The Role of Sectoral Composition in the Evolution of the Skill Wage Premium. Sara Moreira. Northwestern University

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1 The Role of Sectoral Composition in the Evolution of the Skill Wage Premium Sara Moreira Northwestern University February 208 PRELIMINARY AND INCOMPLETE Abstract The main contribution of this paper is to consider sector-specific productivity and factor-specific productivity effects in the analysis of the evolution of the skill wage premium. I use data on wages and employment for 30 industries that compose the U.S. economy to examine the evolution of employment and skill-intensity by sector. To guide the empirical work, I construct a simple two-sector model that explores the role of heterogenous technology in the process of structural change. I show that the college premium depends crucially on industry-specific elasticities of substitution between skilled and unskilled labor and on the nature of the factor augmenting technology. The model allows me to derive closed form expressions for the aggregate elasticity of substitution and estimate its evolution. I find that the degree of skilled-unskilled labor substitutability is diminishing as sectors with low flexibility e.g. services) are becoming increasingly important.. Introduction This paper is motivated by two empirical regularities of the U.S. labor market over the past four decades. The first regularity is the simultaneous rise in the aggregate relative supply of more educated workers and in the college premium Figure ). To explain the rise of quantities and prices of high-skilled workers, the literature explored the contribution of demand and technological factors in shaping wage inequality across skill groups. The leading I am grateful to Erik Hurst, Steve Davis and seminar participants at the University of Chicago for their comments and suggestions. sara.moreira@kellogg.northwestern.edu. Address: 22 Campus Drive, Evanston, IL

2 theory is that technological changes favor skilled workers. The second empirical regularity is that there is substantial diversity across sectors both in terms of changes in employment levels and in terms of skill intensity. Figure 2 documents the continuous structural change process by which labor is reallocated away from manufacturing and toward services. Overall, nonethe evidence suggests that in the past decades, technological change may have been originated in particular sectors of the economy and favored specific production inputs. Such empirical regularities call for research that studies the mechanisms through which change in the composition of an economy shape the degree of inequality among workers in the labor market. Much debate exists concerning the causes of increase in the skill wage premium. Several prominent explanations have been offered. The leading explanation builds on Tinbergen s 974, 975) characterization of the evolution of the wage structure as a race between technological development and access to education. The hypothesis is that wage structure changes are attributed to an increased rate of growth of the relative demand for more-skilled workers driven by skill-biased technological changes, largely associated with the spread of computers and microprocessor-based technologies in the workplace Acemoglu and Autor, 20). Thenone canonical model underlying Acemoglu and Autor 20)none is characterized by a single sector economy, in which technology is represented by an aggregate production function with constant elasticity of substitution where two distinct skill groups are imperfect substitutes, and factor-augmenting changes can be biased Katz and Murphy, 992). In this paper, I build on this hypothesis and construct a framework that allows technological growth to differ across sectors. My objective is twofold. First, I identify the mechanisms through which changes in the relative aggregate supply of skilled workers and differential labor augmenting technology changes the allocation of workers across sectors both at an aggregate level and by skill). Second, nonei assess the relative importance of composition effects on the evolution of the skill wage premium and present some empirical evidence. I start by developing a tractable model of structural change with only two sectors. I use this approach to clearly lay out the mechanisms through which differences in technologies shape the distribution of skill and unskilled workers across sectors. I use the two-sector model to obtain analytical expressions for the equilibrium relative wage and the equilibrium relative skill quantities supplied to each sector. These equilibrium relations are easily generalizable to a n sector economy. In then sector model, I decompose the change in the skill wage premium into movements in the aggregate supply of skilled and unskilled workers, shifts in sectoral biased labor augmenting technology, and neutral sectoral factor demand. When I generalize to the n sector model, the aggregate elasticity of substitution between skill levels I use sector and industry interchangeably. 2

3 is no longer constantand it becomes a function of the labor elasticities of substitution in each sector, the output elasticity of substitution across sectors and the composition of the economy. I also empirically test the importance of heterogeneity in production technologies.none In particular, I predict the evolution of skill wage premiumnone under the model and test if it better predicts the evolution of the skill premium relative to a standard approach that uses a constant elasticity of substitution and appoximates SBTC by a time trend as in Katz and Murphy 992))none. To address the importance of changes in the composition of the economy, I create a counterfactual evolution of demand holding the composition of the economy fixed. I also look at the impact of changes in the composition of the economy when measuring the effect of variation in labor supply. By combining these two effects, I can get an estimate of skill wage premium in which the allocation of workers across industries is kept constant over time. The theoretical framework of the model is based on a model of structural transformation of a closed economy where an unique final good is produced using two intermediate imperfect substitute goods. The inputs in the production of the intermediate goods are skilled and unskilled labor units. While the technology for the final sector is not subject to shocks normalization), the technology of the two intermediate sectors changes over time, to allow for potential skill-biased technical change and differences in Hicks-neutral labor productivity growth. The equilibrium evolves over time according to an exogenous technological change and changes in the relative supply of skilled versus unskilled labor. Both labor market and goods markets operate in a competitive, frictionless environment. Overall, the labor reallocation within and across sectors depends on differences in factor proportions, unequal substitutability between skilled and unskilled workers within each intermediate goods sector, and differences in factor-augmenting technology. If the sectors only differ in skill intensities or Hicks neutral labor augmenting technology), then the increase in the aggregate supply of skilled labor cannot lead to reversals in the skill intensity rank of sectors over time. Differences in the growth of SBTC and in the degree of skill substitutability can explain skill reversals. In particular, flexible sectors can switch from being relatively unskilled to relatively skilled as the skill-intensity of the economy increases. Overall, I show that the limiting equilibrium of this class of economies depend on whether the products of the two sectors are gross substitutes or complements meaning whether the elasticity of substitution between these products is greater than or less than one) and whether skilled and unskilled labor are gross substitutes or complements under skill biased technical change. The empirical estimation of the model entails several challenges. Due to lack of data, I assume that the labor market consists of two types of workers, skilled and unskilled, that are identified as high school and college graduates, respectively. This assumption implies 3

4 that the college premium relative wage of college graduates to high school graduates) is the summary measure of the market valuation of skills. Data comes from the series of consecutive March Current Population Survey, which provides reasonably comparable data on prior year s annual earnings, weeks worked, and hours worked per week for almost five decades, with information on the industry in which the interviewee worked. I gathered information on 30 broad industries that add up to the whole U.S. economy for the period I estimate the primary elasticities of substitution between skill and unskilled workers allowing for substantial variability in technology. This paper constitutes the first attempt to estimate the nature of factor bias technology and the elasticities of substitution between skill levels of labor inputs separately for each industry. There is, however, some literature on the elasticity between labor and capital for both aggregate and industry level Young, 202, Balisteri, McDaniel and Wong, 2003). I rely on those papers when discussing econometric methods. I use the estimates for the skilled-unskilled labor substitutability within each sector, to compute aggregate elasticities of substitution and estimates of shifts in aggregate labor demand. To construct the weights, I use the CPS sample averages on shares of employment by skill level and implied expenditure shares with college educated and high school workers for the period The preliminary evidence shows substantial differences in composition and time patterns. In particular, while education levels have increased within industries despite the fact that more educated workers became relatively more expensive education intensive industries have grown faster than those industries that are more dependent on unskilled labor. I also find that noneover the last 45 years, the aggregate elasticity of substitution has decreased substantially. Its levels have dropped from around.4 during the 970s to about 0.9 in the most recent years, implying that skilled and unskilled workers are becoming gross complements. Previous research had used aggregated data to estimate elasticities of substitution and most estimates are somewhere between.4 and 2 e.g. Heckman et al., 998) In future work, I will directly calibrate the model of structural transformation to the growth experience of the United States. I undertake a simple calibration of the benchmark model to provide a preliminary investigation of its potential. This paper is related to studies that examine the sectoral composition of the economy in explaining shifts in relative labor demand Katz and Murphy 992), Murphy and Welch 993), Berman, Bound, and Griliches 994), and Autor, Katz, and Krueger 998)). A common approach is to conceptualize relative demand shifts as coming from two types of changes: those that occur within industries i.e., shifts that change the relative factor intensities within industries at fixed relative wages) and those that occur between industries i.e., 4

5 shifts that change the allocation of total labor between industries at fixed relative wages). Within industry shifts could represent pure skill-biased technological change, changes in the relative prices of non-labor inputs, and outsourcing, whereas between-industry shifts are often associated with sectoral differences in productivity growth and shifts in product demand across industries from domestic or net-trade sources). The quantitative assessment of the contributions of different sources of relative labor demand shifts require strong assumptions about sectoral production functions and consumer preferences. For example, Katz and Murphy 992) use a simple framework based on the fixed-coefficients input requirements index, which measures the percentage change in the demand for skilled/unskilled labor as the weighted average of percentage employment growth by industry where the weights are the industrial employment distribution for skilled/unskilled labor in a base period. 2 main limitation of these approaches is that they impose perfect-complementarity between skilled and unskilled labor i.e. Leontief technology) and assume that sector technologies are held fixed except for factor neutral technological change. The authors acknowledge that these limitations is likely to imply an understatement of the true between-industry shift of the relative demand for skilled labor. Murphy and Welch 993) and Juhn 994) propose and implement adjustments for this bias under the strong assumption of unit own-price and zero cross-price elasticities of consumer demand. 3 The An alternative set of papers allows for within-sector factor-biased technological changes Berman, Bound, and Griliches 994), Autor, Katz, and Krueger 998)). Assuming Cobb-Douglas industry production functions and Cobb-Douglas consumer preferences, a standard shift-share decomposition of the growth of the aggregate share of skilled labor in total costs can be used to separate the growth in the relative demand for skilled work into a skill-biased technological change component withinsector effect) and a product demand shift component between-sector effect). 4 Murphy and Welch 200) revisit the hypothesis by computing some additional statistics of changes in employment between industries and conclude that we have argued consistently that we have 2 Katz and Murphy 992) empirical analysis explores industry and occupation variation by identifying demand shifts between cells of 50*3 industries-occupations as overall variation and shifts in employment among the 50 industries as between variation. This implies that shifts in employment among occupations within industries are classified as within variation. 3 Empirical analyses of the magnitude of between-industry and between-occupation shifts in relative labor demand using adjusted and unadjusted) versions of the index indicate strong and rather steady betweenindustry and between-occupation demand shifts favoring more-educated workers and high-wage workers. The results indicate that measured changes in the allocation of labor demand between sectors is important, but still much lower than the necessary variation to explain the evolution of the skill wage premium. Thus substantial within-industry and within-occupation demand shifts favoring the more skilled was indicated as a key driving force in the large secular increase in the relative demand for skilled workers. 4 The findings in Autor, Katz, and Krueger 998) suggest the within-sector effect is predominant and the rate of within-industry relative demand growth for college graduates appears to have increased from the 960s to the 970s and remained at a higher level in the 980s and 990s. 5

6 been too quick in discarding the simple early evidence of shifting industrial composition in favor of a less tractable view of non-neutral technical change page 278). Another stream of literature that relates to this contribution are the studies on structural change focused on the dynamics of capital. Two important contributions are Acemoglu and Guerrieri 2008) and Ngai and Pissarides 2007). These papers emphasize the potential nonbalanced nature of economic growth resulting from differential productivity growth across sectors Kuznets facts). Ngai and Pissarides 2007) work through differences in the rates of TFP growth across sectors, while assuming that the sectoral similar production functions are Cobb Douglas and the final sector has an elasticity of substitution less than one. The paper shows that the sector with lower TFP will use increasing fractions of capital and labor. Acemoglu and Guerrieri 2008) look at the role of distributional shares of labor and capital. In this paper, the aggregate elasticity of substitution is also assumed to be less than one, sectors have Cobb-Douglas production functions, but the input shares are different. They show that as capital accumulates, the fractions of capital and labor allocated to the less capital-intensive sector increases. Recently, Alvarez-Cuadrado and Van Long 20) also look at differences in elasticities of substitution between capital and labor within sector. The focus is this paper very different from this literature on structural change as they focus on the dynamics in the presence of capital. I explore the role of technological heterogeneity in the evolution of labor utilization across sectors, not capital. However, the theoretical model and comparative statics exercises developed here are similar to this stream of the literature. In this draft, I start by presenting the conceptual framework. Section 2 derives the theoretical model with two-sectors. Section 3 starts by generalizing the model to a n-sector model. I also discuss extensively the empirical methodology. Section 4 is dedicated to the description of data. I also present some evidence on the evolution of total employment, skill premium and skill intensity by industry. The preliminary results are presented in Section 5, where I show the estimates for the primary elasticities of substitution. In the last section, I discuss falsifiability tests and the necessary next steps to accomplish the research objective. 2. A framework for analyzing the role of heterogeneity in production To study the importance of heterogeneity in production for the evolution of the skill wage premium, I developed a simple model where the sectors have potentially different technologies. The aggregate relative demand for skilled labor will depend on the industry specific demand functions. The central tool used to derive factor demand functions is the production 6

7 function, which represents the heuristic device that describes the maximum output that can be produced from different combinations of inputs using a given technology. I next discuss the simplifying assumptions used to define industry specific production functions. Building on the specification of the technologies, I then develop the general equilibrium model of structural change that allows me to study the mechanisms behind the allocation of workers across sectors. From the model, I determine measures of substitutability between skilled and unskilled work and of skill biased technical change at an aggregate level. 2. Assumptions on the nature of technology Without loss of generality, the production function of unit i can be defined as Q it = Q i L it, X it, T it ) where Q it is a vector of outputs, L it is a vector of heterogeneous labor inputs, X it is a vector of other inputs e.g. types of capital, and materials) and T it denotes the state of technology. In order to simplify the analysis this paper limits its scope to functions that map a vector of inputs to a single output. I will further simplify the analysis at several levels. First, I will be working with labor value added functions. Sato 976) shows that a value added production function exists if there is perfect competition and if we can write the gross output production function as 5 : Q it = Q i F i L it, T it ), G i X it )) where F i L it, T it ) = P qitq it P xit X it P it There is some evidence on the aggregate level that supports this as I will show in Section 5. 6 Second, because I focus on explaining the evolution of the college premium, I reduced the dimensionality of the vector of labor inputs to two groups skilled and unskilled corresponding to college equivalents and high school equivalents. This implies that I am implicitly assuming that all heterogeneous types of labor are perfectly substitutable, except the distinction by schooling level. Section 4 offers further discussion of this assumption. Third, I assume a technology with constant returns to scale and constant elasticities of substitution. Basu and Fernald 997) study the validity of the assumption of constant returns to scale at several aggregation levels and conclude that although estimates of returns 5 Note that the conditions of Leontief s theorem need to be applied here. The theorem states that aggregation of labor inputs in a single index is possible if and only if the marginal rates of substitution of the variables in the aggregate production function are independent of the variables that are not included. 6 As discussed in Section 4 I face several data limitations. At the same time, there are substantial difficulties in defining proper concept of elasticity os substitution in the case of multiple inputs. 7

8 to scale vary widely across relatively disaggregated industries, the average industry appears to produce with constant returns. Constant elasticities of substitution are often used as a simplification of the analysis of production, especially in the case of one output and two factors of production Arrow, Chenery, Minhas and Solow 96)). Finally, I take a stance on the nature of technological progress, by assuming factor augmenting labor technology. Under these assumptions, the labor value added function of industry i is defined as none F i L it, T it ) Y it = β i A it L it ) i i ] + β i ) B it H it ) i i i i where H it and L it are skilled and unskilled labor respectively), i 0, ) is the elasticity of substitution and β i 0, ) is the distributive shares of sector i. The variables A it and B it are exogenous factor-augmenting technology shocks associated with unskilled labor and skilled labor, respectively. Under this specification the technologies of the different sectors differ in three elements: i) distribution parameters, β i, ii) the efficiency coefficients, A it s and B it s, and iii) the elasticity of substitution, i. Reducing the production of a sector to a simple mathematical function will mask many of the complex interactions that characterize a real economy. I follow the standard macroeconomic practice of applying firm-level theory to relatively aggregated data. Thus each sector will be treated as a representative firm equivalent to assuming that all firms in each sector have the same production technology) and the market is presumed to be in competitive equilibrium with all firms operating at lowest possible cost taking factor prices as parameters and neither losing money nor making excess profits. If this stripped down model can produce reasonably accurate predictions, then it can be a useful forecasting tool regardless of how well it describes reality. 2.2 A Model of structural change Final goods sector In this economy, there is a unique final good C), which is produced under perfect competition by combining two intermediate goods, Y and Y 2, according to a CES technology with constant elasticity of substitution ρ 0, ): C t = α Y t ) ρ ρ ] + α) Y 2t ) ρ ρ ρ ρ ) where α 0, ) is the redistributive share. In each period, the representative firm in the final sector maximizes the following condition, taking the prices of the final good, p t, and 8

9 intermediate goods, p t and p 2t as given: max {p t α Y t ) ρ ρ Y t,y 2t 0 ] + α) Y 2t ) ρ ρ } ρ ρ p t Y t p 2t Y 2t 2) Intermediate goods sectorss As described above, both intermediate-goods sectors i =, 2 operate in a competitive environment using two inputs: Skilled labor, H, and unskilled labor, L: none Y it = β i A it L it ) i i ] + β i ) B it H it ) i i i i for i =, 2 3) where i 0, ) is the elasticity of substitution and β i 0, ) is the distributive shares of sector i. The variables A it and B it are exogenous factor-augmenting technology shocks associated with unskilled labor and skilled labor, respectively. In each period, the representative firm of each sector maximizes its profit condition, given prices of the intermediate goods, p t and p 2t, and wages of skilled and unskilled labor, r t and w t respectively), max {p it β i A it L it ) i i L it,h it 0 ] + β i ) B it H it ) i i } i i w t L it r t H it for i =, 2 4) Households The economy is populated by an infinitely lived representative household of constant size. Without loss of generality, I normalize the population size to one. In each period, the household is exogenously endowed with L t units of unskilled time and H t units of skilled time, which are supplied inelastically. 7 The household has preferences over intertemporal consumption of the final good that are given as follows: δ t log C t t=0 where C t is consumption at time t and δ is the discount rate. Because I abstract from intertemporal decisions such as investment and savings) the problem of the household is effectively a sequence of static problems. At each date, households find their endowments of skilled and unskilled time, and given prices, they maximize the per-period utility ln C t 7 This is the assumption used in all empirical literature. See for example Katz and Murphy 992), Autor, Katz and Krueger 998) or more recently Acemoglu and Autor 20). 9

10 subject to the budget constraint. In this setting the agent s problem is simply: Market Clearing max C t 0 log C t s.t p t C t = w t L t + r t H t 5) The demand for labor from the two sectors must equate the exogenous supply of labor by households at each date Equilibrium L t + L 2t = L t H t + H 2t = H t 6) A competitive equilibrium is a set of prices {p t, p t, p 2t, w t, r t }, allocations of time {L t, L 2t, H t, H 2t }, and allocations of goods {C t, Y t, Y 2t } such that: i) given prices, the final product allocations none{y t, Y 2t } solve firm s problem in 2); ii) given prices, time allocations per sector none{l t, H t } and none{l 2t, H 2t } solve the firm s problem in 4) for each sector; iii) given prices and endowments H t and L t, household s allocations {C t } solve the household s problem in 5); iv) markets clear, i.e. equations 6) hold. The solution of the household problem is trivial, once she knows the price of final good and how much she is endowed with skilled and unskilled units of time C t = w tl t + r t H t p t Next, I solve the problem of the representative firm producing the final good. The first order conditions of this problem imply the following conditional demand conditions for the intermediate goods: ) ρ αp Y t = t Ct p t ) ρ Y 2t = α)pt 7) Ct p 2t In each intermediate sector, the conditional demand functions of skilled and unskilled labor are: L it = β i p it w t ) i Ait ) i Y it for i =, 2 H it = βi )p it r t ) i Bit ) i Y it for i =, 2 Free mobility of skilled and unskilled labor implies equalization of the value of the marginal products of each factor across sectors normalizing p t = ) 8) 0

11 ) αβ A t ) ) Y t C ρ t L t Y t ) α β ) B t ) ) Y t C ρ t H t Y t ) = w t = α)β 2 A 2t ) 2 ) 2 Y 2t 2 C ρ t L 2t Y 2t ) = r t = α) β 2 ) B 2t ) 2 ) 2 Y 2t 2 C ρ t H 2t Y 2t 9) The objective of solving the model is twofold. I determine the sectoral allocation of the two types of labor and I study how the skill wage premium is affected by technology. Before proceeding, I define some useful notation. The skill wage premium is defined as: ω t r t w t I also define the shares of labor allocated to sector l t L t L t, h t H t H t, the skill intensity indicators for the economy and in sectors and 2 and the relative factor efficiency s t H t L t, s t H t L t, s 2t H 2t L 2t, b 0t A t A 2t, b t B t A t, b 2t B 2t A 2t. 2.3 The allocation of workers across sectors The technologies of the intermediate sectors differ in three elements: i) distribution parameters, β i, ii) the efficiency coefficients, A it s and B it s, and iii) the elasticities of substitution, i. To explore the role of each element in the technology of the different sectors in the structural transformation of the aggregate technology, I will study specific cases in which I only allow one feature to be distinct across sectors in each case. The distribution parameters and the elasticities are defined as time-invariant parameters, while the A it s and the B it s evolve according to the nature of the labor augmenting technology. For the time invariant parameters, the analysis consists of doing comparative statics to determine l and h when the relative endowment of skilled workers changes. To study the impact of the efficiency coefficients on the evolution of l and h, it is particularly interesting to evaluate the role of Hicks-neutral labor augmenting technology, measured by the evolution of b 0t over time when b t and b 2t are constant, and the role of pure skill-biased labor technical change, measured

12 by the evolution of b t and b 2t when b 0t is constant. To determine the sectoral allocation of the two types of labor, I use equations 9. After some manipulation, I uncover the following equilibrium relations see Appendix A. for details): 8 where Φ.) = l l ) 2 Φ 2.) = h h ) 2 α α Y Y ρ 2 ρ 2 = α α β A ) β 2 A 2 ) 2 2 β β 2 Y ) B ) B 2 ) 2 2 ρ 2 ρ Y2 Y Y L 2 2 ρ 2 ρ 2 H 2 2 ] ρ β A l L) + β ) B h H) ρ ) ] ρ 2 β 2 A 2 l )L) β 2 ) B 2 h )H) 2 ρ 2 ) 2 = 0 0) = 0 ) Equations 0) and ), implicitly define functions with arguments l, h, A, A 2, B, B 2, L, H). These equations together with the definition of production function in equations 3) completely determine the equilibrium of the economy. with some simple algebra. Proposition. Comparative statics can be computed Assume that the technology is such that the elasticity of substitution is the same within sectors = 2 =, and technology is not labor augmenting i.e. A = B = A 2 = B 2 ). Suppose that sector is less skill intensive than sector 2 i.e. the cost share of skilled labor is higher in sector 2 than in sector ), then the fractions of skilled and unskilled workers allocated to the less skill-intensive sector increases as the aggregate skill-intensity increases if > ρ h s > 0 and l s > 0 and the reverse if < ρ. Moreover, as the aggregate skill-intensity increases, the sectoral skill intensities will grow noneat the same pace s s 2 ) s = 0 independently of the relation between and ρ. Proof: See Appendix A.2 Proposition indicates that if the share of skilled workers increases in the economy and the flexibility of substituting between intermediate goods is lower than the substitutability within 8 For the sake of simplicity of exposition, I dropped the time subscripts. 2

13 intermediate sectors, then the less skill-intensive sector will grow relatively more, in terms of the employment of both skilled and unskilled labor. The proposition predicts the opposite insofar as the elasticity of substitution between labor inputs is lower than the elasticity of substitution between intermediate goods. To understand the intuition behind this result, consider the special case where =, i.e. the intermediate sectors have Cobb-Douglas production functions, and as a consequence the cost shares of unskilled labor are constant and equal to the distributional parameters β and β 2, for sectors one and two, respectively. Suppose that the skilled-unskilled labor ratio increases, while h and l remain constant. Under these conditions, sector 2 will grow more because it uses relatively more skilled work under the proposition β 2 > β ). Once we allow for reallocation across sectors, we have that if the sectors are gross complements then the relative price of the intermediate goods will shift in favor of sector, implying a larger fraction of resources allocated to the less skilled intensive sector. In the case of gross substitutability of intermediate goods, an increase in the share of skilled hours in the economy will induce the skill-intensive sector to grow more. Under the technology that is implicit in Proposition, the increase in the supply of skilled labor combined with differences in skill intensities has no impact on the relative skill intensity across sectors because the shares of skilled l and l ) and unskilled employment h and h ) move at the same pace. Under this assumption, there are no reversals in the skillintensity rank across sectors. This pattern may not be consistent with the data. One way to allow for different patterns of sectoral reallocation is to allow for differences in the degree of skill substitutability. Proposition 2. Assume that the technology is such that the flexibility to substitute between skilled and unskilled labor of sector is lower than that of sector 2 i.e. 2 > ), the technology is not labor augmenting i.e. A = B = A 2 = B 2 ), and the distributional parameters are the same i.e. β = β 2 ). When the aggregate skilled-unskilled ratio increases, the fraction of skilled labor allocated to the less flexible sector decreases, while the fraction of unskilled labor increases, h s < 0 and l s > 0 As a consequence, the flexible sector will become relatively more skill intensive than the less flexible sector none Proof: See Appendix A.3 s s 2 ) s Everything else constant, when the economy becomes relatively more skilled, the wage < 0 3

14 skill premium decreases, implying that both sectors will demand relatively more skilled work. Skill intensity, s and s 2, will increase in both sectors. However, the sector with greater elasticity of substitution between types of labor, will increase its demand relatively more. Overall, the share of skilled work used by the flexible sector will increase while the share of unskilled work will decrease. Given full employment, the converse is true for the sector with lower ability to substitute. The diversity in the degree of skill substitutability may have an important role in explaining any observed reversals in skill rankings. In fact, under Proposition 2, the flexible sector can move from relatively unskilled to skilled as the skill-intensity increases. Below, I show how differences in sectoral rates of TFP growth drive structural change. The result is relatively general as it does not depend on the distributional parameters or the substitutability between skill levels in the intermediate sectors. At the same time, the basic features of the impact is recovered by setting = 2. Proposition 3. Assume that the technology is such that the elasticity of substitution is the same within sectors = 2 =, and the technology is not biased but sector has a higher Hicks-neutral growth i.e. A = B > A 2 = B 2 ). Then, the share of employment of skilled and unskilled workers allocated to the fast-growing intermediate sector sector ) increases as its relative level of total factor productivity increases if ρ > h A A 2 ) > 0 and l A A 2 ) > 0 and the opposite if ρ <. Moreover, as the aggregate skill-intensity increases, the sectoral skill intensities will grow noneat the same pace s s 2 ) s = 0 irrespective of whether ρ is above or below one. Proof: See Appendix A.4 The fractions of skilled and unskilled labor allocated to a sector either increase or decrease, but within each intermediate sector, both move in the same direction. To understand this result, note that if there is no substitution between intermediate outputs, then the sector with the highest TFP-growth will expand more. Once we allow for substitutability, it will depend on whether the intermediate goods are gross substitutes or gross complements. If the sectoral outputs are highly complementary in the production of the final good, the relative price of the good produced in the relatively stagnant sector will decrease more than propor- 4

15 tionally, inducing a larger fraction of both skilled and unskilled workers to be reallocated into that sector. As in the case of distributional parameters, Hicks-neutral labor augmenting technology has no impact on the skill intensity ranking across sectors. Now, I turn to explore the role of skill-biased technological change. Proposition 4. Assume that the technology is such that the elasticity of substitution is the same within sectors = 2 =, the technology is skilled-augmenting in sector i.e. B > 0, B 2 = A = A 2 = 0) and > ρ. Then, the fraction of skilled time allocated to the fast-growing sector will increase, while the fraction of unskilled labor will fall Proof: See Appendix A.5 h B > 0 and l B < 0 This proposition claims that over time, bias-augmenting technology can induce a sector to employ more of a skill type and less of the other. To understand the intuition, suppose that when the efficiency of skilled workers increases in sector, there is no sectoral reallocation. The sluggish sector will maintain its productivity, while the fast-growing sector will be able to produce more and its relative price will go down. The degree of substitutability between the sectoral outputs in the production of the final good and the degree of substitutability between skilled and unskilled workers within sectors will play major roles, once the sectors readjust their inputs,. If the sectoral outputs are highly substitutable in the production of the final good, the relative price of the good produced in the fast-growing sector will decrease more than proportionally. Meanwhile in the labor inputs market, sector increases its demand for skilled workers, relatively more if it is very flexible in substituting unskilled by skilled labor. Overall, employment of skilled workers will increase and employment of unskilled will fall in the flexible sector if > ρ, which may lead to skill-intensity rank reversals. Note that the result is not robust to different parametrizations of the elasticities within and across sectors. For example, when =, both skilled an unskilled shares in the fast-growing sector decrease if the elasticity across sectors is lower than, resembling Proposition 2. Several other propositions could be analyzed in more detail. For example it would be interesting to understand how the fraction of skilled time would evolve across sectors when the sectors have different elasticities of substitution and one of the sectors has greater Hicksneutral technological growth. This is left for future versions of this work. 5

16 2.4 Skill wage premium and shift in aggregate labor demand The model yields predictions concerning the evolution of the skill wage premium in response to technology shocks. After some simple manipulation of equations See Appendix A.6 for details), one can show thatnone Φ 3.) = ω 2 t s t ω t β2 β 2 2t β β ) 2 b 2 s t ) b t + ω t + ω 2 t β β β2 β 2 ) ) ρ b t ) 2 b 2 2t ) 2 ρ 2 β β ) ρ) 2 2 ) ρ) 2 ) b 0t ) ρ α ρ = 0 α This expression implicitly defines the skill-wage premium ω as a function of all exogenous parameters in the model noneb 0, b, b 2, s). Solving for the skill wage premium fully determines the equilibrium of the economy. To study the evolution of the skill wage premium, it is essential to study a measure of substitutability between skilled and unskilled work at an aggregate level. 2) One of the advantages of using a model of sectoral change is that it uncovers how the aggregate elasticity can change with the process of structural change. For the two-sectors case, the aggregate elasticity of substitution is a weighted average of the primary elasticities of the model proof in Appendix A.7): Σ t = θ Ht l t + θ Lt h t ) + 2 θ H2t l 2t + θ L2t h 2t ) + ρ θ H2t θ Ht ) l t h t ) 3) where θ Hit and θ Lit are the shares of wage bill with skilled and unskilled workers respectively) in industry i and year t. Under a model of structural change the elasticity of substitution across inputs is no longer constant. It is now a function of substitutability within intermediate sectors and across-sectors. The elasticity Σ t is a convex combination of, 2 and ρ, because their weights sum up to one. A particularly interesting case is that the aggregate elasticity of substitution is positive even if the elasticity of substitution in each sector is zero, because it accommodates the substitutability between sectors in the production of the final good. The aggregate elasticity between skilled and unskilled labor will, in general, change over time since the weights change over time with relative employment and relative skill intensities of the two sectors. There are a few exceptions. For example, if the primary elasticities are equal ρ = = 2, the aggregate elasticity will be constant. This elasticity will also be constant if the sectoral factor shares and elasticities are similar across sectors because the coefficient in the elasticity of substitution in the final sector will be zero. In fact, the effect of substitutability across sectors results from differences in factor intensities s t and s 2t ) and differences in relative size l t and h t ). The skill biased augmenting technology at an aggregate level is a function of within sectors 6

17 skill-biased technical change and differences in neutral factor demand across sectors. When b t and b 2t are fixed, the term b 0t will capture relative technological change that is Hicks neutral. In this model, the skill wage premium varies ω) with the aggregate skill premium and with the technology. By total differentiation of equation 2 we obtain 9 ω t = Σ t s t + Σ t l t h t ) ρ) b 0t + Σ t ) θ Lt h t + θ Ht l t ) + ρ) θ Ht l t h t )] b t + Σ t 2 ) θ L2t h 2t + θ H2t l 2t ) ρ) θ H2t l t h t )] b 2t The technological change is represented by the variation in the Hicks-neutral labor augmenting technology across sectors A t A 2t ), and by skill-biased labor technical change within sectors b and b 2t ). Changes in A t A 2t affect the equilibrium unless ρ =. In fact, the growth of the total factor productivity i.e. positive changes in A t or A 2t, keeping everything else constant, including B A and B 2 A 2 ) increases the skill wage premium if the final goods are gross substitutes, and reduces the skill wage premium if goods are gross complements. When A it and B it i =, 2) grow at different rates, the evolution of the skill premium will also depend on the value of the sectoral elasticities of substitution. Using this model, the relative demand shift favoring high skilled relative to less skilled is given by d t = b 0t l t h t ) ρ) + 2 b it i ) θ Lit h it + θ Hit l it ) + ρ) θ Hit l it h it )] i= Relative wages vary positively with changes in relative demand d t ) and the strength of this positive relation varies inversely with the aggregate elasticity of substiution. Likewise, the negative impact of changes in relative supplies s t ) on relative wages depends inversely on the magnitude of the aggregate elasticity of substitution between the two skill groups ω t = Σ t d t s t ) This model of structural change with two sectors shows how the process of technical change can be originated in different sectors. At the same time, the unit threshold for elasticities of substitution between types of labor and between intermediate-sector goods, will be very important in determining the impact of input biased augmenting technology. Propositions three and four shed some light on the reasons behind this result. The contribution of the growth in the relative efficiency of skilled labor relatively to unskilled labor in one sector 9 I define x as dx x. 7

18 to the whole shift in aggregate demand increases with the elasticity of substitution between skilled and unskilled labor within each sector and the importance of the intermediate sector in total employment l it and h it ) if i > and ρ <. Consider the case of ρ =, then changes in demand are only a function of the growth of the relative efficiency of skilled labor relatively to unskilled labor, while the Hicks-neutral component has no impact. The relative demand for skilled labor will increase in two potential scenarios: i) increase in the efficiency of skilled labor relatively to unskilled labor and the inputs are gross substitutes b i > 0 and i > ), and ii) reduction in the efficiency of skilled labor relatively to unskilled labor when the two inputs are gross complements b i < 0 and i < ). The interesting aspect is that the sectors can be in different scenarios and both can contribute positively to the evolution of the skill wage premium. 2.5 Generalization to a n sector economy The model of structural change with two sectors is limited when there is substantial heterogeneity across sectors in terms of their technology. I generalize the model presented above to an economy with n intermediate sectors. For a sufficiently large n, I minimize the aggregation problems that result from aggregating firms/sectors with very distinct technologies. Nevertheless, the results for an economy with two sectors are important because they clarify the mechanisms underlying the evolution of the parameters of interest and offer tractable expressions for the aggregate elasticity between skilled and unskilled workers and for the relative demand shifts. Proposition 5. Assume that the economy with n sectors operates in an environment similar to the economy with two sectors described in Section 2.2. I further assume that the technology of the final good has constant elasticity of substitution across any two intermediate outputs ρ), whereas the elasticities of substitution between skilled and unskilled work i ) are potentially different across sectors. Then, the aggregate elasticity of substitution for the n-sector economy generalizes tnoneo ) n n Σ t = i θ Hit l it + θ Lit h it ) + ρ θ Hit l it + θ Lit h it ) 5) i= i= and, the aggregate shift in labor demand is given by d n n t = ρ) A it l it h it )+ b it i ) θ Lit h it + θ Hit l it ) + ρ) θ Hit l it h it )] i= i= 6) 8

19 nonewhere θ Hit and θ Lit are the expenditure shares of skilled-labor and unskilled-labor, respectively, in industry i and year t. The terms l it and h it represent the shares of total employment of unskilled and skilled labor used by sector i in year t, respectively. Note that this is a simple generalization of the two sector model elasticity. 3. Empirical methodology Katz and Murphy 992) and most subsequent contributions are not informative about the nature of the skill biased technical change hypothesis. This model allows us to determine the contribution of each sector to the shift in aggregate demand. In this paper, I test the extent to which the model presented above outperforms the canonical model of Katz and Murphy 992) in matching the evolution of the skill wage premium in the last four decades. Moreover, I contribute to the literature on the between-within components of demand shifts, by explicitly using a model to obtain the aggregate relative demand for skilled labor out of the heterogeneous sectoral demands. This exercise involves the construction of counterfactuals that allow me to split the impact of composition effects. 3. Counterfactuals The model developed in Section 2 provides a framework to perform three interesting exercises. First, it can be used to predict the shift in demand each year noneln d t ) lnd t ) and, together with nonelns t ) lns t ), predict the evolution of skill wage premiumnone ln ω ) t ) = Σt dt ln ω t d t ) st ln s t and test to what extent it fits the observed evolution of the skill wage premium. In particular, I determine if the model predicts a greater portion of the evolution of the skill premium than the standard approach which uses a constant elasticity of substitution over time and approximates the SBTC by a time trend. Second, equation 6 allows us to decompose the shift in aggregate relative demand of skill workers between the contribution of each sector. Sector i total contribution is given by c it == d t ρ) l it h it ) A it + d t i ) θ Lit h it + θ Hit l it ) + ρ) θ Hit l it h it )] b it where the first component comes from the fact that Hicks neutral differences across sectors 9

20 can impact the shift in relative demand at an aggregate level even though it has no impact of sector specific demand). The second part results from skill biased technical change it impacts the aggregate and sectoral relative demand for skilled labor). Third, the proposition can be used to study the importance of changes in the composition of the economy. I create the counterfactual evolution of demand where one holds fixed the composition of the economy: ) d t = n A it ρ) li h ) n i + b it i ) θlit ht + θ ) Hit li + ρ) θhit lt h )] t i= i= where l i and h i are fixed over time in the empirical analysis I fixed this to the average values over the period under analysis), and θ Lit and none θ Hit are computed such that the quantities are hold fixed, while the prices are allowed to vary. 0 The effect of composition in the aggregate demand SBTC can be computed as d ) t d t. I also look at the impact of changes in the composition of the economy when measuring the effect of variation in labor supply. To do so, I must create a counterfactual aggregate elasticity in which the composition of the economy is fixed: Σ ) t = n i θhit li + θ ) Lit hi + ρ i= The effect of composition on the skill premium is given by 3.2 Identification ω t ω ) t = Σ d t t Σ ) t ) d t n θhit li + θ ) ) Lit hi i= ) ) s Σ t Σ ) t t To compute the counterfactuals above I need to identify for each industry A it, b it, i s, and ρ. The impossibility theorem of Diamond, McFadden and Rodriguez 978) implies that one must impose the form of technological change to estimate the elasticity of substitution between inputs. Otherwise it is impossible to consistently estimate the contribution of changes in demand schedule differentiated shifts in factor augmentation technology) and the curvature of the factor demand. nonei follow the standard practice 0 The share of skilled labor is given by ωtsit ωt si +ω ts it and so the counterfactual share will be given by +ω t s i. The explanation for this result is that the correlation between biased-productivity change and factor price change can bias the elasticity of substitution.none This issue is a controversial question in the empirical literature that looks at the elasticity of substitution between capital and labor. Antras 2004) shows that if the researcher imposes Hicks neutral technology, the observed variation in the aggregate data will be associated with factor variation rather than technical change and, as a result, bias the elasticity of substitution towards 20

21 in the literature of an aggregate sector, by imposing exponential growth in productivity: A it = A i0 e λl i t+ul it and Bit = B i0 e λh i t+uh it, where λ L i and λ H i are the rates of unskilled and skilled augmentation, respectively, and u L it and u h it are shocks. Note that when λ H i λ L i > 0 sector i has net skilled labor augmentation technology. Constant growth rates mirror the usual assumption in the literature e.g. Katz and Murphy, 992). Goldin and Katz????) argue that the bias in technological growth has increased at a constant rate over the last century. Acemoglu and Autor 20) show that log quadratic trends fit the data better. 2 As in the economy described in Section 2, the n sectors operate in a competitive, frictionless labor market. The hiring decisions of the amounts of labor of different skill levels depend on the first-order conditions for profit maximization. After incorporating the specification for the nature of labor augmenting technology in equations 8, the following relation arises ln s it = i ln β i ) + i ) ln A i0 + i )λ H i β i B i0 λ L i )t i ln ω t + i )u L it u H it ) + vit L vit H ) where vit L and vit h are unaccounted shocks e.g. optimization errors made by firms), s it is the skill-intensity of sector i in year t, and ω t is the skill wage premium in year t. The equation can be transformed to fit the linear regression equation ln s it = γ 0i + γ i t + γ 2i ln ω t + e it 7) where e it = i )u L it u H it ) + v L it v H it ) is the error term. This simple approach allows me to identify i and λ H i λ L i. To consistently estimate γ 0i, γ i and γ 2i, we need the error term to be uncorrelated with ln ω t. This is very unlikely. Efficiency shocks u L it and u h it may be in the information set of firms and will be taken into account when choosing input demands. Optimization errors made by firms are less likely to be correlated to factor demand. The general equilibrium model developed in Section 2 helps us understand that factor demand will be correlated to the error terms and as such ln ω t will be endogenous. At an aggregate level, my model assumes that the supply of labor is exogenous and inelastic, while at an industry level the supply for labor is perfectly elastic at the equilibrium price of inputs. Such equilibrium depends on the sum of factor demands by all sectors inclusive industry i) and as such ln ω t must be an endogenous regressor. In fact, from the theory of simultaneous equations models, these demand equations will not be identified unless the estimation uses some set of exogenous variables that shift the supply of skilled and unskilled labor. The key unity in the U.S. economy where the share of labor and capital were broadly constant until the 990s). 2 A related contribution was made bynone Len-Ledesma, McAdam and Willman 200). The paper tests for different types of factor augmenting technology using the Box-Cox 964) specification applied to a CES on capital and labor. Their findings suggest that the growth rates of technical progress show an asymmetrical pattern. While the growth of labor-augmenting technical progress is almost exponential, that of capital is hyperbolic or logarithmic. 2