Sectoral composition and macroeconomic dynamics

Sectoral composition and macroeconomic dynamics Jaime Alonso-Carrera Departamento de Fundamentos del Análisis Económico and RGEA Universidade de Vigo Jordi Caballé Unitat de Fonaments de l’Anàlisi Economica and MOVE Universitat Autònoma de Barcelona Xavier Raurich Departament de Teoria Econòmica and CREB Universitat de Barcelona April 5, 2011 Abstract We analyze the transitional dynamics of a model with heterogeneous consumption goods. In this model, convergence is driven by two di¤erent forces: the typical diminishing returns to capital and the sectoral change inducing the variation in relative prices. We show that this second force a¤ects the growth rate if the two consumption goods are not Edgeworth independent and if these two goods are produced with technologies exhibiting di¤erent capital intensities. Because the aforementioned dynamic sectoral change arises only under heterogeneous consumption goods, the transitional dynamics of this model exhibits striking di¤erences with the growth model with a single consumption good. We also show that these di¤erences in the transitional dynamics can give raise to large discrepancies in the welfare cost of shocks between the economy with a unique consumption good and the economy with multiple consumption goods. JEL classi…cation codes: O41, O47. Keywords: multi-sector growth models, transitional dynamics, consumption growth. Financial support from the Government of Spain through grants ECO2009-09847, ECO200906953 and ECO2008-02752; PR2009-0162 and SM2009-0001; the Generalitat of Catalonia through the Barcelona GSE Research Network and grants SGR2009-00350 and SGR2009-1051; and the Xunta de Galicia through grant 10PXIB300177PR is gratefully acknowledged. Alonso-Carrera also thanks the Research School of Economics (Australian National University) for its hospitality. Caballé also thanks the …nancial support from the ICREA Academia program. The paper has bene…ted from comments by participants in the World Congres of the Econometric Society (Shanghai), DEGIT (Los Angeles), ESEM (Milan), SAEe (Granada), ASSET (Padova), Australasian Workshop in Macroeconomic Dynamics and seminars in UPV (Bilbao), IAE-CSIC (Barcelona), Australian National University, University of Melbourne, Monash University, Macquarie University, University of Wollongong, National University of Ireland (Maynooth), Geary Institute (UCD), Universidad de Murcia, Universidad de las Islas Baleares, Universitat Rovira i Virgili and Universidade do Minho. Corresponding Author: Jordi Caballé. Universitat Autònoma de Barcelona. Departament d’Economia i d’Història Econòmica. Edi…ci B. 08193 Bellaterra (Barcelona). Spain. E-mail: [email protected] 1. Introduction The literature on economic growth has generally taken the standard model of capital accumulation with a single …nal consumption good as the canonical framework to study the growth pattern of an economy. In particular, this model has been widely used for the analysis of the dynamic e¤ects of shocks in fundaments and for the normative and positive characterization of macroeconomic policy. The main feature of this model is that the economic dynamics is fully driven by the evolution of the return to capital. As the seminal contribution of Ramsey (1928) stated, the optimal intertemporal allocation of consumption and investment leads the growth of consumption expenditure to depend on the net interest rate only. In this paper, we claim that this result does not apply for models that allow for several heterogeneous consumption goods. More precisely, the aforementioned benchmark model may be unsuitable to study the dynamic e¤ects of those shocks featuring a permanent e¤ect on the sectoral composition of consumption. To illustrate this point, we characterize the properties of the transitional dynamics of a growth model where individuals derive utility from consumption of two heterogeneous goods. The recent growing interest for the analysis of structural change and international trade has made popular the use of multi-sector growth models with heterogeneous consumption goods.1A typical by-product of this literature is that the dynamics of the aggregate variables are identical to those predicted by the model with a single consumption good: the growth rate of consumption expenditure only depends on the marginal product of capital. According to this result, the process of convergence would be only determined by the return to capital with independence of the number of consumption goods. We argue instead that this isomorphism between the two types of models is a consequence of some restrictive assumptions imposed on these multi-sector models, namely, either the utility function is additively separable in the amounts of consumption of the di¤erent goods or these consumption goods are produced by means of technologies with identical capital intensities. By relaxing these assumptions, we …rst prove that the rate of growth of expenditure depends not only on the interest rate, but also on the growth rate of relative prices of goods. Therefore, the process of convergence in a general multi-sector growth model is driven by two forces: the return to capital and the dynamic adjustment of relative prices arising from the change in the sectoral composition. Our main purpose in this paper is to analyze how the presence of the later force modi…es the dynamic behavior of the economy. The e¤ect of the interest rate on consumption growth is measured by the intertemporal elasticity of substitution (IES, henceforth). On the contrary, the growth e¤ect of the variation in the relative price of goods is jointly determined by the IES and the Edgeworth elasticity between goods.2Therefore, the relative importance of these two forces in determining the intertemporal allocation of consumption expenditure crucially depends on this Edgeworth elasticity. In fact, we show that the growth rate of relative prices increases (decreases) the rate of growth of expenditure when the 1Examples include, among many others, Echevarria (1997), Konsamung et al. (2001), Ngai and Pissarides (2008), or Perez and Guillo (2010). 2The Edgeworth elasticity between two goods is de…ned as the elasticity of the marginal utility of one good with respect the consumption level of the other good. 2 two consumption goods are Edgeworth substitute (complementary). The intuition of this result is that the increase in the relative price of one good reduces the demand of this good, which increases (decreases) the demand of the corresponding substitute (complementary) goods. As was mentioned before, previous multi-sector growth models found in the literature impose assumptions that prevent the relative prices of consumption goods from displaying the aforementioned growth e¤ects. Some authors assume that the consumption goods are Edgeworth independent (see, e.g., Echevarria, 1997; Laitner, 2000; or Perez and Guillo, 2010) or use a technology yielding a constant relative price between goods (Kongsamunt et al., 2001; or Steger, 2006). Two exceptions are the multi-sector growth models considered in Rebelo (2001) and Ngai and Pissarides (2007). In the later model, the growth of prices a¤ects the rate of growth of expenditure. However, since the capital intensities are identical across sectors, the variation in prices arises only from exogenous, unbiased technological changes in sectoral productivities. On the contrary, in our model the dynamics of prices is endogenous as we consider di¤erent capital intensities across sectors. In this way, the dynamic adjustment of prices directly determines the response of the economy to changes in fundamentals. While Rebelo (2001) does also consider a model where prices are endogenous and the goods are not Edgeworth independent, he does not analyze the corresponding transitional dynamics. Therefore, to the best of our knowledge, the present paper is the …rst analyzing the transitional dynamics of a growth model with heterogeneous consumption goods when the two aforementioned forces driving the transition are operative. In order to study the transitional dynamics when the variation of prices displays the aforementioned growth e¤ects, we analyze a three sector growth model with a homothetic utility function whose argument is a composite good combining two di¤erent consumption goods. These goods are produced by means of constant returns to scale technologies that use physical and human capital as inputs. Furthermore, technologies exhibit di¤erent capital intensities across sectors. As was explained before, the last assumption makes the relative price between the two consumption goods not constant along the transition. To gain some intuition about this result, suppose that human capital becomes relatively scarcer than physical capital. Then, the consumption good produced in the physical capital intensive sector becomes less costly and the relative price of this consumption good decreases. Note that if the consumption goods were produced with technologies with the same capital intensity then the imbalances between the two capital stocks would not modify the relative price between these consumption goods. Finally, we assume in our analysis that the two consumption goods are not Edgeworth independent so that this dynamic adjustment of the relative price results in a modi…cation of the growth rate of consumption expenditure. As occurs in multi-sector growth models with two types of capital, the transitional dynamics will be governed by the imbalances between the two stocks of capital. However, the existence of two di¤erent forces governing the transition yields two interesting di¤erences with respect to the transitional dynamics obtained in the standard growth model with a unique consumption good. First, in growth models with a unique consumption good, convergence in the consumption growth rate occurs from below (above) if the initial value of the ratio of physical to human capital is larger (smaller) than its stationary value. We will show that this behavior may be reversed by 3 introducing heterogeneous consumption goods. In particular, we provide a condition that implies that convergence is from above when the initial value of the capital ratio is larger than its stationary value and from below otherwise. It should be noticed that, when this condition is satis…ed, the initial e¤ect on consumption growth of a shock in one of the capital stocks will be the opposite of the one obtained in a model with a single consumption good. As an example, consider an economy su¤ering a negative shock in human capital. Then, if there is a unique consumption good, this economy will experience a decrease in the growth rate of consumption. In contrast, in our model with heterogeneous consumption goods, the economy will display an increase in the growth rate of consumption expenditure. Second, while the growth rate of consumption expenditure exhibits a monotonic behavior when the diminishing returns to capital is the only force governing the transition, it may exhibit instead a non-monotonic behavior in our model. AlvarezCuadrado et al. (2004) mention evidence of non-monotonic behavior of the consumption growth rate. Steger (2000), among others, has accounted for this non-monotonic behavior by means of the introduction of a minimum consumption level that makes preferences non-homothetic. In contrast, in our model the non-monotonic behavior is explained by the presence of the aforementioned two di¤erent forces acting on the transitional dynamics. In fact, the growth rate exhibits a non-monotonic behavior when these two forces exhibit opposite growth e¤ects. The two di¤erences we have just mentioned imply that the patterns of growth along the transition crucially depend on the parameters values of our model. More precisely, we show that the capital intensity ranking across sectors and the value of the Edgeworth elasticity determine the nature of the transition. We will simulate the economy in order to analyze the transitional dynamics and show that the two forces governing the rate of growth of expenditure have opposite growth e¤ects. As a consequence, in the simulated economy this growth rate exhibits a non-monotonic convergence towards the steadystate and, moreover, the sign of the growth e¤ects of a shock in one of the capital stocks depends on the value of the Edgeworth elasticity. We also use the simulated model to study the growth and welfare e¤ects of technological shocks. This analysis allows us to compare the e¤ects of these shocks in the economy with a single consumption good with the e¤ects in the economy with heterogeneous consumption goods. Regarding the welfare cost of shocks, we show that they will strongly depend on the sectoral composition of the composite consumption good when these shocks cause large e¤ects on the unitary cost of this composite good. These large e¤ects occur when we consider shocks that modify the long-run value of relative prices. In this case, the shocks result in a large distortion in the intratemporal decision concerning the sectoral composition of consumption, which translates in turn into sizeable additional welfare e¤ects. We then conclude that the existing literature, by considering speci…c models where the force linked to the dynamics of the relative prices between goods is not operative, obtain biased results about the e¤ects of those shocks. The paper is organized as follows. Section 2 presents the ingredients of the model. Sections 3 and 4 characterize the equilibrium dynamics of relative prices and of the growth rate of expenditure, respectively. Section 5 develops the numerical analysis concerning the transitional dynamics and the e¤ects of technological shocks. Section 6 presents some concluding remarks, while the Appendix contains the proofs of all the 4 results of the paper. 2. The economy Let us consider a three-sector growth model in which the output in each sector is obtained from combining amounts of two types of capital, kand h, which we dub physical and human capital, respectively. The …rst sector produces an amount y1of commodity using the following production function: y1=A1(s1k)(u1h)1=A1u1hz 1; where s1and u1are the shares of physical and human capital allocated to this sector, z1=s1k/u1his the physical to human capital ratio, A1>0is the sectoral total factor productivity (TFP), and 2(0;1) measures the intensity of physical capital in this sector. We interpret this sector as the one producing manufactures and assume that the commodity y1can be either consumed or added to the stock of physical capital. The law of motion of the physical capital stock is thus given by _ k=A1u1hz 1c1k; (2.1) where c1is the amount of good y1devoted to consumption, and 2[0;1] is the depreciation rate of the physical capital stock. To ease the notation we omit the time argument of all the variables. The second sector produces a consumption good y2by means of the production function y2=A2(s2k)(u2h)1=A2u2hz 2;(2.2) where s2and u2are the shares of physical and human capital allocated to this sector, respectively, z2=s2k/u2his the physical to human capital ratio, A2>0is the sectoral TFP, and 2(0;1) measures the intensity of physical capital in this sector. We interpret this sector as the one producing food and services devoted to consumption, such as cultural or entertainment goods. Thus, the output of this sector can only be devoted to consumption, which we denote by c2;so that y2=c2in equilibrium. Finally, the third sector produces a commodity y3by means of the production function y3=A3[(1 s1s2)k][(1 u1u2)h]1=A3(1 u1u2)hz 3; where z3= (1 s1s2)k/(1 u1u2)his the physical to human capital ratio, A3>0is the sectoral TFP, and 2(0;1) measures the intensity of physical capital in this sector. This commodity is devoted exclusively to increase the stock of human capital and, therefore, we identify this sector with the education sector. The accumulation of the human capital stock is thus given by _ h=A3(1 u1u2)hz 3h; (2.3) where 2[0;1] is the depreciation rate of human capital. The economy is populated by an in…nitely lived representative agent characterized by the instantaneous utility function U(c1; c2) = c 1c1 21 1;(2.4) 5 where the parameter 2[0;1] measures the share of good c1in the composite consumption good, m=c 1c1 2;and  > 0is the (constant) elasticity of the marginal utility of this composite consumption good. Note that this utility function is homothetic, strictly concave, and increasing. The representative agent is endowed with kunits of physical capital and hunits of human capital. Let wbe the rate of return on human capital (i.e., the real wage per unit of human capital) and rthe rate of return on physical capital (i.e., the real interest rate). We assume perfect sectoral mobility so that the wage and interest rate are independent of the sector where the representative agent allocates the units of physical and human capital. Therefore, the budget constraint of the consumer is given by wh +rk = (c1+pc2)+(Ik+phIh);(2.5) where pis the relative price of good c2measured in units of good c1,phis the relative price of human capital measured in units of physical capital (or consumption good c1). Finally, Ihand Ikare the gross investment in human and physical capital, respectively, Ik=_ k+k; (2.6) and Ih=_ h+h: (2.7) 3. Dynamics of relative prices In this section we …rst solve the problems of consumers and …rms and then we derive the system of di¤erential equations characterizing the competitive equilibrium. We use these equations to …nd the long-run equilibrium and to study how the introduction of a second consumption good modi…es the equilibrium dynamics of relative prices. The representative agent maximizes Z1 0 etU(c1; c2)dt; (3.1) subject to (2.5), (2.6), and (2.7), where  > 0is the subjective discount rate. The solution to this optimization problem is given by the following equations derived in the Appendix: p=1 c1 c2;(3.2) _ph ph =rw ph +; (3.3) _c1 c1 =r (1 ) (1 ) _p p;(3.4) and the transversality conditions lim t!1e tp(1)(1)ck= 0;(3.5) and lim t!1e tp(1)(1)ch= 0:(3.6) 6 Equation (3.2) tells us that the price ratio pis equal to the marginal rate of substitution between the two consumption goods. Equation (3.3) shows that the growth of the price phis determined by the standard non-arbitrage condition between the investments in physical and human capital. Finally, equation (3.4) characterizes the growth rate of consumption good c1:From this equation we can easily obtain the growth rate of total consumption expenditure, which is de…ned as c=c1+pc2. Note that equation (3.2) implies that c=c1 =pc2 1:(3.7) Hence, the growth rate of consumption expenditure ccoincides with the growth rate of c1(the consumption expenditure in the good y1;which is the numeraire). We then obtain from (3.4) that _c c=r (1 ) (1 ) _p p:(3.8) Equation (3.8) tells us that the growth rate of consumption expenditure is driven by both the interest rate and by the change in the relative price of the two consumption goods. The e¤ect of a rise in the interest rate on the rate of growth of cis summarized by the intertemporal elasticity of substitution IES = 1=: On the contrary, the growth e¤ect of a rise in the growth rate of the relative price is jointly determined by the IES and Edgeworth elasticity (i.e., the elasticity of the marginal utility of the consumption good c1with respect to the consumption good c2) which is given by " c2@2U=@c1@c2 @U=@c1=(1 ) (1 ): By using (3.8), we see that the growth rate of the relative price pdirectly a¤ects the growth rate of consumption expenditure cwhen "6= 0;i.e., when the two consumption goods are not Edgeworth independent. Under the instantaneous utility function (2.4), the Edgeworth elasticity "is determined by the parameters and : In particular, the two consumption goods are Edgeworth independent when = 1 because in this case the utility function is additively separable in the two goods c1and c2. The previous literature on multisectoral growth models commonly uses a logarithmic speci…cation for preferences and this explains why it does not obtain the growth e¤ect of the variation in relative prices. The intuition on the aforementioned growth e¤ect of the dynamic adjustment of relative prices is as follows. Equation (3.8) is the Euler equation equating the market return from investing one unit of the numeraire y1and the growth of the marginal utility arising from consuming one additional unit of this commodity. When the two consumption goods are Edgeworth independent, then the marginal utility of one consumption good does not depend on the other consumption good. In this case, the growth rate of total consumption expenditure only depends on the interest rate. In contrast, when the two consumption goods are not Edgeworth independent a change in the consumption of good c2alters the marginal utility of consumption good c1: Thus, in this case, the growth of the marginal utility of one good will depend on the 7 growth of both consumption goods. As follows from equation (3.2), the consumption of these goods depends on the relative price. Actually, the concavity of the utility function implies that an increase in the relative price preduces the amount consumed of good c2. This reduction implies an increase (reduction) in the marginal utility of consumption good c1and in the amount of good c1consumed when the two goods are Edgeworth substitute (complementary).3 After having presented the equilibrium conditions on the demand side of our economy, we will now move to the supply side and we will characterize how the dynamics of relative prices is determined. This dynamics depends on the technologies used by the di¤erent sectors and on the market structure. In particular, …rms maximize pro…ts in each sector and, thus, the competitive factors payment must satisfy simultaneously the following equations: r=A1z1 1;(3.9) r=pA2z1 2;(3.10) r=phA3z1 3;(3.11) w= (1 )A1z 1;(3.12) w=p(1 )A2z 2;(3.13) and w=ph(1 )A3z 3:(3.14) Combining the system of equations (3.9) to (3.14) when 6=, we obtain zi= ip1 ;for i= 1;2;3;(3.15) where 1=  1 11 A2 A11  ; 2= 11  1;(3.16) and 3= 11  1:(3.17) From the previous set of equilibrium conditions we obtain the following well-known result, which has important consequences for the equilibrium dynamics of our economy. Proposition 3.1. The relative price pof consumption goods is constant over time for all initial values of the capital ratio z=k=h if and only if at least one of the following conditions holds: (i) =, (ii) =: 3Note that the e¤ect of relative prices on expenditure growth appears because only the good c1can be used as physical capital. If the equilibrium mix of the two consumptions goods could be devoted to investment in physical capital, then the relative price would not a¤ect the growth rate of consumption expenditure c(see Acemoglu and Guerrieri, 2008). 8 Let us …rst consider the condition =; which means that the two consumption goods c1and c2are produced by means of technologies with the same capital intensity. We see that under this condition, equation (3.16) implies that 2= 1when 6= and then, from equation (3.15), we get z1=z2. Therefore, by combining equations (3.9) and (3.10), it follows that the relative price between the two consumption goods remains constant and equal to p=A1 A2:This obviously means that the growth rate of consumption expenditure only depends on the interest rate (see equation (3.8)). Therefore, the transitional dynamics of our model when =coincides with the transitional dynamics of the two-sector growth model with a unique consumption good, which was analyzed by Uzawa (1965) and Lucas (1988). Let us now consider the condition =: Under this condition the two capital goods kand hare produced by means of technologies with the same capital intensity. Observe that in this case conditions (3.9), (3.11), (3.12) and (3.14) imply that z1=z3 and, thus, the relative price between the two capital stocks is constant and given by ph=A1 A3:Equation (3.3) implies that the wage to interest rate ratio w=r remains constant when phis constant. Then, from combining (3.9) and (3.12) we immediately see that z1is constant when phis constant. Therefore, both the interest rate rand z2 are constant as follows from (3.9) and (3.11). Finally, equation (3.10) shows that in this case the relative price pbetween the two consumption goods remains constant. In fact, it is easy to see that the three sectors are using Ak technologies when =:4 Therefore, the transition dynamics in this case coincides with the transition in Ak growth models with several consumption goods (see, e.g., Rebelo, 1991). We have just established the conditions under which the growth rate of consumption expenditure depends not only on the interest rate, but also on the growth rate of the relative price p: This new dependence requires that the consumption goods be not Edgeworth independent and to be produced by means of technologies with di¤erent capital intensities. These arguments then explain why the previous multisector growth models do not …nd a direct e¤ect of relative prices on consumption growth. Some of these models consider logarithmic preferences so that they implicitly assume that consumption goods are Edgeworth independent. Other models assume that consumption goods are produced with technologies that share the same capital intensity. Obviously, in this case the variation of relative prices could still a¤ect directly the growth rate of consumption expenditure under exogenous and biased technological change, that is, when the sectoral TFPs grow at exogenous growth rates that are di¤erent across sectors (see, e.g., Ngai and Pissarides, 2007). However, if technologies exhibit di¤erent capital intensities, the relative price between consumption goods appear as an endogenous channel for the propagation of shocks in fundamentals. In the rest of the paper, we will illustrate the consequences of this endogenous mechanism and, hence, we will assume that 6=and 6=: Note that relative prices would also a¤ect the growth rate of consumption expenditure when =; that is, when services and human capital are produced with 4Note that the technology that produces commodity y1can be rewritten as follows y1=b A1u1h; where b A1=A1(z 1)is constant. The technology that produces commodity y2can be rewritten as y2=b A2u2h; where b A2=A2z 2is constant and, …nally, the technology that produces commodity y3 can be rewritten as y3=b A3(1 u1u2)h; where b A3=A3(z 1)is constant. Since goods y1and y2 are produced with linear technologies, their relative prices are constant and given by p=b A1 b A2 : 9 two dynamic forces for a given capital intensity ranking across sectors and expenditure share (see the expression of in equation (4:3)). We then consider three di¤erent values for ": 0:7;0:95 and 1:2:We set the values of and that jointly replicate those values for "and a long-run growth rate equal to 2%:In the low elasticity economy we obtain = 2 and = 0:016;whereas we get = 2:357 and = 0:0089 for the economy with "= 0:95, and …nally we get = 2:7143 and = 0:0017 for the high elasticity economy. Observe that this calibration implies reasonable values for the IES:0:5, 0.4243 and 0:3684. We next simulate the response of each of the three parameterized economies to imbalances in the capital ratio, i.e., when z06=z:In order to show how important is the growth e¤ect of price variation, we compare the response of these baseline economies with the response of the corresponding economy with a unique consumption good. In order words, we compare the dynamic behaviors of the economy with = 0:3and the economy with = 1: 5.1. Transitional dynamics The expression of in equation (4:3) implies that it takes positive values when  <  and " > 0:Thus, the value of is positive under our empirically plausible values of the fundamental parameters. In this case, the two aforementioned forces governing the transition display opposite growth e¤ects. In our numerical examples, we show that, if the force associated with the variation of prices is the dominating then the transition is going to be di¤erent from that of models with a single consumption good. Figures 2, 3 and 4 show that this is the case when the Edgeworth elasticity is high (i.e., when the value of is high). These …gures show the dynamic response of some relevant variables to imbalances in the capital ratio. In particular, each of these …gures contains six panels. Panels (i), (iv), (v) and (vi) display, respectively, the growth rate of consumption expenditure, the growth rate of GDP, the relative price of consumption goods and the speed of convergence of the state variable zas a function of the deviations of the capital ratio with respect to its stationary value. Note that, following Reiss (2000),we de…ne the non-asymptotic speed of convergence of the ratio of capitals as _z/(zz). Panels (ii) and (iii) display, respectively, the time path of the growth rate of consumption expenditure when the state variable is initially below its long-run value and when it is initially above. Furthermore, all panels compare the transitional dynamics of the baseline economy with heterogeneous consumption goods (continuous line) with the transition in an equivalent economy with a unique consumption good, i.e., with = 1 (dashed line). We parametrize the counterfactual economy with = 1 so that it replicates the same empirical facts used to calibrate our benchmark economy with two heterogenous consumption goods.We observe that the di¤erences between the two economies under consideration are quite signi…cant in the three parametric scenarios. Hence, the direct e¤ect of the price adjustment on the intertemporal allocation of consumption expenditure also has important quantitative consequences for macroeconomic dynamics. [Insert Figures 2, 3 and 4] The …rst three panels of Figures 2, 3 and 4 illustrate numerically the results in 16 Proposition 4.4. We observe that the dynamic adjustment of consumption expenditure is non monotonic under the higher values of in the economy with two consumption goods (= 0:3):Moreover, when is high, the introduction of heterogeneous consumption goods reverses the transition. This occurs because determines the value of the Edgeworth elasticity "provided a value for the consumption share:When the Edgeworth elasticity "is high, the growth e¤ect of changes in the interest rate is low in comparison with the growth e¤ects of changes in the growth of the relative price. In this case, even if the initial values of the economy are close to the corresponding steady-state values, the transition is di¤erent from the one arising in an economy where the transition is governed only by the diminishing returns to capital. The signi…cant e¤ects of the price variation on the intertemporal allocation of consumption expenditure and savings have important quantitative consequences for the dynamic behavior of the other macroeconomic variables. As an illustration, Figures 2, 3 and 4 shows that the paths of the GDP growth rate, the relative price of goods and the speed of convergence also depend on the value of the parameter : This parameter measures the weight of the human capital intensive good in the composite consumption good. Thus, a reduction in makes the composite good more intensive in physical capital, which explains the results displayed in these three …gures. Intuitively, there are two non-competing ways of increasing in relative terms the stock of the scarce capital and, thus, of adjusting the imbalances in the capital ratio: (i) To decrease the accumulation of the relatively abundant capital; and (ii) to decrease the consumption expenditure. The more intensive in physical capital is the composite consumption good, the larger is the relative importance of the second way when z < z. The growth rate of GDP is then a decreasing function of if z < z:On the contrary, the more intensive in physical capital is the composite good, the larger is the relative importance of the …rst procedure when z > z:This implies that the growth rate of GDP is an increasing function of if z > z:Therefore, the dynamic adjustment of any imbalance in the capital ratio is faster when the composite consumption good is more physical intensive. This fact explains why the non-asymptotic speed of convergence always decreases with (see Panel (vi)). We …nally illustrate the implications of the di¤erences in the transitional dynamics across the alternative parametric scenarios by computing the welfare e¤ects of the initial imbalances in the capital ratio.11 Table 1 reports the time-invariant increase (decrease) in consumption required to compensate the welfare costs (gains) of having an initial capital ratio smaller (larger) than the stationary ratio. We again show the results for our baseline economy with = 0:3and for the economy with a single consumption good (i.e., = 1):The last column of this table compares the di¤erences in welfare costs between these two economies and shows that they are large. In particular, the welfare cost is approximately 20% larger in the economy with two consumption goods, whereas the welfare gain is 17% larger. These results follow again from the fact that the composite consumption good in the economies with a low value of is more intensive in physical capital. Obviously, in these economies the unitary cost of the composite good is more sensitive to the relative endowment of physical capital. 11 As in Lucas (1987), we measure the welfare cost of the imbalances in the capital ratio by the percentage increase in composite consumption good mnecessary to obtain the same discounted sum of utility as in the situation where the capital ratio is initially equal to its stationary value. 17 [Insert Table 1] By repeating the previous numerical exercises we obtain that the reported di¤erences in welfare between the two economies are extremely robust to both the size of shocks and the value of . The insigni…cant e¤ect of is explained by analyzing the dynamic behavior of the composite good m=c 1c1 2;which is the fundamental variable for welfare analysis. By using conditions (3.2), (3.7) and (3.27), we obtain _m m=1 A1z1 1(1 )(p):(5.1) Obviously, the growth rate of malso depends on the forces driving the intertemporal allocation of consumption expenditure c: the diminishing returns to capital and the growth rate of prices. However, observe that the net e¤ect of these two forces does not depend in this case on the value of : This occurs because the direct e¤ect of the variation in the relative price on the growth rate of mdoes not depend on the Edgeworth elasticity ". This then explains the insigni…cant e¤ect of on the welfare comparison between the economy with = 0:3and the economy with = 1: Next, we complement the analysis in this subsection by studying how the response of the economy to shocks in fundamentals depends on the value of : Given the previous conclusion about the independence of welfare e¤ects on ; we will only present the results for the case of = 2;which is associated with the value "= 0:7for the Edgeworth elasticity: 5.2. Comparative dynamics and welfare We now proceed to study the dynamic adjustments and the welfare costs from two di¤erent shocks: a sectoral biased technological shock and a sectoral unbiased technological shock. For that purpose, we assume that the economy is initially in a BGP and, unexpectedly, one of these shocks is introduced in a permanent basis. The aim of this analysis is to compare the e¤ects of these shocks in the baseline economy with two consumption goods (= 0:3) with the e¤ects in the economy with a unique consumption good (= 1): We …rst analyze the e¤ects of a biased technological shock that consists of reducing the TFP of the manufacturing sector A1by a 15%. We explain these e¤ects by using Figure 5, which summarizes how the economy responds to the shock; and Table 2, which provides the welfare cost of this shock. Observe that the rate of growth of expenditure initially su¤ers a strong decline and then it increases until it converges to its new long-run, which is smaller than the one before the shock. In the economy with a single consumption good, the growth rate only depends on the interest rate, which instantaneously falls due to the technological shock. This reduces investment and, as a consequence, the stock of physical capital declines during the transition. The reduction in the stock of physical capital implies that the interest rate increases during the transition. Note that the behavior of the interest rate fully explains the initial strong reduction in the rate of growth of expenditure and also its posterior increase during the transition. On the contrary, in the economy with two consumption goods, the rate of growth of expenditure also depends on the growth of the relative 18 price pof consumption goods. This price decreases instantaneously because the shock directly a¤ects the sector producing manufactures, whereas it increases during the transition because the continuous reduction in the stock of physical capital rises the cost of producing services, which is relatively intensive in this capital. This behavior of the relative price phas a positive e¤ect on the rate of growth of expenditure as the Edgeworth elasticity in the benchmark economy satis…es " > 0. The presence of this positive growth e¤ect in the economy with two consumption goods explains both the smaller initial reduction in the rate of growth of expenditure and its larger values along the transition. [Insert Figure 5 and Table 2] The …rst row of Table 2 reports the welfare cost of the considered permanent reduction in the TFP of the manufacturing sector. The main result is that the welfare cost is a 45:6% larger in the economy with a unique consumption good. This large di¤erence arises from the fact that the response of the composite good mto the shock is larger, the larger is the share of manufactures in the composite good. Figure 5 illustrates the dynamic adjustment of that good. Panel (iii) reports deviations of the composite good to physical capital ratio m=k from its initial stationary value. From this panel we conclude that the initial reduction in the value of mis smaller in the economy with = 0:3:The intratemporal substitution between goods in this economy reduces the impact of the shock in the level of the composite good. On the contrary, as Panel (iv) shows, the growth rate of composite good increases during the transition and, what is more interesting, it is smaller in the economy with = 0:3due to the negative e¤ect of the increase in the relative price p(see equation (5:1)). However, the larger recovery of the amount of the composite good in the economy with = 1 is not enough to outweigh its larger instantaneous reduction. In other words, the initial di¤erence in the response of the composite good in the two economies explains the larger welfare cost in the economy with = 1. Figure 6 displays the dynamic e¤ects of an unbiased technological shock consisting of a 5% decrease in the TFP in each sector. We observe that the dynamic adjustment in this case is qualitatively similar to the one accruing from a biased technological shock when = 1. Moreover, the di¤erences between the two economies are now quantitatively insigni…cant because of the smaller incidence of the price adjustment on the rate of growth of expenditure. Since each sectoral TFP falls in the same proportion, the responses of the relative price pand of consumption composition are both smaller when the shock is unbiased. This explains the small discrepancies between the two economies under consideration concerning the dynamic response of the rate of growth of expenditure and the level of composite consumption. Finally, this implies that the welfare cost associated with the unbiased shock is very similar in the two economies. As the second row of Table 2 shows, the welfare cost in the economy with = 0:3is less than 2% larger than in the economy with = 1: At this point, we should also mention that the di¤erences in the e¤ects of the unbiased shock between the two economies only arise because the depreciation rates of both capital stocks are di¤erent, which makes the shock distort the optimal allocation of capital among sectors. If =, then the stationary value of pis not a¤ected by the 19 unbiased shock as it can be derived from (3.15) and (3.26). Moreover, in this case we obtain that the welfare cost in the two economies would coincide. As can be seen from Figure 6, even if some di¤erences arise in the dynamic adjustment of both the growth rate of expenditure and the amount of composite good between the two economies, the larger recovery of the composite good in the economy with = 1 will fully o¤set its larger instantaneous reduction. Therefore, in spite of displaying identical welfare costs, the time-path of the welfare cost associated with a shock is di¤erent across the two economies even if the technological shock is unbiased. We can thus conclude that the discrepancy in the welfare cost of shocks between the two economies under consideration only arises when these shocks have permanent e¤ects on the relative prices and on the sectoral composition of consumption in the economy with two goods. [Insert Figure 6] 6. Concluding remarks We have analyzed the transitional dynamics of an endogenous growth model with two consumption goods. We have shown that the growth rate of expenditure not only depends on the interest rate, but also on the growth rate of the relative price of consumption goods. Convergence in this case may be determined by two di¤erent forces: the diminishing returns to capital and the growth of prices. In particular, this result arises when the two consumption goods are not Edgeworth independent and the technologies producing the two consumption goods have di¤erent capital intensities. These growth e¤ects of relative prices yield interesting di¤erences with respect to the transitional dynamics obtained in the standard growth model with a unique consumption good. We illustrate these di¤erences using a growth model with two capital stocks that we identify with human and physical capital. First, we show that in contrast with the standard growth model, convergence in the growth rate may occur from above if the initial value of the ratio of physical to human capital is larger than its stationary value and may occur from below otherwise. Second, we show that the growth rate of consumption expenditure may exhibit a non-monotonic behavior when the two aforementioned dynamic forces have opposite growth e¤ects. These di¤erences in the transition have other noteworthy implications. First, economies with the same interest rate may exhibit di¤erent growth rates of consumption along the transition. Therefore, our model provides an additional explanation to the cross-country di¤erences in the growth rates. Rebelo (1992) shows that the introduction of a minimum consumption requirement also implies that the growth rates do not equalize. This occurs because the minimum consumption makes preferences non-homothetic so that the IES is no longer constant along the transition. In this framework, convergence is driven by the interest rate and by the time-varying IES. More recently, Steger (2006) shows that, if there are heterogeneous consumption goods and a unique capital stock, then the IES is not constant and the growth rates do not equalize. Obviously, he derives this result when preferences are non-homothetic. In contrast, we show that, when there are heterogeneous consumption goods, the growth rates are di¤erent even with a constant IES because of the e¤ect of the growth of the relative prices along the transition. 20 The previous remark can be illustrated in a di¤erent way. By combining (3.8), (3.3), (3.19) and (4.3) we obtain that the rate of growth of consumption expenditure satis…es _c c=(ph) = 1 r+w ph+ (): This equation shows that the rate of growth of total expenditure depends both on the interest rate and on the wage rate when 6= 0:This implies that cross-country di¤erences in the growth rates will also be explained by wage di¤erentials when 6= 0 (i.e., when there are several consumption goods that are Edgeworth dependent and produced by technologies with di¤erent capital intensity). Moreover, for values of  close to the IES ;interest rate di¤erentials will not explain cross country di¤erences in the growth rates. According to our results, the welfare cost of shocks will also depend on the sectoral composition of the composite consumption good. The relationship between the welfare cost of shocks and the sectoral composition of consumption expenditure will be particularly strong when the shocks permanently modify the value of relative prices. In this case, the e¤ect of these shocks on the cost of the composite consumption good will depend on its sectoral composition. We have shown that biased technological shocks that increase the gap between the return on physical and human capital cause large and permanent e¤ects on prices. We have also shown that the welfare cost of these shocks depends on the intensity of the direct growth e¤ect of dynamic price adjustment. Therefore, this growth e¤ect of relative price is an unexplored channel a¤ecting the persistence and propagation of shocks. We summarize our analysis by saying that the results obtained in aggregate growth models with a single consumption good cannot be generalized to more disaggregated models with heterogeneous consumption goods. In these disaggregated models, the welfare costs of shocks depend on the value of the parameters measuring the sectoral composition of consumption and on the physical capital intensities of the sectors producing these consumption goods. Therefore, the empirical estimation of the sectoral composition parameters should be an important concern for future research on the assessment of the welfare cost of macroeconomic shocks. A natural extension of our paper is to introduce a minimum consumption requirement in one of the consumption goods. The price of this good will be high in the initial stages of development since the minimum consumption requirement will induce a high marginal utility of this good. Then, as the economy develops, the price will fall sharply until convergence is attained. Therefore, it seems that the introduction of a minimum consumption may accelerate the change of prices and, hence, the introduction of this consumption requirement may increase the e¤ect of the growth of the relative price on both the growth rate of consumption expenditures and on the welfare cost of shocks. 21 References [1] Acemoglu, D. and Guerrieri V. (2008). “Capital Deepening and Nonbalanced Economic Growth,”Journal of Political Economy 116, 467-498. [2] Alvarez-Cuadrado, F., Monterio, G. and Turnovsky, S. (2004). “Habit Formation, Catching-up with the Joneses, and Economic Growth,” Journal of Economic Growth 9, 47-80. [3] Bond E., Wang P. and Yip C. (1996). “A General Two-Sector Model of Endogenous Growth with Human and Physical Capital: Balanced Growth and Transitional Dynamics,”Journal of Economic Theory 68, 149-173. [4] Caballé J. and Santos M. (1993). “On Endogenous Growth with Physical and Human Capital,”Journal of Political Economy 101, 1042-1067. [5] Echevarria, C. (1997). “Changes in Sectoral Composition Associated with Economic Growth,”International Economic Review 38, 431-452. [6] Kongsamunt, P., Rebelo, S. and Xie, D. (2001). “Beyond Balanced Growth,” Review of Economic Studies 68, 869-882. [7] Lucas, R. E. (1987). “Models of Business Cycles,”Basil Blackwell. [8] Lucas, R. (1988). “On the Mechanics of Economic Development,” Journal of Monetary Economics 22, 3-42. [9] Mulligan C. and Sala-i-Martín X. (1993). “Transitional Dynamics in Two-Sector Models of Endogenous Growth,”Quarterly Journal of Economics 108, 737-773. [10] Ngai, R. and Pissarides, C. (2007). “Structural Change in a Multi-sector Model of Growth,”American Economic Review 97, 429-443. [11] Perez, F. and Guillo, D. (2010). “Reexamining the Role of Land in Economic Growth,”Manuscript. [12] Perli R. and Sakellaris P. (1998). “Human Capital Formation and Business Cycle Persistence,”Journal of Monetary Economics 42, 67-92. [13] Ramsey, F.P. (1928). “A Mathematical Theory of Saving,”Economic Journal 38, 543–559. [14] Rebelo, S. (1991). “Long-run Policy Analysis and Long-run Growth,”Journal of Political Economy 99, 500-521. [15] Rebelo, S. (1992). “Growth in Open Economies,”Carnegie-Rochester Conference Series on Public Policy 36, 5-46. [16] Reiss, J. P. (2000). “On the Convergence Speed in Growth Models,” FEMM Working. Paper 22/2000. 22 [17] Steger, T.M., (2000). “Economic Growth with Subsistence Consumption,”Journal of Development Economics 62, 343-361. [18] Steger, T.M., (2006). “Heterogeneous Consumption Goods, Sectoral Change and Economic Growth,”Studies in Nonlinear Dynamics and Econometrics 10, No. 1, Article 2. [19] Uzawa, H. (1965). “Optimum Technical Change in an Aggregative Model of Economic Growth,”International Economic Review 60, 12-31. 23 A. Appendix Solution to the consumer’s optimization problem. The Hamiltonian function associated with the maximization of (3.1) subject to (2.5), (2.6) and (2.7) is H=etU(c1; c2) + (wh +rk c1pc2IkphIh) + 1(Ikk) + 2(Ihh); where ,1, and 2are the co-state variables corresponding to the constraints (2.5), (2.6) and (2.7), respectively. The …rst order conditions are et 2 6 4 c 1c1 21 c13 7 5= 0;(A.1) et 2 6 4 (1 )c 1c1 21 c23 7 5p = 0;(A.2) =1;(A.3) ph=2;(A.4) r 1=_1;(A.5) w 2=_2:(A.6) Combining (A.1) and (A.2), we obtain (3.2) and _c2 c2 =_c1 c1 _p p:(A.7) Using (A.3) and (A.4), we obtain ph1=2; which implies that _ph ph +_1 1 =_2 2 ; and (3.3) follows from using (A.5) and (A.6). Combining (A.1), (A.3) and (A.5), we obtain r+=+ [(1 )1] _c1 c1+ (1 ) (1 )_c2 c2; and (3.4) follows from using (A.7). Finally, the transversality conditions (3.5) and (3.6) follow from combining (A.1) and (3.2). Proof of Proposition 3.2. The uniqueness of pfollows from the monotonicity of (p), which can be shown using (3.26), 0(p) = "(1 )A1 1 1p1  #"+  1 ' !p1+ #>(<) 0 if  < (>); 24 and the fact that lim p!0(p) = 1(1)and lim p!1(p) = 1(1)when  < (>): Combining (3.20), (3.21) and (3.22), we obtain u1=z3z z3z1 + 1 pA2z 2!z2z3 z3z1qz (A.8) and 1u1u2=zz1 z3z1 + 1 pA2z 2!z1z2 z3z1qz: (A.9) In a steady state, equations (3.25) and (3.24) simplify to 1u 1u 2=g+ A3(z 3); A1u 1(z 1) zq=g+: By using (A.8) and (A.9), the previous two equations can be rewritten as the following system of two equations: z+ 1 pA2(z 2)! | {z } 1 (z 1z 2)qz=g+ A3(z 3)(z 3z 1) + z 1 | {z } 2 ; z 3+ 1(z 2z 3)(z 3z 1) A1(z 1)qz=(z 3z 1)g+ A1(z 1)+ 1 | {z } 3 z: The steady state values of zand qare the unique solution of this system of equations and they are equal to z= 12(z 2z 3) + 1(z 1z 2)z 32(z 3z 1) A1(z 1) 1(z 2z 3) + 13(z 1z 2)z 3z 1 A1(z 1) ; and q=23z 3 12(z 2z 3)2(z 3z 1) A1(z 1)+1(z 1z 2)z 3; where the steady-state values of zi; i =f1;2;3g;satisfy z i= i(p)1 as follows from (3.15). 25 Figure 3. Transitional dynamics with = 2:357 —Economy with = 0:3- - - Economy with = 1 32 Figure 4. Transitional dynamics with = 2:7143 —Economy with = 0:3- - - Economy with = 1 33 Figure 5. Dynamic e¤ects of a biased technological shock when = 2 —Economy with = 0:3- - - Economy with = 1 34 Figure 6. Dynamic e¤ects of an unbiased technological shock when = 2 —Economy with = 0:3- - - Economy with = 1 35 Table 1. Welfare cost of imbalances in the capital ratio z0= (0:75) z   = 0:3(a) = 1 (b) a=b 2 7:0608% 5:8959% 1:1976 2:357 7:0619% 5:8954% 1:1979 2:7143 7:0634% 5:8959% 1:1980 z0= (1=0:75) z   = 0:3(a) = 1 (b) a=b 27:6300% 6:5049% 1:1730 2:357 7:6284% 6:5039% 1:1729 2:7143 7:6275% 6:5031% 1:1729 Table 2. Welfare cost of technological shocks (= 2) Type of shock = 0:3(a) = 1 (b) a=b Sectoral biased: A1=0:15A114:3821% 26:4386% 0:5440 Sectoral unbiased: A1 A1 =A2 A2 =A3 A3 =0:05 13:5843% 13:3788% 1:0154 36