Lee Ohanian
UCLA
Macroeconomic Analysis
1 Notes on the Solow-Swan (Neoclassical) Growth
Model
In 1961, President John F. Kennedy asked his chief economic adviser, Professor
Robert Solow of M.I.T., whether the Soviet Union would catch up to U.S. income
levels, as Kennedy and his Secretary of Defense, Robert McNamara, feared that
if the U.S.S.R. had enough economic resources, then they may have the ability
to overwhelm the U.S. with military power. Note that this was the time of
the “cold war”, in which the U.S. and the U.S.S.R. were both building military
strength.
Solow assured President Kennedy that this would not happen – and Solow
was right. How did he know this? By the end of these notes, we will be able to
understand why Solow was so sure that the U.S.S.R. would not catch up to the
U.S. Solow used his recently minted growth model to figure this out.
1.1 Solow Model with Constant Level of Technology and
a Constant Population
The Solow growth model, sometimes called the neoclassical growth model has 4
equations, a constant returns to scale production function, an equation for con-
sumption, an equation for investment, and an equation that shows how the stock
of capital changes over time as a consequence of investment and depreciation.
We will assume for the time being that everyone in the economy works, and
the population size is fixed.
The savings rate, which is the fraction of income saved, is given by s, which
is between 0 and 1, and the depreciation
rate of capital, which is given by δ, and is also between 0 and 1:
0 < s ≤ 1 (1) 0 < δ ≤ 1 (2) The production function, and the division of output between consumption and investment, which is also equal to savings in this economy, is given by: F (Kt, ALt) = Yt = Ct + It (3) 1 张熠华� 张熠华� 张熠华� 张熠华� The production function, F (K,AL), takes the inputs capital (K) and labor (L), and the state of technology, which is represented by A, and produces output. Think of the production function as a recipe for taking inputs, and making output. It is convenient to represent the economy’s state of technology as a number that multiplies the number of workers: A ∗ L. This is called labor- augmenting technology, because it explicitly makes workers more productive. For now, we assume that the state of technology, A, is fixed over time. For now, we will also assume that the population is constant over time. We will discuss technological change and population growth later. Investment is used to increase the economy’s capital stock. Investment is equal to the savings rate multiplied by output: It = sYt (4) Consumption is therefore given by: Ct = (1− s)Yt (5) The relationship between the capital stock and investment is given by: Kt+1 = It + (1− δ)Kt (6) Specifically, the amount of capital in the economy for the next period is equal to today’s level of investment, I and the capital in place today that is left over after depreciation, (1− δ)K. We can also write this equation as representing how the capital stock changes between dates t and t+1. Using the Greek capital letter ∆ to denote a change in a variable, we have: ∆Kt+1 = It − δKt (7) These are all the equations in the simplest version of the Solow model. The production function is constant returns to scale, which means that a proportional change in both the capital and labor inputs changes output by the same proportion. If we call the proportion factor x, then this means: xY = F (xK, xAL) (8) Note that if F was decreasing returns, then doubling the inputs would less than double output. For increasing returns, doubling the inputs would more than double output. However, we will assume typically in this course that the production function is constant returns to scale. Now, let’s divide the inputs by AL. We call this the production function in intensive form. y = Y AL = F ( K AL , 1) = F (k, 1) = f(k) (9) Specifically, we call lower case variables the intensive form variables, and we will call f(k) = F (k, 1). Next, suppose that A = 1. Then y is per person output, 2 also called per capita output. This is how economists typically measure living standards within a country. Note also that k is per person capital stock. We assume that the production function is characterized by diminishing marginal product of capital. Specifically, this means that the marginal product of capital is very high when the capital stock is small, and the marginal product is low when the capital stock is large. We represent this as d 2f dk2 < 0. The figure on the website shows a graph of the production function. You can see that the slope of the production function (note that this slope is the marginal product of capital) is very steep when the capital stock is low, and then the slope is much flatter for large levels of the capital stock. The model equations for the intensive form are given by: yt = f(kt) = ct + it (10) it = syt, ct = (1− s)yt (11) kt+1 = it + (1− δ)kt (12) We can say more about the production function. Specifically, economists often use the Cobb-Douglas production function in studying macroeconomics. The Cobb-Douglas function is given as: Y = Kα(AL)1−α (13) One reason we use this function is because it has the property that the share of income paid to the two factors of production, capital and labor, is constant over time. This coincides fairly well with actual data, which shows fairly stable income shares paid to capital and labor. To see that the income shares are constant over time, suppose that the prices of the factors were equal to their marginal products. You may recall this will be the case in a competitive economy. Denoting w as the wage rate in the economy, then the share, or the percent, of income paid to labor will be equal to: wL Y (14) For the Cobb-Douglas function, this becomes: MPL ∗ L Y = (1− α)Kα(A1−α)L−α ∗ L Kα(AL)1−α (15) Note that this simplifies to the term (1− α). As an exercise, show that the share of income paid to capital, when the factor price of capital is equal to its marginal product, is equal to α. Empirically, we can use the NIPA accounts to measure the size of α, which is around 1/3 for many countries. This means that our production function is given by: 3 张熠华� 张熠华� 张熠华� 张熠华� 张熠华� 张熠华� 张熠华� 张熠华� 张熠华� Y = K1/3(AL)2/3 (16) When we represent the production function in intensive form, we get: y = k1/3 (17) We are now in a position to describe the dynamics of this model. Note that this economy will grow as long as investment, i is bigger than depreciation, δk. Therefore, growth will occur for all levels of k such that the following holds: i = sk1/3 > δk (18)
Graphically, this can be seen in the second figure on the website, which graphs
depreciation, δk and investment, sk1/3 on the vertical axis, while k is on the
horizontal axis. Note that in this example, both technology and labor are con-
stant.
Note that in this graph there is a region in which the capital stock grows
over time, which is for all k < k∗ Notice that in this case, the investment curve
lies above depreciation. There is also a region in which the capital stock shrinks
over time, which is for all k > k∗. Note that in this case, the investment curve
lies below depreciation. When k = k∗, we call this the steady state capital stock.
At this level of capital, the economy does not change over time. Rather, it is an
economy in which, consumption, investment, output, and the capital stock are
all constant over time. We call these levels of consumption, investment, output,
and the capital stock as their steady state levels. The steady state occurs when
investment offsets depreciation.
We can solve for the steady state level of the capital stock, k∗, as a function
of model parameters. At the steady state, investment equals depreciation:
sk∗1/3 = δk∗ (19)
Re-arranging terms, we get:
k∗ =
(s
δ
) 3
2
(20)
Note that if we replace the exponent 1/3 with α, then we obtain a formula
for the steady state capital stock given by:
k∗ =
(s
δ
) 1
1−α
(21)
Note that the steady state capital stock is increasing in the savings rate, and
is decreasing in the depreciation rate. Moreover, this means that an economy
can’t permanently grow just by accumulating physical capital. The model in-
dicates that the economy can grow for a finite period of time, which is when
4
k < k∗, but eventually depreciation will become suffi ciently large that invest-
ment will only be able to offset depreciation. Note that this steady state result
holds for any savings rate, including the extreme case in which the savings rate
is 100 percent. As an exercise, verify that this is the case.
The result of the steady state in the Solow model is the reason that Solow
could tell President Kennedy with such certainty that the Soviet Union level of
income would not catch up to the U.S. level. This is because it was well known
that Soviet growth was being driven by a very high level of capital accumulation.
Solow’s model tells us that capital accumulation will not be a perpetual source
of economic growth. Solow was right, as most estimates of the Soviet economy
show that its growth roughly stopped sometime in the 1980s. By around 1990,
the Soviet Union broke up into independent countries.
The Solow model is the simplest modern model of growth. Note that the
concept of a steady state is central in more recent growth models. We will
continue to use this concept throughout the course, including its use in models
with technological change. We will see that in the case of economies with tech-
nological change, the steady state result becomes what we call a steady state
growth path.
1.2 Solow Model with Population Growth
To introduce population growth into this model, suppose that the population
(and the labor force) grows at the constant rate n :
Lt+1 = (1 + n)Lt (22)
There are no other changes relative to the model in the previous section.
Next, we put the model into intensive form as before, in which we divide all
variables by ALt, which yields as before the variables yt, ct,it, kt. However, note
what happens to Kt+1 when we divide by ALt :
Kt+1
ALt
=
Kt+1
ALt
ALt+1
ALt+1
= kt+1(1 + n) (23)
kt+1(1 + n) = it + (1− δ)kt (24)
Now, subtract nkt+1 from both sides of the equation. Further, suppose that
kt+1 is not too different from kt. Then the above equation is approximately
given by:
∆kt+1 ≈ it − (δ + n)kt (25)
The main point is that with this approximation, the Solow model with pop-
ulation growth can be studied exactly as before, but with a higher depreciation
rate. Keep in mind that this is not physical depreciation, Rather, population
growth means that the capital carried over from period t to period t + 1 must
be distributed among more people. Thus, for a given amount of capital that the
economy carries forward from period t to period t + 1 (it + (1− δ)kt), there is
5
张熠华�
张熠华�
张熠华�
张熠华�
less capital per worker tomorrow, relative to today, because of the population
growth.
Thus, the Solow model with population growth also generates a steady state
level of output, consumption, investment and capital per person. Note that the
overall level of these variables grows at the same rate as population growth once
the economy is in its per-capita steady state. In this case, total output is given
by Yt = y∗ALt. Since we are in a steady state, per capita output, y, is constant
at the level y∗, and Y grows at the same rate as the population.
1.3 Solow Model with Technological Change
Now, we will complete the discussion of the Solow model by adding technological
change. Specifically, let’s allow the technological term A to grow over time.
Assume that this growth is given by:
At+1 = (1 + g)At (26)
A reasonable empirical value of g is in the range of .015-.02, which means
that the technological frontier grows at about 1.5 to 2 percent per year. We will
discuss a bit later how we measure the technology parameter A. For now, we
focus on extending the Solow model to incorporate perpetually growing techno-
logical change.
The per-capita variables in this model will be characterized by a steady state
growth path, in which the variables y, c, i, and k will grow at a constant rate,
as opposed to a steady state, in which the per-capita variables stop growing.
To analyze the Solow model with technological change, we proceed in exactly
the same way as in the original case of the model in section 1.1. Let’s assume
that there is no population growth to keep the model as simple as possible.
We form the intensive form of the variables as before, by dividing each of
the variables by AtL. The only difference is that the technology parameter A
has a time subscript.
There are no other changes relative to the model in the previous section.
Next, we put the model into intensive form as before, in which we divide all
variables by AtL, which yields as before the variables yt, ct,it, kt. However, note
what happens to Kt+1 when we divide by AtL :
Kt+1
AtL
=
Kt+1
AtL
At+1L
At+1L
= kt+1(1 + g) (27)
kt+1(1 + g) = it + (1− δ)kt (28)
Now, subtract gkt+1 from both sides of the equation. Further, suppose that
kt+1 is not too different from kt. Then the above equation is approximately
given by:
∆kt+1 ≈ it − (δ + g)kt (29)
6
张熠华�
张熠华�
张熠华�
张熠华�
As in the case of the model with population growth, we can analyze the
intensive form variables as before, with a higher deprecation rate. We interpret
the parameter g as follows. Think of A as creating more units of labor input.
That is, effective labor input is AtL, as technology makes workers more produc-
tive. It is as if there were more workers in the economy. Thus, when we carry
capital forward from period t to period t + 1, there is less capital per effective
worker, because the term At+1 > At.
Note that a steady state will exist for the model in intensive form, just as
before. We can construct per-capita variables by multiplying the intensive form
variables by At. Note that when the intensive variables are in steady state, the
per-capita variables will all grow forever at rate g.We call the case in which the
intensive form variables are constant, and the per-capita variables grow at the
constant rate of technology, is a steady state growth path, or a balanced growth
path.
1.4 There are many Steady States – Which One is Best?
In the Solow model, there is a steady state for every savings rate, 0 < s ≤ 1. What savings rate should society choose? At some level, the Solow model is not really set up to answer this question, because it does not include a utility function. Rather, the savings rate is a parameter, and not a choice variable for the consumers. The optimal growth model, which we will study afterwards, will provide a specific answer to this question, as it will have a utility function in which consumers optimally make savings and consumption decisions. However, some economists have addressed the issue of choosing a savings rate within the Solow model by focusing on what is known as the Golden Rule. The Golden Rule savings rate is the one that maximizes steady state consumption. We can solve for the Golden Rule savings rate in the simplest Solow model (no population growth and no technological change) by maximizing steady state consumption. This is given as: max{f(k∗)− δk∗} (30) This yields the first order condition: f ′(k∗) = δ (31) We therefore choose the savings rate, s, such that the marginal product of capital is equal to the depreciation rate. To solve for the savings rate that delivers this level of the capital stock, note that: sf(k∗) = δk∗ (32) Since the Golden Rule requires that MPK = δ then we have: s = f ′(k∗)k∗ f(k∗) (33) 7 张熠华� 张熠华� 张熠华� 张熠华� Noting that the marginal product of capital will be the competitive price of capital services, then we have that the Golden Rule savings rate is equal to capital’s share of income, which is around 1/3. As I indicated above, we will return to the issue of a welfare-maximizing savings rate when we turn to the optimal growth model. 1.5 The Effect of a Change in the Savings Rate Suppose that the savings rate changes. What happens is that output will rise for two reasons. One is that a higher savings rate, ceteris paribus, will raise output because a greater share of production is being invested, which means a larger capital stock. The second reason is that the higher savings rate in turn creates more output, which in turn creates more investment. The steady state impact of a higher savings rate is given by the following expression, in which the change in output is given by the increase in the capital stock, multiplied the marginal product of capital: ∂y∗ ∂s = ∂y∗ ∂k∗ ∂k∗ ∂s (34) We know what the impact of a change in capital is on output - it is the marginal product of capital, which is ∂y ∗ ∂k∗ . What is the impact of a change in the savings rate on the capital stock? We can see the steady state impact from the following steady state expression, and its derivative with respect to s: sf(k∗(s)) = δk∗(s) (35) Next, differentiate to get: f(k∗) + s[f ′(k)k′(s)] = δk′(s) (36) (37) Re-arranging, we get: k′(s) = f(k∗) δ − sf ′(k) (38) Thus, the steady change in output due to a change in the savings rate is given by: ∂y∗ ∂s = f ′(k) δ − sf ′(k) y∗ (39) The text shows how to turn this derivative into an elasticity that has a relatively simple interpretation. 8 张熠华� 张熠华� 张熠华� 张熠华� 张熠华� Figure 1.5 in the text graphically shows the effect of a one time change in the savings rate on the time paths of consumption, investment, output, and the capital stock. Note that an initial increase in the savings rate means a greater fraction of output is devoted to investment, which in turn means that the higher investment immediately leads to an offsetting drop in consumption. After the first period, consumption rises. Note that if the initial savings rate was below the Golden Rule savings rate, then a higher savings rate that is closer to the Golden Rule rate will mean a new steady state level of consumption that is higher than the previous steady state consumption level. Figure 1.5 shows the new steady state level of consumption being below the old steady state level, however. This means that the text considers the case in which the new savings rate is further away from the Golden Rule than the old savings rate. 1.6 Growth Accounting Our discussion indicates that economic growth can arise from changes in inputs or technology. We can proceed to determine the relative importance of these different sources of growth as follows. In particular, we will be able to residually measure technological change by subtracting the log of the production function over two time periods. Specifially, we will be constructing the Solow Residual. Note that the Solow Residual is equal to output growth, net of capital stock growth scaled by its importance in the production function, and net of labor growth scaled by its importance in the production function. Thus, the Solow Residual is the component of output growth that is not accounted for by changes in the capital and labor inputs: Yt = K α t (AtLt) 1−α (40) ln(Yt) = α ln(Kt) + (1− α) ln(Lt) + (1− α) ln(At) (41) ln(Yt)−ln(Yt−1) = α[ln(Kt)−ln(Kt−1)]+(1−α)[ln(Lt)−ln(Lt−1)]+(1−α)[ln(At)−ln(At−1)] (42) Note that the log differences in the variables in the equation above are equal to their respective percentage changes. We thus have the same formula as above, but just in discrete time. We can make this operational by obtain data on real output, real capital stock, and labor input. Plug in the numbers, and that allows us to construct the relative importance of inputs and technology for growth, but it also allows us to construct a measure of the growth in technology. There is one caveat in terms of interpreting growth accounting results. In particular, suppose that the economy is on a balanced growth path, and that 9 张熠华� 张熠华� 张熠华� � labor input is constant. In this case, we would interpret growth as fraction α of growth from capital, and fraction 1 − a from technology. But note that capital is growing only because of technological change, not because there is an independent source of growth of the capital stock, which is what occurs when the k is below its steady state level. However, we can adjust our formula to take into account this dependence of capital on the technology as follows. Let’s first divide by the population, which we will call N, so that we can express all variables in per capita terms. Y N = ( K N )α( AL N )1−α (43) Next, let’s do a bit of algebra by exponentiating both sides by 1 1−α . Next, divide both sides by (Y/N) α 1−α and we get: Y N = A 1 1−α ( K Y ) α 1−α ( L N ) (44) Note that with this re-writing of the production, all growth in the economy is driven by growth in the technology term A when the economy is on its steady state growth path, (and assuming that the share of the population working is constant). This is because the capital stock and output grow at the same rate along the steady state path, and therefore K/Y is constant along that path. The handout growth accounting uses this equation and data to examine growth accounting in a number of countries. 1.7 Convergence This model predicts that countries that have the same time paths of the technol- ogy variable A will ultimately converge to the same steady state growth path. This also implies that income differences in countries that have similar paths for A will ultimately disappear as they all move on to the same steady state growth path. Accordingly, the model predicts that countries that have the same A time path, but that have low income at a point in time, will catch up to the other countries with the same A path. We tend to see this in the data, as in figure 1.7 of the text. 1.8 Takeaway The Solow neoclassical growth model is a very simple and parsimonious model in which the key economic force is diminishing marginal product of capital. For this reason, the model, with no technological change, possesses a steady state, in which the economy ultimately stops growing. This is because depreciated capital - δk - ultimately becomes suffi ciently large as to equal investment. The reason is because depreciated capital is a linear function (δk), whereas investment is a strictly concave function (skα, α < 1). The strict concavity of the savings 10 张熠华� 张熠华� function means that the marginal product of capital is declining in the amount of capital. The model can be augmented to include population growth. The model with population growth can be analyzed the same way as in the simple case above. The only difference is that we add the population growth rate to the depreciation rate when constructing the equation that shows how the capital stock per worker changes over time. If the model is augmented to include technological change that raises the productivity of the factors of production, then the model will display persistent economic growth in per-capita output, as opposed to a steady state in per- capita output. We can still analyze the model the same way as in the simplest model by presenting the variables in their intensive form, in which we divide the variables by employment multiplied by the technology term, AtL. We add the growth rate of technological progress to the depreciation rate, just as we added in the population growth rate to the depreciation rate in the model with population growth. The model with technological change in intensive form does have a steady state for the intensive form variables. To construct the per-capita variables we simply add back in the technological progress by multiplying all variables by At. When the intensive form variables are equal to their steady state values, then the per-capita variables are said to be on a steady state growth path, in which output, consumption, investment, and the capital stock all grow at a constant rate, which is the growth rate of technological progress. 11