Industrial dynamics and the neoclassical growth model.

Blankenau, William F. ; Cassou, Steven P.

I. INTRODUCTION

Economists have long pondered the impact of a changing industrial composition on the overall economy. Questions such as whether the decline in the manufacturing sector is a worrisome trend are only a recent variant of this question. Not long ago the discussion was more focused on particular parts of the manufacturing sector such as the textile industry. (l) Similarly, the prominence of recent advances in computing and information technologies has cast these industries as possible catalysts of improved economy-wide performance. However, despite the urgency to some of these inquiries, the foundation for these questions is quite old and is often traced to Schumpeter (1942) who described the evolution of the industrial sector as exhibiting a process of creative destruction through the introduction of new goods and technologies.

With all the changes occurring at the industry level, one might think that the overall economy would also show signs of this churning. However, the aggregate goods market data trends have remained very stable over time. In fact, these aggregate trends are so well recognized and so consistent over just about any subinterval of time that they have become the cornerstone for the idea of balanced growth and the neoclassical growth model. (3)

Recent work has taken steps toward reconciling changes in the sectoral composition of output with this remarkable aggregate stability. Collectively, this work explores the modeling restrictions required for this reconciliation. Kongsamut, Rebelo, and Xie (200l) consider balanced growth in the face of a persistent reallocation of labor from agriculture to manufacturing and services. The key to aggregate stability in their model is a knife-edge relationship between a productivity parameter and a preference parameter. This relationship is employed also in Meckl (2002). Foellmi and Zweimuller (2002) build an endogenous growth model, which also accommodates structural change in employment along a balanced growth path. In their model, dynamic differences in income elasticities across sectors give rise to sectoral dynamics where expanding and declining industries coexist. Balanced growth is achieved by assuming a particular willingness to substitute across goods in a hierarchical preferences structure. Most recently, Ngai and Pissarides (2007) build a model where structural change results from different rates of technological progress across industries and balanced growth arises because of a unit elastic intertemporal elasticity of substitution.

This paper also considers trends in industrial composition in a structure that allows balanced growth. However, we focus not only on trends in sectoral employment but also in the skill composition of this employment. We begin by documenting labor and output trends in the 13 major industrial classifications used by the U.S. Commerce Department. Labor trends are documented using U.S. Current Population Survey data from 1968 to 2004, while output trends are derived from U.S. national income account data from 1968 to 1999.

For each industry classification, we disaggregate employment to reveal the trends in unskilled and skilled labor employment. This disaggregation allows us to dichotomize industries as initially relatively high skilled or low skilled. We find that the ratio of skilled workers to unskilled workers has grown in each industry. The absolute increase in the ratio was largest in the initially skilled industries and the ratio has grown faster in the unskilled industries. Furthermore, industries with a relatively high use of skilled workers have accounted for an increasing share of output over time.

With these empirical facts in hand, we build a neoclassical growth model consistent with these trends. The model shares some features with the one in Ngai and Pissarides (2007). In particular, good-specific technological changes yield sectoral dynamics, while unit intertemporal elasticity of substitution yields balanced growth. However, our model more closely resembles that of Blankenau and Cassou (2006), which also shares these features. In that paper, it is shown that the dynamics of time allocation across skilled and unskilled labor can be separated from the dynamics of aggregate output. This allows for balanced growth with a trend toward a more educated labor force.

This paper extends the work in our earlier paper by disaggregating the production sector to additionally capture the industry-level trends. In the version of the model used here, the initially high-skilled industries have dynamic changes in their production processes that result in a more rapidly increasing need for high-skilled workers. The equilibrium level of output also grows more in these industries and the growth rate in skilled workers is relatively smaller than in the initially low-skilled industries.

We demonstrate that the changing industrial composition in the labor force is entirely consistent with our modified neoclassical growth model. We conclude that despite there being considerable turmoil at the industrial level, the aggregate economy can perform as in the standard neoclassical growth model, with either transitional dynamics toward balanced growth or long sustained periods of balanced growth.

Beyond this, we uncover a richer set of dynamics in skilled labor trends and show that the neoclassical growth model requires only modest refinements to offer an explanation. This is an innovation in its own right but proves to be of greater importance. An implication of Ngai and Pissarides (2007) is that sectoral labor trends with balanced growth implies the validity of "Baumol's cost disease" where an industry experiencing slow productivity growth consumes an ever-increasing share of labor. Our model suggests a different form of industry-specific technological changes and implies an eventual stabilization of labor ratios.

The paper is organized as follows. Section II summarizes the data facts that this paper matches. In that section, four types of facts are noted, but only two are described in detail. The two that are described in detail mark the point of departure from Blankenau and Cassou (2006) that is pursued here. In particular, in our earlier paper, only the first two data facts of balanced growth and labor market trends were modeled and that paper had nothing to say about industry-level dynamics. Section II thus describes precisely the additional industrial dynamics that this paper is investigating. In Section III, we present an extended version of our earlier paper, which captures these additional industrial trends. Section IV shows several general results implied by this model, and in Section V, we provide several illustrations that simplify the model enough to see more clearly the presence of the dynamics we are interested in. Section VI summarizes and concludes the paper.

II. THE HISTORICAL FACTS

There are four types of historical facts, which this paper endeavors to model: (1) stable aggregate ratios in the goods market, (2) an increasing fraction of the total labor force that is skilled, (3) industry-level labor dynamics, which vary depending on whether the industry has relatively high-skilled workers or relatively low-skilled workers, and (4) an increasing share of output is produced by the relatively high-skilled industries. Two of these facts, stable aggregate ratios in the goods market over time and aggregate labor market trends, have been well documented elsewhere. The stable aggregate ratios in the goods market are so well known by the profession that they are often referred to as the Kaldor facts in tribute to Nicholas Kaldor who studied these ratios and brought them to the attention of the profession. (4) The aggregate labor market trends, such as a rising use of skilled labor, were more recently recognized by the profession and have also drawn considerable attention. (5)

The labor market trends within the various industrial sectors of the economy are less well known. Some of these trends are illustrated in Figure 1. Panel (a) shows the ratio of skilled labor to unskilled labor from 1968 to 2004 in 13 broad industry classifications used by the U.S. Commerce Department. (6) In the graph, we do not identify the industries by name because the individual plots are hard to distinguish when so many lines are drawn with different line styles. Instead, we use a convention of plotting initially high-skilled industries with solid lines and initially low-skilled industries with dashed lines. However, in Table 1, we report information by industry. (7) One readily apparent trend in this figure is that each sector of the economy increasingly uses skilled workers. Thus, the aggregate trend toward increased skill levels holds at the industry level. However, the figure also illustrates that the trend toward skill is not uniform. Industries that were initially skilled generally have larger absolute increases in the ratio of skilled to unskilled workers.

Although not presented exactly in this fashion, similar trends have been documented in a number of labor studies. For instance, using the same industry breakdown as we do, Bound and Johnson (1992) show that between 1973 and 1988, the share of income earned by skilled workers has grown more rapidly in the industries that we have labeled initially skilled, while in a study that focuses only on manufacturing data for the 1980s, Berman, Bound, and Griliches (1994) note that skill upgrading is correlated with investment in computers and to research and development. Furthermore, these trends have been seen to hold true in Organization for Economic Cooperation and Development data by Machin and Van Reenen (1998), Berman, Bound, and Machin (1998), and Haskel and Slaughter (2002).

[FIGURE 1 OMITTED]

Another sectoral labor market trend is illustrated in Panel (b), where the skilled to unskilled labor ratio is normalized by dividing by the initial value of the ratio. This can be used to compare how the ratio of skilled to unskilled labor has grown in the different industries. Panel (b) also makes use of the convention that initially high-skilled industries are plotted with solid lines and initially unskilled industries are plotted with dashed lines. Flatter slopes for the initially high-skilled industries indicate that the ratio has grown more slowly in these industries.

Table 1 further demonstrates these two sectoral trends by showing various values for the ratio of skilled labor to unskilled labor at different dates. As the table shows, the initially high-skilled industries tend to have larger increments in the ratio but lower growth rates in the ratio. It is these two sectoral facts that we will pursue in our model. However, in our model, rather than keeping track of 13 sectors, we simplify the analysis to just 2. Figure 2 shows these trends for industry aggregates where the four initially high-skilled industries are put together into one aggregate and the nine initially low-skilled industries are put together into the second aggregate. As expected, the initially high-skilled industry aggregate has a larger absolute increase in the ratio of skilled to unskilled workers, but a lower growth rate of this ratio. Furthermore, the near linearity of these figures implies that these trends will hold over most subintervals of the data as well. Finally, note that the last two rows of Table 1 support this result by showing the values for the ratio and the normalized ratio at the beginning and end of the observation period. (8)

The trend in output attributed to relatively high-skilled industries and relatively low-skilled industries is also less well known. (9) In Figure 3, we plot the ratio of output between high-skilled and low-skilled industries using the categorization described in Table 1. This plot shows that over time the initially high skilled industries have accounted for an increasing share of overall GDP. This change appears to result from a long-run trend. Similar results can be seen in Yuskavage (1996) and Lum, Moyer, and Yuskavage (2000). (10)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

III. THE MODEL

This section begins by describing the corporate sector and the various production functions. Next the consumer sector is described and the competitive equilibrium concept defined. The model is formulated in intensive form with exogenous growth. (11)

A. The Corporate Sector

There are three types of producers in the corporate sector. One type produces an investment good and the other two produce consumer goods. The assumption that capital goods are built in a separate sector is used in part to deflect concern that the results are connected to any assumption regarding where capital goods are produced. (12) Consumer goods fall into two categories based on the importance of skilled labor in their production as made explicit below. All investment and consumption goods are produced from capital, skilled labor, and unskilled labor. Production technologies for the investment good and all consumer goods exhibit constant returns to scale. In this case, we lose no generality in assuming that a single firm produces each type of good.

First, consider the firm producing the investment good. We use the notation t to identify variables associated with this sector. The firm produces according to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 0 [less than or equal to] [[gamma].sub.t] 1,0 [[less than or equal to] [alpha] [[less than or equal to] 1. The labor aggregate in Equation (1), given by [[[gamma].sub.l][S.sup.[sigma].sub.l,t] + (1 - [[gamma].sub.l])[u.sup.[sigma].sub.l,t)].sup.1-[alpha]/[sigma]], is a constant elasticity of substitution combination of skilled labor, [S.sup.[sigma].sub.l,t], and unskilled labor, [u.sup.[sigma].sub.l,t]. The parameter [sigma] determines the elasticity of substitution (equal to l/1-[sigma]) between the labor types and [[gamma].sub.l] determines the relative importance of each labor type in determining the size of the labor aggregate. Because of this role, we will often refer to [[gamma].sub.l] as the skill intensity parameter. Under this production formulation, the labor aggregate is combined with capital, [k.sub.l,t], to produce units of the investment good, [y.sub.l,t]. Given the Cobb-Douglas specification, [alpha] is the elasticity of output with respect to the capital input and 1 - [alpha] is the elasticity of output with respect to the labor aggregate.

The consumer goods sector consists of two sectors. One industry is initially more skill intensive. We refer to this as the initially skilled sector and denote it by a. We refer to the other as the initially unskilled sector and denote it by b. To match the data, it is essential that the ratio of skilled to unskilled workers increases in each sector. This could be accomplished with a single good in each industry where the production technology of each good undergoes a skill-biased technological change. However, such an approach implies counterintuitive employment of skilled workers over time. (13) Instead, we model each sector more generally as containing many goods with an expanding product space. Our specification allows the possibility of changes in the production technologies of existing products or the introduction of new goods with new production technologies.

The expanding product space fits the Schumpeter notion of creative destruction as new goods and new technologies reduce the importance of existing goods. (14) This leads to a more natural interpretation of technological change and allows a varied set of possible dynamics. In our specification, new goods in each industry have a higher need for skill than the average prior goods in that industry. In addition, new goods that enter industry a have a higher need for skilled labor than the new goods that enter industry b, thus making the skill demand for industry a rise faster than that in industry b. Collectively, these features cause the average skill intensity of each sector to rise, while the trend is more pronounced in the initially skilled industry.

The number of goods (and firms) at time t in sector j is denoted by [n.sub.j,t] and grows according to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [g.sub.j] [greater than or equal to] 0. Our results are not tied to the relative rates of growth for the two sectors, and to emphasize this, we set the growth rates equal unless otherwise specified. The production function for a representative firm is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 0 [less than or equal to] [[gamma].sub.j,[omega],t] [less than or equal to] 1 and [omega] [member of] [0, [n.sub.j,t]]. The j notation indicates the category of good, a or b, and the [omega] notation identifies a particular good in that category. Hence, [y.sub.j,[omega],t] is the output of good [omega] of industry j at time t, [k.sub.j.[omega],t] is the amount of capital employed in its production, and [S.sub.j[omega],t] and [U.sub.j,[omega],t] are the skilled and unskilled labor inputs. The parameters [alpha] and [sigma] play the same roles as in the investment good sector. Note that we allow the skill intensity parameters (i.e., the [gamma] parameters) in the goods sector to differ across time, goods, and category. This will prove pivotal in generating the dynamics of interest. In contrast, [[gamma].sub.l] is fixed. A constant [gamma] parameter for the numeraire good also proves important in reconciling these dynamics with balanced growth. To simplify notation, we hereafter drop the time subscript. Thus, [y.sub.j,[omega]], is the output of good [omega] in industry j [member of] {a, b} at time t and other industry-specific items are similarly defined. (15)

Equations (1) and (2) are clearly generalizations of the Cobb-Douglas production function prevalent in the growth literature. In our model, as in the simpler Cobb-Douglas case, labor receives a constant share of output given by (1 - [alpha]). We follow the growth literature in justifying its use by noting the lack of trend in this share in the United States and other economies. In contrast, the share of this labor income accruing to skilled and unskilled labor is not constant. We can capture a trend in labor shares across education levels with trends in [[gamma].sub.j[omega]] even in the case where [sigma] = 0. However, we opt to consider the more general case where [sigma] [not equal to] 0 for several reasons. First, we want to emphasize that our results are not contingent upon unit elasticity of skilled and unskilled labor as the Cobb-Douglas case might suggest. Second, we want to discuss the importance of this elasticity in determining our results. Aside from these expositional considerations, we note that empirical observations suggest that the true elasticity is in fact greater than 1. (16)

We assume that factors of production are freely mobile. This implies that factor prices are equal across firms and along with the first-order conditions for firm optimal hiring choices give

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [r.sub.t] is the rental rate of capital, [w.sup.s.sub.t] is the wage to a unit of skilled labor, and [w.sup.u.sub.t] is the wage to a unit of unskilled labor. The price of the investment good is [p.sub.l,t]. Our numeraire good is capital. Since a unit of the investment good is the same as an ex-dividend unit of capital, this requires normalizing [p.sub.l,t] = 1. Given this, other prices, [P.sub.j,[omega]], are stated in terms of the investment or capital good.

With the production sector now described, the manner in which this model extends the model in Blankenau and Cassou (2006) can be readily seen. One goal of Blankenau and Cassou (2006) is to demonstrate what modifications to the neoclassical are required to allow for the long-term trend toward more schooling and a larger share of the workforce with skills. These changes in the workforce are not concerned with the industry-specific trends described in Section II and addressed in this paper, which requires the industry disaggregation described above. The current model can be simplified to arrive at the earlier model by setting [n.sub.b] = 0, [for all]t (and thus dropping all j notation) so that the model collapses to the single industry case. However, such a simplification is unsuitable for the current investigation as it allows no industry differences.

B. The Consumer Sector

The economy is populated by an infinitely lived representative household with lifetime utility defined over an index of current consumption, [c.sub.t],

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [rho] > 0. The consumption index is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [x.sub.j,[omega]] indicates the demand for good [omega] in industry j at date t. The parameter [PSI] > 0 is related to the intratemporal elasticity of substitution across goods so the second case in Equation (7) arises in the case of Cobb-Douglas or unit elastic preferences. (17)

The consumer faces a goods constraint at each date. Let [k.sub.t] be the total capital stock per effective labor unit at date t (hereafter "capital") and 0 < [delta] < 1 be the rate at which this capital depreciates. Furthermore, let [g.sub.A] and [g.sub.L] be the exogenous rates of technological progress and population growth, respectively. (18) Then, the goods constraint is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Adding the terms to the right of the equality in the first line to the integral in the second line gives total payments to factors of production; that is, total income. The integral in the third line gives total consumption spending. Income less consumption spending is investment that we denote by it. Thus, Equation (8) reduces to the familiar law of motion for the capital stock given by [[??].sub.t] = [i.sub.t] - ([delta] + [g.sub.A] + [g.sub.L])[k.sub.t].

To arrive at a measure of total output, we follow the convention of weighting the quantities of each good by their market prices. Period t output, then, is

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Similarly, the total capital stock is

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that because the price of capital is 1 regardless of where it is employed, the capital stock is not weighted by prices.

The agent also faces a time constraint. In each period, the agent has an endowment of one unit of time. Since leisure is not valued, the agent allocates time to maximize labor income net of the education cost of acquiring skill. The cost of education could be modeled as a time cost, a goods cost, or a combination of both. Including both costs proves redundant and we prefer to follow Lucas (1988) and many others in modeling a time cost. This is consistent with the upward trend in college enrollment and duration seen throughout most of the post-World War II years in the United States. Milesi-Ferretti and Roubini (1998) show that the nature of the education cost is important in analyzing tax policy. However, we are not considering issues related to taxation and the distinction is not as important.

The essential feature of time allocation is that it is costly to refine the time endowment for the provision of skilled labor. For simplicity, we model the education requirement as linear and contemporaneous. Specifically, to provide a unit of skilled labor, 1/[theta] units of time must be spent in education. Given this, the total amount of skill provided in the labor market, St, is related to time spent in education, [e.sub.t], according to

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and the time constraint is

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With this, the model is fully specified and we are able to define an equilibrium in this economy.

DEFINITION. A competitive equilibrium is a set of infinite price sequences [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [k.sub.0] given such that

(1) Given prices, firms maximize profits subject to production constraints (1) and (2). With factor mobility, this yields Equations (3)-(5).

(2) Given prices, consumers maximize utility Equation (6) subject to resource constraints (8), (11), and (12).

(3) Markets clear:

(a) Capital goods produced equals investment: [y.sub.l,t] = [i.sub.t].

(b) Consumption goods produced equals consumption goods demand: [y.sub.j,[omega]] = [X.sub.j,[omega]] for 0 [less than or equal to] [omega] [less than or equal to] [n.sub.jt] and j= a, b.

(c) Capital input demand equals capital input supplied: Equation (10).

(d) Labor input demand equals labor input supplied: Equations (11) and (12). (19)

IV. IMPLICATIONS OF THE GENERAL MODEL

In this section, we present several general results and organize them into three subsections. The first subsection formulates a generalized version of the separation result from Blankenau and Cassou (2006). (20) This separation result shows that the dynamics of aggregate output can be tracked without knowledge of the sectoral composition of this output or the dynamics of time allocations. This result is useful because it means that we can concentrate on sectoral dynamics and labor market dynamics knowing that any trends in those parts of the economy will be consistent with balanced growth in total output.

The second and third subsections present results on industrial labor market dynamics and industrial output composition, respectively. These subsections explain both intuitively and formally how the model can achieve these dynamics. Later, in Section V, we provide several sample economies to illustrate these results.

A. Separation Theorem

To describe the equilibrium dynamics, it proves convenient to introduce [v.sub.t] as a measure of the share of the capital stock used to produce investment goods. It will be shown shortly that [v.sub.t] also represents the time used in the production of investment goods.

We also will make extensive use of the following z variables that are defined for each good and are related to each good's [gamma] value. Although the [gamma] values are the fundamental distinguishing characteristic for each good and account for all the dynamics presented here, there are many occasions where thinking about things in terms of the z values proves to be advantageous. For instance, it will be shown later that these z terms will provide a nice formulation for relative prices and they will also have a useful interpretation as labor productivity parameters. With that in mind, we define (21)

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Recall that all items with subscript j [member of] {a, b} are time specific. Thus, [z.sub.j,[omega]] differs across both goods and time, while [z.sub.l] is constant.

PROPOSITION 1. Household Allocations.

(a) Capital and labor allocations: There exists a [v.sub.t] such that capital is allocated according to

(14) [k.sub..,t] = [v.sub.t][k.sub.t],

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and time is allocated according to

(16) [u.sub.l,t] = [v.sub.t](1 - [[gamma].sub.l]).sup.1/1-[sigma]]Z.sup.-[sigma].sub.l],

(17) [S.sub.l,t] = [v.sub.t][[gamma].sub.l] [theta]/1 + [theta]].sup.1/1 - [sigma]] [Z.sup.-[sigma].sub.l],

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(b) Dynamics and convergence: The dynamics of [v.sub.t] and [k.sub.t] are governed by (22)

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where Equations (20) and (21) describe a globally stable system converging to a path with [[??].sub.t] = kt = O.

(c) Output: The total value of output is given by

(22) [y.sub.t] = [k.sup.[alpha].sub.t][Z.sub.l](1 - [apha])(1 - [alpha]).

Part (a) of the proposition shows how labor and capital are allocated for production of the investment good and the various consumption goods. These equations show that at each point in time, the same share of capital and time, [v.sub.t], is allocated to provide the investment good and the complement is allocated to providing consumption goods. (23)

There are two key implications of Parts (b) and (c). First, the dynamics of output can be tracked without knowledge of how time is allocated within the consumption goods sectors. To see this, simply note the absence of any consumption market indicator in Equations (20) and (21). In equilibrium, the relative value of output in the investment good or consumption good categories reflects the value of the inputs used in the respective categories. Since [v.sub.t] and 1 - [v.sub.t] are the shares of both labor and capital allocated to producing the investment good and consumption goods, knowing the value of investment is sufficient to find the value of output. Note that to find the value of any particular consumption good, one must know how the 1 - [v.sub.t] units of time and the (1 - [v.sub.t])[k.sub.t] units of capital are allocated across the various goods. However, this allocation does not influence the total value created in the consumption sector.

The second key implication is that [v.sub.t] and [k.sub.t] eventually converge to steady-state levels. At this point, output in intensive form is constant (Equation (22)). Since dynamics persist in the consumption sector, this implies that the allocation of resources across investment and consumption is independent of such dynamics. Equation (9) shows that if output and investment have converged to a steady state, so has the value of consumption goods. However, this is an aggregation of prices and quantities and does not require that prices of consumption goods have stabilized. In fact, the dynamics of prices prove closely related to our results in the next section. To have balanced growth, we need to neutralize the effect of price changes in the consumption goods sector on the savings rate as expressed by v. The unit elastic intertemporal rate of substitution inherent with logarithmic preferences assures the independence of price dynamics and savings.

While we are not able to relax the unit elastic intertemporal rate of substitution assumption and preserve balanced growth, we are able to defend it as a reasonable approximation for our purposes. As mentioned earlier, Kongsamut, Rebelo, and Xie (2001) and Meckl (2002) assume a particular relationship between a technology parameter and a preference parameter. In addition, Blankenau and Cassou (2006) and Ngai and Pissarides (2007) also make the same assumption held in this paper. There are several advantages to this choice. First, at the aggregate level, the resulting framework is precisely the Ramsey-Cass-Koopmans model and thus, both well understood and widely accepted. Second, the value is empirically defendable. While a wide range of estimates populate the literature, a value close to 1 for the intertemporal elasticity of substitution is not uncommon. (24)

Finally, before turning to the labor market and output dynamics, it is useful to elaborate on one of the more intuitive interpretations of [Z.sub.[omega]]. Using Equations (2), (14), (18), and (19), it can be shown that

(23) [y.sub.j],[omega]/[u.sub.j][omega] + [S.sub.j],[omega](1 + 1/[theta]) = [k.sup.[alpha]].sub.t] [Z.sup.(1 - [sigma])(1 - [alpha]).sub.j,[omega]].

The left-hand side is the equilibrium output per unit of time (i.e., labor productivity) for good (j, [omega]). To see this, note that the denominator is the equilibrium amount of time used in the production of good (j, [omega]) inclusive of education costs. Since [k.sub.t] is constant in balanced growth, [Z.sup.(1 - [sigma])(1 - [alpha]).sub.j,[omega]) scales this equilibrium productivity measure and we can think of [Z.sub.j,[omega]] as a determinant of equilibrium labor productivity.

B. Industrial Labor Market Dynamics

In this subsection, we explain how the model is capable of reproducing the empirical fact of industry-specific labor dynamics even along a balanced growth path. The principle insight for this result comes from Equations (18) and (19). These demonstrate that even with [v.sub.t] constant in balanced growth, the allocation of time to each good changes with [[gamma].sub.j,[omega]], or [Z.sub.j]. To show that industry-level labor dynamics persist with balanced growth, we need to demonstrate that the labor aggregates within each industry change through time. Since our data are in terms of the ratio of skilled to unskilled labor, we focus on this aggregate. It is straightforward to show that

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

To interpret how this ratio evolves over time, note that changes arise from two sources. First, the ratio can change over time because new goods in the industry have values for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that move the ratio, and second, it can change if any existing good, [[gamma].sub.j],[omega]] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], has a change in its [[gamma].sub.j,[omega]] value. Furthermore, since [S.sub.j]/[U.sub.j] does not depend on any economic variables and is entirely determined by the [[gamma].sub.j,[omega]] parameters, the key to generating the observed industry-level dynamics is for the time paths of [[gamma].sub.j[omega]] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be such that the firm-level Equations (18) and (19) aggregate in Equation (24) to the proper industry-level dynamics. In Section V, we show that such industry-level dynamics are easy to achieve by constructing several examples, which provide both analytical and numerical demonstrations of the sought after dynamics.

C. Industrial Output Composition

The model is also capable of reproducing a changing industrial output composition even when aggregate growth is balanced. To intuitively understand this possibility, consider a simple situation in which the number of goods in each sector is equal at each date. In particular, each sector starts with the same number of goods and the growth rates for goods in each sector are the same. This means that at each instant of time, new goods appear in pairs, with one new good in each sector. Next focus on the demand curves for these new goods. Since the elasticity of substitution in the utility function is the same for each good, the demand curves for each good will be the same. Consider the situation where [PSI] < 1. (25) This implies that demand functions for all goods are relatively elastic. So to produce the desired changes in the output shares, we need for the marginal goods in sector a to generate relatively greater output values than the marginal goods in sector b. This will arise so long as the supply curve for the marginal good in sector a is further to the right than the supply curve for the marginal good in sector b. But this can be assured through appropriate relative values for the marginal good's [[gamma].sub.j,[omega]]. In Section V, several specific formulations for [[gamma].sub.j,[omega]] processes that achieve this result are provided.

To more formally understand this possibility, consider Proposition 2.

PROPOSITION 2. Let [y.sub.j] be output in sector j as measured by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then,

(25) [y.sub.a]/[y.sub.b] = [Z.sub.a]/[Z.sub.b]

As is common, we measure each sector's contribution to total output by its market value. From this, we conclude that in equilibrium, the relative size of industries and the labor productivities in those industries are closely related. This is also a feature in Ngai and Pissarides (2007). It is an artifact of perfect factor mobility and technology-induced structural change and is likely to arise in other models that share these features.

It is clear from Equation (25) that many kinds of behavior for this output ratio are possible. We are most interested in situations where industry a becomes a larger share of total output over time. We emphasize that all that is needed to generate an upward trend in industry a's share of output is for [Z.sub.a] to grow more rapidly than [Z.sub.b]. This ultimately depends upon the component [Z.sub.j,[omega]] values.

To intuitively trace these dynamics again, it is useful to consider industries with equal numbers of goods and equal rates of good introduction so that goods are introduced in pairs, one for each industry. Begin by focusing on the case in which goods are relatively substitutable: [PSI] < 1. In this case, goods with higher values of [Z.sub.j,[omega]] have higher labor productivity and have greater resources employed in the goods production. (26) With both more resources and higher productivity, the output of goods with higher [Z.sub.j,[omega]] values is higher. Of course, the equilibrium price is lower, but because demand is elastic on net, there is a higher equilibrium value. Because both marginal goods have the same demand equations, the industry that experiences the largest increase in value will be the one which has the marginal good that has the largest productivity. To summarize, in the substitutable good case, the sector that experiences a rising share of output has new goods with relatively greater productivities.

Next focus on the case in which goods are relatively complementary: [PSI] > 1. In this case, goods with higher values of [Z.sub.j,[omega]] have higher productivity but also have fewer resources employed in the goods production. This lower resource input in part offsets the higher productivity. The net effect is a higher output (Equation (36) in the Appendix A), but along with the lower price, this is enough to imply that the value of the output is lower because demand is inelastic. Because both marginal goods have the same demand equations, the industry that adds the smallest increment to its value will be the one which has the marginal good that has the largest productivity. In other words, in the complementary good case, the sector that experiences a rising share of output actually has new goods with relatively lower productivities. This surprising result follows because complementarity in utility implies a preference for keeping consumption levels more equal across goods. Thus, the marginal good that has a high productivity will not see as large a difference in equilibrium production as in the substitutable good case because resources are shifted to other production activities to maintain this more equal consumption preference.

V. ILLUSTRATIONS

There are two sets of sectoral dynamics that we would like our model to exhibit along a balanced growth path. First, the sector that starts out with a higher ratio of skilled to unskilled workers experiences larger increases but slower growth in the ratio of skilled to unskilled labor employed, and second, this sector should account for an increasing share of output. Section IV makes clear that these dynamics can be achieved when [[gamma].sub.j,[omega]] follows appropriate time paths. In this section, we provide several specific formulations, which demonstrate these dynamics more clearly or provide further insight into what is necessary to achieve them.

The first example simplifies the economy considerably and is designed primarily to shed light on Baumol's cost disease, which suggests that an industry experiencing slow productivity growth consumes an ever-increasing share of income. It is shown that this disease is not necessarily present in our setup and distinguishes our results from Ngai and Pissarides (2007) where the disease is present.

The second example is designed to provide more insight into what is necessary to achieve the two dynamic trends. It works with a simple unit elastic utility function and Cobb-Douglas production function and shows that the integrals in Equation (24) can be easily evaluated. Then, with both a general and specific formulation for [[gamma].sub.j,[omega]], it is shown how the desired labor dynamics can be achieved. This example then goes on to show that the unit elastic utility function will not be able to achieve the desired industrial share dynamics except when sector a has a higher rate of new good introduction than sector b. Although this may seem like a negative result, it is actually positive because it shows that to obtain both types of dynamic results when each sector experiences an equal rate of new good introduction, one must move away from the intratemporal unit elastic utility function. The third example simply generalizes this second example slightly to achieve both dynamic results.

A. Output Dynamics and Baumol's Cost Disease

To explore this issue, it is enough to work with a simple model where only sector a experiences technological change and goods within each sector have identical technologies. In this case, we can use Equations (18) and (19) to show the ratio of skilled labor to unskilled labor in sector a is

(26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that the co notation has been dropped since goods are identical. From this, it is clear that [[??].sub.a] > 0 is sufficient to assure that this ratio is increasing through time. Furthermore, Equation (25) in this case reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In discussing the dynamics, we consider only the empirically relevant case where [sigma] > 0. (27) This is for brevity and a symmetric set of results exists with [sigma] < 0. Proposition 3 shows conditions under which the desired dynamics arise.

PROPOSITION 3. Suppose goods within each sector have identical production technologies, [[??].sub.a] > 0, [[??].sub.b] = 0 and [sigma] > 0. Then, the ratio of skilled workers to unskilled workers' grows in industry a and is fixed in industry b. Furthermore, the relative share of output in industry a grows if [PSI] < 1 and [[gamma].sub.a] > 1 + [theta]/1 + 2[theta] or if [PSI] > 1 and [[gamma].sub.a] < 1 + [theta]/1 + 2[theta].

Intuitively, the proposition is a simplified version of the argument made above. An increase in [[gamma].sub.a] yields an overall productivity increase in industry a so long as [[gamma].sub.a] > 1+[theta]/1+2[theta]. This productivity increase shifts the supply curve for the good to the right, increasing the quantity produced and decreasing its price. This first effect serves to increase the total value of output and the second to decrease it.

Whether the net effect on total value is positive or negative depends on the relative size of these effects and hence on the elasticity of demand curve for the item. If demand is relatively elastic ([PSI] < 1), the equilibrium price will fall modestly in response to a supply increase and total revenue from the sector will increase. Since the value of output in the other sector is unchanging, the changing sector grows relative to the other. Thus, the industry experiencing technological change will become a larger part of total output if demand is elastic and vice versa.

There are two keys to make this result work. The first involves the relationship between [Z.sub.a] and the value of total output. If [PSI] < 1, goods are relative substitutes and an increase in [Z.sub.a] increases the value of output as discussed above. The second involves the relationship between [[gamma].sub.a] and [[Z.sub.a]. This is a nonmonotonic relationship. When [[gamma].sub.a] is relatively large (small), [Z.sub.a] is increasing (decreasing) in [[gamma].sub.a]. Thus, we conclude that when [[gamma].sub.a] is sufficiently large, further increases will increase [Z.sub.a] and with [PSI] < 1, this increases the relative value of its output. Alternately, when [[gamma].sub.a] is sufficiently small, increases in [[gamma].sub.a] decrease [Z.sub.a] and with [PSI] > 1, this increases the relative value of its output.

The conditions of Proposition 3 make it easy to generate a dynamic economy where the industry experiencing skill-biased technological change accounts for a growing share of total output. Considering, for example, the following path for [[gamma].sub.a] assuming [PSI] < 1:

(27) [[gamma].sub.] = [[[gamma].sub.a].bar] + ([bar.[[gamma].sub.a]] - [[[gamma].sub.a].bar]) (t/a + t),

where [[[gamma].sub.a].bar] is constant and [bar.[[gamma].sub.a]] = 1. Then, in each period, the skill ratio grows in industry a and its share of output grows. The ratio of output in the two industries, however, converges to

(28) [y.sub.a]/[y.sub.b] = ([theta]/1 + [theta]).sup.(1-[alpha])(1-[psi])/[psi]] [Z.sup.-1.sub.b].

When [[gamma].sub.a] grows according to Equation (27), productivity in industry a begins at [[gamma].sub.a] and approaches 1 asymptotically. With a relatively elastic demand, the size of this industry grows. However, Equation (28) shows that it does not grow without limit. As [[gamma].sub.a] is bounded, so is its relative productivity. Thus, the ratio of the size of industries is limited as well. This result indicates that, in the limit, one industry does not fully dominate the other in terms of the value of output.

Similarly, one industry does not fully dominate the other in terms of resources used. It is straightforward to show that in the limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [L.sub.a] is the share of time devoted to industry a. Since v lies between 0 and 1 and [Z.sub.b] > 0, the limiting value of [L.sub.a] is less than 1. Thus, we see that one sector does not end up consuming all resources. One interpretation of this is that Baumol's cost disease need not be an implication of observed sectoral shifts. Intuitively, we have specified a way in which productivity in one sector always grows more rapidly than in another. With the proper set of preferences, this industry grows relative to the other. In our specification, however, productivity differences are bounded. This gives upper bounds to both the ratio of output across the two industries and the share of resources employed by the growing industry.

Finally, let us note that due to the separation theorem, we have a great deal of flexibility in specifying the process by which [[gamma].sub.a] changes through time. We choose the form above as an example due to its simplicity and because with it, trends in the model match those in the data. This specification is not unique in its ability to match the empirical facts and any number of other specifications for [[gamma].sub.a] are also possible. However, reasonable specifications would place an upper bound on [[gamma].sub.a] less than or equal to one. As such, they would bound the ratio of skilled to unskilled labor and the general findings above would hold.

B. Implications When [PSI] = 1 and [sigma] = 0

Here, we focus on conditions necessary to achieve the kind of industrial sector labor market dynamics seen in Section II. Because the integrals in Equation (24) are not possible to solve generally, we consider the case where intratemporal utility is logarithmic ([PSI] = 1) and production is Cobb-Douglas ([sigma] = 0). By setting [PSI] = 1 and [sigma] = 0, Equations (18) and (19) simplify and can be integrated across sectors to produce

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] d[omega]; that is, [[bar.[gamma]].sub.j] is the average skill intensify in sector j.

For ease of notation, it proves convenient to define [[??].sub.j] [equivalent to] = [[bar.[gamma]].sub.j]/1 - [[bar].[gamma]].sub.j] for j = a, b. This is increasing in [[bar.[gamma]].sub.j] and thus rises as the relative importance of skill in industry j rises. For this reason, we refer to [[??].sub.j] as the skill intensity ratio. The following proposition relates changes in the skill intensity ratio (a technology measure) to changes in the skill ratio (an equilibrium outcome).

PROPOSITION 4. Suppose [PSI] = 1 and [sigma] = 0. If at each date t,

(31) [[??].sub.a] > [[??].sub.b] and [[??].sub.b]/[[??].sub.b] > [[??].sub.a]/ [[??].sub.a],

then at each date t, movement in skill ratios are related according to

(32) ([[??].sub.a]/[U.sub.a]) > ([[??].sub.b]/[U.sub.b])

and

(33) ([[??].sub.b]/[U.sub.b])/([S.sub.b]/[U.sub.b]) > ([[??].sub.a]/[U.sub.a])/[S.sub.a]/[U.sub.a]).

The first requirement in Equation (31) is that the absolute increase in the skill intensity ratio for the initially high-skilled industry exceeds that of the initially low-skilled industry. The second requirement is that the skill intensity ratio for the initially low-skilled industry grows more rapidly than for the initially high-skilled industry. If these conditions are met, the behavior of skill ratios anticipated by the neoclassical growth model are precisely those observed in the data.

Since [[??].sub.b] < [[??].sub.a] by definition, there is clearly some ratio of growth rates that will satisfy this condition at any time t. The right-hand ratio in Equation (31) is clearly growing when the middle condition is satisfied. Thus, Equation (31) will be satisfied in all time periods only if the middle ratio is increasing more rapidly than the right-hand ratio and is also converging to something less than or equal to one.

The observed labor market dynamics then will arise given any process for [[gamma].sub.j,[omega]] that satisfies Equation (31). In the following section, we demonstrate that if one is willing to rely on numerical solutions, it is not difficult to arrive at processes for [[gamma].sub.j,[omega]] that meet the requirements. Before turning to this, though, we demonstrate that some analytical results are available. We need a process for [[gamma].sub.j,[omega]] for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is tractable and can be shown to satisfy Equation (31) at each moment. One candidate can be interpreted as a vintage capital structure because a good that enters at date t has a particular set of input elasticities for capital and labor, which never change for the rest of time. Under this formulation, the production elasticity for skilled labor for good [omega] in sector j at time t is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

where 0 < [PHI] [less than or equal to] 1 and [bar.[[gamma].sub.j]] > [[[gamma].sub.j].bar] with both constant. Notice that [[gamma].sub.j] gives the lower limit for the sector j skilled labor elasticity and [bar.[[gamma].sub.j]] gives the upper limit. In this formulation, the elasticity for any product (j, [omega]) does not change over time and is only a function of the product's type. Also note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that at any point in time, newer goods are more skill intensive.

We assume that, at each date, the two sectors have the same number of goods. Since we wish to construct the example with industry a as the higher skilled industry, we assume [gamma].sub.a] > [[gamma].sub.b]. The relationship between ([bar.[[gamma].sub.a]] - [[gamma].sub.a]) and ([bar.[[gamma].sub.b]] - [[[gamma].sub.b].bar]) is less important and for simplicity, it is easiest to just assume that they are equal. Together these imply ([bar.[[gamma].sub.a]] > ([bar.[[gamma].sub.b]]. We also need to make sure that the growth rates for labor inputs within each sector have the proper relationships, which requires that [n.sub.0] is large and that .5 > ([bar.[[gamma].sub.a]] > ([bar.[[gamma].sub.b]] These assumptions are sufficient to establish the following result.

PROPOSITION 5. Suppose [PSI] = 1 and [sigma] = 0, [[gamma].sub.j,[omega]] is described by Equation (34), that no is sufficiently large, .5 > ([bar.[[gamma].sub.a]] > ([bar.[[gamma].sub.b]] and (([bar.[[gamma].sub.a]] - [[[gamma].sub.a].bar]) = ([bar.[[gamma].sub.b]]- [[[gamma].sub.b].bar]). Then, labor market dynamics match those in Equations (32) and (33).

One downside of the assumptions in this subsection is that they imply that the only way for the ratio of output between the relatively high-skilled and the relatively low-skilled sectors can increase over time is for sector a to have a higher rate of new good creation. This can be seen by noting that when [PSI] = 1, Equation (25) reduces to

[y.sub.a]/[y.sub.b] = [n.sub.a]/[n.sub.b].

Thus, one industry eventually becomes inconsequential. In addition, it can be shown that in the limit, all resources used in the production of consumption goods are consumed by the growing industry. By relaxing the assumptions that [PSI] = 1 and [sigma] = 0, in the following section, we are able to explain observed dynamics even with the relative product space in each industry constant and avoid these implications.

C. Complete Dynamics

The previous subsection showed how the model can match the labor market facts, but the implications for the industrial output shares were lacking. In this section, we show that by relaxing the assumptions that [PSI] = 1 and [sigma] = 0, we can obtain a model with many goods in which sector a not only becomes increasingly skilled but also produces an increasing share of total output even when new good creation is equal in the two sectors.

At this level of generality for the model, analytical results are not available. So to make headway, we simulate the model numerically. To implement this, we set

(35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where 0 < [[gamma].sub.j] < [[bar.[gamma]].sub.j] > 1,0 [less than or equal to] [PHI] 0 [less than or equal to] [lambda]. This is a parsimomous yet general specification for the [[gamma].sub.j,[omega]] process, To understand the implications of this functional form, first focus on the case where [PHI] > 0, [lambda] = 0. With these restrictions, any good, once introduced has the same production technology forever. However, new goods (with a higher m) are more skill intensive. Thus, the introduction of new goods increases the average skill intensity in each industry. Next, consider the case where [lambda] > 0, [PHI] = 0. Here, each new good in industry j has [[gamma].sub.j,[omega]] = [[bar.[gamma]].sub.j]. To see this, note that when a good is introduced, it is the frontier good and [n.sub.j] = [omega]. As time passes and the frontier grows, the skill intensity of each existing good falls. This reflects the possibility that as a good has been in production longer, simplification of the production process allows the producer to substitute lower cost unskilled labor for skilled labor. Aside from seeming to be a natural process, it is one supported by empirical evidence. (28) With this as the only source of dynamics, there would be a gradual decrease in the need for and employment of skilled labor. However, this is countered by a growing product space. As the share of the product space with relatively high-skilled labor needs grows, the equilibrium share of the workforce with skill grows with it. In the general case with [lambda], [PHI] > 0, the two sources of dynamics work in tandem generating a rich set of dynamics where the frontier goods are increasing in skill content and existing goods are going through a process of "simplifying by doing" yet aggregate output is growing at a steady pace.

To demonstrate that this functional form yields results consistent with the empirical facts, we choose parameter values that can generate dynamic simulations, which match the data described in Section II. There is considerable flexibility for many parameters, so alternative parameterizations can also match the data well. For this purpose, we employ the following set of parameters: (29)

Parameter [sigma] [alpha] [PSI] [theta] [[bar.[gamma].sub.a]]
 [[bar.[gamma].sub.b]] [[[gamma].bar].sub.a]
 [[[gamma].bar].sub.b] [n.sub.j,O][n.sub.j]/
 [n.sub.j] [lambda] [PHI]

Value .4 .4 .5 10 .6 .4 .3 .1 1 .02 1 1

Since the model is no longer analytically tractable, we simulate the model numerically. The results of this exercise are summarized in Figure 4.

Panel (a) demonstrates that this model captures the desired industrial share dynamics. It shows that the value of output in industry a grows more rapidly than industry b and thereby accounts for an increased share of total output. Panels (b) and (c) demonstrate that the model captures the labor trend dynamics. In Panel (b), we see that both industries experience increases in the employment of skilled labor relative to unskilled labor and that the absolute increase in this ratio is largest in industry a. Panel (c) then computes this ratio normalized by its initial level. With this normalization, it is clear that the ratio grows more rapidly in industry b.

These dynamics are similar to the observed dynamics in both industrial output and labor market trends, which were described in Section II. While the results are particular to our specification of the process governing [[gamma].sub.j],[omega] we note that many plausible processes could yield similar results. For example, setting either [PSI] or [PHI] equal to zero and recalibrating other parameters yields pictures very similar to those above. If we remove the In operator in Equation (35), results are again similar so long as [PHI] > 0. (30) In fact, we find that the lessons learned in the simpler cases can provide guidance for what is needed to produce the observed dynamics in the more complicated cases. If we set the initial value of [[gamma].sub.j],[omega] sufficiently large, [sigma] > 0, [PSI] < 1, and specify an appropriate process for [[gamma].sub.j],[omega] the results can be recreated. Furthermore, with [sigma] < 0 and the initial value of [[gamma].sub.j],[omega] sufficiently small, similar dynamics arise.

A key point is that there is a great deal of flexibility in choosing the functional form for skill intensities, each of which is consistent with balanced growth in aggregate output. This allows us to specify forms which align with both data and economic intuition. While the above specification is succinct and general, it is but one of many possible specifications. Furthermore, while our choice of parameters matches the data nicely, it is not the unique choice which would accomplish this. In future empirical work, it will be useful to estimate the time paths for [gamma] at this industry level.

VI. SUMMARY AND CONCLUSIONS

This paper shows that it is possible to have a host of industrial sector dynamics within a structure that aggregates up to the standard neoclassical growth model. This demonstration is important because data show that some industrial sectors gain in size in the overall economy and others decline in size in the overall economy. It is sometimes speculated that industries in decline point to an ailing economy. What the demonstration here shows is that it is possible to have these dynamic industrial changes occurring yet the overall economy remains balanced and healthy.

[FIGURE 4 OMITTED]

We achieve these results by modifying a special case of the Ramsey-Cass-Koopmans model of exogenous growth. Other than our assumption of unit elastic intertemporal elasticity of substitution, our model at the aggregate level is a simple restatement of this venerable workhorse of growth theory. At the industry level, however, it is a rich generalization of the one- and two-sector growth models often built on this framework. The richness of this generalization is made possible by a theorem indicating that any sort of sectoral output and labor dynamics can be made consistent with balanced growth in aggregate.

This freedom in modeling industrial dynamics is restricted by empirical observations. We uncover several features of industrial dynamics that the model should encompass. Some of these, we feel, have not been well documented in prior literature. In particular, we show that the skill-intensive industries have been growing more rapidly and have experienced larger absolute increases, but slower growth rates, in the ratio of skilled to unskilled labor. Guided by these observations, we specify processes of technological change that recreate these results in a competitive equilibrium.

To understand the workings of our model more fully, we specify some simple versions of the model where the intuition is apparent before turning to numerical results for the full model. In the full model, the dynamics are indeed quite rich. We specify a process whereby the skill content of new goods grows through time, while goods once introduced can become more or less skill intensive as learning occurs. All the while, observed industrial dynamics persist along a balanced growth path.

We note that our separation theorem can be taken as supportive of much earlier work in growth theory, which ignores salient trends that seem to loom large for the macroeconomy. The trend toward decreased agricultural output and then manufacturing as shares of output are historical examples. The explosion of computing and information technologies and products may serve as current examples. However, we see the separation as having perhaps a more meaningful implication. Our results indicate that researchers interested in industrial sector trends may benefit from conducting analysis within the context of general equilibrium models that preserve the stylized fact of balanced growth. By digging deeper into industrial dynamics and considering the equilibrium consequences of such dynamics, new insights arise. In the example provided here, the dynamics required to reproduce the empirical observations lead to interesting long-run implications for industrial dynamics. We show that a continuation of the technological change consistent with recent experience is an eventual leveling out of the industrial composition of output and the skill composition of labor.

ABBREVIATION

GDP: Gross Domestic Product

doi:10.1111/j.1465-7295.2009.00192.x

APPENDIX A

Proof of Proposition 2

Substitute Equations (15), (18), and (19) into Equation (2) and use the definition of [Z.sub.j,[omega]] to get

(36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

An analogous derivation yields an expression for [y.sub.t,t. Substitute this along with Equation (36) into Equation (3) to obtain

(37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

and along with Equation (13), this yields Equation (25).

Proof of Proposition 3

The first statement is clear upon taking the time derivative of Equation (26). To verify the second statement, note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With [sigma] > 0, [PSI] < 1, this is positive so long as the bracketed expression is positive. This requires [[gamma].sub.a] > 1 + [theta]/ 1+2[theta]. With [sigma] > 0, [PSI] > 1, this is positive so long as the bracketed expression is negative. This requires [[gamma].sub.a] > 1 + [theta]/ 1+2[theta].

Proof of Proposition 4

Before starting the proof, it will be useful to find expressions for the time derivative of the skilled to unskilled labor ratio and the growth rate of the skilled to unskilled labor ratio. To this end, note that for any industry j, Equations (29) and (30) give

(39) [S.sub.j]/[U.sub.j] = ([theta]/1 + [theta]) [[bar.[gamma]].sub.j])/(1 - [[bar.[gamma]].sub.j]).

so that

(40) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Equations (39) and (40) gives

(41) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We are now ready to prove the theorem. First note that

(42) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, note that the right-side inequality in Equation (31) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which upon substitution of Equation (42) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Combining this with Equation (40) gives Equation (32). Second, note that the left-side inequality in Equation (31) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which upon substitution of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and Equation (42) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Combining this with Equation (41) gives Equation (33).

Proof of Proposition 5

First, note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Differentiate to verify. This gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that

(43) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Next, note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

so that

(44) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since this derivative is positive, it shows that the average skill intensity ratio is always increasing in each sector. Next, we need to note a few relationships for [[[gamma].bar].sub.j] and [[bar.[gamma]].sub.j] terms. It is straightforward to show the following hold: [[[gamma].bar].sub.a]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We need to show first that Equation (43) implies that at each date t, [[??].sub.a] > [[??].sub.b]. But this follows since [[[gamma].bar].sub.a] + [[bar.[gamma]].sub.a] [n.sup.[PHI].sub.j] > [[[gamma].bar].sub.b] + [[bar.[gamma]].sub.b][n.sup.[PHI].sub.j] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now we are ready to verify Equation (31). Consider the inequality on the right side of Equation (31) first. Plugging Equation (44) into Equation (42), we must show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But this follows because ([[bar.[gamma]].sub.a] - [[[gamma].bar].sub.a]) = ([[bar.[gamma]].sub.b]- [[[gamma].bar].sub.b] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Next focus on the left inequality in Equation (31). From Equations (42)-(44), we need to show

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because we have assumed that no is sufficiently large, ([[gamma].sub.j] + [[bar.[gamma]].sub.j] [approximately equal to] [[bar.[gamma]].sub.j][n.sup.[PHI].sub.j] and ((1- [[[gamma].bar].sub.j])+ (1 - [[bar.[gamma]].sub.j])[n.sup.[PHI].sub.j]) [approximately equal to] (1 - [[bar.[gamma]].sub.j])[n.sup.[PHI].sub.j].

Thus, it is enough to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But this follows because .5 > [[bar.[gamma].sub.a] > [[bar.[gamma]].sub.b].

APPENDIX B

Data Appendix

Labor data from 1968 to 1991 come from Current Population Surveys: March Individual Level Extracts, 19681992, Second ICPSR Version. (31) The remaining data are taken from the March Supplement of the Current Population Survey as made available through Data Ferret (http://dataferrett.census.gov).

Classification of industries is not consistent across time within or across these data sets. Table A1 shows how we have reconciled the different classifications. For the 19681991 period, the variable used for industry categorization is v57 labeled "Industry." For most years, this variable takes a value from 1 to 51. However, the industries associated with particular values vary through the years. For example, from 1968 to 1971, a value of 33 or 34 for v57 indicates an individual in the finance, insurance, and real estate industry, whereas from 1972 to 1982, this industry is indicated by a value of 35 or 36.

For 1991-2002, the variable indicating industry is A_DTIND. For 1992-2002, this variable is labeled "Current Status-Industry Detailed Recode," and for 2003-2004, it is labeled" Industry and Occupation-Main Job Detailed Industry." Again, the variable generally takes a value from 1 to 51.

In each year, data were adjusted using the appropriate population weights provide by the Current Population survey. Since data are provided sporadically, military is excluded.

TABLE A1

Disaggregated Industry Mapping

 1968-1971 1972-1982 1983-1988

Agriculture 1 1-2 1
Mining 2 3 2
Construction 3 4 3
Manufacturing durable goods 4-14 6-17 (a) 4-17
Manufacturing nondurable goods 5-24 18-27 18-27
Transportation 25-26 28-29 28
Communications 27 30 29
Utilities and sanitary services 28 3l 30
Wholesale trade 29 32 31
Retail trade 30-31 33-34 32
Finance, insurance, and real
 estate 33-34 35-36 33-34
Private household 32 37 35
Business services 35 38 36
Personal services 37 40 38
Entertainment and recreational
 services 38 41 39
Hospitals 40 43 40
Medical services 39 42 41
Education 42 45 42
Social services 41 44 43
Other professional 43 46 44
Forestry and fisheries 44 47 45
Public administration 45-48 48-51 46
Auto and repair services 36 39 37

 1989-1991 1992-2002 2003-2004

Agriculture 1-2 1-2 1
Mining 3 3 3
Construction 4 4 4
Manufacturing durable goods 4-18 5-18 5-13
Manufacturing nondurable goods 19-28 19-28 14-20
Transportation 29 29 23
Communications 30 30 25-31
Utilities and sanitary services 31 31 24, 39
Wholesale trade 32 32 21
Retail trade 33 ** 33 22, 45-46
Finance, insurance, and real
 estate 34-35 34-35 32-35
Private household 36 36 50
Business services 37 37 37-38
Personal services 39 39 48
Entertainment and recreational
 services 40 40 44
Hospitals 41 41 41
Medical services 42 42 42
Education 43 43 40
Social services 44 44 43
Other professional 45 45 36, 49
Forestry and fisheries 46 46 2
Public administration 47-51 47-50 51
Auto and repair services 38 38 47

Notes: For economy of presentation, we aggregate the 23 industries
of Table Al into the 13 industries of Table 1. This mapping is
provided in Table A2.

(a) 5 is ordnance and is not included since this is not available
in other years.

TABLE A2

Aggregated Industry Mapping

Aggregated Industries in Table 1 Components from Table A1
Educational and health services Hospitals, medical services,
 and education
Professional and business Business services, other
 services professional, and social
 services
Financial services Finance, insurance, and real
 estate
Public administration Public administration
Leisure and hospitality services Entertainment and recreational
 services
Information Communications
Manufacturing Manufacturing durable goods and
 manufacturing nondurable goods
Mining Mining
Other services Personal services, utilities,
 and sanitary services
Wholesale and retail trade Retail trade and wholesale trade
Transportation Transportation
Construction Construction
Agriculture and forestry Agriculture, forestry, and
 fisheries

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(1.) The origin of this concern is probably local news outlets or trade publications that have more focused readerships. Headlines such as "A house divided: Manufacturing in crises," from the November 1, 2005, issue of Industry Week or "Textile trade deficit hits all-time high," from the March 7, 2005, edition of the Southwest Farm Press indicate the feeling behind these changes. While academic studies, such as Crandall (1993), Sachs and Shatz (1994), or Fisher and Rupert (2005), offer support to the decline in manufacturing, they offer more objective viewpoints about the costs and benefits of the changing industrial structure.

(2.) For instance, Krueger (1993) has asked how computers have impacted wages and Greenwood and Yorukoglu (1997) have discussed the advantages new information technologies have for productivity.

(3.) Kaldor is credited with bringing these facts to the attention of the profession through numerous reports in the 1950s and 1960s. Solow (1970) reflects on his work in growth and credits Kaldor (1961) as a source for the empirical facts. More recently, these facts have inspired Lucas (1988) and Romer (1987) in their work on endogenous growth.

(4.) See, for instance, Kaldor (1961) for one of his presentations of the facts. The first five of these facts indicate that in industrial countries output, employment and capital grow at a steady rate, while the capital/output ratio and factor shares are constant. These facts have been reviewed and the data series extended by Solow (1970), Romer (1987, 1990), and Blankenau and Cassou (2006).

(5.) See, for instance, Denison (1985), Jorgenson, Gollop, and Fraumeni (1987), Jones (2002), and Blankenau and Cassou (2006).

(6.) We define people who had four or more years college as being skilled, while people with less are described as unskilled. This definition is common in the economics literature. See, for example, Carneiro and Heckman (2003). These trends are robust to alternative methods for defining skilled workers and unskilled workers. For example, expanding the class of skilled workers to include associate college degree holders produced largely the same results.

(7.) For further information on the data, see Appendix B.

(8.) Leisure and hospitality services and information are marginal industries, which could have been just as easily regarded as initially high skilled. Grouping them with the initially high-skilled industries does not change the aggregate result that the initially high-skilled industries have larger absolute increases in the ratio of skilled to unskilled labor yet have smaller growth rates in this ratio. A table with this breakdown can be obtained from the authors.

(9.) Some popular press single industry anecdotes, however, are well known. For instance, the fact that the manufacturing industry has accounted for a declining share of GDP is widely reported and discussed in mainstream media.

(10.) See, for instance, Table 15 of Yuskavage (1996) where industry growth rates are reported for several intervals of time or Chart 2 in Lum, Moyer, and Yuskavage (2000) where percentages of GDP accounted for by private services-producing industries, private goods producing industries, and the government are plotted from 1963 to 1998.

(11.) An appendix describing an aggregate form of the model and its conversion into the intensive form can be obtained from the authors upon request.

(12.) We assume that only one type of capital good exists to keep the formulation simple. It is possible to have a variety of capital goods and aggregate them in a fashion analogous to what is done in the consumption good sector, but this only adds modeling structure and does nothing to change the main findings.

(13.) In particular, firms producing a given single product require an increasing percentage of skilled workers.

(14.) In our specification, however, new goods do not replace old goods.

(15.) The richness of the dynamic structure requires some compromise of precision to keep the notation succinct. First, since [n.sub.a,t] may not equal [n.sub.b,t], the support of [omega] [member of] [0, [n.sub.j,t]] may differ by industry. It would be more precise then, for example, to write [y.sub.j,[omega](j),t] indicating the sector specificity of [omega]. However, no later confusion arises with our abbreviated notation of [y.sub.j[omega],t]. Furthermore, it will turn out that any item indexed by j [member of] {a, b} is in general time specific even along the balanced growth path. In this sense, the time notation is redundant for these items.

(16.) See Katz and Murphy (1992), Blankenau (1999), and Blankenau and Cassou (2008).

(17.) With log preferences intertemporally, the term 1/[n.sub.a] + [n.sub.b] plays no role in the dynamics of the model. We include it to eliminate love for variety and clarify that our results are not driven by this consideration.

(18.) Recall that the model is written in intensive form. The conversion from levels to intensive form generates the [g.sub.A] and [g.sub.L] terms in the standard way. Details are available from the authors.

(19.) Equations (11) and (12) appear as both constraints to the consumer and labor market clearance conditions because we have assumed there to be a single representative agent.

(20.) Since this result is a generalization of our earlier result, it is presented here with no proof. Readers interested in the formal proof can obtain one by contacting the authors.

(21.) For [sigma] = 0, it can be shown that [z.sub.l] [equivalent to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A similar expression arises for [Z.sub.j.[omega]].

(22.) Because - [[??].sub.t]/1 - [??].sub.t] = 1-[v.sub.t]/1 - [[??].sub.t], -[[??].sub.t]/ 1 - [v.sub.t] can be interpreted as the growth rate of 1 - [v.sub.t].

(23.) Equation (14) shows that [v.sub.t] is the share of capital allocated to investment. Since 1/[theta] units of time must be spent in education to provide a unit of skilled labor, [S.sub.l,t](1 + 1/[theta]) units of time are required to provide [s.sub.l,t] units of skilled labor to investment. From Equations (16) and (17), note that [u.sub.l,t] + [s.sub.l,t](1 + 1/[theta]) = [v.sub.t].

(24.) See, for example, Beaudry and van Wincoop (1996), Vissing-Jorgensen (2002), Zhang (2006), and the calibration exercises of Prescott (1985) and Jones, Manuelli, and Siu (2000).

(25.) The discussion for [PSI] > 1 is similar, while the case in which [PSI] = 1 results in the industrial shares that are unchanging over time. The result when [PSI] = 1 is discussed more fully in example 2.

(26.) It can be shown that the share of time and capital used for good [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(27.) Katz and Murphy (1992) estimate [sigma] = .291, Blankenau (1999) finds [sigma] = .414, and Blankenau and Cassou (2008) find [sigma] = .284.

(28.) See Adler and Clark (1991), for example.

(29.) Some of the parameters were chosen with the following facts in mind. Katz and Murphy (1992) and Blankenau (1999) provide estimates of o. Researchers often set [alpha] in the range .3-.4. A value of 10 for [theta] is consistent with 4 yr of college for a 40-yr career.

(30.) In this case, simplifying by doing is too rapid and the skilled ratio falls without setting [PHI] > 0 to counter this.

(31.) Chief Investigator: Robert Moffit, University of Michigan. Published by the Inter-university Consortium of Political and Social Research in 1999.

WILLIAM F. BLANKENAU and STEVEN P. CASSOU *

* We would like to thank seminar participants at the V Workshop on International Economics in Malaga, T2M conference in Toulouse, Durham University, England, Universidad del Pais Vasco, Universidad de Oviedo, Universidad Carlos III de Madrid, University of Kansas, University of Missouri, Kansas State University, Indiana University, the 2006 North American Summer Meeting of the Econometric Society at the University of Minnesota, and the 2006 Midwest Macroeconomics meeting at Washington University in St. Louis for helpful comments on earlier drafts of the paper. We would like to offer particular thanks to Gonzalo Fernandez de Cordoba, the editor of this journal, and two anonymous referees for their insights as well. Cassou would also like to acknowledge the support and hospitality of Universidad del Pais Vasco and Spanish Ministry of Education and Science, grant number SEJ2006-12793/ECON. 2006-2009.

Blankenau: Department of Economics, 327 Waters Hall, Kansas State University, Manhattan, KS 66506. Phone (785) 532-6340, Fax (785) 532-6919, E-mail blankenw@ksu.edu

Cassou. Department of Economics, 327 Waters Hall, Kansas State University, Manhattan, KS 66506. Phone (785) 532-6342, Fax (785) 532-6919, E-mail scassou@ksu.edu

TABLE 1

Skilled to Unskilled Labor Ratios Across Industries

 Skilled to Unskilled Ratio

 1968 Value 2004 Value Change

Initially skilled
 Educational and health services 0.562 0.874 0.312
 Professional and business services 0.334 0.682 0.348
 Financial services 0.180 0.582 0.403
 Public administration 0.177 0.622 0.445

Initially unskilled
 Leisure and hospitality service 0.105 0.487 0.382
 Information 0.101 0.636 0.535
 Manufacturing 0.083 0.279 0.197
 Mining 0.068 0.159 0.092
 Other services 0.061 0.216 0.155
 Wholesale and retail trade 0.056 0.177 0.122
 Transportation 0.046 0.167 0.121
 Construction 0.045 0.111 0.066
 Agriculture and forestry 0.026 0.159 0.134

Aggregates
 Initially skilled aggregate 0.363 0.741 0.378
 Initially unskilled aggregate 0.064 0.217 0.153

 Normalized Ratio

 1968 Value 2004 Value Change

Initially skilled
 Educational and health services 1.000 1.556 0.556
 Professional and business services 1.000 2.042 1.042
 Financial services 1.000 3.242 2.242
 Public administration 1.000 3.523 2.523

Initially unskilled
 Leisure and hospitality service 1.000 4.632 3.632
 Information 1.000 6.278 5.278
 Manufacturing 1.000 3.384 2.384
 Mining 1.000 2.355 1.355
 Other services 1.000 3.546 2.546
 Wholesale and retail trade 1.000 3.186 2.186
 Transportation 1.000 3.607 2.607
 Construction 1.000 2.473 1.473
 Agriculture and forestry 1.000 6.251 5.251

Aggregates
 Initially skilled aggregate 1.000 2.040 1.040
 Initially unskilled aggregate 1.000 3.404 2.404