Generic Analysis of Endogenous Growth Models

The neoclassical theory of economic growth, as formulated

by Solow

Solow (1956), and Cass Cass (1965)-Koopmans Koopmans (1965), attributes virtually all long-run growth to technological

progress. The level of technology is taken to be an exogenously

growing factor outside the model. So the baseline model of growth

says that growth is caused mostly by a factor that is not in the model.

It is important to understand why Solow and others made this

assumption. The answer is that models of perfect competition

are the simplest existing models of firm behavior, with

well-understood implications. But models with perfect competition require

constant returns to scale, and say that all factors of production are

paid their marginal product; and Euler’s Theorem says that the sum of the factor payments

exhausts all output:

Thus, the perfectly competitive firm has no money left over with which

to finance basic research, invent patentable technologies, or do anything

other than meet the payroll of production workers and pay the cost of

capital. Thus, no firm can afford to pay for technological research,

and there is therefore no alternative to the assumption that

technological progress occurs exogenously.

This unsatisfactory situation persisted for a long time, but a famous

paper by Paul Romer Romer (1986) led the way to the formulation

of a new generation of models that allow a role for investment in

knowledge to affect growth.

The Romer paper spawned a great deal of further theoretical work by

a host of others, but a penetrating

paper by Sergio Rebelo Rebelo (1991) provided a succinct summary

of the key feature of all of these models. This section summarizes

the Rebelo point as interpreted by Barro and Sala-i-Martin

Barro & Sala-i-Martin (1995).

1The Key Point¶

Rebelo’s key observation is as follows. Consider the class of models

with a Cobb-Douglas aggregate production function in capital and labor:

where no restriction is made on the $\labShare$ and $\kapShare$ coefficients.

(Recall that Solow, Cass, and Koopmans all assumed $\labShare+\kapShare=1$;

Rebelo is relaxing this restriction). Now suppose for simplicity that

in this economy saving is a constant proportion of gross income.

In continuous time, the growth of the capital stock is given by

Suppose further that population growth is constant at

and as usual define per-capita variables as the aggregate normalized

by population, e.g. $k_t = K_t/L_t$. Then the aggregate per-capita accumulation

equation can be rewritten

2Characteristics of the Steady State¶

If this model has a steady-state growth rate, that rate must

satisfy

for some constant $\gamma$. (The value of $A$ does not affect the conclusions

from here on, and so we will assume without loss of generality that $A=1$). From (5), this implies

that

Take the time derivative of this equation to obtain

Using this equation, we can construct a complete catalog of the

possible circumstances under which steady-state growth $\gamma$ can

be different from zero endogenously.

2.1Possibilities for Steady State Growth¶

1. $\labShare+\kapShare = 1$ (Constant Returns Models)

   1. $\{\labShare, \kapShare\} < 1 \rightarrow \gamma = 0$ (Solow case)

2.  {math}`\labShare = 0, \kapShare = 1`: Rebelo {cite:t}`rebelo:long` {math}`AK` growth model

2. $\labShare+\kapShare > 1$ (Increasing Returns Models)

   1. $\{\labShare, \kapShare\} < 1$

      1. $\popGro > 0 \rightarrow \gamma = \overbrace{\left(\frac{\labShare+\kapShare-1}{(1-\kapShare)}\right)}^{>0} \popGro$

    2.  {math}`\popGro = 0 \rightarrow \gamma = 0`

2.  {math}`\labShare>0, \kapShare = 1`

    1.  {math}`\popGro>0 \rightarrow 0 = \popGro(\labShare+\kapShare-1)` which is not satisfied for any {math}`\gamma`

    2.  {math}`\popGro=0 \rightarrow 0 = 0` which can be satisfied for any {math}`\gamma`

So whatever the details of endogenous growth models may be, in the end

any model that generates perpetual growth without exogenous

technological progress must be mathematically reducible to a form like

that of either 1.b., 2.a.i., or 2.b.ii.

It is worth delving a bit further into why the Solow case cannot

generate perpetual growth. The answer can be understood using

the Solow growth accounting framework, which says that there are only three sources

of long-run growth: technology, labor, and capital. Thus, there are only

two potential sources of growth of output per unit of labor: Technology

and an increase in the capital/labor ratio. But if $\kapShare < 1$,

the gross marginal product of capital approaches zero as the capital/labor

ratio approaches infinity; subtracting out depreciation, eventually the

net marginal product of capital becomes negative. Thus, capital accumulation

can sustain growth only so long.

The bottom line is that there are only two configurations of the model

that are capable of generating perpetual growth in a way that makes any

sense: 1.b. (the Rebelo $AK$ model) and 2.a.ii.

What this means is that any model that aims to permit perpetual

long-run growth must ultimately boil down to a structure in which

there are constant returns to scale for some set of factors of

production that can jointly be accumulated forever.