The response of aggregate production to fertility-induced changes in population age distribution.

• Author: Denton, Frank T.

1. Introduction

   Differences in fertility can induce marked differences in the age distribution of a population, and the age distribution has implications for the economy - for patterns of consumption and investment, the capacity to produce output, the level of per capita income, and other characteristics. That is true both between countries and through time. In the U.S., Canada, and some other developed nations, the transition from the "baby boom" of the 1950s and early 1960s to the subsequent "baby bust" produced major shifts in population structure, and further shifts are in prospect as a result of natural aging. To take the Canadian example, the total fertility rate fell from a record high of 3.9 children per woman in 1959 to a record low of 1.6 in 1987.(1) Primarily in consequence of that, and of the continuing low rates, the percentage of population under 15 fell from 33.9 at the 1961 census to 20.9 at the 1991 census, while the percentage 65 and over increased from 7.6 to 11.6. Assuming that fertility rates remain near their recent levels, the percentage 65 and over is projected to reach 14.3 by 2010 and 23.1 by 2040.(2) The economic consequences of "population aging" are a matter of general concern.

   Our purpose in this paper is to explore one important aspect of changes in age distribution, namely the way in which they alter input availability and output capacity, and hence average real income levels. Much of the literature on the economic effects of age distribution has been concerned with the expenditure side of the economy.(3) Our interest, on the other hand, is in long-run supply effects. We specify a multilevel aggregate production process, assign plausible values to its parameters, and obtain steady-state solutions under a range of alternative fertility assumptions. The central issue can be put as follows: Abstracting from all other considerations, does an economy with an "old" or a "young" population have a markedly different capacity for generating output and income per capita than one with a less extreme age distribution?

   There are two considerations at the heart of this issue. One is the relationship between age distribution and the ratio of available labor to population; that can be dealt with in a relatively straightforward manner. The other is the degree of substitutability or complementarity in production between different age groups in the labor force, and more generally between different age-sex groups. The latter consideration has been emphasized by Easterlin in his well known work on the relationship between fertility and the economy [7]. It has also been addressed econometrically by a number of authors [2; 9; 10; 13].(4) Unfortunately, the estimated elasticities of substitution that have been reported in the literature are somewhat difficult to compare, owing to differences in methods and definitions. Moreover, to the extent that comparisons are possible the estimates are found to vary appreciably. Some of the variation may be attributed to differences in the data used, but whatever the reason the result is a degree of uncertainty about how an economy actually functions with respect to the substitution or joint use of male and female workers from different age groups. For that reason, as well as others, we have broadened the basic question of how much age distribution matters to include the associated question of how much more or less it matters in differently functioning economies - in economies with different parameter configurations, that is.

   Our present interest in long-run, or steady state effects on output should not be taken to suggest that shorter-run effects are not important, and worth studying. Our focus reflects merely the particular aspect of the relationship between age distribution and macroeconomic performance that we have chosen to explore in this paper, and the particular model that we have found useful for that purpose. An adaptation of the model to deal with the output implications of actual or projected age distributions in the United States, Canada, or other countries over the next few decades might be a worthwhile line of investigation. However, that is for the future; our goal here is the more limited one.(5)

   We proceed as follows. In the next section we outline the framework of the analysis and describe the aggregate production model to be used. We go on to define the alternative demographic scenarios for which steady state solutions are to be obtained, and the procedures for assigning values to the parameters of the model. We then present and discuss the steady state solutions themselves. We summarize the findings in the final section of the paper.

2. A Model of Production

   We assume an economy that produces a single good using inputs of capital and labor. Labor input is classified according to tasks performed and the age and sex of the workers who perform them. Workers of different ages and sexes have different skills and are assigned to tasks in accordance with those skills. Within each age-sex group the total of labor services provided is determined by the size of the population, together with the rate of participation and the group's natural rate of unemployment. The production model thus has a hierarchical structure, with population at the base and aggregate production at the top.

   Aggregate Production

   Q = [([Delta][K.sup.-[Rho]] + (1 - [Delta])[L.sup.-[Rho]]).sup.-s/[Rho]]. (1)

   Q is output and K and L are aggregate inputs of capital and labor services. The function is a standard two-input CES function with allowance for possible nonconstant returns to scale (if s [not equal to] 1).

   Labor Input Aggregation: Level 1

   [Mathematical Expression Omitted].

   There are I individual tasks performed by workers and the labor associated with the different tasks is combined to obtain total labor input by the use of a CES aggregator function. Tasks are indexed by i = 1, 2, . . ., I. The [Theta] parameters sum to 1.

   Labor Input Aggregation: Level 2

   [Mathematical Expression Omitted].

   Workers are divided into J age-sex groups. [L.sub.ij] is the labor provided by age-sex group j for the carrying out of task i. The total amount of labor associated with task i is obtained by aggregating over all age-sex groups, again using a CES function. There are I such functions. The [Phi] parameters sum to 1.

   Task Assignment

   [L.sub.ij] = [[Xi].sub.ij][[Lambda].sub.ij][E.sub.j] (i = 1, 2, . . ., I; j = 1, 2, . . ., J). (4)

   [E.sub.j], the total amount of labor provided by age-sex group j, is allocated to tasks by a linear assignment function. [[Lambda].sub.ij] is the proportion of group-j labor assigned to task i and [[Xi].sub.ij] is a scaling factor that makes [L.sub.ij] equal to 1 in the demographic scenario chosen as reference case. The scaling represents simply a convenient choice of units. (Note that when [L.sub.ij] is equal to 1, L and [L.sub.i] are also equal to 1, in equations (2) and (3).) There are IJ individual assignment functions, incorporating IJ allocation parameters ([[Lambda].sub.ij]). The allocation parameters are intended to capture the idea that different age-sex groups have different combinations of skills. One might think of a typical member of the jth group as spending a certain fraction of his/her labor time on task 1, a certain fraction on task 2, and so on.

   Employment Determination

   [E.sub.j] = [e.sub.j][n.sub.j]N (j = 1, 2, . . ., J). (5)

   The amount of labor services supplied by the jth age-sex group is the product of the employment ratio ([e.sub.j]), the size of the total population (N), and the jth group's proportion of the total population ([n.sub.j]). The employment ratio can be written as [e.sub.j] = [p.sub.j](1 - [u.sub.j]), where [p.sub.j] is the group's rate of participation (in hours per year or some other appropriate measure) and [u.sub.j] is the rate of unemployment. We take [u.sub.j] to be the natural unemployment rate, and assume it constant across steady states. In principle, the rate of participation could vary with wage and fertility levels. However, to avoid complicating the model we assume [p.sub.j], and hence [e.sub.j], also to be constant across steady states. The argument in support of this assumption is provided in Denton, Mountain, and Spencer [6].

   Technical Progress

   We are interested in comparisons across states of the economy associated with alternative steady-state demographic scenarios. Assuming that technical progress is neutral, we take the level of progress to be the same in every state at the time of comparison. That...