Technology adoption over the life cycle and aggregate technological progress.

• Author: Swanson, Charles E.

1. Introduction

   Technology becomes useful only when individuals adopt it for productive activity. The manner in which adoption occurs is therefore important for a complete understanding of the evolution of technological change. We study a model with exogenously occurring technological progress and endogenous adoption of it. Individuals with N-period lives optimally allocate their time between leisure, work and adoption of new technology. We show that optimal behavior is characterized by a sequence of four phases of life which, described in their order of occurrence, are: (1) adoption only (schooling); (2) work and adoption (career path); (3) work but no adoption (end-of-work-life easing); (4) no work or adoption (retirement).

   The presence of the third phase, a period in which older workers choose not to adopt new technology, has several implications for the effects of changes in aggregate exogenous technology. Among these is the implication that a wave of innovation will have a smaller effect when the number of workers in this phase is relatively large. This is tested using patent data as a proxy for innovation and the Solow residual as a measure of technological progress; some support for this proposition is found.

   The model also implies that older (third phase) workers will appear more productive because each hour of non-leisure time will be devoted entirely to productive work, rather than being divided between work and technology adoption time. The data provides some support for this proposition as well: measured productivity is positively correlated with the relative size of the oldest cohort of workers.

   The technology we study is characterized by two components, a maximum level, or frontier, and an adoption-ease parameter, both of which are assumed to grow exogenously. Individuals allocate their time endowment among work, technology adoption and leisure. Technology is skill-enhancing; thus the time spent adopting technology enhances current and future individual skills and wages.

   The process of invention and adoption has received renewed attention in the recent literature. While much work, such as Jovanovic and MacDonald [6] and Rustichini and Schmitz [12], studies specific aspects of the transmission of technology through particular industries, we are concerned primarily with the decision making of individuals in the acquisition of new technology for themselves and the relationship of these decisions to aggregate growth rates. Our work is much like the studies by Caballe and Santos [2] and Lucas [8], but is perhaps closer to Rios-Rull [11] who, like us, has finite lived individuals and decisions that differ over the life-cycle.

   Our paper is most closely related to Parente and Prescott [10] and uses some of their results. Parente and Prescott take technology formation as given and focus exclusively on its adoption. They study a model in which the cost (barriers) to technology adoption shrinks continuously. We use this same idea and add to it the notion that there is a limit to what can be adopted, a frontier, which grows with time.

   We assume technological advance is skilled-labor augmenting, a simplifying assumption which is restrictive, but which has some empirical validation. A recent study by Kahn and Lim [7] suggests that enhancements to skill-labor are by far the most important source of technical progress.

   The following section presents the discrete time version of the model. Section III describes its main properties. Section IV contains a numerical specification and solution. Section V describes some tests of the model based on patent data. Section VI contains the conclusion. The appendix contains a continuous-time version of the model and proofs of its various properties.

2. The Model

   We consider a growing economy in which finite-lived individuals choose consumption, leisure, work hours and adoption time over the course of their lifetimes. Technology is labor-augmenting, has two components, and expands perpetually in both. The first, denoted by [z.sub.t], indicates the frontier or the maximum skill level that an individual can possess. The second, denoted by [b.sub.t], indicates the ease with which new technology is adopted.(1) Both components of technological change grow at an exogenous rate; we only study the adoption of this technology.

   Skill Acquisition and Individuals' Decisions

   An individual of age i at date t who devotes [e.sub.i,t] units of time to skill acquisition will increase skill by [e.sub.i,t][b.sub.i,t] units, as long as this does not place the individual beyond the frontier; an individual's knowledge cannot exceed society's. Formally,

   [s.sub.i+1,t+1] = min{[z.sub.t+1], [s.sub.i,t] + [b.sub.t+1][e.sub.i+1,t+1]}, (1)

   where [s.sub.i,t] is the skill level of the individual of age i in period t.

   Wage income at age i in date t is determined by the individual's skill level, [s.sub.i,t], the aggregate wage (per skill-hour or efficiency unit), [w.sub.t], and hours worked, [h.sub.i,t]. The present value of income for the individual born in period t is

   [y.sub.t] = [summation of] (1/[(1 + r).sup.i])[w.sub.t+1][s.sub.i,t+1][h.sub.i,t+i] where i = 0 to T, (2)

   where r is the constant interest rate and [w.sub.t] is the market wage measured in consumption units per skill-hour.

   The individual maximizes

   [summation of] u([c.sub.i,t+i], [l.sub.i,t+i]) where i = 0 to T (3)

   subject to:

   [l.sub.i,t+i] = 1 - [h.sub.i,t+i] - [e.sub.i,t+i] (4)

   and

   [C.sub.t] [less than or equal to] [y.sub.t], (5)

   where

   [C.sub.t] = [summation of] (1/[(1 + r).sup.i])[c.sub.i,t+i] where i = 0 to T (6)

   is the present value of consumption, and [c.sub.i, t] and [l.sub.i, t] are consumption and leisure of those of age i at date t. Equation (6) assumes a steady state and hence a constant interest rate; we limit our analysis to the steady state case to emphasize the adoption decision process. The thrust of the argument would not be affected if notation were expanded so as to incorporate non-constant interest rates.

   The period utility function has the functional form

   u([c.sub.i,t+i], [l.sub.i,t+i]) = log([c.sub.i,t+i]) + a log([l.sub.i,t+i]). (7)

   Aggregate Capital and Labor

   The number of age i individuals at date t is [n.sub.i, t] and each holds [k.sub.i, t] units of capital. Total capital at date t is therefore

   [K.sub.t] = [summation of] [n.sub.i,t][k.sub.i,t] where i = 0 to T. (8)

   Total labor input at t, measured in efficiency unit hours, is the sum over all age cohorts i of the efficiency unit contribution of the cohort, which is, for cohort i, the product of population, hours and the skill level of each member. Total labor input at date t is therefore

   [H.sub.t] = [summation of] [h.sub.i,t][s.sub.i,t][n.sub.i,t] where i = 0 to T. (9)

   Aggregate output is determined by the production function

   [Mathematical Expression Omitted], (10)

   and aggregate capital accumulates according to

   [K.sub.t+1] = [K.sub.t](1 - [Delta]) + [I.sub.t], (11)

   where [I.sub.t] is gross aggregate investment in the homogenous capital good.

   The market wage is the marginal product of labor:

   [w.sub.t] = [F.sub.H]([K.sub.t], [H.sub.t]) = (1 - [Alpha])[([K.sub.t]/[H.sub.t]).sup.[Alpha]] (12)

   and the real interest rate is the marginal product of capital, less depreciation:

   [r.sub.t] = [Alpha][([H.sub.t]/[K.sub.t]).sup.1 - [Alpha]] - [Delta].(13)

   Along the steady state growth path, [w.sub.t] and [r.sub.t] are constant; this justifies omitting the subscript t on these variables as is done in the rest of the paper. Note that a constant value for [w.sub.t] implies that the wage per skill-hour is constant; the skill-corrected wage w[s.sub.i, t] for each fixed cohort i will grow through time at the economy's growth rate.

   Capital and Money

   Savings, which appear implicitly in the budget constraints of equations (2), (5) and (6), can be held as capital or as government money whose supply is constant. In a riskless world the real return on paper money is equal to the real return on capital so individuals do not care how they divide their savings between capital and money. The presence of money allows for an easy determination of the real interest rate. If the growth parameters [z.sub.t] and [b.sub.t] increase at rate g, the aggregate labor supply [H.sub.t] will grow at rate g, as will the capital stock and total output. The real value of the money supply will therefore rise at rate g, as long as the division of savings between capital and money remains constant. If the value of the aggregate money stock rises at rate g and the nominal money supply is fixed, each unit of money will rise in value at rate g. Thus, the real rate of return on money is g. Since money and capital have the same real rate of return, the real interest rate is

   r = g. (14)

   In other words, the presence of a fixed stock of money implies that the economy will be on the golden rule growth path.

3. Lifecycle Behavior

   We proceed by describing the steady state behavior of individuals under the assumption that both types of technology (that is, the frontier and the ease of adoption) grow at the same constant rate and that population growth is zero. We then use this behavior to draw inferences about how shifts in the aggregate rates of innovation are likely to affect aggregate rates of measured technology growth.

   The Steady State

   Assume that [z.sub.t] and [b.sub.t] both grow at the same exogenous rate g. To illustrate the meaning of this...