THE SOLOW MOOEL 21
that there is no international trade in the model because there is only
a single good: I'll give you a 1941 Joe DiMaggio autograph in exchange
Charles Jones for ... your 1941 Joe DiMaggio autograph? Another assumption of the
model is that technology is exogenous- that is, the technology avail-
Introduction To Economic Growth 2nd Edition able to firms in this simple world is unaffected by the actions of the
firms, including research and development (R&D). These are assump-
tions that we will relax later on, but for the moment, and for Solow, they
Chapter 2 serve well. Much progress in economics has been made by creating a
very simple world and then seeing how it behaves and misbehaves.
The Solow Model Before presenting the Solow model, it is worth stepping back to con-
sider exactly what a model is and what it is for. In modern economics, a
model is a mathematical representation of some aspect of the economy.
It is easiest to think of models as toy economies populated by robots. We
specify exactly how the robots behave, which is typically to maximize
All theory depends on assumptions which are not quite their own utility. We also specify the constraints the robots face in seek-
true. That is what makes it theory. The art of success- ing to maximize their utility. For example, the robots that populate our
ful theorizing is to make the inevitable simplifying economy may want to consume as much output as possible, but they are
assumptions in such a way that the final results are .
limited in how much output they can produce by the techniques at their
not very sensitive. -ROBERT SOLOW (1956), P. 65.
disposal. The best models are often very simple but convey enormous
insight into how the world works. Consider the supply and demand
n 1956, Robert Solow published a seminal paper on economic framework in microeconomics. This basic tool is remarkably effective
growth and development titled "A Contribution to the Theory of Eco- at predicting how the prices and quantities of goods as diverse as health
nomic Growth." For this work and for his subsequent contributions to care, computers, and nuclear weapons will respond to changes in the
our understanding of economic growth, Solow was awarded the Nobel economic environment.
Prize in economics in 1987. In this chapter, we develop the model pro- With this understanding of how and why economists develop mod-
posed by Solow and explore its ability to explain the stylized facts of els, we pause to highlight one of the important assumptions we will
growth and development discussed in Chapter 1. As we will see, this make until the final chapters of this book. Instead of writing down util-
model provides an important cornerstone for understanding why some ity functions that the robots in our economy maximize, we will sum-
countries flourish while others are impoverished. marize the results of utility maximization with elementary rules that
Following the advice of Solow in the quotation above, we will make the robots obey. For example, a common problem in economics is for
several assumptions that may seem to be heroic. Nevertheless, we hope an individual to decide how much to consume today and how much to
that these are simplifying assumptions in that, for the purposes at hand, save for consumption in the future. Another is for individuals to decide
they do not terribly distort the picture of the world we create. For ex- how much time to spend going to school to accumulate skills and how
ample, the world we consider in this chapter will consist of countries much time to spend working in the labor market. Instead of writing
that produce and consume only a single, homogeneous good (output). these problems down formally, we will assume that individuals save a
Conceptually, as well as for testing the model using empirical data, it is constant fraction of their income and spend a constant fraction of their
convenient to think of this output as units of a country's gross domes- time accumulating skills. These are extremely useful simplifications;
tic product, or GDP. One implication of this simplifying assumption is without them, the models are difficult to solve without more advanced
20
, 22 2 THE SOLOW MODEL THE BASIC SOLOW MODEL 23
mathematical techniques. For many purposes, these are fine assump- rent capital until the marginal product of capital is equal to the rental
tions to make in our first pass at understanding economic growth. Rest price:
assured, however, that we will relax these assumptions in ｾｨ｡ｰｴ･ｲ＠ 7.
aF Y
w = aL = (1- a)y,
fj ｾＱ＠
t..• ｾ＠ THE BASIC SOLOW MODEL i!F y
r =-=a-
aK K.
The Solow model is built around two equations, a production function
Notice that wL + rK = Y. That is, payments to the inputs ("factor
and a capital accumulation equation. The production function describes
payments") completely exhaust the value of output produced so that
how inputs such as bulldozers, semiconductors, engineers, and steel-
there are no economic profits to be earned. This important result is a
workers combine to produce output. To simplify the model, we group
general property of production functions with constant returns to scale.
these inputs into two categories, capital, K, and labor, L, and denote out-
Notice also that the share of output paid to labor is wL/Y = 1 - a and
put as Y. The production function is assumed to have the Cobb-Douglas
the share paid to capital is rK/Y = a. These factor shares are therefore
form and is given by
constant over time, consistent with Fact 5 from Chapter 1.
y = F(K,L) = Kau-a. (2.1) Recall from Chapter 1 that the stylized facts we are typically in-
terested in explaining involve output per worker or per capita output.
where a is some number between 0 and 1. 1 Notice that this produc- With this interest in mind, we can rewrite the production function in
tion function exhibits constant returns to scale: if all of the inputs are equation (2. 1) in terms of output per worker, y = Y /L, and capital per
doubled, output will exactly double. 2 • =
worker, k K/L:
Firms in this economy pay workers a wage, w, for each umt of
labor and pay r in order to rent a unit of capital for one period. We (2.2)
assume there are a large number of firms in the economy so that perfect
competition prevails and the firms are price-takers. 3 Normalizing the This production function is graphed in Figure 2.1. With more capital
price of output in our economy to unity, profit-maximizing firms solve per worker, firms produce more output per worker. However, there are
the following problem: diminishing returns to capital per worker: each additional unit of capital
we give to a single worker increases the output of that worker by less
maxF(K,L)- rK- wL. and less.
K,L
The second key equation of the Solow model is an equation that
According to the first-order conditions for this problem, firms will ｨｩｾ･＠
describes how capital accumulates. The capital accumulation equation
labor until the marginal product of labor is equal to the wage and w1ll is given by
1Charles Cobb and Paul Douglas (1928] ｰｲｯｾ［＾ｳ･､＠ this functional form in their analysis of k = sY- dK. (2.3)
u.s. manufacturing. Interestingly, they argued that this production function, with a value
for a of 1/4, fit the data very well without allowing for technological progress. . This kind of equation will be used throughout this book and is very
zRecall that if F(aK, aL] = aY for any number a > 1, then we say that the productwn
important, so let's pause a moment to explain carefully what this equa-
function exhibits constant returns to scale. If F(aK, aL] > a Y, then the production func-
tion exhibits increasing returns to scale, and if the inequality is reversed the production tion says. According to this equation, the change in the capital stock,
function exhibits decreasing returns to scale. k-;-is equal to the amount of gross investment, s Y, less the amount of
.3You may recall from microeconomics that with constant returns to scale the number of depreciation that occurs during the production process, dK. We'll ;now
firms is indeterminate-i.e., not pinned down by the model. discuss these three terms in more detail. 1. I
, 24 2 THE SOLOW MODEL THE BASIC SOLOW MODEL 25
.1 A COBB-DOUGLAS PRODUCTION example, we often assume d = .05, so that 5 percent of the machines
FUNCTION and factories in our model economy wear out each year.
To study the evolution of output per person in this economy, we
rewrite the capital accumulation equation in terms of capital per person.
Then the production function in equation (2.2) will tell us the amount
of output per person produced for whatever capital stock per person
is present in the economy. This rewriting is most easily accomplished
by using a simple mathematical trick that is often used in the study of
growth. The mathematical trick is to "take logs and then derivatives"
(see Appendix A for further discussion). Two examples of this trick are
given below.
Example 1:
k = K/L ===? logk = logK -logL
k k
===}-=---
t
k K L.
k
Example 2:
The term on the left-hand side of equation (2.3) is the continuous
y = k" ===?logy= a logk
time version of Kt+ 1 - K1 , that is, the change in the capital stock per . ic
"period." We use the "dot" notation 4 to denote a derivative with respect ===? ｾ＠ = ak.
to time:
Applying Example 1 to equation (2.3) will allow us to rewrite the
. dK capital accumulation equation in terms of capital per worker. But before
K=(it·
we proceed, let's first consider the growth rate of the labor force, L/L.
The second term of equation (2.3) represents gross investment. Fol- An important assumption that will be maintained throughout most of
lowing Solow, we assume that workers/consumers save a constant frac- this book is that the labor force participation rate is constant and that
tion, s, of their combined wage and rental income, Y = wL + rK. The the population growth rate is given by the parameter n. 5 This implies
economy is closed, so that saving equals investment, and the only use that the labor force growth rate, L/L, is also given by n. If n = .01, then
of investment in this economy is to accumulate capital. The consumers the population and the labor force are growing at one percent per year.
then rent this capital to firms for use in production, as discussed above. This exponential growth can be seen from the relationship
The third term of equation (2.3) reflects the depreciation of the cap-
ital stock that occurs during production. The standard functional form
used here implies that a constant fraction, d, of the capital stock depre- Take ｾｯｧｳ＠ and differentiate this equation, and what do you get?
ciates every period (regardless of how much output is produced). For
5 0ften, it is convenient in describing the model to assume that the labor force participation
4 Appendix A discusses the meaning of this notation in more detail. rate is unity-i.e., every member of the population is also a worker.
