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McCallum, Bennett T.. "Neoclassical vs. endogenous growth analysis: an overview." Economic Quarterly. Federal Reserve Bank of Richmond. 1996. HighBeam Research. 22 Apr. 2018 <https://www.highbeam.com>.
McCallum, Bennett T.. "Neoclassical vs. endogenous growth analysis: an overview." Economic Quarterly. 1996. HighBeam Research. (April 22, 2018). https://www.highbeam.com/doc/1G1-19175386.html
McCallum, Bennett T.. "Neoclassical vs. endogenous growth analysis: an overview." Economic Quarterly. Federal Reserve Bank of Richmond. 1996. Retrieved April 22, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-19175386.html
After a long period of quiescence, growth economics has in the last decade (1986-1995) become an extremely active area of research - both theoretical and empirical.(1) To appreciate recent developments and understand associated controversies, it is necessary to place them in context, i.e., in relation to the corpus of growth theory that existed prior to this current burst of activity. This article's exposition will begin, then, by reviewing in Sections 1-4 the neoclassical growth model that prevailed as of 1985. Once that has been accomplished, in Section 5 we shall compare some crucial implications of the neoclassical model with empirical evidence. After tentatively concluding that the neoclassical setup is unsatisfactory in several important respects, we shall then briefly describe a family of "endogenous growth" models and consider controversies regarding these two classes of theories. Much of this exposition, which is presented in Sections 6-8, will be conducted in the context of a special-case example that permits an exact analytical solution so that explicit comparisons can be made. Finally, some overall conclusions are tentatively put forth in Section 9. These conclusions, it can be said in advance, are broadly supportive of the endogenous growth approach. Although the article contends that this approach does not strictly justify the conversion of "level effects" into "rate of growth effects," which some writers take to be the hallmark of endogenous growth theory, it finds that the quantitative predictions of such a conversion may provide good approximations to those strictly implied.
1. BASIC NEOCLASSICAL SETUP
Consider an economy populated by a large (but constant) number of separate households, each of which seeks at an arbitrary time denoted t = 1 to maximize
u([c.sub.1]) + [Beta]u([c.sub.2]) + [[Beta].sup.2]u([c.sub.3]) + . . ., (1)
where [c.sub.t] is the per capita consumption of a typical household member during period t and where [Beta] = 1/(1 + [Rho]) with [Rho] [greater than] 0 the rate of time preference. The instantaneous utility function u is assumed to be well behaved, i.e., to have the properties u[prime] [greater than] 0, u[double prime] [less than] 0, u[prime](0) = [infinity], u[prime]([infinity]) = 0. The analysis would not be appreciably altered if leisure time were included as a second argument, but to keep matters simple, leisure will not be recognized in what follows. Instead, it will be presumed that each household member inelastically supplies one unit of labor each period.
It is assumed that the number of individuals in each household grows at the rate [Nu]; thus each period the number of members is 1 + [Nu] times the number of the previous period. In light of this population growth, some analysts postulate a household utility function that weights each period's u([c.sub.t]) value by the number of household members, a specification that is effected by setting [Psi] = 1 in the following more general expression:
u([c.sub.1]) + [(1 + [Nu]).sup.[Psi]] [Beta]u([c.sub.2]) + [(1 + [Nu]).sup.2[Psi]] [[Beta].sup.2]u([c.sub.3]) + . . . . (1[prime])
With [Psi] = 0, expression (1[prime]) reduces to (1) whereas [Psi] values between 0 and 1 provide intermediate assumptions about this aspect of the setup. Most of what follows will presume [Psi] = 0, but the more general formulation (1[prime]) will be referred to occasionally.
Each household operates a production facility with input-output possibilities described by a production function [Y.sub.t] = F([K.sub.t], [N.sub.t]), where [N.sub.t] and [K.sub.t] are the household's quantities of labor and capital inputs with [Y.sub.t] denoting output during t. The function F is presumed to be homogeneous of degree one so, by letting [y.sub.t] and [k.sub.t] denote per capita values of [Y.sub.t] and [K.sub.t], we can write
[y.sub.t] = f([k.sub.t]), (2)
where f([k.sub.t]) [equivalent to] F([k.sub.t], 1). It is assumed that f is well behaved (as defined above).
Letting [v.sub.t] denote the per capita value of (lump-sum) government transfers (so - [v.sub.t] = net taxes), the household's budget constraint for period t can be written in per capita terms as
f([k.sub.t]) + [v.sub.t] = [c.sub.t] + (1 + [Nu]) [k.sub.t+1] - (1 - [Delta])[k.sub.t]. (3)
Here [Delta] is the rate of depreciation of capital. As of time 1, then, the household chooses values of [c.sub.1], [c.sub.2], . . . and [k.sub.2], [k.sub.3], . . . to maximize (1) subject to (3) and the given value of [k.sub.1]. The first-order condition necessary for optimality can easily be shown to be
(1 + [Nu])u[prime]([c.sub.t]) = [Beta]u[prime]([c.sub.t+1])[f[prime]([k.sub.t+1] + 1 - [Delta]], (4)
and the relevant transversality condition is(2)
[Mathematical Expression Omitted]. (5)
The latter provides the additional side condition needed, since only one initial condition is present, for (3) and (4) to determine a unique time path for [c.sub.t] and [k.sub.t]+1. Satisfaction of conditions (3), (4), and (5) is necessary and sufficient for household optimality.(3)
To describe this economy's competitive equilibrium, we assume that all households are alike so that the behavior of each is given by (3), (4), and (5).(4) The government consumes output during t in the amount [g.sub.t] (per person), the value of which is determined exogenously. For some purposes one might want to permit government borrowing, but here we assume a balanced budget. Expressing that condition in per capita terms, we have
[g.sub.t] + [v.sub.t] = 0. (6)
For general competitive equilibrium (CE), then, the time paths of [c.sub.t], [k.sub.t], and [v.sub.t] are given by (3), (4), and (6), plus the transversality condition (5). In most of what follows, it will be assumed that [g.sub.t] = [v.sub.t] = 0, in which case the CE values of [c.sub.t] and [k.sub.t] are given by (4) and
f([k.sub.t]) = [c.sub.t] + (1 + [Nu]) [k.sub.t+1] - (1 - [Delta])[k.sub.t], (7)
provided that they satisfy (5).
Much interest centers on CE paths that are steady states, i.e., paths along which every variable grows at some constant rate.(5) It can be shown that in the present setup, with no technical progress, any steady state is characterized by stationary (i.e., constant) values of [c.sub.t] and [k.sub.t].(6) (These constant values imply growth of economy-wide aggregates at the rate [Nu], of course.) Thus from (4) we see that the CE steady state is characterized by f[prime](k) + 1 - [Delta] = (1 + [Nu])(1 + [Rho]) or
f[prime](k) - [Delta] = [Nu] + [Rho] + [Nu][Rho]. (8)
This says that the net marginal product of capital is approximately (i.e., neglecting the interaction term [Nu][Rho]) equal to [Nu] + [Rho], a condition that should be kept in mind. If the more general utility function (1[prime]) is adopted, the corresponding result is f[prime](k) + 1 - [Delta] = (1 + [Rho])[(1 + [Nu]).sup.1 - [Psi]]. Thus with [Psi] = 1, i.e., when household utility is u(c) times household size, we have f[prime](k) - [Delta] = [Rho].
It can be shown that, in the model at hand, the CE path approaches the CE steady state as time passes. Given an arbitrary [k.sub.1], in other words, [k.sub.t] approaches the value [k.sup.*] that satisfies (8) as t [right arrow] [infinity]. This result can be clearly and easily illustrated in the special case in which u([c.sub.t]) = log [c.sub.t], [Mathematical Expression Omitted], and [Delta] = 1.(7) (Below we shall refer to these as the "LCD assumptions," L standing for log and CD standing for both Cobb-Douglas and complete depreciation.) In this case, equations (4) and (7) become
[Mathematical Expression Omitted] (9)
and
[Mathematical Expression Omitted]. (10)
Since the value of [Mathematical Expression Omitted] summarizes the state of the economy at time t, it is a reasonable conjecture that [k.sub.t+1] and [c.sub.t] will each be proportional to [Mathematical Expression Omitted]. Substitution into (9) and (10) shows that this guess is correct and that the constants of proportionality are such that [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. These solutions in fact satisfy the transversality condition (TC) given by (5), so they define the CE path. The [k.sub.t] solution can then be expressed in terms of the first-order linear difference equation
log [k.sub.t+1] = log[[Alpha][Beta]A/(1 + [Nu])] + [Alpha] log[k.sub.t], (11)
which can be seen to be dynamically stable since [absolute value of [Alpha]] [less than] 1. Thus log[k.sub.t] converges to [(1 - [Alpha]).sup.-1] log[[Alpha][Beta]A/(1 + [Nu])]. For reference below, we note that subtraction of log [k.sub.t] from each side of (11) yields
log[k.sub.t+1] - log[k.sub.t] = (1 - [Alpha])[log[k.sup.*] - log[k.sub.t]], (12)
where [k.sup.*] = [[Alpha][Beta]A/[(1 + [Nu])].sup.1/(1-[Alpha])], so 1 - [Alpha] is in this special case a measure of the speed of convergence of [k.sub.t] to [k.sup.*].
It might be thought that the complete-depreciation assumption [Delta] = 1 renders this special case unusable for practical or empirical analysis. But such a conclusion is not inevitable. What is needed for useful application, evidently, is to interpret the model's time periods as pertaining to a span of calendar time long enough to make [Delta] = 1 a plausible specification - say, 25 or 30 years. Then the parameters A, [Beta], and [Nu] must be interpreted in a corresponding manner. Suppose, for example, that the model's time period is 30 years in length. Thus if a value of 0.98 was believed to be appropriate for the discount factor with a period length of one year, the appropriate value for [Beta] with 30-year periods would be [Beta] = (0.98)(30) = 0.545. Similarly, if the population growth parameter is believed to be about one percent on an annual basis, then we would have 1 + [Nu] = (1.01)(30) = 1.348. Also, a realistic value for A would be about 10k(1-[Alpha]), since it makes k/y = 3/30 = 0.1. So the LCD assumptions could apparently be considered for realistic analysis, provided that one's interest is in long-term rather than cyclical issues.(8)
2. TECHNICAL PROGRESS
Since the foregoing model approaches a steady state in which per capita values are constant over time, it may seem to be a strange framework for the purpose of growth analysis. But in the neoclassical tradition, growth in per capita values is provided by assuming that steady technical progress occurs, continually shifting the production frontier as time passes. With technical progress proceeding at the rate [Gamma], the production function would in general be written as [Y.sub.t] = F([K.sub.t], [N.sub.t], [(1 + [Gamma]).sup.t]).(9) It transpires, however, that steady-state growth is only possible when technical progress occurs in a "labor-augmenting" fashion, i.e., when
[Y.sub.t] = F([K.sub.t], [(1 + [Gamma]).sup.t][N.sub.t]).(10) (13)
But then with F homogeneous of degree one, we have
[Mathematical Expression Omitted], (14)
where [y.sub.t] [equivalent to] [Y.sub.t]/[(1 + [Gamma]).sup.t][N.sub.t] and [Mathematical Expression Omitted] are values of output and capital per "efficiency unit" of labor. Alternatively, [y.sub.t] = [y.sub.1][(1 + [Gamma]).sup.t]f([k.sub.t]/[k.sub.1][(1 + [Gamma]).sup.t]). The household's budget constraint, when expressed in terms of these variables (and with [v.sub.t] [equivalent to] 0), becomes
[Mathematical Expression Omitted]. (15)
Maximizing (1) subject to (15) gives rise to the following first-order condition, analogous to (4):
[Mathematical Expression Omitted]. (16)
In addition, we have the transversality condition
[Mathematical Expression Omitted]. (17)
Since there are no additional equilibrium conditions, presuming that [g.sub.t] = [v.sub.t] = 0, competitive equilibrium time paths of [c.sub.t] and [k.sub.t] are determined by (15) and (16), given the initial value of k, and the limiting condition (17).
Now, in order for steady growth of both [c.sub.t] and u[prime]([c.sub.t]) to be possible, it will be assumed that agents' preferences are such that the function u[prime]([c.sub.t]) has a constant elasticity.(11) For reasons of symmetry, the function is usually written as
[Mathematical Expression Omitted], (18)
which has an elasticity of marginal utility of -[Theta] and reduces to u([c.sub.t]) = log [c.sub.t] in the special limiting case in which [Theta] [approaches] 1.(12) Using (18), then, we rewrite (16) as
[Mathematical Expression Omitted]. (19)
Finally, we define [Mathematical Expression Omitted], which implies that [Mathematical Expression Omitted], so we can rewrite (19) once more as
[Mathematical Expression Omitted]. (20)
The latter shows that [k.sub.t] will be constant in the CE steady state, and (15) then implies that the same will be true for [Mathematical Expression Omitted]. Thus we see that the per capita variables [k.sub.t], [c.sub.t], and [y.sub.t] will all grow at the rate [Gamma]. Thus equation (20) shows that the (constant) value of [Mathematical Expression Omitted], which equals the marginal product of capital in unadjusted units, will satisfy
[Mathematical Expression Omitted]. (21)
We can approximate [(1 + [Gamma]).sup.[Theta]] with 1 + [Gamma][Theta], assuming [Gamma] is small in relation to 1. …
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