitz and G. W. Huffman [988], "Investment, Capacity Utilization and Real Business Cycle", American Economic Review, June 988, 78, 402-7. [9] McGrattan, E., R., Rogerson and R. Wright, [997], "An Equilibrium Model of the Business Cycle with Household Production and Fiscal Policy", International Economic Review, September, May 997, 38(2), pp.267-290. 0

[0] Raffo, A. [2007], "Net Exports, Consumption Volatility and International Business Cycle Models", RWP 06-0. [] Uhlig, Harald [999], "A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models Easily", in Ramon Marimon and Andrew Scott (eds). Computational Methods for the Studies of Dynamic Economies. Oxford: Oxford University Press.

Table. Benchmark Parameters Preferences = 0:99 Domestic Shares c=y = 0:8 = :0 i=y = 0:2 Trade Shares im = 0:5 Technology = 0:36 Productivity 0:906 0:088 0:088 0:906 = 0:025 = 2 = 0:00852 = :5 ;2 = 0:2580 2

Table 2. Properties of BKK Model with Home Production Std Dev relative to Output Correlation with Output DA C I NX TOT NXQTY Data :2 0:8 2:76 0:5 0:2 0:4 BKK 0:98 0:58 2:76 0:50 0:64 0:39 " = 0 0:97 0:56 2:76 0:46 0:62 0:38 " = 0:25 0:98 0:56 2:76 0:50 0:60 0:37 " = 0:5 0:98 0:58 2:76 0:54 0:58 0:07 " = 0:75 :02 0:60 2:76 0:5 0:52 0:36 " = 0:95 :06 0:70 2:76 0:50 0:5 0:45 BKK with GHH :09 0:79 2:76 0:5 0:45 0:43 Note. DA = Domestic Absorption, C = Consumption, I = Investment, TOT = Terms of Trade, NX = Net Exports over GDP, NXQTY = Real Net Exports over Real GDP. Statistics for the model refer to averages of 00 simulations of length 00 quarters after applying HP lter with smoothing parameter equal to 600. In all simulations, capital adjustment costs are included to reproduce the volatility of investment relative to output. 3