A Simple Theory of the Aggregate Production Function

A Simple Theory of the Aggregate Production Function Javier A. Birchenall & Rish Singhania University of California at Santa Barbara Kang H. Cao US Department of Transportation Preliminary and Incomplete. Please do not cite. May 11, 2012 Abstract This paper reconciles a key microeconomic fact sectoral heterogeneity in production technologies and a key macroeconomic fact the relative constancy of the aggregate labor share. The model economy s aggregate production function has two distinct features: aggregate capital-labor substitution occurs by substitution within and across sectors, and the aggregate labor share is a weighted average of the sectoral labor shares. This measurement of the labor share has a direct empirical counterpart in national income data. We apply the theory to study the recent decline in the aggregate labor share in Continental Europe. We ask: How much of this decline can be accounted for by an increase in labor taxes? How much of this decline can be accounted for by an increase in globalization? We find that a 10% increase in labor taxes reduce labor s share of national income by 5 percentage points. Offshoring also reduces labor s share of national income. However, at current offshoring levels, the magnitude of this reduction is quantitatively insignificant. 1

1 Introduction We present a simple way to reconcile sectoral heterogeneity in production technologies, a key microeconomic fact, with relatively constant aggregate factor shares, a key macroeconomic fact. Economists know these facts by heart, but we know little about the reasons for the relative constancy of aggregate factor shares and even less about the relationship between microlevel heterogeneity and the aggregate economy. We analyze an economy with a large number of intermediate sectors. Using a constant returns to scale sector-specific production function, each sector combines capital and labor to produce an intermediate good. Capital and labor are homogeneous and traded in competitive markets. Intermediate goods are then combined using a Dixit-Stiglitz aggregator to produce a final consumption good. We derive an aggregate production function with two distinctive features. First, substitution between capital and labor takes place through two separate channels: substitution between labor and capital within each sector, and substitution between the intermediate goods produced in the different sectors. The conventional aggregate production function provides no explicit account for sectoral heterogeneity. As we show, ignoring this channel of substitution may misrepresent the response of aggregate factor shares to policy changes. Second, the aggregate labor share in our model economy is a weighted average of sectoral labor shares. The weights correspond to the sector s contribution to value added. This measurement of the aggregate labor share has a direct empirical counterpart in sectoral input-output data. Thus, we can apply the same measurement methodology as input-output data to identify the sources of change in aggregate labor shares in our model economy. Our model can reproduce fairly constant aggregate factor shares and, in a special case (Proposition 2), we obtain an exact aggregate Cobb-Douglas production function. The intuition behind these results is fairly simple. If intermediate goods have Cobb-Douglas production functions, changes in the 1

0.80 0.75 Aggregate Labor Share 0.70 0.66 0.61 0.56 1970 1975 1980 1985 1990 1995 2000 2005 2010 Year France Germany Italy UK US Figure 1: Aggregate Labor Shares - EU KLEMS aggregate labor share are only due to changes in value added weights. Thus, if value added weights vary little then the aggregate labor share will be relatively constant. In a Dixit-Stiglitz aggregator, the elasticity of substitution between intermediate goods determines how responsive value-added weights are. If the intermediate good elasticity approaches one, then value added weights tend to a constant. Hence, inspite of sectoral heterogeneity, we obtain an exact aggregate Cobb-Douglas production function. In contrast, if intermediate goods are perfect substitutes, it is optimal to concentrate all factors in a single sector, which effectively eliminates any sectoral heterogeneity in the economy. 1 Between these two extremes, we find numerically a wide range of values of the elasticity of substitution between intermediate goods and several parameterizations consistent with relatively constant aggregate factor shares. We apply the previous microeconomic structure to two problems in aggregate economics. First, we examine the distributional consequences of tax 1 This result is well-known and stems from the fact that under constant returns to scale the most efficient sector absorbs all resources. This result can be seen as analogous to Ramsey s classical conjecture regarding heterogeneity in time preference rates. In an economy with exponential discounting and time preference heterogeneity, the most patient individual will eventually hold all the capital. 2

policies. Since the mid-1970s, OECD countries have experienced a steady decline in labor s share of income. Figure 1 depicts aggregate labor shares in Germany, France, Italy, the United States, and the United Kingdom. Labor shares in continental Europe have declined more than 7 percentage points since the mid-1970s while the aggregate labor share in the US and the UK have remained virtually unchanged. Our policy experiment illustrates how tax policies alter the shape of the aggregate production function. We show later on (Table 3) that changes in aggregate labor shares are negatively correlated with changes in labor taxes. That is, the decline in aggregate labor shares has been larger where labor taxes have increased more, i.e., in continental Europe. We also show that the relative importance of high-labor intensive sectors to value added has consistently declined in Europe over time. In our numerical examples, the observed changes in effective tax rates are able to explain the majority of the observed declines in labor shares in continental Europe. Our framework also incorporates explicitly the sectoral heterogeneity needed to capture the observed substitution from labor- to capital-intensive sectors. The second application examines the impact of a decline in the price of labor-intensive intermediate goods on the aggregate labor share. Over the past two decades, markets around the world have become increasingly integrated raising concerns about the impact of globalization on income inequality. 2 category that has played and increasingly important role in international trade is trade in intermediate goods, see, e.g., Feenstra and Hanson (2003) and Grossman and Rossi-Hansberg (2010). Our numerical analyses show that having access to lower prices for labor-intensive intermediate goods lowers the 2 Several papers have examined empirically the relationship between openness (proxied by a number of direct and indirect measures) and the aggregate labor share; see, e.g., Ortega and Rodriguez (2001); Harrison (2002); Guscina (2006); Jayadev (2007); Jaumotte and Tytell (2007); Rodriguez and Jayadev (2010). This literature has found a robust negative relationship between openness measures and the aggregate labor share of developed countries. The existing literature, however, has focused primarily on the impact of trade on the bargaining power of labor relative to capital; see, e.g., Rodrik (1997); Ortega and Rodriguez (2001); Harrison (2002); Bental and Demougin (2010). A 3

aggregate labor share but that these changes are significant only at levels of trade that exceed those currently seen in developed countries. Related literature. Our paper is most closely related to Acemoglu and Guerreri (2008). These authors considered an economy with two intermediate sectors, each with a Cobb-Douglas production function. In their model, intermediate goods are also aggregated using a Dixit-Stiglitz aggregator. Acemoglu and Guerreri (2008) fully characterized the equilibrium allocation of labor and capital, and noted, using a calibration exercise, a slow convergence to the stationary allocation when the elasticity of substitution between intermediate inputs is lower than one. One of our main objectives is to highlight the key reason for the slow convergence in Acemoglu and Guerreri (2008). We show that the relative constancy of aggregate factor shares seen in their numerical results is primarily a result of aggregation. We also show that relatively constant aggregate factor shares can be obtained for a larger class of environments and parameterizations. For example, in our simulations, the aggregate factor shares are relatively constant even when the elasticity of substitution between intermediate goods is above one. We also examine aggregate labor shares under labor-augmenting technical change, capital-skill complementarities, sectoral CES production functions, and heterogeneous capital. Houthakker (1955-56) is an early theoretical study which, based on the aggregation of explicit microeconomic heterogeneity, also obtained an aggregate Cobb-Douglas production function. The underlying microeconomic structure in Houthakker (1955-56), however, has a series of counterfactual implications. We show that sectoral value-added weights and sectoral labor shares should exhibit positive trends not seen at the level of disaggregation we study. Our simulations are also related to Fisher (1971) and Fisher et al. (1974). These studies examined the statistical fit of an aggregate Cobb-Douglas function to 4

simulated data generated from firms with heterogeneous technologies. 3 Although not as a central theme, we revisit their numerical results. We show that their assumption of random capital accumulation plays a central role in their findings. Relaxing this assumption leads to a rapid concentration of labor and capital in a single sector. This rapid convergence means that in the absence of random capital accumulation, sectoral heterogeneity in Fisher (1971) and Fisher et al. (1974) is lost fairly rapidly. Our paper is also related to studies on the relative performance of labor market outcomes in Europe and the US; e.g., Blanchard (1997); Bentolila and Saint-Paul (2003); Prescott (2004); Rogerson (2009). Blanchard (1997) argued that the decline in the aggregate labor share in continental Europe is due to an adverse shift in labor demand and suggested biased technological change or a shift in the workers ability to capture rents. Our focus on labor taxes provides a parsimonious approach to examine shifts in labor market conditions consistent with observed changes in hours of work; see, e.g., Prescott (2004) and Rogerson (2009). Bentolila and Saint-Paul (2003) studied the relationship between sectoral labor shares and capital-output ratios, technological change, and non-competitive wage setting. Their analysis, however, did not focus on the relationship between micro-level heterogeneity and the aggregate economy. The paper proceeds as follows. Section 2 presents the theory. Section 3 presents the quantitative results. Sections 4 and 5 examine the applications 3 There is a large literature on the exact aggregation of production functions; see, e.g., Fisher (1969) for an early contribution and Felipe and Fisher (2003) for a recent survey. Our framework is consistent with an exact Cobb-Douglas production function but our analysis can be better described as approximate aggregation of production functions. It is known that heterogeneous production functions cannot be reconciled theoretically with an aggregate Cobb-Douglas production function on the basis of exact aggregation. Exact aggregation requires all microeconomic units to have the same technology of production (among other stringent requirements), see Fisher (1969, Theorem 1). There is also a large literature that examines technological change and the relative constancy of aggregate labor shares; see, e.g., Acemoglu (2001) and Jones (2003). Our emphasis differs from that of existing papers since we consider sectorally heterogeneous technologies. These papers, in contrast, take the aggregate production function as a primitive in order to focus on the choice of technology. Finally, we do not focus on measurement aspects. It is particularly difficult to distinguish the returns to capital from the returns to labor in the data. Judgements must be made about the assignment of self-employment, imputations for physical capital, and the appropiate place for human capital. Krueger (1999), Gollin (2002), and Gomme and Rupert (2004) discuss several problems associated with the measurement of factor shares. 5

just mentioned. Section 6 concludes. The Appendix contains supplemental results. 2 A simple theory In this section we present a simple theory of the aggregate production function. We adopt the following notation throughout the paper: lower case letters refer to sectoral-level variables and the corresponding upper case letters refer to aggregates. Environment. Time is discrete, indexed by t = {0, 1,..., }. There is an infinitely-lived representative household. Consumption at date t is given by C t. Preferences over consumption are: U(C) β t u(c t ), (1) t=0 with a discount factor β (0, 1). We assume that the period utility function u(c t ) is well-behaved. In our quantitative section, we will restrict attention to iso-elastic utility functions. Physical capital and labor are given by K t and L t, respectively. The representative household supplies labor inelastically. Population is increasing exogenously at a rate n > 0. There is an initial endowment of capital, K 0 > 0, and capital depreciates at a constant rate δ (0, 1). Households supply capital and labor to firms, which in turn produce one final consumption good. Total output in the economy is Y t. The aggregate feasibility condition in period t is: C t + K t+1 (1 δ)k t Y t. (2) Output is produced by combining intermediate goods from a large number of sectors, each indexed by i [0, 1]. Let y t (i) denote the amount of interme- 6

diate good i produced in period t. Intermediate goods are combined using a Dixit-Stiglitz aggregator to produce the final good: [ 1 ] ε/(ε 1) Y t = y t (i) (ε 1)/ε, (3) 0 where ε (0, ) is the elasticity of substitution between the intermediate goods. In the limit, as ε, intermediate goods become perfect substitutes. If ε 0, intermediate goods are perfect complements and we define (3) to be Y t min i [0,1] {y t (i)}. Let k t (i) and l t (i) denote the amount of capital and labor used in sector i, respectively. Each intermediate good is produced with a production function given by: y t (i) = f(i, k t (i), l t (i)). (4) We index f(i, k t (i), l t (i)) by i because production functions are sectorspecific. We assume that the sectoral production functions f(i, k t (i), l t (i)) are time-invariant, display constant returns to scale in k t (i) and l t (i), and are strictly increasing, continuous, and strictly concave. Labor and capital are homogeneous and perfectly mobile across sectors. The resource constraint for labor is: 1 0 l t (i)di A t L t, (5) and the resource constraint for capital is: 1 0 k t (i)di B t K t. (6) The parameters A 1 and B 1 represent technical change. The case A > 1 and B = 1 represents labor-augmenting technical change and the case A = B > 1 represents neutral technical change. We examine these cases separately in the next section. 7

The aggregate production function F (B t K t, A t L t ) is the maximum amount of the final good that can be produced given effective labor, A t L t, and effective capital, B t K t. The aggregate production function is the solution to the following program: F (B t K t, A t L t ) { 1 ε/(ε 1) max f(i, k t (i), l t (i)) di} (ε 1)/ε, (7) k t(i),l t(i) 0 subject to (5) and (6). In the Lagrangian associated with the variational problem, let w t and r t be the multipliers associated with (5) and (6), respectively. In a competitive equilibrium, w t and r t will be the equilibrium wage rate of labor and the rental price of capital. Properties of the solution. We examine the properties of the solution to the static and dynamic problems separately. The next proposition characterizes the aggregate production function. Proposition 1 There exists a solution to the program (7). Moreover, the aggregate production function displays constant returns to scale, is weakly increasing, continuous, and weakly concave. Proof. The proof is standard; see, e.g., the proof of Proposition 1 in Prescott (2003). The feasible set is compact and non-empty and, given the assumptions on f(i, k t (i), l t (i)), the objective function is continuous. Therefore, a solution exists. Re-scaling labor and capital by a common factor re-scales the objective function by the same factor, which verifies homogeneity. The solution is weakly increasing since an increase in effective labor or capital enlarges the feasible set. Continuity follows from the Theorem of the Maximum. Concavity follows since the objective function is strictly concave and the feasible set is convex. The function F (B t K t, A t L t ) summarizes the aggregate production possibilities in our two-factor economy. Notice that the shape of the aggregate production function does not have to be the same as the shape of individual 8

production functions. For example, the aggregate production function will not generally be Cobb-Douglas even if all individual sectors have Cobb-Douglas technologies with heterogeneous technology parameters. Two special cases are of interest. First, if sectoral technologies are identical there is no aggregation problem. Thus, F (B t K t, A t L t ) = f(b t K t, A t L t ). Second, if intermediate inputs are perfect substitutes and production functions f(i, k t (i), l t (i)) are constant returns to scale, the most productive sector absorbs all labor and capital: { F (B t K t, A t L t ) = max f(i, B t K t, A t L t ) }. i [0,1] A competitive market. The program in (7) can be decentralized in a perfectly competitive economy. Assume that the intermediate goods are traded in perfectly competitive markets. Let P t be the price of the final good and p t (i) be the price of intermediate good i. Profits for the producer of the final good are where Y t is given by (3). { Π t max P t Y t y t(i) 1 0 } p t (i)y t (i)di, The demand function for intermediate good i is such that y t (i)/y t = (p t (i)/p t ) ε, where the price of the final good, as a function of the price of intermediate goods, is ( 1 1/(1 ε) P t = p t (i) di) 1 ε. (8) 0 As a normalization, we assume P t = 1. Intermediate producers rent labor and capital in competitive markets. Let w t and r t denote the market price of labor and capital, respectively. Profits 9

for the producers of intermediate goods are π t (i) where y t (i) is given by (4). max {p t(i)y t (i) w t l t (i) r t k t (i)}, k t(i),l t(i) The profit-maximizing conditions for l t (i) and k t (i) are: p t (i) f(i, k t (i), l t (i))/ l t (i) = w t, and p t (i) f(i, k t (i), l t (i))/ k t (i) = r t, for all i [0, 1]. These first order conditions equalize the value of the marginal product of labor and capital to their respective prices. The aggregate labor share. Let s t (i) be the labor share in sector i. Under the assumption that labor is paid its marginal product, the labor share of sector i is s t (i) w tl t (i) p t (i)y t (i) = f(i, k t(i), l t (i)) l t (i) l t (i) f(i, k t (i), l t (i)). (9) Let v t (i) be the value-added weight of sector i. That is, v t (i) is the relative weight of sector i in total income, i.e., p t (i)y t (i)/p t Y t. Then, v t (i) p t(i)y t (i) P t Y t = ( ) 1 1/ε yt (i), (10) where the equality follows from our previous normalization of the price of final goods, i.e., P t = 1. Finally, let S t be the aggregate labor share, S t w t L t /Y t. The aggregate labor share associated with the aggregate production function (7) is trivially given by: S t = 1 0 Y t v t (i)s t (i)di. (11) The aggregate labor share for our model economy is the weighted average of sectoral labor shares with the weights given by the value-added weights. This measurement is the same of traditional input-output methodology. An implication of (11) is that the aggregate labor share can vary due to changes between- and within-sectors. Let v t+1 (i) denote the value-added weight of 10

sector i in period t + 1, and let s t+1 (i) be the labor share of sector i. One way to describe the time variation in the aggregate labor share is as follows: S t+1 S t = 1 [v t+1 (i) v t (i)] s t (i)di + 1 0 0 [s t+1 (i) s t (i)] v t+1 (i)di. (12) The first term in (12) represents changes in between sectors weighted by the labor share of the sector in period t. The second term is the change within sectors weighted by the average value-added share in period t + 1. This decomposition is not unique (as other values can be used as a base to examine between- and within-sector changes), but it provides sufficient conditions to obtain an invariant aggregate labor share: Proposition 2 Suppose the sectoral production functions are Cobb-Douglas. If the elasticity of substitution between intermediate goods tends to one, then the aggregate labor share tends to a constant. Proof. Suppose the different sectors have heterogeneous Cobb-Douglas production functions, f(i, k t (i), l t (i)) = k t (i) α(i) l t (i) 1 α(i). Thus, s t (i) = 1 α(i) for all t. The within component in (12) will be identically equal to zero and all movements in the aggregate share will be due to changes in value-added weights. Next, suppose that ε 1. Then, the final good s production function becomes Cobb-Douglas, and p(i)y t (i)/y t is constant. That is, the value-added weights are constant, v t+1 (i) = v t (i) = v(i). The between component in (12) will also equal zero. Thus, as ε 1, S t 1 s t (i)di = 1 0 0 [1 α(i)]di, for all t, (13) which is the equal-weighted average of sectoral labor shares. Several features are important. First, sectoral technologies have the same Cobb-Douglas shape but they are all heterogeneous in their technological parameters. That is, labor shares differ across sectors. Second, the previous 11

result will apply even in an economy with heterogeneous capital goods but homogeneous labor. The reason is that (11) will also hold under heterogeneous capital goods as long as labor is traded in a competitive market and compensated according to its marginal product. We will study such an environment in the next section. Third, all intermediate goods are equally important in the production of the final good. However, the previous result will also hold if the relative importance of the intermediate goods is different. Let ψ(i) represent the relative importance of intermediate good i, with 1 ψ(i)di = 1. Suppose also that (3) 0 is of the form [ 1 ] ε/(ε 1) Y t = ψ(i){y t (i)} (ε 1)/ε di. 0 The prices of final and intermediate goods will change accordingly and the value added weights (10) will become v t (i) = ψ(i) (y t (i)/y t ) 1 1/ε. In this case, (13) will yield an aggregate labor share equal to 1 ψ(i)[1 α(i)]di, for all t, 0 which is also a constant. Fourth, we have taken the sectoral aggregator (3) as a primitive to the problem. Following Jovanovic (1996), we can consider one additional level of disaggregation such that (3) would be an indirect production function resulting from a more primitive program. Let z t (i) denote the number of plants to be operated for each of the possible intermediate goods. All plants within a given sector operate the same technology, but the scale of operations determines total output from the sector. Output from an intermediate sector that uses z t (i) plants is g(z t (i))y t (i). The plant-allocation problem is given by: Y t max z t(i) 1 0 g(z t (i))y t (i)di, subject to: 1 0 z t (i)di = 1. (14) From Jovanovic (1996, Lemma 1), it is possible to see that if g(z t (i)) = z t (i) 1/(1 ε), then Y t, the value function of the program (14), will be of the form (3). Thus, whereas Proposition 2 applies to sectoral data, it is possible 12

to take an even more disaggregated view of the production function. Proposition 2 differs from Acemoglu and Guerreri s (2008) results. They consider an economy with two sectors, each with a constant sectoral labor share, e.g., s t (1) = 1 α(1) and s t (2) = 1 α(2). In their model, as time goes by, sectoral heterogeneity disappears. The relative value-added weight between sectors i and j is v t (i)/v t (j) = (y t (i)/y t (j)) 1 1/ε, see (10). Thus, as long as ε > 0 (and regardless of the magnitude of ε), a growing economy with sectoral heterogeneity in production technologies will converge to a one-sector model. In Acemoglu and Guerreri (2008), as t, S t tends to 1 α(2) or 1 α(1) because value-added weights tend to 0 or 1, depending on whether ε is larger or smaller than one, see Acemoglu and Guerreri (2008, Proposition 1). In contrast to Acemoglu and Guerreri (2008), sectoral heterogeneity does not disappear in Proposition 2. Decomposition (12) suggests that we should also obtain relatively invariant aggregate shares when the elasticity of substitution between intermediate goods is in the neighborhood of one. Our numerical results confirm this intuition. Proposition 2 may resemble Houthakker s (1955-56) classical and influential derivation of an aggregate Cobb-Douglas production function. The approaches, however, are very different. While the aggregate labor share is constant, the underlying microeconomic structure in Houthakker (1955-56) has a series of counterfactual implications. In Appendix D, we apply the measurement of the aggregate labor share in (11) to the microeconomic structure in Houthakker (1955-56). We show that the assumptions in Houthakker (1955-56) imply that sectoral labor shares and value-added weights should grow at the rate of wage growth. These predictions seem counterfactual, at least at the one level of aggregation we examine in this paper. Houthakker (1955-56), however, may provide a better description of microeconomic production functions at more disaggregate levels. 13

Optimal accumulation. The dynamic accumulation program consists of maximizing the discounted present value of lifetime utility, U(C) in (1), subject to the aggregate feasibility condition (2). Suppose that the aggregate production function is at least once differentiable and recall that r t+1 is the rental rate of capital in period t + 1. The Euler equation associated with the capital accumulation program is: u c (C t ) = β(1 + r t+1 δ)u c (C t+1 ), for all t. (15) To ensure the existence of an optimal program, the discount rate β must be smaller than a critical value; see, Brock and Gale (1969, Lemma 2). For a period utility of the form u(c t ) = C γ t /γ, it is sufficient to assume β[a(1 + n)] γ < 1. This assumption serves to bound the rate of growth of feasible sequences. It is also possible to write an equivalent recursive formulation of the dynamic accumulation program. The recursive formulation of the dynamic program is available in Appendix B. 3 Quantitative results In this section we use the framework developed above to examine the shape of the aggregate production function F (B t K t, A t L t ). We focus on the time series behavior of the aggregate labor share S t and the resulting aggregate elasticity of substitution between capital and labor. Functional forms. We consider sectoral CES production functions: f(i, k t (i), l t (i)) = [α(i)k t (i) (σ(i) 1)/σ(i) + (1 α(i))l t (i) (σ(i) 1)/σ(i) ] σ(i)/(1 σ(i)), (16) where α(i) is a distributional parameter and σ(i) is the elasticity of substitution. If σ(i) = 1, we obtain the Cobb-Douglas production function, f(i, k t (i), l t (i)) = k t (i) α(i) l t (i) 1 α(i). The period utility function is u(c t ) = 14

C γ t /γ. We parameterize our benchmark case as follows: the number of time periods is T = 75 and the number of sectors is I = 50. We will consider a longer time horizon in the robustness section. The growth rate of labor is n = 0.01, the depreciation rate of capital is δ = 0.05, and the intertemporal elasticity of substitution parameter in consumption is γ = 0.5. These values reflect typical assumptions in neoclassical growth models; see, e.g., Barro and Sala-i-Martin (2003, pp. 66-67). We consider two values for the elasticity of substitution between the intermediate goods. First, ε = 0.76, which is the value used by Acemoglu and Guerrieri (2008, pp. 20). Second, we assume ε = 1.5, which is commonly used in the trade literature to measure the preference for variety in consumption. Notice that Proposition 2 applies to small values of ε and that these previous values do not necessarily represent small deviations from zero. As a robustness check, we will examine even larger values of the parameter ε. The benchmark values are summarized in Table 1. The production function in (16) has two parameters: the elasticity of substitution between capital and labor, σ(i) [0.75, 1.25], and the distributional parameter α(i) [0.2, 0.5]. Both parameters are uniformly distributed across sectors. The midpoint of s t (i) = 1 α(i) is 0.65 which coincides with the cross-sectional average value of the labor shares in our data. The range of α(i) is based on actual sectoral labor shares once imputed labor shares equal to 1 are excluded. When σ(i) = 1, s t (i) becomes sector i s labor share. In the benchmark CES case, α(i) is negatively related to σ(i). Later sections examine different assumptions about how α(i) and σ(i) are related across sectors. In terms of technical change, we consider two scenarios. First, we assume neutral technical change at rates B = A = 1.01, which we take directly from the calibration in Acemoglu and Guerrieri (2008). We also consider laboraugmenting technical change at a rate A = 1.01 and B = 1. It is know from Acemoglu and Guerrieri (2008) that under neutral technical change, the 15

Table 1: Benchmark Parameters. Description Symbol Value A. Non-production parameters Time periods T 75 Number of firms I 50 Growth rate of labor n 0.01 Depreciation rate δ 0.05 Elasticity of intertemporal substitution γ 0.50 B. Production parameters Elasticity of substitution - intermediate goods ε 0.76 Distributional parameter α(i) [0.20, 0.50] Elasticity of substitution - capital and labor σ(i) [0.75, 1.25] Labor Augmenting Technical Change A 1.01 Capital Augmenting Technical Change B 1.01 model economy exhibits non-balanced growth. In a one-sector model with labor augmenting technological change, Uzawa s Theorem implies balanced growth and constant factor shares; see, e.g., Acemoglu (2010, pp. 60). There is no analogous result for an economy with labor augmenting technical change and production heterogeneity at the sectoral level, which is the framework we examine here. Therefore, we present numerical results for both forms of technological change. Evaluation. We evaluate the performance of our numerical results in two main ways. We first plot the aggregate labor share S t in (11). This is the most transparent way to present our results. We also compute its average value, S = T 1 T t=1 S t, and the average change, S = T 1 T t=1 (S t+1 S t ). Since there is reallocation across sectors, we also examine the value-added contribution of the largest sector. This sector is referred to as the dominant sector by Acemoglu and Guerreri (2008). Let v t arg max i {v t (i)} be the value-added weight of the dominant sector in period t. We compute the average change in its value added by v = T 1 T t=1 (v t+1 v t ). This measure is indicative of 16

how fast sectoral reallocation takes place over time. Then, we examine the aggregate elasticity of substitution between capital and labor using the definition of this elasticity, as well as a series of least squares projections. That is, we evaluate the discrete analog of the elasticity of substitution formula: σ t = d ln[k t /L t ]/d ln[f L,t /F K,t ], and report the average of σ t, which is σ. 4 We also compute the aggregate elasticity of substitution between capital and labor based on: ˆσ arg min σ ln(y t /L t ) b L 0 σ ln w t b L t t, (17) ˆσ arg min σ ln(y t /K t ) b K 0 σ ln r t b K t t, (18) ˆσ arg min σ ln(k t /L t ) b R 0 σ ln(w t /r t ) b R t t. (19) The term b j 0 is a constant and b j t, j = L, K, R, is added to allow for a time trend usually included to take technological change into account. We compute b j t but also present values of ˆσ when b j t is assumed equal to 0. The logic behind using expressions (17) to (19) is that they are the pricing predictions in a competitive economy with an aggregate CES production function. These methods represent alternative ways to estimate the aggregate elasticity of substitution between capital and labor in the data; see, e.g., Antràs (2001) and León-Ledesma et al. (2010). Main results. The easiest way to see our results is to look at the time series of the aggregate labor share. As a benchmark, we show the behavior of the aggregate labor share under neutral and labor augmenting technological change. We also present results for the aggregate elasticity of substitution between capital and labor from our numerical simulations. If an aggregate Cobb-Douglas production function fits the generated data, the aggregate labor shares should be constant and the aggregate elasticity of substitution 4 We represent the aggregate elasticity of substitution between capital and labor by σ. Since sectoral elasticities are indexed by i, the previous departure from our convention should not lead to confusions. 17

0.90 1.00 Aggregate Labor Share 0.73 0.57 Aggregate Labor Share 0.73 0.47 0.40 0 25 50 75 0.20 0 25 50 75 0.90 1.00 Aggregate Labor Share 0.73 0.57 Aggregate Labor Share 0.73 0.47 0.40 0 25 50 75 0.20 0 25 50 75 Figure 2: Aggregate Labor Shares - Neutral Technical Change between capital and labor should be 1. Figure 2 plots the time series of the aggregate labor share under neutral technical change. This figure has two panels which show our results for the two different values of ε. Within each panel, we organize our results based on individual production functions, e.g., for sectoral production functions with the Cobb-Douglas and CES functional forms. The aggregate labor share in each figure is shown by the bold solid line. The dotted lines show the crosssectional maximum and the minimum labor shares across sectors. Given these values, one can easily infer the behavior of the cross-sectional distribution of labor shares over time. The main result is evident in Figure 2. Under both values of ε, the aggregate labor share is relatively constant. Despite the wide range of sectoral labor shares s t (i), the aggregate value S t, remains relatively unchanged over time. The average change in the aggregate labor share, in all four cases, is in the 18

fourth-decimal place. To put this change in perspective, the value of effective capital at the end of our simulations, relative to its initial value, in specification C-D is B T K T /BK 0 = 33.9. Thus, while labor, capital, and income exhibit rapid growth, the factor shares are relatively constant. Further, notice that the standard deviation of S t under a sectoral Cobb- Douglas specification with ε = 0.76 is 0.0008 whereas the same statistic for ε = 1.5 is 0.0016, which is twice the former value. Thus, as ε increases, labor shares become more variable. Notice also that sectoral CES production functions yield more variable aggregate labor shares in all specifications. These results are expected due to Proposition 2. Our statistical tests confirm the visual impression of relatively constant labor shares. The aggregate elasticity of substitution between capital and labor, when computed from F (B t K t, A t L t ) directly, ranges from 0.89 to 1.04. The least square projections in (17)-(19) give us estimates of this elasticity ranging from 0.99 to 1.16. In both cases, the elasticities are fairly close to 1, which make the function F (B t K t, A t L t ) indistinguishable from an aggregate Cobb-Douglas production function. Figure 3 plots the time series of the aggregate labor share under labor augmenting technical change. Once again, the aggregate labor share is relatively constant and the aggregate elasticity of substitution computed directly from F (B t K t, A t L t ) ranges from 1.02 to 1.05. After we account for the time trend, this aggregate elasticity ranges from 0.99 to 1.07. These values are in fact very close to those obtained under neutral technical change, which points to the difficulty of identifying the nature of technical change based solely on the shape of the aggregate production function. There is an important difference between the two forms of technological change at the sectoral level. Under neutral technical change and sectoral CES production functions, the distribution of sectoral shares fans out whereas the same distribution under labor augmenting technical changes converges to a 19

Neutral technical change Table 2: Numerical Results Labor-augmenting technical change ε = 0.76 ε = 1.5 ε = 0.76 ε = 1.5 C-D CES C-D CES C-D CES C-D CES A. Time series properties 0.6557 0.6403 0.6397 0.6054 0.6522 0.6426 0.6377 0.6095 S t (0.0008) (0.0029) (0.0016) (0.0153) (0.0006) (0.0017) (0.0004) (0.0059) 0.0001-0.0002 0.0001-0.0005 0.0000-0.0001 0.0000-0.0003 S (0.0001) (0.0002) (0.0001) (0.0174) (0.0001) (0.0003) (0.0001) (0.0006) v 0.0000-0.0041 0.0018-0.0040 0.0021-0.0041 0.0017 0.0000 (0.0000) (0.0349) (0.0104) (0.0348) (0.0075) (0.0349) (0.0100) (0.0000) 1.0102 1.0460 1.0138 0.8932 1.0201 1.0359 1.0281 1.0483 σ t (0.0290) (0.0441) (0.0551) (0.9397) (0.0231) (0.0535) (0.0493) (0.0528) B. Wage equation, (17) ˆσ ˆσ(b L 0 = 0) 0.9918 (0.0000) 1.0172 (0.0000) 0.9985 (0.0004) 1.1628 (0.0243) 0.9918 (0.0000) 1.0171 (0.0000) 0.9983 (0.0003) 1.0717 (0.0008) 2.2805 1.8305 1.7413 1.7757 2.4593 2.0209 1.9575 2.4016 (0.0804) (0.0485) (0.0481) (0.0385) (0.1754) (0.1197) (0.1221) (0.1555) C. Interest rate equation, (18) ˆσ ˆσ(b K 0 = 0) 0.9918 (0.0000) 1.0172 (0.0000) 0.9985 (0.0004) 1.0547 (0.0325) 0.9918 (0.0000) 1.0171 (0.0000) 0.9983 (0.0003) 1.0713 (0.0008) 0.3123 0.5659 0.5649 0.5506 0.9918 1.0171 0.9971 1.0746 (0.0421) (0.0274) (0.0278) (0.0285) (0.0000) (0.0000) (0.0003) (0.0007) D. Relative factor-price equation, (19) ˆσ ˆσ(b R 0 = 0) 0.9918 (0.0000) 1.0172 (0.0000) 0.9985 (0.0004) 1.1088 (0.0266) 0.9918 (0.0000) 1.0171 (0.0000) 0.9983 (0.0003) 1.0714 (0.0008) 0.9918 1.0170 0.9910 1.0036 1.5034 1.3730 1.3457 1.5799 (0.0000) (0.0000) (0.0005) (0.0122) (0.0611) (0.0426) (0.0442) (0.0601) 20

0.90 1.00 Aggregate Labor Share 0.73 0.57 Aggregate Labor Share 0.77 0.53 0.40 0 25 50 75 0.30 0 25 50 75 0.90 1.00 Aggregate Labor Share 0.73 0.57 Aggregate Labor Share 0.77 0.53 0.40 0 25 50 75 0.30 0 25 50 75 Figure 3: Aggregate Labor Shares - Labor Augmenting Technical Change stationary distribution. This convergence to a stable distribution is indicative of balanced growth. The difference in the distribution of sectoral labor shares thus provides an indirect way to identify the nature of aggregate technical change. 5 Sensitivity analysis. We first examine how sensitive the results are to alternative parameterizations. Optimal capital accumulation limits the number of periods to T = 75. To expand the number of periods, we consider a fixed savings rate at a rate φ = 0.20. Capital is accumulated according to K t+1 = φy t + (1 δ)k t. Proposition 2 is a consequence of aggregation and it does not require an optimal capital accumulation framework. Under fixed savings rates, we can extend the analysis to T = 150. Next we consider two different values of the savings rate, φ: φ = 0.10 and 5 The elasticity of substitution and the nature of technical change cannot be separately identified in aggregate time series, as shown by Diamond et al. (198$). Recently, León-Ledesma et al. (2010) revisited this result and suggested a modified identification strategy, which does not involve the use of sectoral data advocated here. 21

Aggregate Labor Share 0.660 0.657 0.653 Aggregate Labor Share 0.660 0.650 0.640 0.650 0 50 100 150 0.630 0 50 100 150 Aggregate Labor Share 0.660 0.650 0.640 0.630 0 50 100 150!" Aggregate Labor Share 0.650 0.617 0.583 0.550 0 50 100 150 Figure 4: Aggregate Labor Shares - Robustness to φ φ = 0.30. The resulting aggregate labor shares are displayed in Figure 4. In this figure, φ = 0.20 is indicated by the solid line. The aggregate labor share when φ = 0.10 is indicated by the dashed line, and the case φ = 0.30 is shown by a dashed line marked with crosses. It is evident from Figure 4 that our results for C-D are not sensitive to changes in φ. When individual production functions are CES, we examine the importance of σ(i), the elasticity of substitution between labor and capital in sector i, in determining the behavior of the aggregate labor share. We consider two experiments on σ(i). First, we alter the relative distribution of α(i) and σ(i). Second, we restrict all σ(i) to be less than one, σ(i) [0.25, 0.75]. Our results are displayed in Figure 5. The distribution of labor shares across firms is crucial to understanding the behavior of the aggregate labor share in the CES case. Hence, we indicate the maximum and the minimum labor shares by dotted lines. In the benchmark, firms with high values of α(i) have low σ(i), 22

Aggregate Labor Share 1.00 0.70 0.40,-/.10324 Aggregate Labor Share 0.80 0.70 0.60! #"%$'&)(#"%* 0.10 0 50 100 150 0.50 0 50 100 150 1.00 +,- 56037. 0.80 Aggregate Labor Share 0.67 0.33 Aggregate Labor Share 0.70 0.60! #"%$'&)(#"%* 0.00 0 50 100 150 0.50 0 50 100 150 Figure 5: Aggregate Labor Shares - Robustness to σ α(i) and σ(i) are negatively related. Here, we reverse this correlation. Figure 5 shows that the aggregate labor share under this parametrization is relatively stable. When σ(i) [0.25, 0.75], the aggregate labor share is also stable. 4 Policy experiments and the labor share In this section we apply our previous framework to study the distributional effect of labor taxes. Prescott (2004) has observed that hours worked per adult are markedly lower in continental Europe compared to the US, see also Rogerson (2009). Both of these previous papers argue that tax policies can account for a large fraction of these differences. Figure 1 has previously shown that aggregate labor shares differed markedly since the mid-1970s. To examine these differences, we rely primarily on the EU KLEMS database (http://www.euklems.net), which covers most countries 23

in the European Union, in addition to Japan and the United States. This database includes detailed measures of output and inputs at the industry level for 28 sectors, roughly at the 2-digit SIC level. We calculate aggregate labor shares as the weighted average of each sector s labor shares, with weights equal to the value-added shares; e.g., as in (11). We compute the value added as the value of output subtracting the value of intermediate inputs, and making the necessary adjustments to account for self-employment. 6 Table 3 reports the average value of aggregate labor shares for the G-7 countries during 1975-79 and 2003-07. We take the mid-1970s as the starting point because disaggregate information by industries in EU KLEMS is not available before that period. Aggregate labor shares also appear to have reached a peak at that time, see Bentolila and Saint-Paul (2003, Table 1). To assess the validity of the EU KLEMS data, we also report the aggregate labor shares from the OECD Economic Outlook (http://stats.oecd.org/). Table 3 also reports the effective marginal tax rates on labor income as calculated in Prescott (2004) for a period roughly coincident with that of the labor shares. Table 3 shows differences in aggregate labor shares even among countries with similar technologies. The table also shows a decline in aggregate labor shares in continental Europe. This decline has been larger in the countries where labor taxes have increased more. The US and the UK had virtually no change in labor taxes and no change in labor shares. France, Italy, and Japan have large increases in labor taxes accompanied by large declines in labor shares. The correlation between the change in aggregate labor shares in EU KLEMS and the change in labor taxes is 0.78. The same correlation is 0.74 in the OECD database. These correlations suggest that tax policies strongly influence aggregate labor shares. 7 6 We follow Gollin (2002) and scale up the total compensation by the average employee compensation in the economy. This method is based on the assumption that proprietor s labor incomes equal the average employee compensation. 7 Jaumotte and Tytell (2007) and the IMF World Economic Outlook (2007, Chapter 5) examined empirically the impact of globalization, technological change, and tax policies on aggregate labor shares. Their econometric study for a panel of 18 developed countries over 1982-2002 found that labor taxes were 24

Table 3: Aggregate labor share and labor taxes in G-7 countries. Aggregate labor share Effective labor EU KLEMS OECD income tax 1975-79 2003-07 1975-79 2003-07 1970s 1990s Germany 0.69 0.62 0.75 0.67 0.52 0.59 France 0.74 0.63 0.79 0.67 0.49 0.59 Italy 0.69 0.59 0.81 0.67 0.41 0.64 Canada 0.64 0.60 0.66 0.61 0.44 0.52 United Kingdom 0.72 0.72 0.71 0.69 0.45 0.44 Japan 0.73 0.56 0.72 0.58 0.25 0.37 United States 0.66 0.63 0.68 0.66 0.40 0.40 To examine this issue, we follow Prescott (2004) closely. Suppose that labor-leisure choices are based on a representative household with preferences given by u(c t, H t ) = ln C t + ɛ ln(1 H t ), where H t denotes aggregate hours of work and the parameter ɛ > 0 specifies the value of non-market productive time for the household. Suppose that taxes are levied on the producers of intermediate goods, although who pays the tax is not relevant for the allocation. The household s budget constraint is C t + K t+1 = w t H t + (1 + r t δ)k t + T t, where T t represents transfers in period t. The household s first order condition is standard, ɛc t /(1 H t ) = w t. The resource constraint for hours of work is analogous to (5) with H t in place of A t L t. The resource constraint for capital is (6). Profits for the producers of intermediate goods are π t (i) {( ) ( ) ( ) } pt (i) wt rt max y t (i) l t (i) k t (i), k t(i),l t(i) 1 + τ c 1 τ h 1 τ k strongly negatively associated with aggregate labor share, even after controlling for country fixed-effects and measures of globalization. 25

where τ c, τ h, and τ k are the consumption, labor, and capital tax rates, respectively. The tax τ h > 0 raises the cost of labor. The effective marginal tax rate on labor income can be written as 1/(1 τ) = [1/(1 τ h )]/[1/(1 + τ c )], which is identical to the one in Prescott (2004, Equation (5)). Suppose that sectoral production functions are CES, (16). The first order condition for labor can be written as (1 τ)(1 α(i))(l t (i)/y t (i)) 1/σ(i) = w t. Similarly, sectoral labor shares are s t (i) w t l t (i)/y t (i) = (1 τ)(1 α(i)) (l t (i)/y t (i)) 1 1/σ(i). Using (11) and simple rearrangements to the household s first order condition, it is possible to write the key equilibrium relationship for aggregate hours of work as follows H t 1 H t = (1 τ) ɛ ( Yt C t ) 1 ( ) 1 1/σ(i) ( ) 1 1/ε lt (i) yt (i) (1 α(i)) di. (20) y t (i) 0 The previous expression has a new element compared to a similar expression in Prescott (2004, Equation 8). 8 The last term represents the before tax aggregate labor shares, which are no longer constant, as assumed by Prescott (2004). Within our framework, the factor shares respond to changes in taxes in two different ways. First, an increase in the cost of labor prompts sectors to substitute labor for capital. This form of substitution is driven by the elasticity of substitution σ(i), and it is channeled as changes in sectoral labor shares, s t (i). Second, labor market policies change the distribution of resources across sectors. Labor taxes trigger a reallocation toward capitalintensive sectors, channeled as changes in value-added weights, v t (i). These substitution possibilities depend on the parameter ε. To examine the importance of the alternative channels of substitution in the model, we compute (20) and aggregate labor shares in the benchmark parameterization. We let effective taxes range from τ [0, 0.20], which are roughly consistent with the observed changes in Table 3. For a given tax rate, 8 Prescott (2004) emphasized the intratemporal distortions introduced by the tax τ and the intertemporal factors that alter consumption, C t /Y t. Y t 26

0.66 0.66 Aggregate Labor Share 0.58 0.51 Aggregate Labor Share 0.58 0.51 0.43 0.00 0.05 0.10 0.15 0.20 0.43 0.00 0.05 0.10 0.15 0.20 Figure 6: Response of Aggregate Labor Shares to Labor Taxes 35 35!" 31 31 # Hours Hours 27 27 23 0.00 0.05 0.10 0.15 0.20 23 0.00 0.05 0.10 0.15 0.20 Figure 7: Response of Aggregate Hours to Labor Taxes we compute numerically the steady state values of capital, based on the Euler equation (15), and then we solve numerically the equilibrium hours and the aggregate labor share. In our simulations, there are no transitional dynamics. Figure 6 plots the before tax aggregate labor share as a function of the tax rate, τ. (After tax labor shares would decline mechanically due to a tax increase.) We present four specifications: C-D and CES with ε = 0.76 and ε = 1.5. In all specifications, the aggregate labor share declines as the tax increases. The predicted decline in aggregate labor shares is large and consistent with the observed changes in Table 3. 27

Figure 7 considers the response in aggregate hours to tax changes in our framework and compares it with the results obtained using constant aggregate factor shares, as assumed by Prescott (2004). We assume a constant aggregate labor share of 0.65. Figure 7 shows that the assumption of constant aggregate labor shares underpredicts the response in aggregate hours to changes in taxes. Hours of work always decline with taxes but hours decline more in our model economy than under constant factor shares. The decline in hours in our simulations is robust to the alternative values of ε, although hours decline more under ε = 1.5, marginally. While Prescott (2004) considered an aggregate Cobb-Douglas production function, he noted that the factor shares were important for the previous equilibrium relationship. A common criticism to Prescott (2004) is the use of large aggregate elasticities of labor supply. An alternative interpretation of our previous finding is that if aggregate labor shares decline due to an increase in labor taxes, (20) the value of the elasticity of labor supply needed to account for changes in hours of work is lower than in the case of constant aggregate labor shares. To examine the importance of the alternative channels of substitution in the data, we apply the measurement (11) and the decomposition (12) to the EU KLEMS data. The decomposition (12) uses as a base for the change in value-added weights the labor share of sectors in period t, s t (i), and as a base for the change in labor shares the value-added weights in period t + 1, v t+1 (i). It is possible to substitute the previous bases by s t+1 (i) and v t (i) respectively. In our analyses, we use the average values [s t (i) + s t+1 (i)]/2 and [v t (i) + v t+1 (i)]/2 as bases. Figure 8 shows these decompositions for a sample of countries. 9 The solid line represents the total change since the initial year in the sample and the dashed line the changes if the sectoral labor shares are assumed invariant; i.e., 9 The labor shares have declined markedly in France and Japan but these countries have experienced important changes in the self-employment category and we do not display the previous decomposition for that reason. The plot for Canada is very similar to the one of the US. These figures are available upon request. 28