Long-run economic growth

Computer Technology, Human Labor, and Long-Run Economic Growth

Stuart W. Elliott Carnegie Mellon University

Heinz School Working Paper 98-23 Revision: August 1998

This research was supported by a grant and a visiting scholarship from the Russell Sage Foundation. Early exploratory funding was also provided by the Sloan Foundation. Useful suggestions and criticisms about earlier versions of this material were provided by Ashish Arora, Harvey Brooks, Peter Cappelli, Hirsh Cohen, Richard Cyert, Richard Day, Amanda Ellsworth, Gary Fields, Jon Fincham, Robert Gibbons, Claudia Goldin, Robert Heilbroner, Mark Kamlet, Lawrence Katz, Rachel Kranton, Lester Lave, Frank Levy, Richard Murnane, Paul Osterman, Arnold Packer, Michael Piore, Paul Romer, Herbert Simon, Robert Solow, Lowell Taylor, Eric Wanner, and participants at seminars at Carnegie Mellon, MIT, the Milken Institute, and the Russell Sage and Sloan Foundations. Research assistance was provided by Amanda Lehman and Rebecca Hanson.

1998 by Stuart W. Elliott. All rights reserved.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 2

Abstract

Over the coming century, computer technology is likely to become capable of reproducing many of the skills now performed by human labor. This paper describes three models of the aggregate economic changes that occur when capital becomes capable of performing human work skills. The basic model, with a single sector and homogeneous labor, projects output growth rates over the next few decades that are substantially above historical growth rates in industrialized countries, assuming plausible increases in computer skill. The projected output growth is accompanied by structural changes reflecting the reduced role of labor, with wage growth lagging output growth and the labor share of output decreasing. Resource limits do not substantially affect the levels of output and wage growth in the near future. The 2-type model, with fixed skill differences between different workers, produces similar growth in output and average wages over the next several decades. However, the worker skill differences produce large increases in wage inequality between types of workers. The 2-sector model, with different skill requirements for different economic sectors, also produces similar growth in output and wages over the next several decades. For the three models, asymptotic growth in output and wages is substantially reduced by resource limits, worker skill differences, and sector skill differences, even though those constraints do not substantially reduce growth over the next few decades. The models produce patterns of change in the labor share and capital-output ratio that are consistent with broad trends in economic data. JEL Classification: O33, O41, J24 Keywords: computers, technology, productivity, growth, labor demand, wage inequality.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 3

Computer technology offers the potential of reproducing many of the human skills that are currently used in the economy. As portions of this potential are realized, there will be an increase in the range of skills provided by computers and a corresponding decrease in the range of skills provided by human labor. Such shifts in capabilities from humans to computers are a fundamental aspect of the economic impact of computer technology. With each passing year, it becomes increasingly important to be able to model the changing nature of computers as they move further beyond performing the simple tasks, such as database manipulation and word processing, that have been the focus of office machinery for the past century. This paper addresses the theoretical challenge of analyzing the economic impact of computers by developing three models of aggregate economic growth and labor demand in which the skills of capital increase over time. To make the nature of the modeling problem clear, it is helpful to consider the changes that have occurred over the past two decades in computer speech recognition. In the mid-1970s, research systems for continuous speech recognition operating on milliondollar computers processed speech 10 times more slowly than the rate at which people speak, using vocabularies of 100s of words, and needing to be trained to individual speakers (D. Raj Reddy, 1976). Today, such systems process speech as fast as people can speak on high-end personal computers (PCs), using vocabularies of 10,000s of words, and being able to adjust automatically to the idiosyncrasies of different speakers (BBN, 1994). Thus 20 years of research have transformed speech recognition from a skill that was impossible for computers to perform at any practical level into a skill that is now poised for substantial commercial application. Based on the rate of diffusion of past innovations (Frank M. Bass, 1969; Edwin Mansfield, 1989) it is reasonable to expect that speech recognition technology will be applied throughout the economy over the next several decades. However, the skill of speech recognition is used in different work tasks and by different occupations than the

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 4 skills related to databases and documents that computers have previously performed. Because of these differences, it is likely that the impact of computer speech recognition on the economy will be different than the impact of these earlier computer skills. If we want to project the impact of computers from the 1990s into the next several decades, or if from the vantage of 2010 or 2020 we want to understand the impact of computers for the previous several decades, we must recognize a technological shift as computers begin to take over speech tasks. Of course, speech recognition is not the only new computer skill that will be applied over the next several decades. A review of computer science research reveals that the entire range of human skills is on the long-run research agenda (Jon Doyle and Thomas Dean, 1997). Some of these capabilities, like speech recognition, are beginning to be applied already, whereas others will not begin to have an economic impact until one or more decades from now. Even though these skills will all be performed using computers, they represent substantially different technologies with different potential impacts throughout the economy. An analysis of the impact of computers on productivity and labor demand requires theoretical tools that can take into account these changes in their capabilities. Historically other technologies have allowed capital to reproduce portions of the human skill set. In particular, domesticated animals and a variety of motive technologies have approximated some physical human skills, involving physical force combined with simple control. Similarly, various mechanical and electronic calculators, predecessors of computers, have approximated some cognitive human skills involving arithmetic and very simple reasoning. Because of these technologies, the distribution of skills used by humans in the economy has changed dramatically over the past several centuries, as capital has provided perfect substitutes for some human skills and close substitutes for others.1

1

There is no indication that the set of basic human skills has changed substantially over the past several centuries -- or, indeed, over the past several millennia. New technologies often require new tasks to be performed that are collected to form new occupations, but these new tasks and occupations draw on an unchanging set of basic human skills involving reasoning, language, perception, and movement.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 5 Although this reproduction of human skills is one important aspect of general technological change, I focus here on computer technology in particular. I do this because it is computer technology that has the potential to reproduce most of the skills that are today still performed by humans in the economy. The paper uses a general framework for analyzing capital's reproduction of human skill that can be applied to earlier technological change, but the model is specified in a way that matches stylized facts about computer technology. Section I presents the basic model. Section II describes a sensitivity analysis of the basic model that shows the effect of future computer technology on economic growth and labor demand. Section III describes the 2-type model, in which there are two fixed types of labor and the new computer skills affect only one of these types. This extension is used to model the impact of computers when the workers displaced by them can move into some occupations but not others. Section IV describes the 2-sector model, in which the new computer skills are used in only one sector of the economy. This extension is used to model the impact of computers that can be applied to some industries but not others. Section V discusses the application and testing of the models historically. Section VI concludes.

I. A Basic Model of Increasing Computer Skill

A. Overview and Related Literature

The key innovation of the models in this paper is to describe aggregate production as a function of skill inputs rather than as a function of capital and labor. Some of the skill inputs can be provided only by capital, others only by labor, and still others by either capital or labor. Change in computer capabilities is described as an increase in the portion of the skill inputs that can be provided by either capital or labor and a corresponding

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 6 decrease in the portion that can be provided only by labor. The pace of this change in computer capabilities is determined exogenously. Under certain assumptions about the cost structure of computers, the optimal use of capital and labor usually turns out to be a corner solution in which capital provides all of the skill inputs that could be provided by either capital or labor. To generate predictions about growth and labor demand, the resulting skill-based aggregate production function is placed into a standard 1-sector growth theory framework. Using standard growth theory parameter values, the model's equilibrium growth path is derived for different assumptions about future computer technology. Since there are no externalities, the model uses simple optimization by a social planner, which produces the same outcome as would be obtained using perfectly competitive firms and households. The model is in some sense an elaboration of a simple model due to Herbert A. Simon (1977), with extensions allowing a more general treatment of the tradeoffs between labor and capital and the inclusion of capital accumulation. The model differs from most other economic models of technological change (e.g., Robert M. Solow, 1957) by specifically focusing on non-neutral change that increases the capabilities of capital while leaving the capabilities of labor unchanged. This difference is what captures the notion that computers, and technology more generally, allow capital to duplicate an increasing portion of human work skills over time. This approach is similar to a recent model of technical progress by Joseph Zeira (1997), except that his model is specified not in terms of skill inputs but in terms of intermediate goods that can be produced with different mixes of capital and labor. The model differs from recent models of endogenous growth (e.g., Robert E. Lucas, 1988; Paul M. Romer, 1990) by focusing on exogenously determined change. Given the economic incentives that currently exist for developing new computer applications, it seems a reasonable first approximation to assume that the effect of economic incentives on technological discovery is already at ceiling. As a result, unless there is a

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 7 substantial decline in these economic incentives, the pace of development over the next few decades will be determined largely by engineering constraints and by the logic of technological discovery. Finally, the model differs from the literature on the economic impact of computers by its explicit focus on changes in computer skill and on the problem of projecting the impact of future change. Much work has already been done to explain the impact computers have had or not had on productivity (e.g., Erik Brynjolfsson and Lorin Hitt, 1996; Daniel E. Sichel, 1997), and to investigate the relation computers have had or not had to the shift in relative labor demand towards skilled workers (e.g., David H. Autor et al., 1997; Timothy F. Bresnahan, 1997). However, the existing literature treats computer technology as a homogeneous black box, providing no way for results using 1970s or 1980s data to be generalized and applied to other time periods in which computer technology is capable of radically different skills. In some cases, a static view of computer skills encouraged by a black box approach has lead researchers to suggest that the economic impact of computers may decrease in the future since they conclude that the higher payoff applications have already been adopted (Sichel, 1997; Robert J. Gordon, 1998). Such a suggestion betrays a lack of appreciation for the range of computer skills that are in the technological pipeline and underlines the importance of developing a model of computer technology that allows economists to talk about the changing nature of their capabilities.

B. The Basic Model

The model's aggregate production function is specified with a constant returns CES function that includes a continuum of skill inputs and a continuum of nonproducible natural resource inputs:

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 8 σ

(1)

1 σ −1 1− β σσ− 1  σ −1 σ Yt = A  ∫ Si, t di + ∫ Ri di  1− β 0 

where Yt is aggregate output, the function Si,t gives the density of the continuum of skill inputs, Ri gives the density of the continuum of natural resource inputs, σ is the elasticity of substitution among inputs, β determines the relative importance of natural resources, and A is the production function constant. A continuum of skill inputs is used to avoid specifying the differential use of different skills and to simplify the description of incremental technological change. A continuum is used for natural resource inputs as well to make the model tractable. The skill inputs to the production function can be provided by capital, labor, or both, for different portions of the skill continuum. The portion of the skill continuum that can be provided by both capital and labor depends on the state of computer technology. Capital and labor are both denominated in terms of human-equivalent years of skill input. In caricature, computer technology is modeled in terms of PCs, and it is assumed that PCs can be substituted 1-for-1 for humans for portions of the skill continuum that computers become capable of performing. The capabilities of PCs increase over time without affecting their cost. This costless increase in computer capabilities comes from a combination of faster processing and better programming, reflecting two stylized facts about computer technology: 1) the historical exponential increase in processing power available per dollar2 ; and 2) the near-zero marginal cost of copying programs.3 2

Since the 1950s, computer prices have decreased at an annual rate of about 20% (Robert J. Gordon, 1989; Dale W. Jorgenson and Kevin Stiroh, 1995). Indeed, it appears that this trend may go back to at least the 1930s, taking into account more primitive computing devices (Hans Moravec, 1988). The trend has lasted through a number of changes in basic processing technologies, including mechanical, relay, vacuum tube, transistor, and integrated circuit technologies. Current assessments project that the trend will continue for at least 10 more years (Ted Lewis, 1996; Philip E. Ross, 1996; Gary Stix, 1995). Prospects for continuing the trend past this point probably involve another switch in basic processing technology, perhaps to quantum, optical, or DNA computers (Joel Birnbaum, 1997). 3 Obviously PCs would require robotic attachments to perform human skills involving perception and movement, and such attachments would entail some extra cost. This minor detail is ignored.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 9 Two types of distributions of capital and labor over the skill continuum are possible in equilibrium. The first type is the corner solution in which capital provides all of the portion of the skill continuum that can be provided by both capital and labor, as well as the portion that can be provided only by capital. The corner solution will be optimal when the marginal productivity of labor is greater than the marginal productivity of capital at this most extreme distribution of capital and labor across the skill continuum. When that condition is satisfied, the budget constraints for the skill inputs in the social planner's optimization problem can be given as: αt

(2a)

∫ Si , t di = Kt ,

if µt > 1

0

1− β

(2b)

∫ Si , t di = L ,

αt

if µt > 1

where Kt and L are the available inputs of capital and labor, and αt is the dividing line between the portion of the skill continuum that can be provided by only capital or by both capital and labor and the portion of the skill continuum that can be provided by only labor. The condition necessary to produce the corner solution is specified in terms of µt which is the equilibrium ratio of the marginal productivity of labor to the marginal productivity of capital at time t. Since the model assumes a competitive equilibrium and computer technology is denominated in terms of PCs, this ratio will also be referred to as the laborPC price ratio. L is assumed for simplicity to be constant. The second type of distribution of capital and labor over the skill continuum occurs when the marginal productivities of labor and capital are equal in equilibrium. In this case, the inputs from capital and labor can be treated as a single pool: capital provides the portion of the skill continuum that only capital can provide and labor provides the portion that only labor can provide, but both capital and labor are used on the portion of the skill continuum

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 10 that both can provide. The budget constraint for the skill inputs in the social planner's optimization problem can be given as: 1− β

(2c)

if µt = 1

∫ Si , t di = Kt + L , 0

For the versions of the model considered in this paper, it turns out that this type of distribution of capital and labor occurs only in extreme cases in which computers take over a large portion of the skill continuum. In principle, there is also a third type of distribution of capital and labor: the corner solution in which labor provides all of the portion of the skill continuum that both capital and labor can provide. However, capital today already provides a large amount of skill input that human labor could provide instead. Increasing the capabilities of capital still further will not make the use of capital less attractive. As a result, this possible corner solution is not of practical interest and will be ignored. If capital and labor are each assumed to be homogeneous, then in equilibrium they will each be allocated evenly across their respective portions of the skill continuum. Similarly, if the aggregate quantity of natural resources, R, is assumed to be homogeneous, then in equilibrium it will be allocated evenly across the continuum of natural resource inputs.4 With these assumptions, the production function (1) can be rewritten in a reduced form using aggregate Kt, L, and R, that takes into account the optimal distribution of these quantities across the skill and natural resource continua:

(3a)

 K  Yt * = Aα t  t    αt  

σ −1 σ

  L + (1 − α t − β )   1 − αt − β 

σ −1 σ

 R + β   β

σ −1 σ

σ

 σ −1  ,  

if µt > 1

4

Aggregate natural resources are assumed to be constant and to provide a fixed flow of productive services over time.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 11

(3b)

  K + L Yt * = A(1 − β ) t   1− β   

σ −1 σ

 R + β   β

σ −1 σ

σ

 σ −1  ,  

if µt = 1

where (3a) makes use of the budget constraints (2a) and (2b), and (3b) makes use of the budget constraint (2c). When the capabilities of capital change, any redistribution of capital and labor over the skill continuum is assumed to be both instantaneous and costless. With the reduced form production function (3a), the ratio of the marginal productivities of labor and capital can be easily calculated:  K (1 − α t − β )  σ ∂Yt * ∂L =  t µt ≡  , Lαt ∂Yt * ∂Kt   1

(4)

if µt > 1

The economy begins in steady-state equilibrium at t = 0 with µ0 > 1 and computer capabilities specified by α0. Computer capabilities increase monotonically over time on some exogenously-determined path αt to the final level α∞. As long as the economy is operating at the corner solution in which capital provides all of the skill inputs it is capable of providing, then each infinitesimal increase in α will initially decrease µ until K has a chance to respond. The subsequent increases in K, to move the economy toward steadystate equilibrium, will increase µ. If an increase in α would push µ below 1, as calculated in (4), then the appropriate reduced form production function will switch from (3a) to (3b). The latter will continue to be the appropriate reduced form production function unless further increases in K once again increase µ above 1, as calculated by (4). While (3b) is in effect, any additional increases in α have no direct effect on output, though such increases make it still more difficult to reach the point where the corner solution (3a) is again in

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 12 effect. Obviously, if α and K are both changing smoothly over time, the net movement of

µ is theoretically ambiguous and depends on the parameters that control how quickly α and K change. In the growth model, the aggregate production function is used in a budget constraint that determines changes in the capital stock: (5)

K˙ t = Yt − Ct − δKt

where Ct is consumption and δ is the rate of capital depreciation. The social planner optimizes over the stream of consumption C to maximize utility: Ct1−θ − 1 − ρ t e dt 0 1−θ

∞

(6)

U (C ) = ∫

where ρ is the rate of time preference and θ is the inverse of the intertemporal elasticity of substitution. Direct solution of the dynamic optimization problem posed by (5) and (6), where consumption is the control variable and capital is the state variable, with output given by (3a) or (3b) in terms of the aggregate capital and labor inputs rather than the underlying skill inputs, yields the following differential equation for consumption:

(7)

 C  ∂Y * C˙t = t  t − δ − ρ  θ  ∂Kt 

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 13

II. Sensitivity Analysis of the Basic Model's Projections of Output Growth and Labor Demand

The behavior of the model developed in the previous section depends on the values of seven parameters. Section A describes the parameter values that are used in the analysis, section B describes the asymptotic behavior of the model, and section C describes the behavior along the equilibrium path.

A. Parameter Assumptions

1. The path of change in the capital parameter αt The path of exogenous change in computer technology that determines αt is described in terms of another parameter at, which is the portion of the current human skill input that future computers will be able to provide: (8)

α t = α 0 + (1 − α 0 − β ) at

where α0 is the initial value of the parameter defining the capabilities of capital and will be discussed in the next section. This rescaling of the technology parameter is done only for convenience so that technological change can be described directly in terms of the current human skill distribution. The path of at is described with a logistic function of t using a single parameter:

(9)

1.1 at =  − 0.1 z 1− .04 t 1 + 10 

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 14 where z is the portion of the current human skill distribution that computers will ultimately be able to provide at t = ∞. This logistic function is constructed so that the technology reaches 90% of its ultimate capability at t = 50. Thus, for example, an assumption of z = 0.8, meaning that 80% of current human skill input could ultimately be provided by computers, would imply a50 = 0.72. Obviously other functions could be used to describe the technology path. Ideally, at should be determined by data on the current human skill distribution that has been combined with projections from computer scientists about which portions of the skill distribution are plausible for future PCs to provide and with economic analyses of the plausible diffusion rate for such future computer skills. However, such an estimate of at is a large project in its own right and goes far beyond the limits of the current paper. Here, the analysis approximates the set of possible at paths by using (9) along with four different values for z that span the feasible range: 0.25, 0.5, 0.75, and 1.0. Preliminary work on projecting future computer skills (not discussed here), suggests that the intermediate values of 0.5 and 0.75 are probably the most plausible, given the substantial range of computer technology currently under development.

2. The initial capital parameter α0 and the natural resource parameter β These two parameters are chosen so that the share of output going to labor at t = 0 is 0.7, to roughly approximate the labor share in the current economy. If η is the portion of the non-labor share of output that goes to nonproducible natural resources, then the reduced form production function (3a) yields the following two share ratios: 1

(10a)

 σ  K0  3(1 − η)  α0 =  7  1 − α0 − β   L 

(10b)

 σ  R 3η  β =  7  1 − α0 − β   L 

1

σ −1 σ

σ −1 σ

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 15

These are combined with (4) to solve for α0 and β in terms of µo, the initial labor-capital marginal productivity ratio: (11a) α 0 =

1 σ −1 7 η  3ηµ0 L  σ −1 µ0 + +1 3(1 − η) 1 − η  7R  σ −1

η  3ηµ0 L  1 − η  7R  (11b) β = σ −1 7 η  3ηµ0 L  σ −1 µ0 + +1 3(1 − η) 1 − η  7R  It is not clear what portion of the non-labor share of output goes to nonproducible natural resources. Calculations by Edward F. Denison (1974) and by Laurits R. Christensen et al. (1980) suggest that the factor share for land in the current economy is roughly 0.04 or 0.1, respectively. However, typically neoclassical growth models omit natural resources, implicitly assuming that resources are not fixed and can themselves be produced using other economic inputs. In support of this assumption, the historical record suggests that ordinary economic responses have been effective in keeping the real prices of natural resources in the U.S. relatively stable over the past 120 years, a period during which total real GDP increased by over 50 times.5 The analysis here uses two different assumptions about natural resources: η = 0 and η = 1/3. The first produces initial output shares of 0.3 for physical capital and 0 for natural resources, whereas the second produces initial shares of 0.2 and 0.1.

5

Harold J. Barnett, 1979; William J. Baumol et al., 1989; Angus Maddison, 1982; U.S. Department of Commerce, 1992.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 16 3. The production function constant A and the initial labor-capital marginal productivity ratio µ0 The choice of one of these parameters determines the value of the other, given the values of L, R, η, δ, and ρ. This can be seen by obtaining the initial steady-state equilibrium level of capital from (3a) and (7): σ

(12)

  σ −1 1 σ −1 σ −1 1  1−α − β σ L σ + βσ R σ  ( 0 )  K0 = α 0  σ −1   δ + ρ   − α0    A   

and substituting from (10a) and (10b) to produce: 1

(13)

 α0  1−σ A = (δ + ρ )   0.3(1 − η) 

where α0 is determined by µ0, as shown in (11a). Of the two parameters, it is more meaningful to set a value for µ0. Since capital is denominated in terms of PCs, if we assume that the current economy is in competitive equilibrium so that factors are paid their marginal products, then µ0 should be set to equal the current ratio of the cost of employing a human worker to the cost of employing a PC. The average cost for full-time workers including salary and benefits, but omitting training and personnel administration, was about $31,000 in 1990 (U.S. Dept. of Commerce, 1992, Table 649). In comparison, the annual operating cost of a PC has been estimated to be about $8,000, including maintenance and operator training (Steve Lohr, 1996). These numbers suggest crudely that hiring a human worker is about 4 times more expensive than

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 17 hiring a PC. Since this cost difference increases if the computer is used for multiple shifts, the analysis assumes µ0 = 10.

4. The input elasticity of substitution σ The use of a simple aggregate CES production function imposes a common elasticity of substitution among all inputs. Given this constraint, the analysis should use a common value that reasonably approximates the different aggregate elasticities of substitution between capital and labor, between capital and natural resources, and between labor and natural resources. It is also important to consider the aggregate elasticity of substitution between different types of labor, which is generally higher than the elasticity of substitution between capital and labor: as capital takes over an increasing portion of labor input, the points of marginal tradeoff between capital and labor are likely to occur in regions of the skill continuum that are today near the points of marginal tradeoff between types of human labor with different skills. To cover the range of plausible estimates for the various elasticities of substitution (Daniel S. Hamermesh, 1993), the analysis will use three different estimates of a common value: σ = 0.5, 1.0, and 1.5.

5. The growth model parameters δ, ρ, and θ Plausible values for these parameters are taken from the growth literature (Robert J. Barro and Xavier Sala-i-Martin, 1995). The rate of time preference, ρ, is assumed to be 0.02. The inverse of the intertemporal elasticity of substitution, θ, is assumed to be 3. Two values are used for the rate of capital depreciation, δ : 0.05 and 0.10. The first is taken from the growth literature. The second is a higher value to allow for the higher rate of depreciation of computer technology, which is estimated to be around 0.25

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 18 (Sichel, 1997). Ideally, the model should separate computers from other capital and apply appropriate rates of depreciation to each. The single intermediate value of 0.10 is used here to roughly approximate a more complex model with two depreciation rates.

6. The quantities of labor L and natural resources R The value of L is normalized to 1, so the various quantities of output, capital, and consumption can be understood to be per capita quantities. The value of R turns out to be irrelevant because of the way that αt, α0, β, and A are defined. This can be seen by substituting (8), (11a), (11b), and (13) into (3a) and (3b).

B. The Model's Projections of Asymptotic Output Growth and Labor Demand

The sensitivity analysis for the asymptotic case considers the changes in output and labor demand resulting from different combinations of possible values for 3 parameters: the portion of current human skill input that PCs will ultimately be able to perform, z, the elasticity of substitution among inputs, σ, and the portion of the non-labor share of output that is due to nonproducible natural resources, η. Although the sensitivity analysis also considers multiple values for a fourth parameter, the rate of depreciation, δ, that parameter does not affect either the steady-state equilibrium levels of capital or the comparisons between initial and asymptotic levels of output and so it can be ignored in this section. The economy reaches its asymptotic equilibrium after the full potential range of new computer technology is available and all profitable increases in capital have been made. At the asymptote a∞ = z, so (8) implies: (14)

α ∞ = α 0 + (1 − α 0 − β )z

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 19

The asymptotic steady-state level of capital, K∞, can be calculated using (7), either (3a) or (3b), and substitutions from (11a), (11b), (13), and (14). A value for K∞ can be calculated using (3a) and inserted into (4) to calculate the implied value for µ∞, with substitutions from (11a), (11b), and (14). If that implied value for µ∞ is greater than 1 then it is appropriate to use (3a); otherwise it is appropriate to use (3b). In cases where z = 1, it is necessary for any employed labor to perform skills that capital could also perform, so (3b) is the only appropriate form of reduced-form production function. The growth of capital from the initial equilibrium at t = 0 to the asymptotic equilibrium at t = ∞ is shown in Table 1 as the ratio of these two levels. These growth ratios span a large range, from 1.926 to infinity. For the range of parameter values considered here, the asymptotic growth in capital increases with z and σ, and generally decreases with η. Table 2 shows the asymptotic labor-PC price ratio, µ∞. This ratio equals 1 only when z = 1; that is, only when computer technology is able to reproduce the entire human skill continuum. In all other cases, the asymptotic price ratio indicates that it is appropriate to use (3a) as the reduced-form production function, implying that the economy operates at the corner solution where capital provides all of the skill inputs that can be provided by either capital or labor. Note that the asymptotic value of the price ratio is not only greater than 1 (unless z = 1), but is also greater than the initial equilibrium value of 10. For the range of parameter values considered here, the asymptotic price ratio increases with z (unless z = 1) and σ, and decreases with η. Table 3 shows the growth in output as a ratio between the asymptotic and initial levels. These growth ratios are similar to the growth ratios for capital, spanning a large

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 20 range from 1.508 to infinity. The asymptotic growth in output increases with z and σ, and decreases with η. The output growth ratios are smaller than those for capital, except when both are infinite, implying an increase in the capital-output ratio. The size of the increase in the capital-output ratio is shown in Table 4. In the cases when capital and output become infinite, the asymptotic capital-output ratio is calculated using (3a) or (3b) to derive the appropriate equation for the capital-output ratio and then taking the limit as capital goes to infinity. For some of these cases, the capital-output ratio decreases. For the range of parameter values considered here, the change in the capital-output ratio increases with z and

σ, except when capital and output become infinite, and it also increases with η. Table 5 shows the growth in consumption as a ratio between the asymptotic and initial levels. This ratio ranges from 1.439 to infinity. Although consumption grows substantially, a comparison with Table 3 shows that consumption growth is smaller than output growth. This is not surprising, since the increase in the capital-output ratio requires a larger portion of total output to be devoted to replacing depreciated capital, thus leaving less available for consumption. As for capital and output, the asymptotic growth in consumption increases with z and σ, and decreases with η. Table 6 shows the growth in wages as a ratio between the asymptotic and initial levels.6 This ratio ranges from 0.100 to infinity. Wages are calculated as the marginal product of labor using (3a) or (3b), as appropriate. Not surprisingly, wages increase as a result of the increased role and supply of capital except in the case in which capital is able to provide the entire range of human skill inputs, thus forcing wages down to the level of rental payments to capital. However, when wages increase they do not increase by as 6

Note that the asymptotic change in wages is the same as the asymptotic change in the labor-PC price ratio, obtained from the values in Table 2 by dividing by the initial price ratio of 10, except for the cases

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 21 much as consumption increases. Asymptotic wage growth is increasing in z (except when z = 1), increasing in σ, and decreasing in η. Table 7 shows labor's asymptotic share of output. This share is decreasing in z and

σ, resulting in a substantial decrease from the initial share of 0.7 for all combinations of parameter values considered here. The impact of η depends on whether the elasticity of input substitution is greater or less than 1. There are four primary conclusions to take away from these asymptotic results of the basic model. 1) For plausible parameter values the model projects large amounts of future growth in output. 2) This projected output growth is accompanied by a structural change in the economy towards a higher capital-output ratio and a lower labor share, reflecting the larger portion of the skill continuum being provided by capital. 3) Although the labor share decreases, the model projects substantial growth in labor demand, as reflected in the growth in asymptotic wages and the increase in the asymptotic labor-PC price ratio, except when capital becomes capable of providing the entire range of human skill input. 4) The level of future computer technology makes a large difference in the projected growth of both output and wages, suggesting that it is important to determine the plausible range of values for this parameter.

C. Projections of Output Growth and Labor Demand along the Equilibrium Path

It is hard to assess the practical importance of the asymptotic growth ratios in the previous section because they provide no information about the rate of growth: even growth that goes asymptotically towards infinity might occur so slowly that it barely affects the economy over the near-term future. To understand what impact new computer

when capital becomes infinite. This is the case because the marginal productivity of capital is the same in both the initial and asymptotic steady-state equilibria, except for the cases when capital becomes infinite.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 22 technologies may have on the economy over the next few decades, it is necessary to use the model to project change along the equilibrium growth path. This section describes the behavior of the model at t = 25. Recall that the change in computer technology is described with a logistic function for which 90% of the ultimate capability is achieved at t = 50. This function is symmetric around t = 25, making that point the time of fastest technological change and the time when 45% of the ultimate computer capability has been achieved. Units of time in the model correspond to years in the real economy, so the behavior of the model at t = 25 can be thought of as a projection of the behavior of the economy several decades from now. Since new technologies typically take several decades to diffuse through the economy (Bass, 1969; Mansfield, 1989), this projection involves the economic application of new computer skills, such as speech recognition, that are just now becoming commercially viable. The optimal equilibrium path is estimated numerically using equations (5) and (7), with substitutions from (3a), (8), (9), (11a), (11b), and (13). The economy is assumed to start from the initial steady-state equilibrium level of capital described in (12). Movement from the initial steady-state equilibrium to the equilibrium path that leads to the asymptotic steady-state equilibrium involves an instantaneous increase of consumption at t = 0 to a level above the initial steady-state equilibrium. Numerical estimation involves choosing a level of consumption for t = 0 that is as high as possible without leading to consumption growth that eventually outpaces output growth and destroys capital investment. The growth path is estimated only out to t = 40. This short horizon is chosen so that the estimate can be performed using only (3a), thus avoiding the complexities of a simulation that can choose dynamically between (3a) and (3b) as appropriate. In some cases when z = 1, a simulation using (3a) produces a labor-PC price ratio that falls below 1 by t = 50, thus making (3a) inappropriate. Given the short horizon, in some cases the estimate for initial consumption will be slightly too high, resulting in falling levels of capital

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 23 for some t > 40. However, the impact of these overestimates on performance at t = 25 is small. Although a falling level of capital in later time periods indicates that the equilibrium path has been incorrectly calculated, the level of capital falls initially along the equilibrium path for all combinations of parameter values. This results because the instantaneous increase in consumption at t = 0 occurs before there can be any increase in output, requiring a reduction in capital investment below the level required to fully replace depreciation. Despite this initial fall in capital, output growth is positive starting at t = 0 because improved computer technology and instantaneous capital redistribution allow cheaper capital inputs to be substituted for more expensive labor inputs. Table 8 shows the growth rate of capital at t = 25, which ranges from 0.007 to 0.109. As noted above, capital falls at t = 0 , with a growth rate ranging from -0.003 to -0.039. By t = 25, capital has been increasing fast enough and long enough to have recovered from its initial fall and to have increased beyond its initial steady-state equilibrium level. The ratio of capital at t = 25 to the initial steady-state level of capital ranges from 1.023 to 4.229. Both the growth rate of capital at t = 25 and the ratio of its level to the initial steady-state level are generally increasing in all 4 parameters. Table 9 shows the value of the labor-PC price ratio at t = 25, which ranges from 3.535 to 8.930. These values confirm that the use of (3a) in the simulation is appropriate. Although the asymptotic values of this ratio in Table 2 are above 10, except when z = 1, the values at this intermediate point along the equilibrium path are all below 10. In all cases, the value of the labor-PC price ratio begins to fall starting at t = 0. In most of the combinations of 4 parameter values considered, the labor-PC ratio is still falling slowly at t = 25. The labor-PC price ratio at t = 25 is decreasing in z and σ (except when z = 1), and increasing in δ and η, though the effect of resources is small.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 24 Table 10 shows the growth rate of output at t = 25, which ranges from 0.008 to 0.106. Although output growth is positive all along the equilibrium path, it is lower initially at t = 0, ranging from 0.001 to 0.019, because the initial pace of technological change is small and the level of capital is falling. The ratio of output at t = 25 to the initial steady-state level of output ranges from 1.134 to 4.375. Both the growth rate and the growth ratio of output at t = 25 are increasing in z, σ, and δ, whereas the effect of η is small and of mixed direction. The growth rates in Table 10 can be put in perspective by comparing them to the long-run growth rate of per capita output in industrialized countries over the past century, which has been about 0.019 (Barro and Sala-i-Martin, 1995). For 38 of the 48 combinations of parameter values, the growth rate resulting from the substitution of new computer skills for human labor, in a model ignoring all other sources of technological and economic change, is larger than the long-run growth rate experienced in industrialized countries from all sources of change. Of course, other sources of technological and economic change could be added to the model by the usual means, and would result in even higher projected growth rates. Table 11 shows the change in the capital-output ratio at t = 25 by comparing it to the capital-output ratio at t = 0. Although the capital-output ratio increases asymptotically, a net increase does not show up for a long time. By t = 25, the capital-output ratio shows a net increase only when δ = 0.10 and η = 1/3. For the other cases, the capital-output ratio has not yet recovered from its initial decrease. Indeed, when δ = 0.05 and η = 0, the capitaloutput ratio is still falling at t = 25. Table 12 shows the growth rate of consumption at t = 25, which ranges from 0.007 to 0.095. The growth ratio of consumption at t = 25 to consumption at the initial steadystate equilibrium ranges from 1.105 to 3.494. Not surprisingly, both the growth rate and the growth ratio of consumption at t = 25 are smaller than the corresponding values for total

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 25 output, with the growth rate of consumption averaging 87% of the output value and the growth ratio of consumption averaging 91% of the output value. Both the growth rate and the growth ratio of consumption at t = 25 are increasing in z, σ, and δ, whereas the effect of η is small and of mixed direction. Table 13 shows the growth rate of the wage at t = 25, which ranges from -0.018 to 0.040. The growth ratio of the wage at t = 25 to the initial wage ranges from 0.915 to 1.796. Both the growth rate and growth ratio of the wage are increasing in σ and δ, and are generally increasing in z except for high levels of z when the elasticity of input substitution is low; the effect of η is mixed but generally small. Note that the value of δ seems to be more critical for the growth rate of the wage than it is for the growth rates of capital, output, and consumption, particularly for the higher values of z. The initial growth rate of the wage is mixed, ranging from -0.005 to 0.007. The initial growth rate is increasing in σ, decreasing in z when σ = 0.5, mixed when σ = 1.0, and increasing in z when σ = 1.5. Table 14 shows the share of output going to labor at t = 25, which ranges from 0.641 to 0.287, having fallen from its initial value of 0.7. The labor share is decreasing in z and generally decreasing in σ. The impacts of both δ and η on labor share are mixed and generally small. A comparison of tables 7 and 14 shows that labor share at t = 25 is below its asymptotic value when σ = 0.5, η = 0, and z = 0.5 or 0.75. In these cases, labor share must increase in later periods. For example, when z = 0.75 and δ = 0.05, labor share reaches its lowest point near t = 50, when it is 0.273, and then rises to 0.452 at t = 100, 0.521 at t = 150, and 0.532 at t = 200. Labor share can also show this same reversal when

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 26

σ = 0.5 and η = 1/3, but the effect is weaker and the value does not decline below the asymptotic value until after t = 25. The model's performance along the equilibrium path at t = 25 both amplifies and modifies the four conclusions drawn in the previous section. 1) The model projects growth rates of output and consumption from new computer skills that are large compared to longrun historical growth rates from all sources of change. 2) The structural change of a decreasing labor share arrives relatively early. In contrast, the increasing capital-output ratio arrives relatively late, since the ratio initially falls and takes some time to recover and move towards its higher asymptotic value. In some cases, labor share also shows a period of movement opposite to its asymptotic direction, when the initial decrease goes below the asymptotic value so that labor share then experiences a long period of increase towards the asymptote. 3) The medium-term effect of computers on labor demand is mixed, with wages generally rising but the labor-PC price ratio initially falling. In addition, although wages increase in many cases, they decrease for the higher levels of future computer technology unless there is either a high level of capital depreciation or a high substitutability of inputs. This result suggests that the development of a more complex treatment of depreciation may be more important for analyzing the impact of computers on labor demand than for analyzing their impact on output. 4) The level of future computer technology makes a large difference in the projected growth rates of output and wages, underlining the conclusion from the asymptotic results that it is important to determine the plausible range of values for this parameter. 5) The impact of the natural resources parameter on growth several decades from now is generally small, even though its impact on asymptotic growth can be large. This result suggests that it is acceptable when analyzing medium-term growth to use a simpler model that omits natural resources entirely.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 27

III. A 2-Type Model of Increasing Computer Skill

This section extends the basic model to include two types of labor. The purpose of this extension is to analyze the impact of new computer skills in a world in which human labor is not homogeneous and so cannot change costlessly from one portion of the skill continuum to another. The extension involves the extreme assumption that the cost of occupational change is so high that there is no movement of labor between the portions of the skill continuum performed by the two different types of labor. This extreme assumption serves as a useful contrast to the equally extreme assumption of the basic model, in which labor is homogeneous and can move costlessly along the human portion of the skill continuum. Section A describes the 2-type model, section B describes the parameter values, section C describes the asymptotic behavior, and section D describes the behavior along the equilibrium path.

A. The 2-Type Model

In the 2-type model, the types of labor differ by the portions of the skill continuum that they provide: type 1 labor provides a portion of the skill continuum that computers will also begin to provide, whereas type 2 labor provides a portion of the skill continuum that computers will never be able to provide. Type 1 labor is incapable of performing the skills that type 2 labor performs.7 The two types can be thought of as low-skill and high-skill labor, respectively, since some of the skills of the type 1 workers are easy enough for computers to reproduce, and since the type 1 workers cannot provide type 2 skills. To fix ideas it may be helpful for the current technological era to think about type 2 labor in terms

7

It is irrelevant whether or not type 2 labor is able to perform the skills that type 1 performs.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 28 of professional occupations, which typically require higher levels of human skill acquired through special education, training, or talent. The 2-type model uses the same constant-returns CES aggregate production function (1) as in the basic model, except that natural resources are omitted (β = 0) for simplicity. The budget constraints for the skill inputs of the social planner's optimization problem are as follows: αt

(16a)

∫ Si , t di = Kt ,

if µ1,t > 1

0

1− ϕ

(16b)

∫ Si , t di = λL ,

αt

1− ϕ

(16c)

∫ Si , t di = Kt + λL ,

if µ1,t > 1

if µ1,t = 1

0

1

(16d)

∫ Si , t di = (1 − λ ) L

1− ϕ

where µ1,t is the low-skill labor-PC price ratio, λ is the proportion of labor that is low-skill, and ϕ is the dividing line between the portion of the skill continuum that can be provided by capital or low-skill labor and the portion of the skill continuum that can be provided by high-skill labor only. If capital and the two types of labor are each assumed to be homogeneous, then in equilibrium they will each be allocated evenly across their respective portions of the skill continuum. As a result, the production function can be rewritten in one of two reduced forms, similar to the reduced forms used in the basic model:

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 29

 K  (17a) Yt * = Aα t  t    αt  

σ −1 σ

 λL  + (1 − α t − ϕ )   1 − αt − ϕ 

σ −1 σ

 (1 − λ ) L  + ϕ   ϕ 

σ −1 σ

σ

 σ −1   

if µ1,t > 1   K + λL  (17b) Yt * = A (1 − ϕ ) t   1−ϕ   

σ −1 σ

 (1 − λ ) L  + ϕ   ϕ 

σ −1 σ

σ

 σ −1  ,  

if µ1,t = 1

The appropriate reduced form is determined by the low-skill labor-PC price ratio, in the same way as for the basic model. The low-skill labor-PC price ratio for (17a) can be calculated as for the basic model:  K (1 − α t − ϕ )  σ ∂Yt * ∂ (λL) ≡ =  t  , ∂Yt * ∂Kt λL α t   1

(18)

µ1, t

if µ1,t > 1

The same growth model equations used in the basic model, (5)-(7), are used for the 2-type model.

B. Parameter Assumptions for the 2-Type Model

The basic model equations (8)8 and (9) are used in the 2-type model to describe the path of change of the capital parameter αt. Note, however, that z ≤ λ, since z is the portion of total current human skill distribution that computers will ultimately be able to provide and by assumption computers can never provide any of the skill input provided by high-skill labor. The sensitivity analysis here assumes λ = 0.75, and examines three different values for z that span the feasible range: 0.25, 0.5, and 0.75. 8

Again, natural resources are omitted from the 2-type model, so β = 0.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 30 For simplicity, the 2-type model also assumes that all labor receives the same wage at the initial equilibrium, which allows (11a)9 to be used to determine the initial capital parameter α0. Furthermore, the high-skill labor parameter ϕ can be determined from α0 and the portion of labor that is high-skill: (19)

ϕ = (1 − λ )(1 − α 0 )

All other parameter assumptions in the 2-type model are the same as for the basic model.

C. The 2-Type Model's Projections of Asymptotic Output Growth and Labor Demand

Table 15 compares the asymptotic output growth ratios for the basic and 2-type models. When z = 0.25 there is only a small impact on asymptotic growth, reflecting the fact that in this case the 2-type model imposes only a minor restriction on the optimal allocation of labor: in the basic model the density of workers along the human portion of the skill continuum increases to 1.33 times its original value, whereas in the 2-type model there is an increase to 1.5 times for low-skill labor and no increase for high-skill labor. In contrast, when z = 0.75, asymptotic growth in the 2-type model is only 25% of the growth in the basic model. In this case, the value of 75% of the work force is effectively lost by the inability of the low-skill workers to move into high-skill occupations. Although those low-skill workers are employed in the 2-type model, they are competing directly with capital and making a negligible impact on total output. The comparison between the basic and 2-type models in the asymptotic growth of capital and consumption are similar to the comparison for the asymptotic growth in output and so are not shown separately. As a result of this similarity, the asymptotic change in the

9

Since natural resources are omitted, η = 0.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 31 capital-output ratio is essentially the same for the basic and 2-type models. There is one case in which this is not true: when σ = 0.5 and z = 0.75, the growth in capital in the 2type model is somewhat less than 25% of the growth in the basic model, so the increase in the capital-output ratio is only 1.496 in the 2-type model compared to 1.553 in the basic model. This lower level of capital reflects the contributions of low-skill labor to capital's portion of the skill continuum. Although this also occurs when σ = 1.0 and z = 0.75, in that case the increase in capital is so great that the contribution of low-skill labor is negligible. Table 16 compares the asymptotic wage growth ratios for the basic and 2-type models.10 Since the basic model assumes that labor is homogeneous and can move costlessly between low-skill and high-skill occupations, the comparison growth ratios for the basic model are equal across skill levels. Compared to the basic model, wage growth ratios in the 2-type model are smaller for low-skill workers and generally higher for highskill workers, reflecting the fact that labor is no longer being divided equally along the human portion of the skill continuum. In the case when z = 0.75 things are a bit different because low-skill work becomes part of the capital portion of the skill continuum. As a result, low-skill workers are paid the same rate as capital and so show a substantial fall in wages, whereas high-skill workers are now evenly divided along the entire human portion of the skill continuum and so show the same asymptotic wage growth as in the basic model. When wages are averaged over the two skill types, the comparison between the basic and 2-type models in asymptotic growth is similar to that shown in table 15 for output: when z = 0.25 the fixed skill differences of the 2-type model cause only a small reduction in asymptotic average wage growth, whereas when z = 0.75 they reduce average wage growth to about 25% of its value for the basic model.

10

As for the basic model, the asymptotic changes in the wages are the same as the asymptotic changes in the labor-PC price ratios -- except when capital becomes infinite -- and so a table showing the asymptotic labor-PC price ratios is omitted.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 32 Since the initial wages of low-skill and high-skill labor are equal, a comparison between the asymptotic growth ratios in the 2-type model shows the skill-based wage inequality that results asymptotically from new computer technology. When z = 0.25, the asymptotic wages for high-skill labor are 31% to 125% larger than the asymptotic wages for low-skill labor. The skill-based wage inequality increases to very high levels when z = 0.75. A more complete model of skill-based wage inequality could begin with unequal wages at t = 0 and would result in larger levels of asymptotic wage inequality. It is important to be clear about the nature of the skill difference that has been introduced in the 2-type model and the degree of skill upgrading that would be required to return the model to wage equality. For example, when z = 0.5, two-thirds of the type 1 workers are replaced by computers, requiring those workers to move into new occupations. In the 2-type model, it is assumed that these displaced low-skill workers are unable to move into high-skill occupations but that they can and do move costlessly into other lowskill occupations. The critical difference between the 2-type and basic models is not that displaced workers in the 2-type model don't move into new occupations but that none of them move into high-skill occupations. When z = 0.5, the asymptotic wage equality of the basic model requires that one-third of the low-skill workers, representing 25% of the entire labor force, move from low-skill occupations to high-skill occupations. Table 17 compares the asymptotic labor shares for the basic and 2-type models. For the basic model, the total labor share in Table 7 is divided so that 75% goes to initially low-skill labor and 25% goes to initially high-skill labor. Asymptotic output share is lower for the low-skill labor in the 2-type model than in the basic model, and it is higher for the high-skill labor. The total labor share of output is unchanged between the basic and 2-type models, except for the z = 0.75 case when low-skill labor takes over part of the capital portion of the skill continuum and so captures a small piece of the output that goes to capital in the basic model.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 33 There are two basic conclusions to take away from these asymptotic results about the 2-type model. 1) The impact of fixed worker skill differences on asymptotic growth in output, capital, consumption, average wages and total labor share is relatively small, except when computers take over the whole low-skill portion of the skill continuum. 2) However, even for moderate levels of future computer technology, the addition of worker skill differences can cause large inequalities among workers in wages and output shares. This inequality is produced in the 2-type model because low-skill workers are prevented from moving into high-skill occupations, even though they are capable of moving into other lowskill occupations.

D. The 2-Type Model's Projections of Output Growth and Labor Demand along the Equilibrium Path Table 18 compares the growth rate of output at t = 25 for the basic and 2-type models. The difference between the 2-type and basic model output growth rates averages -0.001, ranging from -0.006 to 0.000. The 2-type output growth ratios at t = 25 average 99.5% of the corresponding basic growth ratios, ranging from 97.0% to 100.1%. The impact of the 2-type constraint on consumption growth at t = 25 closely follows the pattern of its impact on output growth. The difference between the 2-type and basic model consumption growth rates averages -0.001, ranging from -0.007 to 0.000. The 2-type consumption growth ratios average 98.8% of the corresponding basic growth ratios, ranging from 94.7% to 99.9%. The impact of the 2-type constraint on capital growth at t = 25 is also small, though with a slightly different pattern than the pattern for output and consumption growth. The difference between the 2-type and basic model capital growth rates averages 0.000, ranging from -0.001 to 0.001. The 2-type capital growth ratios average 101.2% of the corresponding basic growth ratios, ranging from 100.3% to 103.5%.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 34 Because the 2-type constraint tends to increase capital growth and decrease output growth at t = 25, it results in larger capital-output ratios. The 2-type capital-output ratios average 101.8% of the corresponding basic capital-output ratios, ranging from 100.2% to 105.7%. Although the 2-type constraint increases capital-output ratios compared to the basic model, the effect is too small to cause a net increase in the capital-output ratios; like the basic model when η = 0, the 2-type model shows substantial decreases in the capitaloutput ratio at t = 25, even though the asymptotic results show that this ratio must eventually increase. The 2-type capital-output ratios at t = 25 average 85.9% of the corresponding initial ratios for these simulations, ranging from 64.6% to 98.9%. Table 19 compares the growth rate of wages at t = 25 for the basic and 2-type models. The 2-type wage growth is lower for the low-skill workers and higher for the high-skill workers than the corresponding growth in the basic model. For the low-skill workers, the difference between 2-type and basic model wage growth rates averages -0.014, ranging from -0.048 to -0.002. The 2-type wage growth ratios for the low-skill workers average 87.1% of the corresponding basic model growth ratios, ranging from 64.8% to 97.2%. For the high-skill workers, the difference between the 2-type and basic model wage growth rates averages 0.019, ranging from 0.005 to 0.047. The 2-type wage growth ratios for high-skill workers average 139.5% of the corresponding basic model growth ratios, ranging from 108.3% to 218.3%. The skill difference in the wage growth rates at t = 25 averages 0.033, ranging from 0.007 to 0.092. The ratio of the high-skill wage growth ratio to the low-skill wage growth ratio averages 1.670, ranging from 1.114 to 3.306. The difference in wage growth rates and the high-to-low ratio of wage growth ratios are both increasing in z, decreasing in σ, and unaffected by δ. The growth rate of the average wage at t = 25 in the 2-type model averages 0.007, ranging from -0.022 to 0.029. The growth ratio of the average wage averages 1.176,

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 35 ranging from 1.015 to 1.541. The difference between the 2-type and basic models in the average wage growth rate at t = 25 averages -0.005, ranging from -0.025 to 0.000. Although the average wage growth rate is always lower in the 2-type model at t = 25, the average wage growth ratios are about same, with the 2-type growth ratio averaging 100.2% of the basic growth ratio, ranging from 98.9% to 104.1%. The labor-PC price ratio for low-skill labor at t = 25 averages 5.741, ranging from 3.511 to 8.065. The 2-type low-skill labor-PC price ratio is always lower than the laborPC price ratio in the corresponding basic model, with the 2-type ratio averaging 89.1% of the basic ratio, ranging from 72.3% to 97.3%. Table 20 compares the labor shares for the basic and 2-type models at t = 25. Output share in the 2-type model is lower for the low-skill labor and higher for the highskill labor than in the basic model: the share of low-skill labor in the 2-type model averages 87.5% of the share of initially low-skill labor in the corresponding basic model, ranging from 66.8% to 97.2%, whereas the share of high-skill labor in the 2-type model averages 140.5% of the share of initially high-skill labor in the corresponding basic model, ranging from 108.3% to 223.0%. For low-skill workers, the labor share comparisons of the 2type and basic models are decreasing in z and increasing in σ, with a negligible impact of δ. For high-skill workers, the labor share comparisons are increasing in z and decreasing in

σ, with a negligible impact of δ. The impact of the 2-type constraint on total labor share at t = 25 is small. When

σ = 0.5, total labor share is slightly larger in the 2-type model, ranging from 100.3% to 106.3% of the corresponding basic model total labor share. When σ = 1.0, total labor share is identical in the 2-type and basic models. When σ = 1.5, total labor share is slightly smaller in the 2-type model, ranging from 99.1% to 99.9% of the corresponding basic model total labor share.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 36 The 2-type model's performance along the equilibrium path at t = 25 amplifies the two conclusions drawn in the previous section from the asymptotic analysis. 1) The impact of worker skill differences on growth in output, capital, consumption, average wages, and total labor share is small. Even in the most extreme case when z = 0.75 and σ = 0.5, the growth ratios for output, consumption, and average wages are reduced by only a few percent, even though the corresponding growth rates are reduced substantially. 2) The introduction of worker skill differences causes substantial inequalities among workers in wages and output shares. Although the inequalities at t = 25 are smaller than the enormous inequalities that the model projects asymptotically, in many cases they are already large compared to existing levels of skill-based wage inequality in the economy. Obviously, the model's projected levels of wage inequality at t = 25 would be even larger if the high-skill workers begin at t = 0 with higher wages than those of the low-skill workers.

IV. A 2-Sector Model of Increasing Computer Skill

This section extends the basic model to include two sectors of the economy. The purpose of this extension is to analyze the impact of new computer skills in a world in which different industries do not all require the same mix of skills. The extension involves the extreme assumption that the portion of the skill continuum that capital will take over from labor is entirely concentrated in one sector of the economy, with the other sector requiring only skills that capital will never be able to provide. This extreme assumption serves as a useful contrast to the equally extreme assumption of the basic model, in which all parts of the economy have the same skill requirements. Section A describes the 2-sector model, section B describes the parameter values, section C describes the asymptotic behavior, and section D describes the behavior along the equilibrium path.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 37

A. The 2-Sector Model

In the 2-sector model, the sectors differ by the skill requirements of their production functions: the portion of the skill continuum that capital takes over from labor is used only in sector 1, so that sector 2 uses only skills drawn from the portion of the skill continuum that capital will never be able to provide. Sector 1 may or may not also require some skills that capital will never be able to provide. As in the 2-type model, it is convenient to think about the contrast in the 2-sector model in terms of "lower" and "higher" skill requirements. Sector 1 uses lower skills that will become partly or wholly provided by computers, whereas sector 2 uses higher skills that will not become provided by computers. For the current technological era, sector 2 might be thought of as the subset of service industries that employ workers in professional occupations. Labor in the 2-sector model is homogeneous, as it is in the basic model, so there is no barrier preventing workers in lowskill jobs in sector 1 from moving to high-skill jobs in sector 2; the skill difference separating the two sectors is a barrier for computers not for humans. The 2-sector model uses two constant-returns CES aggregate production functions that are similar to the production function (1) used in the basic model, except that the skill continua for the two different sectors represent different skills. As in the 2-type model, the 2-sector model omits natural resources (β = 0) for simplicity. The budget constraints for the skill inputs of the social planner's optimization problem are as follows: αt

(20a)

∫ S1, i , t di = Kt ,

if µt > 1

1

if µt > 1

0

(20b)

∫ S1, i , t di = λt L ,

αt

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 38 1

(20c)

∫ S1, i , t di = Kt + λt L ,

if µt = 1

0

∫ S2, i , t di = (1 − λt ) L

1

(20d)

0

In contrast to the budget constraints for the 2-type model, these equations divide the lower and higher labor skills by a separation between the two skill continua for the different sectors, S1 and S2; they allow labor to move between jobs with different skill requirements, as indicated by the time subscript on λt; and as a result of the free movement of labor they have a single labor-PC price ratio, µt. If capital and the labor in each sector are each assumed to be homogeneous, then in equilibrium they will each be allocated evenly across their respective portions of the skill continuum. As a result, the two production functions can be rewritten in reduced form:

(21a)

 K  Y1, t * = A1 α t  t    αt  

σ −1 σ

(21b) Y1, t * = A1 ( Kt + λt L) , (22)

 λL  + (1 − α t ) t   1 − αt 

σ −1 σ

σ

 σ −1  ,  

if µt > 1

if µt = 1

Y2, t * = A2 (1 − λt ) L

As with the other two models, the appropriate reduced form for sector 1 is determined by the labor-PC price ratio. The labor-PC price ratio for (21a) can be calculated as before:  Kt (1 − α t )  σ ∂Y1, t * ∂ (λt L) µt ≡ =   , ∂Y1, t * ∂Kt  λt L α t  1

(23)

if µt > 1

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 39 Production in sector 1 is used for both consumption and capital investment, whereas production in sector 2 is used for only consumption:11 (24)

Y1, t = C1, t + K˙ t + δKt

(25)

Y2, t = C2, t

where C1,t and C2,t are the consumption values for the 2 sectors. The social planner optimizes over the streams of consumption, C1 and C2, to maximize utility:

∞

(26)

U (C1 , C2 ) = ∫

m(C1, t , C2, t )

1−θ

1−θ

0

− 1 − ρt e dt

where m(C1,t,C2,t) is an instantaneous utility function of the consumption from the two sectors and is represented by a simple CES function:

ζ −1

(27)

ζ

  ζ −1 m(C1, t , C2, t ) ≡  C1,ζt + C2,ζt    ζ −1

where ζ is the elasticity of substitution in the instantaneous utility function. The dynamic optimization problem posed by (26) and (24), with C1,t and λt as control variables and Kt as the state variable, yields three equations:

11

(28)

mt−θ

∂m − ρt e = νt ∂C1, t

(29)

mt−θ

∂Y * ∂m A2 Le − ρt = ν t 1,t ∂C2,t ∂λt

An alternate model in which capital is produced by sector 2 might be more appropriate for some historical periods.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 40

(30)

∂Y1, t * ν˙ −δ = − t ∂Kt νt

where νt is the shadow price on capital from the present-value Hamiltonian. The solution is described in the Appendix.

B. Parameter Assumptions for the 2-Sector Model

The 2-sector model uses equation (8)12 to describe the path of change of the capital parameter αt, but since this parameter affects only the sector 1 part of the economy, equation (9) must be modified so that z can still describe the portion of total current human skill distribution that computers will ultimately be able to provide:

(31)

1.1 z at =  − 0.1 t 1 − . 04  1 + 10  λ0

As with 2-type model, it is now necessary that z ≤ λ0. The sensitivity analysis assumes

λ0 = 0.75, and examines three different values for z that span the feasible range: 0.25, 0.5, and 0.75. As in the basic model, the assumption that labor's initial share of output is 0.7 can be combined with the reduced form production function (21a) and the initial labor-PC price ratio (23) to determine the initial capital parameter α0:

(32)

12

α0 =

1 7 σ −1 µ0 λ0 + 1 3

As in the 2-type model, β = 0 since natural resources are omitted.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 41 which is analogous to (11a) for the basic model. The share ratio and the marginal productivity of capital from (21a) can be combined with (30) and the steady-state equilibrium condition that ν˙t ν t = − ρ to determine the constant A1 in terms of α0:  α (7λ0 + 3)  1 − σ A1 = (δ + ρ ) 0  3   1

(33)

which is analogous to (14) for the basic model. The elasticity of substitution, ζ, between sector 1 and sector 2 consumption, is assumed to be 0.5. A value is chosen in the inelastic range because the two most notable sector-specific substitutions of capital for human labor skill, in agriculture and in manufacturing, led to an increase in consumption smaller than the increase in labor productivity and a net transfer of labor to other industries. Within the inelastic range, a value of 0.5 is chosen for simplicity. The value of the constant A2 can be determined from the initial steady-state equilibrium, using the static optimizing conditions analogous to (28) and (29), and substituting for the derivatives of m using the assumption that ζ = 0.5: −1

(34)

2

 ∂Y *   C  A2 =  1, 0   1, 0  L−1  ∂λ0   1 − λ0 

(Note that C1,0 is the sector 1 consumption in the initial steady-state equilibrium, not the sector 1 consumption at t = 0 after the instantaneous increase necessary to move onto the equilibrium path.) All other parameter assumptions in the 2-sector model are the same as for the basic model.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 42

C. The 2-Sector Model's Projections of Asymptotic Output Growth and Labor Demand

The asymptotic value α∞ is obtained from (8), using the asymptotic value a∞ = z/λ0. The asymptotic value λ∞ is obtained from the t = ∞ version of (34), with the value of A2 derived from (34) and λ0. Table 21 shows the asymptotic output growth ratios for the 2 sectors. Not surprisingly, asymptotic output growth is larger in sector 1 than in sector 2. Since the only input to sector 2 is human labor using skills that computers can never acquire, the only way to increase output in that sector is by moving workers from sector 1 to sector 2. In the extreme cases, when z = 0.75 or σ = 1.5, sector 1 production becomes infinite and all labor that is initially in sector 1 moves to sector 2. The value of λ∞ mirrors the asymptotic output growth for sector 2, ranging from a value of 0.681 when the sector 2 growth ratio is 1.275, to a value of 0 when the sector 2 growth ratio is 4.000. The asymptotic growth ratio for consumption in sector 1 is slightly lower than the growth ratio for output, since sector 1 growth is achieved by a large increase in capital use. The asymptotic growth ratio for consumption in sector 2 is exactly the same as for output, since all sector 2 output is consumed. Table 22 compares the asymptotic growth ratios for aggregate output for the basic and 2-sector models. In the 2-sector model, it is necessary to find a way to aggregate the output from the 2 sectors. Since the growth in the 2 sectors is quite different, the initial and asymptotic relative prices will also be quite different, making it unsatisfactory to use either for the aggregation. Instead, the instantaneous utility function m, given by (27), is used to aggregate the output from the 2 sectors. In cases where the basic model produces very

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 43 high asymptotic growth, the 2-sector model produces smaller growth because of the limitations in the labor-intensive sector 2. However, in cases where the growth in the basic model is smaller, the 2-sector model produces growth of comparable, and sometimes larger, size. Consumption in the two sectors can also be aggregated using m. The asymptotic growth ratio for aggregate consumption is similar to the growth ratio for aggregate output. Asymptotic capital growth in the 2-sector model is larger than for the corresponding values of the basic model. As in the basic model, asymptotic capital growth in the 2-sector model is larger than asymptotic output growth. This is particularly true when capital is compared to aggregate output, since the asymptotic growth ratio of capital can tend towards infinity while producing only moderate growth in aggregate output. Table 23 compares the asymptotic change in the capital-output ratio for the basic and 2-sector models. For the 2sector model, the change is shown both for the aggregate economy, using aggregate output, and for sector 1 by itself. For the aggregate economy, the capital-output ratio increases more for the 2-sector model than for the basic model. Within sector 1, where both capital and output can achieve high levels of growth, the change in the capital-output ratio is identical to that of the basic model, except when the sector is completely automated so that capital and output become infinite. The measurement of wage growth also encounters the problem of aggregating over the 2 sectors: it is easy to obtain the marginal productivity of labor within either sector 1 or sector 2, but the former shows high productivity growth and the latter shows none at all, so the final measurement of wage growth depends on how the sectors are integrated. To solve the aggregation problem, the wage is calculated as the product of the labor share of output (using sector 1 marginal productivities) and the aggregate output obtained from m. The resulting asymptotic wage growth ratios are shown in table 24, along with the comparison ratios from the basic model. Except for the cases where the basic model produces

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 44 explosive growth, the two models produce comparable levels of asymptotic wage growth, with the 2-sector model often producing slightly higher figures. The asymptotic labor-PC price ratio, µ∞, can be calculated from the marginal productivities of labor and capital within sector 1 alone. The ratios are similar to those shown for the basic model in table 2, except that they are larger and become infinite for all cases when sector 1 production becomes infinite (when z = 0.75 or σ = 1.5). Table 25 compares the asymptotic labor share of output for the basic and 2-sector models. For the 2-sector model, labor shares are shown both for the aggregate economy and for sector 1 alone. In all cases, the aggregate labor share in the 2-sector model is higher than that in the basic model, whereas the sector-1 labor share is lower than the labor share in the basic model. Sector 1 begins with a lower labor share of 0.636 at t = 0, compared to 0.7 for the basic model, but still shows a larger reduction in labor share after accounting for this difference in starting points. Note that it is only within sectors that computers always cause an asymptotic decrease in labor share, since in the 2-sector model it is possible for aggregate labor share to increase if computers take over a large proportion of tasks within one sector. It is not necessary for sector 1 production to become infinite for this to occur; for example, when z = 0.7 and σ = 0.5, asymptotic labor share is 0.806. There are three basic conclusions to take away from these asymptotic results about the 2-sector model. 1) The impact of sector skill differences on asymptotic growth in aggregate output, capital, aggregate consumption, and wages is relatively small, except when computers take over a large portion of sector 1 skills so that growth becomes explosive. In the latter case, the limits on increasing sector 2 output substantially reduce the 2-sector growth in aggregate output, aggregate consumption, and wages in comparison to the growth in the basic model. 2) Sector skill differences increase the asymptotic change in the aggregate capital-output ratio in comparison to the basic model. However, the sector 1 change in capital-output ratio is the same as in the basic model, except when growth

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 45 becomes explosive. 3) Sector skill differences increase the asymptotic aggregate labor share of output in comparison to the basic model; for moderate levels of computer skill this results in a smaller fall in aggregate labor share, whereas for high levels of computer skill this results in a net rise in aggregate labor share. Within sector 1, the decrease in labor share is larger than it is in the basic model.

D. The 2-Sector Model's Projections of Output Growth and Labor Demand along the Equilibrium Path Table 26 shows the growth rate of output at t = 25 for the 2 sectors. The 2-sector output growth is higher for sector 1 and lower for sector 2 than the corresponding growth in the basic model. For sector 1, the differences between the 2-sector and basic model output growth rates average 0.004, ranging from 0.001 to 0.009. The 2-sector output growth ratio for sector 1 averages 105.5% of the corresponding basic model growth ratios, ranging from 101.0% to 113.2%. For sector 2, the differences between the 2-sector and basic model output growth rates average -0.010, ranging from -0.028 to -0.001. The 2sector output growth ratios for sector 2 average 83.3% of the corresponding basic model growth ratios, ranging from 63.3% to 96.3%. The differences between sector 1 and sector 2 output growth rates at t = 25 average 0.014, ranging from 0.002 to 0.037. The ratios of sector 1 to sector 2 growth ratios average 1.287, ranging from 1.050 to 1.789. The output growth of sector 2 reflects the movement of labor from sector 1 to sector 2, which is indicated by a decrease in λ25 from its initial value of 0.75. The value of λ25 averages 0.672, ranging from 0.536 to 0.728. The pattern of 2-sector consumption growth at t = 25 is similar to the pattern of output growth. For sector 1, the differences between the 2-sector and basic model consumption growth rates average 0.003, and the consumption growth ratios average 110.1% of the corresponding basic model growth ratios. For sector 2, the differences

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 46 between the 2-sector and basic model consumption growth rates average -0.006, and the consumption growth ratios average 93.3% of the corresponding basic model growth ratios. Table 27 compares the growth rate of aggregate output at t = 25 for the basic and 2sector models. As for the asymptotic results, the output of the 2 sectors is aggregated using the instantaneous utility function m. The differences between the 2-sector and basic model aggregate output growth rates average -0.001, ranging from -0.007 to 0.000. The 2-sector aggregate output growth ratios at t = 25 average 97.4% of the corresponding basic growth ratios, ranging from 91.3% to 99.7%. For aggregate consumption, the differences between the 2-sector and basic model growth rates at t = 25 average 0.003, and the growth ratios average 99.7% of the corresponding basic model growth ratios. Capital growth at t = 25 in the 2-sector model tends to be larger than growth in the basic model. The differences between the 2-sector and basic model capital growth rates average 0.002, ranging from 0.000 to 0.006. The 2-sector capital growth ratios at t = 25 average 102.4% of the corresponding basic growth ratios, ranging from 99.7% to 106.1%. Table 28 compares the change in the capital-output ratio at t = 25 for the basic and 2-sector models. For the 2-sector model, both the aggregate and the sector 1 capital-output ratios are shown. Since capital growth tends to be larger in the 2-sector model, whereas aggregate output growth tends to be smaller, the aggregate capital-output ratios for the 2sector model are larger than for the basic model. The changes in the 2-sector aggregate capital-output ratio average 108.4% of the corresponding changes in the basic model, ranging from 101.4% to 124.0%. However, these results do not hold for sector 1, where output growth is also larger than in the basic model. The changes in the sector-1 capitaloutput ratio average 97.1% of the corresponding changes in the basic model, ranging from 93.2% to 99.5%. As in the 2-type model, the impact of the 2-sector constraint is not large enough to reverse the result from the basic model that the t = 25 aggregate capital-output ratio shows a

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 47 net decrease from its initial value. The aggregate capital-output ratios at t = 25 average 88.6% of the corresponding initial ratios, ranging from 69.1% to 99.9%. The sector-1 capital-output ratio at t = 25 averages 82.1% of corresponding initial ratios, ranging from 59.7% to 97.3%. Table 29 compares the growth rate of wages at t = 25 for the basic and 2-sector models. As for the asymptotic results, wages are calculated using the labor share and aggregate output in order to deal with the problem of aggregating over the 2 sectors. The differences between the 2-sector and basic model wage growth rates average -0.001, ranging from -0.004 to 0.004. The 2-sector wage growth ratios average 96.9% of the corresponding basic growth ratios, ranging from 95.0% to 98.9%. The overall pattern of the labor-PC price ratios at t = 25 is similar for the 2-sector and basic models. The 2-sector price ratios average 101.3% of the corresponding basic price ratios, ranging from 94.0% to 114.5%. Table 30 compares the labor shares for the basic and 2-type models at t = 25. The aggregate labor shares in the 2-sector model average 99.5% of the labor shares in the corresponding basic model, ranging from 95.3% to 105.2%. The sector 1 labor shares in the 2-sector model average 80.4% of the labor shares in the corresponding basic model, ranging from 68.9% to 87.6%. As for the asymptotic results, the lower sector 1 labor shares partly reflect the lower initial starting labor share of 0.636. The pattern of labor share changes for sector 1 of the 2-sector model is similar to the pattern of changes for labor share in the basic model: in all cases labor share shows a decrease compared to its initial value, both at t = 25 and asymptotically, but when σ = 0.5 labor share can fall below the asymptotic value by t = 25, implying that it must eventually increase. In contrast, the aggregate labor share of the 2-sector model at t = 25 is below its asymptotic value in 8 of the 9 cases. This occurs largely because there are 5 cases in which asymptotic labor share is higher than the initial share of 0.7.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 48 The 2-sector model's performance along the equilibrium path at t = 25 amplifies and modifies the three conclusions drawn in the previous section from the asymptotic analysis. 1) The impact of sector skill differences on growth in aggregate output, capital, aggregate consumption, and wages is small. 2) The impact of sector skill differences on the change in the capital-output ratio is also small, for ratios calculated using either aggregate or sector 1 output. For both the aggregate and sector 1 ratios, the 2-sector version of the model continues the pattern of changes from the basic model: the capital-output ratio initially falls, even though in most cases it will eventually rise substantially above its starting value. 3) Sector skill differences preserve the initial fall in aggregate labor share, observed along the equilibrium path at t = 25, even though in many cases the asymptotic aggregate labor share shows only a small decrease or even an increase. As a result, in most cases aggregate labor share in the 2-sector model must show an increase somewhere along the equilibrium path. The pattern of labor share changes in the basic model is more similar to the pattern shown in sector 1 of the 2-sector model, with a net fall in labor share for all cases and a fall below the asymptotic value only when σ = 0.5.

V. Estimating and Testing the Models with Historical Data

The primary motivation for developing this modeling framework was to provide a way to understand the economic impact of future computer technology. But, of course, the substitution of capital for portions of human skill input has occurred historically and any tests of these models must be carried out on historical data. This section briefly discusses some of the issues related to possible historical applications and tests of these models. At the outset, it is important to note that the production function used in these models is no more than a minor elaboration of ordinary aggregate production functions. It

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 49 differs from those functions only by noting the different skills that capital and labor provide to the production process. Such diverse skill provision is exactly what we expect to be going on "inside" ordinary production functions. Production functions make sense not because mere lumps of capital and labor directly produce output but because those inputs are used to perform a variety of tasks that make up the production process, which is what the skill-based production function describes. If capital becomes more capable and so can provide some of labor's tasks more cheaply than labor, then we expect that capital will take over those tasks from labor, just as the skill-based production function describes and as has occurred over the past two centuries as machines have become able to perform tasks requiring simple repetitive motor skills and simple cognitive skills such as arithmetic. Thus to the extent that ordinary aggregate production functions are a sensible approximation of the world, it must be partly because they approximate some underlying skill-based production function, such as the one used in this paper. This argument relieves part of the burden of proof in justifying the skill-based production function as a departure from ordinary production functions. Such relief does not mean that the skill-based production function can be accepted before proving its worth in providing an improved understanding of the economy; rather, it means only that the skill-based production function should have some a priori preference and should not be given up lightly. The basic contribution of this skill-based modeling framework is to provide a way to link technological changes that increase capital's portion of the skill continuum to the resulting impact on output, wages, and the overall structure of the economy. The structural changes in the economy involve shifts in the capital-output ratio, the labor share of output, the occupational mix of the labor force, and the sectoral mix of production. The models provide an integrated description of the path of all of these changes, and so can in principle be tested against the historical pattern of economic changes that resulted when new technology allowed capital to reproduce human skills. Unlike most economic models of technological change, which infer such change from shifts in productivity, the skill-based

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 50 models use an explicit definition of the level of technology that can in principle be directly measured from information on the changing mix of human work tasks. A full test of the models requires data on the portion of the skill continuum that capital provides competitively at each point in time. Obviously a time series for these capital skills is not readily available. For future periods, it will be possible to create such a time series for the U.S., using the skill-based occupational information that the Department of Labor is collecting in the new O*Net database of occupations (American Institutes for Research, 1995). In cases where earlier occupational categories are sufficiently finegrained, it might be possible to detect shifts in capital skills from changes in the occupational mix within industries. In other cases, it might be possible to construct crude measures of changes in capital skills from historical sources describing the components of the production process at different points in time. Of course, the development of such crude measures would be time-consuming. A simpler possibility would be to use measures of labor productivity as a proxy for the unavailable data on capital skills. The path of the capital parameter αt could be calibrated to produce a given level of labor productivity growth within a sector, and then the model's predictions of structural changes could be compared to historical data. This route to testing unfortunately gives up the extra explanatory link provided by a direct measure of technology. Another difficulty with this approach is that it attributes all labor productivity growth to the substitution of capital for human skill input, rather than separating out the neutral technological change that increases productivity while keeping the same mix of skills from capital and labor. An even simpler option for testing the model is to look for the distinctive features of the structural change predicted by the models. Growth in output and wages, and shifts among occupations and industries, can be produced by conventional growth models with

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 51 neutral technological change.13 The distinctive part of the structural prediction of the skillbased models concerns the changes in the capital-output ratio and the labor share of output. The capital-output ratio shows an initial decline, during the early period of technological change, followed by a larger increase later on that eventually produces a net increase compared to the initial value. The models suggest that this basic pattern should be observed both in the aggregate and within individual sectors, though more complex models with several sectors undergoing capital skill change at different times would likely show that the early decline in the capital-output ratio could be easily obscured in the aggregate data. The labor share of output is characterized by a decline during the early period of technological change that slows and sometimes reverses during later periods. In aggregate data, it is possible for the long-run net effect on labor share to be small or positive, if a high portion of human skill input in one sector is taken over by capital. More complex models with several sectors undergoing capital skill change at different times would likely show that the net effect on aggregate labor share could be small along the equilibrium path as well, if one sector's increasing labor share happened to cancel out another sector's decreasing labor share. Thus for labor share, the predicted effects may be visible only within individual sectors. Even within sectors, the labor share may experience substantial periods of steady or increasing values and fail to show the predicted net decrease in a dataset that happens to omit an early period of faster technological change. The stylized facts about the aggregate capital-output ratio and the aggregate labor share are usually understood to be that both have been relatively constant over long periods of time (Nicholas Kaldor, 1963). This indeed seems to be the case for aggregate labor share (Dale W. Jorgenson et al., 1987). For the capital-output ratio the situation is more complicated (Angus Maddison, 1995). The aggregate capital-output ratio in the U.S. rose during the 19th century and declined during the 20th century. However, the 20th century decline conflates a steady continued increase in the ratio of machinery and equipment to 13

Though without providing the possibility of an independent definition of technology.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 52 output, and a substantial decline in the ratio of non-residential structures to output. Since it is more plausible that it is the machinery and equipment portion of capital that has replaced human work skills, rather than the non-residential structures portion of capital, the twocentury increase in the ratio of machinery and equipment to output seems to be consistent with the behavior of the skill-based models. The five other countries described by Maddison show an unambiguously increasing long-run aggregate capital-output ratio, both with a capital measure that uses machinery and equipment alone and with a capital measure that includes structures. The relative constancy of the labor share over long periods of time is potentially a problem for the theory, but only if one believes that all industries share the same distribution of skill requirements so that the entire economy can be viewed as a single sector. If not, then the movements of the labor share may be obscured in aggregate data but visible in industry-level data. To see whether this is the case, table 31 shows the trends in the capital-output ratio and the labor share for 35 industries over the 44-year period from 1948 to 1991. The average over the period for each is also shown. The table is derived from Dale Jorgenson's KLEM dataset.14 This dataset provides indices for the value and prices of capital, labor, energy, and materials used in each of the 35 industries, along with indices for the price of each industry's output. To calculate the labor share of value-added in each industry, the value of labor is divided by the total value of capital and labor. To calculate the capital-output ratio, the value of capital is divided by its price to obtain the quantity of capital, and the total value of capital and labor is divided by the price of output to obtain the quantity of value-added. The trend values given are the coefficients from simple linear regressions with the year as the sole explanatory variable. For the capital-output ratio, the table shows 25 industries with a trend that is positive and significant, 5 industries with a trend that is negative and significant, and 5

14

The methodology for constructing the dataset is described in Jorgenson et al. (1987), and the dataset is available at Jorgenson's internet site at http://kuznets.harvard.edu/~djorgens/.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 53 industries with no significant trend. For labor share, the table shows 12 industries with a trend that is positive and significant, 14 industries with a trend that is negative and significant, and 9 industries showing no significant trend. The table makes it abundantly clear that there is substantial change at the industry level in both the capital-output ratio and the labor share. The large number of industries showing a positive trend in the labor share indicates that the models would provide a better fit to the historical data when σ = 0.5 than when σ = 1.0 or 1.5. Thus far, this appears to be the only way to produce a rising labor share within this modeling framework. Both the capital-output ratio and the labor share have industries showing positive trends and industries showing negative trends, but the mix of these is different. For the capital-output ratio there are many more industries showing positive trends than negative trends, whereas for the labor share the numbers are roughly equal. This difference in the mix of positive and negative trends could likely be produced with a multi-sector model with sectors at different points along their paths of technological change during the period 194891 -- some experiencing a relatively fast period of technological change (corresponding to 0 < t < 50 in the current models) and others experiencing a relatively slow period (corresponding to 50 < t < 100). For example, for the path described in II.C., when z = 0.75, δ = 0.05, and η = 0, labor share falls during the first 50 periods and then rises strongly until sometime after t = 100. In contrast, the capital-share ratio falls during only the first 25 periods, and then rises strongly until sometime after t = 100. In this example, all of the industries experiencing the fast period of change would show a declining labor share, but only some of them would show a declining capital-output ratio. Although the models can produce both positive and negative trends for both the capital-output ratio and the labor share, they do not produce all possible combinations of these trends. In particular, the initial decline in the capital-output ratio occurs when labor

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 54 share is falling, and the later increase in labor share occurs when the capital-output ratio is rising. So the model is not consistent with a decreasing capital-output ratio combined with an increasing labor share. This inconsistent combination is produced by 2 industries: Transportation and Services. Given the mix of positive and negative trends, this small number of inconsistent combinations would be produced by chance pairings and so it does not add any additional support to the models. Obviously, these simple trends cannot provide a real test of the skill-based models of technological change. More work is required to link the path of these industry statistics to the paths produced by the models, and to link the industry trends to the rate of change in capital skills during this period. However, the trends do provide two pieces of encouraging information: 1) Both the capital-output ratio and the labor share show clear long-term trends in many industries and are not correctly described as being relatively stable. 2) The relative mix of positive and negative trends for the capital-output ratio and the labor share is broadly consistent with the mix of trends produced by the skill-based models of technological change for the parameter values that produce periods of increasing labor share.

VI. Conclusion

This paper has developed three models of the aggregate economic changes that occur when capital becomes capable of performing human work skills. The motivation for studying these models has been a desire to understand how to think about the future economic impact of computer technology in a systematic way. The skill-based production function links projections about future computer skills to their impact on the economy. Because of this link, the models make it possible to use technical information that is currently available about computers to draw conclusions about the future economy.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 55 The paper does not make projections about future computer skills, since that is a large task in its own right. However, it is important to note that that task is clearly defined and quite feasible. First, it is necessary to gather information about the distribution of skills currently used by the human labor force, since it is the proportion of skill input replaced by capital, not the number of skills replaced, that determines how much the capital parameter changes. Such skill information is now becoming available with the new O*Net occupational database from the U.S. Department of Labor (American Institutes for Research, 1995). Second, it is necessary to gather information about the computer capabilities that are in the R&D pipeline for the next few decades. Since new technology typically takes several decades to diffuse, it is not necessary to predict those future capabilities from nothing; instead it is sufficient to survey the current level of computer capabilities in the lab. Preliminary work on projecting future computer skills suggests that the intermediate values of 0.5 and 0.75 for the technology parameter z are probably the most plausible. Although the development of the skill-based models was motivated by a desire to understand the future impact of computer technology, the models can be used as well to understand the past impact of all technology. Existing growth models treat technological change as a parameter to be estimated in fitting other economic data, rather than as an independent variable that is operationally defined and that can be measured from historical sources that describe the production process. Of course, the substitution of capital skills for labor skills is only one portion of technological change and so the skill-based approach cannot entirely supplant the other approaches used to describe technological change. However, the skill-based models suggest that this one portion tells a rich story about the structural shifts that occur in the economy over time. The models provide an intriguing hypothesis about the relation between technological change within industries and the resulting paths of the labor share and the capital-output ratio. The crude attempt here to assess that hypothesis encourages further work along these lines. Although it has not been

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 56 explored here in relation to historical data, the models also provide predictions about the paths of changes in productivity, wages, the occupational mix of labor, and the sectoral mix of production. Application of the models to the past can refine them for making projections about the future. The historical discussion in the paper, though crude, is sufficient to show that a value of 0.5 for σ is more consistent with the patterns of past economic change than the values of 1.0 or 1.5. This value for σ, combined with the more plausible values for z, suggests that the plausible growth rates in aggregate output 25 years from now range from 0.015 to 0.036, depending on the model used and the values for δ and η. Much of this range of projected growth rates is larger than the long-run historical rate of growth in industrialized countries, even though the models exclude all other forms of technological change that also increase output. Accompanying this output growth, the models project plausible growth rates in average wages 25 years from now that range from -0.022 to 0.015. The projected wage growth rates are substantially below the projected output growth rates. The higher growth rates assume that 7.3% or 12.7% of the labor force (for z = 0.5 or 0.75, respectively) is able to move from low-skill to high-skill occupations over the next 25 years. This degree of movement is required not only to avoid the lowest levels of average wage growth, but also to avoid increased skill-based wage inequality: in the 2type model when labor movement from low-skill to high-skill jobs is prevented, high-skill wages become 2.04 to 3.31 times larger than low-skill wages in 25 years, even though the model starts with wage equality. Much higher levels of occupational change would be required from t = 25 to t = 50 to avoid substantial additional increases in wage inequality. The models suggest two ways in which the impact of applying computers over the next 50 years may be more extreme than the impact caused by the technological changes of the past 200 years. First, computers may cause a faster shift in skills and a faster increase in productivity than were experienced in the past. Agriculture employed about 80% of the

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 57 labor force in 1820 and capital replaced labor for much of this work over the next 150 years (U.S. Department of Commerce, 1975). The size of this skill shift in agriculture is comparable to a z of about 0.75 for the 1820 economy, but computers are likely to cause this level of change over the next 50 to 75 years rather than the next 150 years. Second, both the speed of arrival of new computer skills and the increasing level of their complexity suggest that low-skill workers may begin to have difficulty moving into high-skill jobs. Initially this would cause low-skill wages to be forced down as in the 2-type model, but eventually low-skill labor could become virtually unemployable. This was avoided in the past, with the movement from agriculture to manufacturing and services, because technology changed slowly enough that workers could change occupations between generations rather than within generations, and because in that era high-skill work required only a high school education not a graduate school education (Claudia Goldin and Lawrence F. Katz, 1997). The problem of the increasing complexity of the skills required by human labor is particularly troubling. There are numerous examples of specialized computer systems that can outperform humans for particular tasks. Continued development in the coming decades will make such systems more general and more robust, to the point where they become able to compete with humans in providing relatively broad, widely-used skills. For example, the reading levels of computers in the lab for well-defined reading tasks (Beth M. Sundheim, 1995) already rival the reading levels of about 80% of the population of the U.S. and other industrialized countries (OECD, 1995). Of course, these computer systems are only prototypes that require further development before they can be easily applied to the reading tasks that are required at work. Such systems are at an earlier stage of development than the speech recognition systems that are now being marketed, so we should not expect that they will be refined and widely diffused until perhaps three or four decades from now. However, as these systems become available, the reading skills of 80% of the labor force will need to increase for those workers to avoid being replaced by computers for this skill.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 58 Although reading is not the only skill used at work, suggesting that workers with uncompetitive reading skills could rely on other skills for employment, it is likely that computers will make similar progress on those other skills over the next few decades as well. Obviously the level of education will need to increase, and with better education the skills of the labor force will improve. However, there is a danger that a substantial and increasing portion of the future labor force will have difficulty acquiring levels of skill in any skill area that are competitive with future computer technology. The models presented in this paper avoid the problem of permanent unemployment by assumption, but it is unlikely that our economy will be able to avoid that problem in the decades ahead.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 59

References American Institutes for Research. Development of Prototype Occupational Information Network (O*NET) Content Model. Utah Department of Employment Security, 1995. Autor, David H., Lawrence F. Katz, and Alan B. Krueger. Computing Inequality: Have Computers Changed the Labor Market? Cambridge, MA: National Bureau of Economic Research, Working Paper 5956, 1997. Barnett, Harold J. "Scarcity and Growth Revisited," in V. Kerry Smith, ed., Scarcity and Growth Reconsidered. Baltimore: The Johns Hopkins University Press, 1979, pp. 163217. Barro, Robert J., and Xavier Sala-i-Martin. Economic Growth. New York, NY: McGraw-Hill, 1995. Bass, Frank M. "A New Product Growth Model for Consumer Durables." Management Science, 15(5), January 1969, pp. 215-227. Baumol, William J., Sue Anne Batey Blackman, and Edward N. Wolff. Productivity and American Leadership: The Long View. Cambridge, MA: MIT Press, 1989. BBN. "BBN Achieves Benchmark in Real-time, Continuous, Speaker-independent, 40,000 Word Dictation." Cambridge, MA: BBN, mimeo, 1994. Birnbaum, Joel. "Evolution and Impacts of Electronic and Non-electronic, Biological and Optical Computing Technologies." ACM97 Conference, March 3, 1997, San Jose, CA. Bresnahan, Timothy F. Computerization and Wage Dispersion: An Analytical Reinterpretation. Unpublished paper, Stanford University, 1997. Brynjolfsson, Erik, and Lorin Hitt. "Paradox Lost? Firm-level Evidence on the Returns to Information Systems Spending." Management Science, 42(4), April 1996, pp. 541-558. Christensen, Laurits R., Dianne Cummings, and Dale W. Jorgenson. "Economic Growth, 1947-73: An International Comparison," in John W. Kendrick and Beatrice N. Vaccara, eds., New Developments in Productivity Measurement and Analysis, NBER Conference on Research in Income and Wealth. Chicago: University of Chicago Press, 1980, pp. 595-698. Denison, Edward F. Accounting for United States Economic Growth 1929-1969. Washington, DC: The Brookings Institution, 1974. Doyle, Jon, and Thomas Dean. "Strategic Directions in Artificial Intelligence." AI Magazine, 18(1), Spring 1997, pp. 87-101. Goldin, Claudia, and Lawrence F. Katz. The Origins of Technology-Skill Complementarity. New York, NY: Russell Sage Foundation, Working Paper 118, 1997. Gordon, Robert J. "The Great Productivity Speed-up and Slowdown: How Much is Mismeasurement?" Chicago, IL: Conference of the American Economics Association, January 4, 1998.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 60 Gordon, Robert J. "The Postwar Evolution of Computer Prices," in Dale W. Jorgenson and Ralph Landau, eds., Technology and Capital Formation. Cambridge, MA: MIT Press, 1989, pp. 77-125. Hamermesh, Daniel S. Labor Demand. Princeton, NJ: Princeton University Press, 1993. Jorgenson, Dale W., Frank M. Gollop, and Barbara M. Fraumeni. Productivity and U.S. Economic Growth. Cambridge, MA: Harvard University Press, 1987. Jorgenson, Dale W., and Kevin Stiroh. "Computers and Growth." Economics of Innovation and New Technology, 3, 1995, pp. 295-316. Kaldor, Nicholas. "Capital Accumulation and Economic Growth," in Friedrich A. Lutz and Douglas C. Hague, eds., Proceedings of a Conference Held by the International Economics Association. London: Macmillan, 1963. Lewis, Ted. "The Next 10,0002 Years: Part 1," Computer, 29(4), May 1996, pp. 64-70. Lohr, Steve. "Sun Offering a Lower-cost Network Computer." New York Times, October 29, 1996, p. D1. Lucas, Robert E., Jr. "On the Mechanics of Economic Development." Journal of Monetary Economics, 22, 1988, pp. 3-42. Maddison, Angus. Phases of Capitalist Development. Oxford: Oxford University Press, 1982. Maddison, Angus. Monitoring the World Economy 1820-1992. Paris: OECD, 1995. Mansfield, Edwin. "The Diffusion of Industrial Robots in Japan and the United States." Research Policy, 18, 1989, pp. 183-192. Moravec, Hans. Mind Children. Cambridge, MA: Harvard University Press, 1988. OECD. Literacy, Economy and Society; Results of the First International Adult Literacy Survey. Paris: OECD, 1995. Reddy, D. Raj. "Speech Recognition by Machine: A Review." IEEE Proceedings, 64(4), April 1976, pp. 502-531. Romer, Paul M. "Endogenous Technological Change." Journal of Political Economy, 98(5), October 1990, Part 2, pp. S71-S102. Ross, Philip. E. "Moore's Second Law." Forbes, March 25, 1996, pp. 116-117. Sichel, Daniel E. The Computer Revolution. Washington, DC: Brookings Institution Press, 1997. Simon, Herbert A. The New Science of Management Decision, rev. ed. Englewood Cliffs: Prentice-Hall, 1977. Solow, Robert M. "Technical Change and the Aggregate Production Function." Review of Economics and Statistics, 39(1), August 1957, pp. 312-320.

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Stix, Gary. "Toward 'Point One'." Scientific American, February 1995, pp. 90-95. Sundheim, Beth M. "Overview of Results of the MUC-6 Evaluation," in Sixth Message Understanding Conference (MUC-6); Proceedings of a Conference Held in Columbia, Maryland, November 6-8, 1995. San Francisco, CA: Morgan Kaufmann Publishers, 1995. U.S. Department of Commerce, Bureau of the Census. Historical Statistics of the United States, Colonial Times to 1970. Washington, DC: U.S. Government Printing Office, 1975. U.S. Department of Commerce, Bureau of the Census. Statistical Abstract of the United States: 1992, 112th ed. Washington, DC: U.S. Government Printing Office, 1992. Zeira, Joseph. Workers, Machines and Economic Growth. Unpublished manuscript, Brandeis University, June 1997.

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 62

Appendix: 2-Sector Model Solution

Equation (28), after substituting for m from (27) with ζ = 0.5, substituting for C2,t from (22) and (25), taking logs, differentiating by t, and substituting from (30), yields:

(A1)

 (θ − 2)C1, t C˙1, t C + C2, t  ∂Y1, t * λ˙t = 1, t − δ − ρ +  C1, t 2C1, t + θC2, t  ∂Kt  2C1, t + θC2, t 1 − λt

Equation (29), after transforming in a similar way and using (21a) to calculate the partial derivative of Y1,t*, yields:

(A2)

(θ − 2)C2,t C˙1,t C1, t + C2, t +

C1, t

 ∂Y *  α˙ t − gt ht =  1, t − δ − ρ  +  ∂Kt  (σ − 1)( gt + 1 − α t )

θC1, t + 2C2, t λ˙t gt λ˙t + , C1, t + C2, t 1 − λt σ ( gt + 1 − α t ) λt

where  (1 − α t ) Kt  gt ≡ α t    λt L  1 σ

and

ht ≡

(1 − σα t ) σ (1 − α t )

σ −1 σ

α˙ t (σ − 1) K˙ t + Kt αt σ

It is straightforward to solve for C˙1, t and l˙t from (A1) and (A2).

it µt > 1

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 63

Table 1: Asymptotic Growth in Capital: K ∞ K 0 η=0

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.939

4.548

174.2

1.925

4.549

78.23

z = 0.50

4.610

43.33

∞

3.894

28.26

∞

z = 0.75

18.96

11,000

∞

8.071

710.4

∞

z = 1.00

∞

∞

∞

11.84

45,000,000

∞

Table 2: Asymptotic Labor-PC Price Ratio: µ ∞ η=0

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

15.08

21.54

128.2

12.79

18.20

62.37

z = 0.50

28.35

100.0

∞

15.71

51.39

∞

z = 0.75

93.11

10,000

∞

12.16

490.0

∞

z = 1.00

1

1

1

1

1

1

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 64

Table 3: Asymptotic Growth in Output: Y∞ Y0 η=0

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.637

2.873

61.24

1.508

2.426

20.77

z = 0.50

3.368

20.00

∞

2.507

10.28

∞

z = 0.75

12.21

4,000

∞

4.410

196.0

∞

z = 1.00

∞

∞

∞

5.786

10,000,000

∞

Table 4: Asymptotic Change in Capital-Output Ratio:

η=0

K∞ Y∞

K0 Y0

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.185

1.583

2.845

1.277

1.875

3.767

z = 0.50

1.369

2.167

1.684

1.553

2.750

2.928

z = 0.75

1.553

2.750

0.868

1.830

3.625

1.445

z = 1.00

0.906

0.665

0.528

2.046

4.500

0.858

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 65

Table 5: Asymptotic Growth in Consumption: C∞ C0 η=0

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.555

2.416

30.43

1.439

2.072

11.19

z = 0.50

3.029

13.64

∞

2.275

7.280

∞

z = 0.75

10.36

2,091

∞

3.800

110.2

∞

z = 1.00

∞

∞

∞

4.777

4,167,000

∞

Table 6: Asymptotic Growth in Wages: W∞ W0 η=0

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.508

2.154

12.82

1.279

1.820

6.237

z = 0.50

2.835

10.00

∞

1.571

5.139

∞

z = 0.75

9.311

1,000

∞

1.216

49.00

∞

z = 1.00

0.368

0.501

0.632

0.100

0.100

0.583

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 66

Table 7: Asymptotic Labor Share of Output: W∞ L Y∞ η=0

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.645

0.525

0.147

0.594

0.525

0.210

z = 0.50

0.589

0.350

0.000

0.439

0.350

0.000

z = 0.75

0.534

0.175

0.000

0.193

0.175

0.000

z = 1.00

0.000

0.000

0.000

0.012

0.000

0.000

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 67

Table 8: Rate of Growth in Capital at t = 25: K˙ 25 K 25

η=0

δ = 0.05

δ = 0.10

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.007

0.010

0.018

0.010

0.016

0.027

z = 0.50

0.016

0.022

0.036

0.021

0.031

0.047

z = 0.75

0.027

0.035

0.052

0.032

0.046

0.063

z = 1.00

0.038

0.048

0.066

0.045

0.060

0.077

z = 0.25

0.011

0.018

0.032

0.015

0.024

0.041

z = 0.50

0.024

0.036

0.057

0.027

0.044

0.066

z = 0.75

0.037

0.053

0.079

0.044

0.062

0.086

z = 1.00

0.053

0.072

0.101

0.058

0.081

0.109

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 68

Table 9: Labor-PC Price Ratio at t = 25: µ 25

η=0

δ = 0.05

δ = 0.10

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

7.627

7.190

6.399

7.785

7.283

6.410

z = 0.50

6.004

5.655

5.012

6.199

5.776

5.066

z = 0.75

4.757

4.594

4.166

4.956

4.735

4.248

z = 1.00

3.722

3.765

3.535

3.915

3.916

3.632

z = 0.25

8.730

8.285

7.539

8.930

8.513

7.776

z = 0.50

7.616

7.086

6.374

7.746

7.408

6.649

z = 0.75

6.520

6.135

5.611

6.845

6.486

5.886

z = 1.00

5.500

5.307

5.013

5.720

5.699

5.328

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 69

Table 10: Rate of Growth in Output at t = 25: Y˙25 Y25

η=0

δ = 0.05

δ = 0.10

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.008

0.012

0.020

0.008

0.013

0.022

z = 0.50

0.018

0.025

0.038

0.017

0.027

0.040

z = 0.75

0.029

0.039

0.055

0.027

0.041

0.057

z = 1.00

0.041

0.053

0.071

0.037

0.055

0.074

z = 0.25

0.010

0.016

0.028

0.010

0.016

0.029

z = 0.50

0.022

0.034

0.054

0.020

0.033

0.054

z = 0.75

0.036

0.053

0.079

0.033

0.052

0.079

z = 1.00

0.053

0.074

0.105

0.045

0.073

0.106

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 70

Table 11: Change in Capital-Output Ratio at t = 25:

η=0

δ = 0.05

δ = 0.10

K 25 Y25

K0 Y0

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.939

0.858

0.786

0.981

0.936

0.923

z = 0.50

0.898

0.779

0.695

0.972

0.897

0.869

z = 0.75

0.868

0.723

0.634

0.972

0.866

0.822

z = 1.00

0.845

0.681

0.588

0.980

0.840

0.781

z = 0.25

0.981

0.937

0.905

1.033

1.048

1.116

z = 0.50

0.960

0.880

0.825

1.050

1.052

1.085

z = 0.75

0.936

0.827

0.755

1.076

1.034

1.026

z = 1.00

0.913

0.778

0.694

1.091

1.008

0.967

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 71

Table 12: Rate of Growth in Consumption at t = 25: C˙ 25 C25

η=0

δ = 0.05

δ = 0.10

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.008

0.011

0.018

0.007

0.011

0.019

z = 0.50

0.016

0.022

0.034

0.015

0.023

0.036

z = 0.75

0.025

0.034

0.049

0.023

0.035

0.052

z = 1.00

0.035

0.047

0.065

0.031

0.048

0.067

z = 0.25

0.009

0.014

0.024

0.007

0.013

0.024

z = 0.50

0.019

0.029

0.047

0.017

0.028

0.047

z = 0.75

0.031

0.046

0.071

0.025

0.044

0.071

z = 1.00

0.045

0.065

0.095

0.035

0.062

0.095

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 72

Table 13: Rate of Growth in Wages at t = 25: W˙ 25 W25

η=0

δ = 0.05

δ = 0.10

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.003

0.005

0.009

0.003

0.006

0.010

z = 0.50

0.003

0.009

0.014

0.002

0.010

0.016

z = 0.75

0.000

0.010

0.017

-0.003

0.012

0.019

z = 1.00

-0.011

0.007

0.017

-0.018

0.009

0.019

z = 0.25

0.006

0.009

0.014

0.005

0.009

0.014

z = 0.50

0.012

0.017

0.025

0.007

0.017

0.025

z = 0.75

0.015

0.024

0.033

0.008

0.023

0.033

z = 1.00

0.014

0.028

0.039

-0.002

0.027

0.040

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 73

Table 14: Labor Share of Output at t = 25: W25 L Y25

η=0

δ = 0.05

δ = 0.10

η = 1/3

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.625

0.621

0.586

0.628

0.621

0.577

z = 0.50

0.546

0.543

0.490

0.549

0.543

0.480

z = 0.75

0.461

0.464

0.407

0.461

0.464

0.397

z = 1.00

0.370

0.385

0.333

0.368

0.385

0.324

z = 0.25

0.641

0.621

0.566

0.639

0.621

0.558

z = 0.50

0.575

0.543

0.460

0.568

0.543

0.451

z = 0.75

0.500

0.464

0.372

0.490

0.464

0.363

z = 1.00

0.417

0.385

0.296

0.400

0.385

0.287

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 74

Table 15: 2-Type Asymptotic Growth in Output: Y∞ Y0

Basic

2-Type

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.637

2.873

61.24

1.579

2.823

60.54

z = 0.50

3.368

20.00

∞

2.526

17.32

∞

z = 0.75

12.21

4,000

∞

3.051

1,000

∞

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 75

Table 16: 2-Type Asymptotic Growth in Wages: W1,∞ W1,0 and W2,∞ W2,0

Basic

Type 1

Type 2

2-Type

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.508

2.154

12.82

1.108

1.882

11.77

z = 0.50

2.835

10.00

∞

0.709

5.774

∞

z = 0.75

9.311

1,000

∞

0.100

0.100

0.384

z = 0.25

1.508

2.154

12.82

2.492

2.823

15.42

z = 0.50

2.835

10.00

∞

6.379

17.32

∞

z = 0.75

9.311

1,000

∞

9.311

1,000

∞

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 76

Table 17: 2-Type Asymptotic Labor Share of Output: W1,∞ L1 Y∞ and W2,∞ L2 Y∞

Basic

Type 1

Type 2

2-Type

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.484

0.394

0.110

0.368

0.350

0.102

z = 0.50

0.442

0.263

0.000

0.147

0.175

0.000

z = 0.75

0.401

0.131

0.000

0.017

0.000

0.000

z = 0.25

0.161

0.131

0.037

0.276

0.175

0.045

z = 0.50

0.147

0.088

0.000

0.442

0.175

0.000

z = 0.75

0.134

0.044

0.000

0.534

0.175

0.000

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 77

Table 18: 2-Type Rate of Growth in Output at t = 25: Y˙25 Y25

Basic

δ = 0.05

δ = 0.10

2-Type

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.008

0.012

0.020

0.008

0.012

0.020

z = 0.50

0.018

0.025

0.038

0.016

0.025

0.038

z = 0.75

0.029

0.039

0.055

0.024

0.038

0.054

z = 0.25

0.010

0.016

0.028

0.010

0.016

0.028

z = 0.50

0.022

0.034

0.054

0.020

0.033

0.053

z = 0.75

0.036

0.053

0.079

0.030

0.051

0.077

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 78

Table 19: 2-Type Rate of Growth in Wages at t = 25: W˙1,25 W1,25 and W˙ 2,25 W2,25

Basic

δ = 0.05 Type 1

δ = 0.10

δ = 0.05 Type 2

δ = 0.10

2-Type

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.003

0.005

0.009

-0.004

0.002

0.007

z = 0.50

0.003

0.009

0.014

-0.016

0.001

0.009

z = 0.75

0.000

0.010

0.017

-0.045

-0.008

0.006

z = 0.25

0.006

0.009

0.014

0.000

0.006

0.012

z = 0.50

0.012

0.017

0.025

-0.009

0.009

0.019

z = 0.75

0.015

0.024

0.033

-0.033

0.005

0.021

z = 0.25

0.003

0.005

0.009

0.016

0.012

0.013

z = 0.50

0.003

0.009

0.014

0.032

0.025

0.025

z = 0.75

0.000

0.010

0.017

0.047

0.038

0.036

z = 0.25

0.006

0.009

0.014

0.020

0.016

0.019

z = 0.50

0.012

0.017

0.025

0.040

0.033

0.035

z = 0.75

0.015

0.024

0.033

0.059

0.051

0.052

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 79

Table 20: 2-Type Labor Share of Output at t = 25: W1,25 L1 Y25 and W2,25 L2 Y25

Basic

δ = 0.05 Type 1

δ = 0.10

δ = 0.05 Type 2

δ = 0.10

2-Type

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.469

0.466

0.439

0.429

0.446

0.427

z = 0.50

0.409

0.407

0.367

0.331

0.368

0.343

z = 0.75

0.345

0.348

0.305

0.233

0.289

0.269

z = 0.25

0.481

0.466

0.424

0.440

0.446

0.412

z = 0.50

0.431

0.407

0.345

0.348

0.368

0.322

z = 0.75

0.375

0.348

0.279

0.251

0.289

0.246

z = 0.25

0.156

0.155

0.146

0.198

0.175

0.159

z = 0.50

0.136

0.136

0.122

0.225

0.175

0.145

z = 0.75

0.115

0.116

0.102

0.257

0.175

0.134

z = 0.25

0.160

0.155

0.141

0.203

0.175

0.153

z = 0.50

0.144

0.136

0.115

0.236

0.175

0.136

z = 0.75

0.125

0.116

0.093

0.276

0.175

0.122

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 80

Table 21: 2-Sector Asymptotic Growth in Output: Y1,∞ Y1,0 and Y2,∞ Y2,0

Sector 1

Sector 2

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.829

3.705

∞

1.275

1.657

4.000

z = 0.50

5.063

57.92

∞

1.862

3.421

4.000

z = 0.75

∞

∞

∞

4.000

4.000

4.000

Table 22: 2-Sector Asymptotic Growth in Aggregate Output: Y∞ Y0

Basic

2-Sector

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.637

2.873

61.24

1.631

2.755

14.33

z = 0.50

3.368

20.00

∞

3.421

10.64

14.33

z = 0.75

12.21

4,000

∞

14.33

14.33

14.33

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 81

Table 23: 2-Sector Asymptotic Change in Capital-Output Ratio: K∞ Y∞

K0 Y0

2-Sector: Aggregate

Basic

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.185

1.583

2.845

1.328

2.129

∞

z = 0.50

1.369

2.167

1.684

2.026

11.80

∞

z = 0.75

1.553

2.750

0.868

∞

∞

∞

2-Sector: Sector 1

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.185

1.583

2.570

z = 0.50

1.369

2.167

0.946

z = 0.75

0.877

0.635

0.487

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 82

Table 24: 2-Sector Asymptotic Growth in Wages: W∞ W0

Basic

2-Sector

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

1.508

2.154

12.82

1.537

2.192

20.48

z = 0.50

2.835

10.00

∞

3.194

9.880

20.48

z = 0.75

9.311

1,000

∞

20.48

20.48

20.48

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 83

Table 25: 2-Sector Asymptotic Labor Share of Output: W∞ L Y∞

2-Sector: Aggregate

Basic

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.645

0.525

0.147

0.660

0.557

1.000

z = 0.50

0.589

0.350

0.000

0.654

0.650

1.000

z = 0.75

0.534

0.175

0.000

1.000

1.000

1.000

2-Sector: Sector 1

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.569

0.424

0.000

z = 0.50

0.502

0.212

0.000

z = 0.75

0.000

0.000

0.000

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 84

Table 26: 2-Sector Rate of Growth in Output at t = 25: Y˙1,25 Y1,25 and Y˙2,25 Y2,25

Sector 1

δ = 0.05

δ = 0.10

Sector 2

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.009

0.013

0.022

0.007

0.008

0.013

z = 0.50

0.019

0.028

0.042

0.015

0.018

0.026

z = 0.75

0.030

0.043

0.060

0.028

0.030

0.040

z = 0.25

0.011

0.018

0.031

0.006

0.009

0.017

z = 0.50

0.025

0.039

0.060

0.015

0.020

0.034

z = 0.75

0.041

0.062

0.088

0.026

0.034

0.051

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 85

Table 27: 2-Sector Rate of Growth in Aggregate Output at t = 25: Y˙25 Y25

2-Sector: Aggregate

Basic

δ = 0.05

δ = 0.10

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.008

0.012

0.020

0.008

0.012

0.019

z = 0.50

0.018

0.025

0.038

0.018

0.025

0.036

z = 0.75

0.029

0.039

0.055

0.029

0.039

0.053

z = 0.25

0.010

0.016

0.028

0.010

0.015

0.026

z = 0.50

0.022

0.034

0.054

0.021

0.032

0.049

z = 0.75

0.036

0.053

0.079

0.036

0.051

0.072

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 86

Table 28: 2-Sector Change in Capital-Output Ratio at t = 25:

δ = 0.05

δ = 0.10

σ = 1.0

σ = 1.5

δ = 0.10

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.939

0.858

0.786

0.947

0.865

0.807

z = 0.50

0.898

0.779

0.695

0.915

0.800

0.737

z = 0.75

0.868

0.723

0.634

0.893

0.758

0.691

z = 0.25

0.981

0.937

0.905

0.999

0.964

0.953

z = 0.50

0.960

0.880

0.825

0.996

0.937

0.912

z = 0.75

0.936

0.827

0.755

0.993

0.910

0.872

2-Sector: Sector 1

δ = 0.05

K0 Y0

2-Sector: Aggregate

Basic

σ = 0.5

K 25 Y25

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.934

0.843

0.763

z = 0.50

0.890

0.758

0.663

z = 0.75

0.860

0.699

0.597

z = 0.25

0.973

0.921

0.876

z = 0.50

0.944

0.855

0.782

z = 0.75

0.917

0.793

0.703

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 87

Table 29: 2-Sector Rate of Annual Growth in Wages at t = 25: W˙ 25 W25

Basic

δ = 0.05

δ = 0.10

2-Sector

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.003

0.005

0.009

0.002

0.005

0.008

z = 0.50

0.003

0.009

0.014

0.002

0.008

0.014

z = 0.75

0.000

0.010

0.017

-0.004

0.009

0.018

z = 0.25

0.006

0.009

0.014

0.005

0.009

0.013

z = 0.50

0.012

0.017

0.025

0.010

0.017

0.025

z = 0.75

0.015

0.024

0.033

0.012

0.024

0.037

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 88

Table 30: 2-Sector Labor Share of Output at t = 25: : W25 L Y25 2-Sector: Aggregate

Basic

δ = 0.05

δ = 0.10

σ = 0.5

σ = 1.0

σ = 1.5

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.625

0.621

0.586

0.621

0.620

0.587

z = 0.50

0.546

0.543

0.490

0.534

0.537

0.492

z = 0.75

0.461

0.464

0.407

0.439

0.453

0.408

z = 0.25

0.641

0.621

0.566

0.638

0.621

0.570

z = 0.50

0.575

0.543

0.460

0.566

0.542

0.470

z = 0.75

0.500

0.464

0.372

0.483

0.462

0.391

2-Sector: Sector 1

δ = 0.05

δ = 0.10

σ = 0.5

σ = 1.0

σ = 1.5

z = 0.25

0.544

0.541

0.503

z = 0.50

0.445

0.445

0.391

z = 0.75

0.340

0.350

0.296

z = 0.25

0.561

0.541

0.480

z = 0.50

0.476

0.445

0.357

z = 0.75

0.381

0.350

0.256

Elliott: Computer Technology, Human Labor, Long-Run Economic Growth -- Page 89

Table 31: Capital-Output Ratio and Labor Share by Industry, 1948-1991

Industry 1-Agriculture, forestry and fisheries 2-Metal mining 3-Coal mining 4-Crude petroleum and natural gas 5-Nonmetallic mineral mining 6-Construction 7-Food and kindred products 8-Tobacco manufactures 9-Textile mill products 10-Apparel and other textile products 11-Lumber and wood products 12-Furniture and fixtures 13-Paper and allied products 14-Printing and publishing 15-Chemicals and allied products 16-Petroleum refining 17-Rubber and plastic products 18-Leather and leather products 19-Stone, clay and glass products 20-Primary metals 21-Fabricated metal products 22-Machinery, except electrical 23-Electrical machinery 24-Motor vehicles 25-Other transportation equipment 26-Instruments 27-Miscellaneous manufacturing 28-Transportation and warehousing 29-Communication 30-Electric utilities 31-Gas utilities 32-Trade 33-FIRE 34-Services 35-Government *=significant at P