Human Capital Investment, Inequality and Economic Growth

Growth, Human Capital Investment and Inequality

We begin with a basic model of economic growth in which aggregate output at date t is determined by the size of the labor force (L), the per-worker stocks of skilled (S) and unskilled (U) human capital embodied in L, physical capital (K), and the state of technology (). Normalizing L=1 expresses all quantities in per-worker units, and we write output per worker as:

1. 
   
   

Corresponding to the three inputs are three factor prices (rental rates), R_S, R_U, and R_K, all measured in real terms. Our assumption that there are only two skill types is obviously limiting, as we will note below, but it serves to make our essential points in a very simple framework while sacrificing little in terms of generality.

The driving force behind growth is technological improvement that raises output produced by given factor quantities, and determines factor prices. We assume that physical capital is elastically supplied in the long run, so that R_K is exogenously determined while R_S and R_U are endogenously determined by demand (technical change) and supply (investment in human capital) forces specified below.¹⁵ On the demand side, the evident long-term increase in measures of the skill premium (R_S/R_U) indicates that the effects of capital deepening and/or biased technological progress have favored skilled labor, so that one or both of the following conditions hold:

(2)

where F_j denotes the marginal product of factor j. We are agnostic as to the relative contributions of (2a) and (2b). For example, (2a) could result from a declining price of physical capital combined with greater ease of substitution of capital with unskilled labor than with skilled labor ( in the usual notation).¹⁶ For simplicity we assume in what follows that rising relative productivity of skilled labor is generated by skill-biased technical change (SBTC), as in (2b), and R_K is assumed to be fixed.

Specifically, we assume that labor inputs S and U only appear in (1) through a single human capital aggregate H(S,U). Allowing for labor-augmenting technical progress A_t, output per worker is

3. 
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With competition, constant returns and capital in perfectly elastic supply at constant price R_K, the rate of growth in output per worker is determined by the growth rates of A and H:

3. 
   
   

which embeds capital deepening because K grows in proportion to A and H. According to (4), for a given rate of labor-augmenting technical progress the growth rate of output per worker depends only on the growth of human capital per worker—the ability to upgrade the average worker’s skills. Forces that limit human capital accumulation, such as the deceleration in growth of educational attainment documented above, correspondingly limit growth.

We place additional structure on H by assuming a constant elasticity of substitution between S and U:

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With (3) and (5), S and U are weakly separable from other factors. Then the equilibrium evolution of relative skill prices must be consistent with firms’ willingness to employ the supplied stocks of skills:

3. 
   .
   

where . Then SBTC is represented by, which raises the relative productivity of type-S skills as in (2b) so long as , which evidence indicates and which we shall assume in what follows. The share of labor income accruing to type-S workers is simply

.

With and a given skill premium , SBTC () raises the skilled income share. But for a given state of technology B_S, a higher skill premium () reduces the skilled share because relative demand is price elastic. This property will prove important in our examination of the relation between inequality and growth, developed below.¹⁷

Condition (6) is familiar in the analysis of changing relative wages. Assume that skill biased technical progress causes the relative demand for S to grow at a steady rate over the long term, so is constant. Then changes in the factor ratio S/U drive the returns to skill—if demand grows faster than supply then the skill premium R_S/R_U will rise, and conversely. For example, Katz and Murphy (1992) apply (6) to the evolution of the male college-high school wage premium in the U.S. from 1963 to 1987, assuming constant relative demand growth. Their estimate of for the elasticity of substitution between college-trained and high school-trained labor does well in tracking the college wage premium, even well outside of the sample period they study—see Murphy and Welch (2001). Autor’s (2002) review of evidence from several studies offers a somewhat higher “consensus” estimate of . The only important point for what follows is that —the relative demand for skilled labor is price elastic.

The Supply of Skills

The (inverse) demand equation (6) determines the relative rental prices of skilled and unskilled human capital for any given stocks, S and U. Our point of departure is to explicitly model behavioral responses on the supply side that determine the relative abundance of skilled human capital, S/U. For each skill type, we specify the supply of skills as being the result of individuals’ wealth maximizing human capital investments and their choices of how a given quantity of human capital should be applied. Then both the overall quantities of skills of each type and their distributions across workers are endogenous.

We maintain the structure of (1) and (3) in which there are just two types of human capital, skilled (S) and unskilled (U)—generalizing to an arbitrarily large hierarchy of skills and associated relative prices is straightforward.¹⁸ We think of S and U as categories of workers, such as those with and without a college education. To save on notation, it will not cause confusion to use S and U to denote both skill types and the average amounts of each type of human capital that enter the production function. For given skill prices R_S and R_U that are expected to apply over working careers individuals choose whether to be skilled or unskilled, given their backgrounds and abilities. Even with only two skill types, this setup will generate a full income distribution because we assume that individuals have heterogeneous “abilities” to invest in human capital, and so they will acquire different quantities of skills and apply them in different ways.

Specifically, given choice of skill type , we assume that individuals make an investment choice of how much human capital, H_j, to acquire. They also choose how intensively to use their human capital, which we denote as T_j. The simplest interpretation of T that it represents simple labor supply (e.g. hours worked as in Figures 7A-B), but we view it more broadly as representing alternative opportunities to apply a given stock of skills. For example, in a world where the rental price of skilled human capital, R_S, is high, skilled (S) individuals may choose to apply their human capital to more remunerative though less pleasant activities, such as business occupations rather than teaching. Then T is a shorthand embedding occupational choice, effort and initiative in the model. The fact that changes in the intensity of skill use occur on margins other than time worked has the important empirical implication that these intensive margin responses will show up in wages and not just earnings.¹⁹

Let a represent an individual’s investment abilities with c.d.f. G(a) in the labor force. For an individual with investment abilities a who has chosen to be of skill type j, we assume that the choices of H_j and T_j solve

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The first term on the right of (7) is total earnings from supplying H_j units of human capital at intensity T_j, which is proportional to the rental price of type-j skill, R_j. Thus human capital supplied by an ability-a individual is . The remaining terms are the costs (disutilities) of acquiring skills and applying them intensively. We assume rising marginal cost of acquiring human capital where is the constant elasticity of marginal cost with respect to H; marginal cost rises faster when is large. Greater intensity of use is also subject to rising marginal cost, with elasticity. The cost of acquiring human capital also depends each individual’s ability to invest, a, through the cost shifter c_j(a). Individuals differ in this ability, and we assume that higher ability individuals are better at investing (they have lower costs of acquiring skills of either type):

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We make the natural assumption that type-S human capital is more costly to acquire, so for all abilities a; additional conditions on these costs appear shortly. We maintain the shorthand of referring to a as an individual’s “ability” to invest, though we don’t think of it as simply an individual’s ability in the usual sense. In fact, in our analysis ability a only affects earnings indirectly, by making it easier to acquire human capital, so more is acquired. We therefore interpret a as a broad index of advantages in acquiring human capital, encompassing much more than just individual talents. For example, it can also embed family or other characteristics (educated or wealthy parents, access to better schools, and so on) that make it cheaper or easier for some individuals to acquire human capital than for others. Then greater human capital investments by one generation will reduce the average costs of investing in the next generation by shifting the distribution of a. For our purposes the important thing is that people differ in characteristics that make the acquisition of human capital more or less difficult.

Given a person’s chosen type j (S,U), the necessary conditions for optimal choices of H and T in (7) are instructive:

Condition (9a) indicates that human capital H is more valuable when it can be used intensively (T is large), so more is acquired. Condition (9b) indicates that intensity of use is greater when H is large, so more human capital is applied. Thus H and T are strong complements because they are multiplied in the first term of (7). This will have important implications below. The solutions for H and T are (in logs):

(10a)

(10b)

The second order condition for a maximum of (7) is , so both H_j and T_j are increasing with Rj and also with ability a, due to condition (8). More able investors acquire more skills (10a) and also apply them more intensively (10b), so earnings exhibit a form of increasing returns in ability.

Now define the following price elasticities of human capital acquisition and intensity of use:

(11)

Total human capital applied is , so for a person of ability a

(12a)

and log earnings are

(12b)

Note from the definitions in (11) that reductions in either cost elasticity (or) increase the price elasticities of human capital supplied (Z) and earnings (E=RZ).

Equations (10a-b) and (12a-b) are the solutions for human capital acquired (H), intensity of use (T), supply (Z) and earnings (E) given an individual’s ability a and choice of a skill type S or U. They can be inserted in (7) to obtain an expression for maximum utility that can be realized from the choice of skill type j:

(13)

Equation (13) is the Given (13), an individual of investment ability a chooses a skill-type to maximize utility. That is, a person of ability a chooses to be skilled (S) if , and conversely. With appropriate conditions on and this choice implies a cutoff level of investment ability a* where only individuals with a >a* choose to be type-S while those with a < a* choose to be type-U. The indifference condition determining a* is , which from (13) implies for marginal individuals. Then earnings are monotonically increasing in ability and a marginal individual would earn identical amounts from either skill type.²⁰ A bit of algebra then yields:

(14) .

The cost of producing type-S human capital must be higher than for type-U, otherwise all would choose S because we assume R_S > R_U. We assume conditions on and so that a greater premium for type-S skill increases relative supply of S by drawing in lower a investors:

(15) .

An increase in the skill premium R_S/R_U “pulls in” lower ability individuals on the margin if the costs of producing type-S human capital fall more rapidly with ability than the costs of producing type-U. That is, we assume that the relative cost of producing type-S human capital is smaller for more able individuals.

Equations (10), (12) and (15) specify three margins by which an increase in the return to skill drives investment in human capital and so expands the relative supply of skills applied in the market. First, in (10a), an increase in R_S expands investment on the intensive margin—all type-S individuals (a > a*) invest more because the value of each unit of H_S is greater. Second, complementarity of H and T reinforces this response in (10b) because each unit of human capital is also applied more intensively—for example by working more or seeking opportunities to apply the larger stock of skills to more valuable uses—which further raises the return to investment. Thus total human capital applied, , rises by even more. These effects “exacerbate” the impact of a change in the skill premium on income inequality—the elasticity of earnings with respect to the premium is strictly greater than unity—because high ability individuals make complementary adjustments in behavior to exploit their price advantage.

The third source of “skilled” labor supply is the extensive margin determined by (14). As R_S rises relative to R_U the share of workers who choose to be type-S rises because greater returns cause individuals on the a* margin to switch from U to S—for example, by attending college or acquiring other forms of type-S skill. The magnitude of this response depends on the distribution of investment abilities, G(a) with density g(a). The aggregate human capital factor ratio is:

where and are the average amounts of human capital applied by persons of each skill type. Using the solution for Z_j(a) in (11) we obtain an expression for the aggregate skill ratio on the supply side.

(16) .

Now let be the hazard of G; then is the percentage increase in the type-S share per unit reduction in a*. Displacement of (16) and substitution of the extensive margin response from (15) yields an expression for growth in the relative supply of “skilled” human capital:

(17)

In (17) the term represents exogenous supply shifts that change the skill ratio over time, such as through changes in the costs and availability of schooling, skill-biased immigration, or long-term changes in the distribution of investment abilities.²¹ Such long term changes may occur, for example, because of changes in the quality of schools or because increased skills acquired by one generation—higher college attendance by the baby-boom generation, for example—affect the ability to produce human capital in their offspring, “bathing” them in cognitive skills, as Heckman (2008) phrased it. Then the distribution of a would change over time. This is related to the effects of human capital in some endogenous growth models, where a greater stock of human capital reduces the cost of producing more; see Becker et al. (1990) and the discussions in Topel (1999) and Acemoglu (2009). An important distinction is that in our analysis the complementarity is assumed to occur at the “micro” level of individuals and families, so that investment responses to a higher skill premium are heterogeneous.

The bracketed price elasticity in (17) is the endogenous supply-side response of human capital (the skill ratio) to an increase in the skill premium. It includes responses on the intensive and extensive margins mentioned above. The intensive margin(s) response to a rising skill premium is > 0: holding constant the share of the labor force that is skilled, a rising price of skill causes greater relative investment by high ability type-S workers (), who also apply their greater skills more intensively than before (). This response is stronger ( is larger) when the cost elasticities and are small; see (11). The terms making up represent the supply response on the extensive margin—individuals who are drawn into the skilled labor pool by higher returns. This elasticity is greater when (i) the hazard is large, which means that persons with the potential to become skilled are abundant relative to the existing stock (i.e. there are many individuals that are close to the margin); (ii) when is large, so that “new” type-S workers are similar to existing ones; and (iii) when the skill premium “moves” the extensive margin a* by a lot (see (15)).

The Supply of Human Capital and Equilibrium Inequality

The bracketed terms in (17) determine the aggregate supply elasticity of relative skills, S/U. The demand elasticity for S/U is , the elasticity of technical substitution between the skill aggregates. We can insert (17) into (6) to obtain an expression for the evolution of the skill premium in terms of demand and supply shifters and the behavioral responses of buyers and sellers:

(18a)

The bracketed term measures growth in net demand for skilled human capital; the skill premium and hence earnings inequality will be rising if growth in relative demand for skill induced by SBTC, , outpaces the exogenous growth in relative supply,. Equation (18a) is a market equilibrium framework for thinking about the determinants of a rising skill premium, which in our analysis is the driving force behind observed increases in wage and income inequality. But the skill premium is not a direct measure of earnings inequality because of the magnifying effects of human capital investment and utilization responses discussed above. To see this, consider two fixed levels of ability and ; for example, at fixed percentiles (say 90 and 10) of the earnings distribution. Then the earnings ratio between these ability levels is

Using (18a),

(18b)

Comparison of (18a) and (18b) illustrates the important distinction between sources of human capital supply response and their implications for earnings inequality. Specifically, greater supply elasticity on the extensive margin mitigates inequality because more workers choose to become skilled in response to a rising skill premium, just as entry by new sellers dampens the impact of rising product demand on price in a competitive industry. In contrast, greater supply elasticity on the intensive margins magnifies earnings inequality (when ) because infra-marginal individuals respond to a higher skill premium by investing in more skills and applying them more intensively, which increases earnings disparities between high and low ability individuals. In our view, this distinction is especially important in light of the long term “stall” in college completion rates among men, which was documented above. The failure of supply from the extensive margin to keep pace with rising demand for skill raised the skill premium, and so created the incentive for the more able to benefit even more, in proportion to the elasticity . This has magnified the impact of SBTC on earnings inequality.

When growth in the supply of skilled labor on the extensive margin is sufficient to keep the skill premium from rising, inequality between individuals of differing abilities will remain unchanged since the intensive margin responses will be neutral across skill groups. But when growth in supply on the extensive margin is insufficient to maintain a fixed skill premium, supply responses on the intensive margin come into play. These responses mitigate the impact of the supply changes on skill prices by increasing the relative supply of the skill type with the rising relative price. However, these same responses exacerbate the impact on inequality since they further increase earnings for the skill group that experienced a rising relative price. When interpreted in at the level of families, this magnifying effect on inequality can play out over generations.

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   Inequality and Growth
   

Our analysis above indicates a central role for the supply of human capital, on differing margins, in determining equilibrium inequality. The next step is to incorporate these outcomes into the model of economic growth given by (4), repeated here:

(19)

Recalling that is the labor income share of skilled workers, displacement of (5) yields:²²

(20)

All factor prices are measured in real terms and capital is in perfectly elastic supply () so productivity growth accrues to human capital because of induced capital deepening:

(21a)

or

(21b)

Using condition (21a) in (20) eliminates price terms, yielding a simple expression for the growth rate of the human capital aggregate:

(22)

According to (22), aggregate human capital per worker grows for two basic reasons. First, technical progress () raises both skill prices and induces skill acquisition and utilization by both S and U workers, with common supply elasticity. Second, SBTC () and supply shifts () raise H by directly increasing the effective amount of type-S human capital. These effects are proportional to the skilled (S) share of labor income, .

The final step is to use (22) in (19), obtaining an expression for growth in output per worker:

(23)

Contemporaneous changes in the skill premium, given by (18), are second order and so they do not appear directly in either (22) or (23). Thus it might appear that factors causing greater income inequality are also of second order importance for economic growth. Yet (23) draws an important distinction between the effects of labor-augmenting but skill-neutral technical progress and skill-biased changes in technology and supply on economic growth. Skill-biased technical progress and exogenous supply growth increase overall productivity growth by augmenting the relative supply of skilled human capital. This human capital deepening impacts overall productivity growth in proportion to the labor income share of the affected skill group, , which is endogenous. From the definition of the skilled labor income share:

(24)

With, the skilled income share declines as the skill premium R_S/R_U increases because relative demand for skilled human capital is price elastic. So, for a given rate of change in skill-biased technology, greater inequality reduces economic growth because a higher skill premium induces substitution away from skilled human capital, which is a source of productivity growth in our model.

How important is inequality as an impediment to productivity growth? The calculation isn’t straightforward because we do not observe a direct estimate of the change in R_S/R_U over time; instead we observe changes in relative wages, which include the behavioral responses of human capital investment and utilizations represented by the elasticity . To get a rough (and probably conservative) estimate of this, consider the labor supply responses of high-wage individuals, as graphed in Figures 7A and 7B. Treating as a pure labor supply (hours) response and using the data from Figures 7A and 7B, Table 2 shows estimates of the ratio

for various intervals in the upper half of the male and female wage distributions. The implied elasticity is largest in high percentiles, where wage gains and hours increases were the biggest. Near the top of the respective distributions the estimates for men and women are remarkably similar, about .09 for both. If is of similar magnitude then a (very) rough estimate is .

We use the college/high school wage premium as an index for changes in R_S/R_U over time. According to Figures 4A and 4B, this premium increased by about 50 log points after its 1979 nadir. Using implies =.50/1.2 = .42. According to (18), this increase would have been mitigated if the endogenous supply of skilled workers had been highly elastic (if had been large) or if exogenous supply growth of skilled human capital was sufficient to offset rising demand. So assume counterfactually that these effects had been large enough to maintain the skill premium at its 1979 level. Then R_S/R_U would be 42 log points lower than it was. Productivity growth in the U.S. has averaged slightly more than two percent per year since 1979, so per year. Assume further that dlnA=0, which means that all productivity growth has been due to SBTC and growth in supply. Defining skill groups in terms of efficiency units of college-educated and high-school educated workers let , so the bracketed growth rate of human capital is 3.3 percent per year. With, as discussed above, (24) implies that the skilled income share would be percent higher—call it =0.70. Then, had inequality not increased in response to SBTC, the growth rate of labor productivity would be ==.0033 per year higher than it was. Over 10 years this reduction in inequality would increase productivity by about 3.4 percent.²³

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