The Solow Model

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

This is another idea for modifying how to teach the Solow model. One thing I’d like to do is go immediately to including productivity – it follows cleanly from the simplest growth model. Second, I think it might be nice to work with the K/Y ratio immediately. In this way, I think you can actually skip using the whole “k-tilde” thing. And, *gasp*, do away with the traditional Solow diagram.

The simplest growth model doesn’t allow for transitional growth, and this due to the fact that it does not allow for capital, a factor of production that can only be slowly accumulated over time. The Solow Model is a standard model of economic growth that includes capital, and will be better able to account for the transitional growth that we see in several countries.

Production in the Solow Model takes place according to the following function

\displaystyle  Y = K^{\alpha}(AL)^{1-\alpha}. \ \ \ \ \ (1)

{K} is the stock of physical capital used in production, and {A} and {L} are defined just as they were in our simple growth model. So the production function here is just a modification of the simple model to include capital. The coefficient {\alpha} is a weight telling us how important capital or {AL} are in determining output.

To analyze this model, we’re going to rewrite the production function. Divide both sides of the function by {Y^{\alpha}}, giving us

\displaystyle  Y^{1-\alpha} = \left(\frac{K}{Y}\right)^{\alpha} (AL)^{1-\alpha} \ \ \ \ \ (2)

and then take both sides to the {1/(1-\alpha)} power, which gives us the following expression

\displaystyle  Y = \left(\frac{K}{Y}\right)^{\alpha/(1-\alpha)} AL. \ \ \ \ \ (3)

In per capita terms, this is

\displaystyle  y = \left(\frac{K}{Y}\right)^{\alpha/(1-\alpha)} A.  \ \ \ \ \ (4)

Output per worker thus depends not just on {A}, but also on the capital-output ratio, {K/Y}.

So to understand the role of capital in economic growth, we need to understand the capital-output ratio and how it changes over time. We’ll start by looking at the balanced growth path, and then turn to situations where the economy is not on the balanced growth path (BGP).

One fact about the BGP is that the return to capital, {r}, is constant. The return to capital is {r = \alpha Y/K}, which depends (negatively) on the capital-output ratio (the return to capital is just the marginal product of capital). If {r} is constant on the BGP, then it must be that {K/Y} is constant on the BGP as well. What does this mean? It means that {K/Y} can have a level effect on output per worker, but has no growth effect. To see this more clearly, take logs of output per worker,

\displaystyle  \ln y(t) = \frac{\alpha}{1-\alpha} \ln\left(\frac{K}{Y}\right) + \ln A(t) \ \ \ \ \ (5)

and then plug in what we know about how {A(t)} moves over time,

\displaystyle  \ln y(t) = \frac{\alpha}{1-\alpha} \ln\left(\frac{K}{Y}\right) + \ln A(0) + gt. \ \ \ \ \ (6)

The capital-output ratio affects the intercept of this line — a level effect — alongside {A(0)}. The slope of this line — the growth rate — is still {g}.

The capital/output ratio is constant along the BGP, and has no effect on the growth rate on the BGP. But what if the economy is not on the BGP? Then it will be the case that {K/Y} affects the growth rate of output per worker, because the {K/Y} ratio will not be constant. More precisely, the growth rate of capital/output is

\displaystyle  \frac{\dot{K/Y}}{K/Y} = \frac{\dot{K}}{K} - \frac{\dot{Y}}{Y}.  \ \ \ \ \ (7)

So the {K/Y} ratio will change if capital grows more quickly or more slowly than output. First, capital accumulates as follows

\displaystyle  \dot{K} = s Y - \delta K \ \ \ \ \ (8)

where {\dot{K}} is the change in the capital stock. {s} is the savings rate, the fraction of output that the economy sets aside to invest in new capital goods, so that {sY} is the total amount of new investment. {\delta} is the depreciation rate, the fraction of the existing capital stock that breaks or becomes obsolete at any given moment.

To find the growth rate of capital, divide through the above equation by {K} to get

\displaystyle  \frac{\dot{K}}{K} = s\frac{Y}{K} - \delta. \ \ \ \ \ (9)

You can see that the growth rate of capital depends on the capital/output ratio itself.

The growth rate of output is

\displaystyle  \frac{\dot{Y}}{Y} = \alpha \frac{\dot{K}}{K} + (1-\alpha)\frac{\dot{A}}{A} + (1-\alpha)\frac{\dot{L}}{L}. \ \ \ \ \ (10)

Now, with (7), and using what we know about growth in capital and output, we have

\displaystyle  \frac{\dot{K/Y}}{K/Y} = (1-\alpha)\left(s\frac{Y}{K} - \delta - g - n \right) \ \ \ \ \ (11)

where we’ve plugged in that {\dot{A}/A = g}, and {\dot{L}/L = n}.

Re-arranging a bit, the capital output ratio is growing if

\displaystyle  \frac{K}{Y} < \frac{s}{\delta + n + g}, \ \ \ \ \ (12)

and growing if the capital/output ratio is larger than the value on the right-hand side. In other words, if the capital stock is relatively small, then it will have a tendency to grow faster than output, raising the {K/Y} ratio. Eventually {K/Y = s/(\delta+n+g)}, the steady state value, and the {K/Y} ratio stops changing.

What is happening to growth in output per worker? If {K/Y < s/(\delta+n+g)} then the {K/Y} ratio is growing, and so output per worker is growing faster than {g}. So the temporarily fast growth in output per worker in Germany or Japan would be because they found themselves with a {K/Y} ratio below their steady state value. How would this occur? It’s easier to see how this works if we re-write the {K/Y} ratio slightly

\displaystyle  \frac{K}{Y} = \frac{K}{K^{\alpha}(AL)^{1-\alpha}} = \left(\frac{K}{AL}\right)^{1-\alpha}. \ \ \ \ \ (13)

From this we can see that the {K/Y} ratio would be particularly low if the capital stock, {K}, were to be reduced. This is what happened in Germany, to a large extent, after World War II. The capital stock was destroyed, so {K/AL} fell sharply. This made {K/Y} fall below the steady state value, which meant that there was growth in the {K/Y} ratio, and so growth in output per worker greater than {g}.

A slightly different situation describes South Korea. There, we can think of there being a level effect on {A}, an advance in productivity. This also makes {K/AL} fall sharply, and again causes growth in {K/Y} and growth in output per worker faster than {g}. But in both this case and in Germany’s, as the {K/Y} ratio grows it approaches the steady state value and growth in output per worker slows down to {g} again.

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The Simplest Growth Model

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

This is an idea for a new way of introducing growth theory. Given that productivity growth is the source of long-run growth, it seems to make sense to start with that, rather than with the Solow model.

Let’s write down a very simple model of economic growth. Let total output {Y} be determined by

\displaystyle  Y = A L \ \ \ \ \ (1)

where {A} is a measure of labor productivity, and {L} is the number of workers. If we divide through by {L}, then we get a measure of output per worker. To keep notation clean, let {y = Y/L} be output per worker, so that now we have

\displaystyle  y = A \ \ \ \ \ (2)

as our model of economic growth. Basically, output per worker is simply equal to labor productivity {A}.

From this we know that the time path of output per worker is simply the same as the time path of labor productivity, {A}. So what determines the time path of labor productivity? We’ll assume that it is growing at a constant rate, meaning that it goes up by the same percent every period of time,

\displaystyle  A(t) = A(0) e^{g t}. \ \ \ \ \ (3)

Here, we’ve written {A(t)} to be clear that we mean labor productivity at any given time {t}. {A(0)} is labor productivity in the initial moment of time. The exponential term says that labor productivity grows at the rate {g} over time.

The exponential term implies, perhaps not surprisingly, exponential growth. You get exponential growth when something goes up by the same percent every period of time. If {g = 0.02}, then we have 2% growth. At time zero, labor productivity is just {A(0)}. When {t=2}, then {A(2) = A(0)e^{.02(2)} = 1.041 A(0)}, or labor productivity is a little more than 4% higher than at time zero. When {t=10}, {A(10)=A(0)e{.02(10)}=1.221}, or labor productivity is more than 22% higher than at time zero.

It may not seem obvious, but output per worker in the U.S. and most other developed nations displays exponential growth. Our model matches that, as

\displaystyle  y(t) = A(0) e^{g t}. \ \ \ \ \ (4)

These countries also tend to have a similar growth rate of about 1.8%, or {g=0.018}. Seeing this in a figure, though, is difficult. Graphing {y} over time for the U.S. gives you a curve that quickly accelerates upwards and is almost off the page. Graphs like this will also make it difficult to compare countries to one another.

For that reason, among others, we like to work with the natural log of output per worker, {\ln{y(t)}}. Taking natural logs of {y(t)} gives us

\displaystyle  \ln{y(t)} = \ln{A(0)} + g t.  \ \ \ \ \ (5)

This is an equation that says the natural log of output per worker is a linear function of time, {t}. If we graph {\ln{y(t)}} against {t}, we get a straight line, similar to the trend line we drew in the figure for U.S. output per worker.

We can calculate the growth rate of output per worker by taking the derivative of (5) with respect to time. This results in the following

\displaystyle  \frac{\dot{y}}{y} = g. \ \ \ \ \ (6)

The value of {A(0)} is fixed, so the derivative of it with respect to time is just zero. The notation {\dot{y}/y} is a shorthand way of writing the growth rate. {\dot{y}} is the absolute change in output per worker at any given moment, and by dividing by {y} we get that change relative to the level of output per worker. This means that {\dot{y}/y} is essentially the percent change in output per worker at any given moment.

That’s it for the simple growth model. Output per worker depends on labor productivity {A(t)}, and labor productivity grows at a constant rate {g}, which means output per worker grows at that same rate. Despite the mechanical simplicity, this model helps us be clear when we are talking about the growth experiences of different countries. It allows us to distinguish between two forces determining output per worker.

  • Level effects: These refer to {A(0)}, the intercept of the line in (5)
  • Growth effects: These refer to {g}, the slope of the line in (5)

Looking at the data over the long run, the general impression we get that the growth rate {g} is similar across countries, and they differ mainly because of level effects. That is, {A(0)_{Japan}} appears to be lower than {A(0)_{US}}, but the growth rate {g} is very similar. Theories of economic growth should be consistent with these facts. Things like investment rates, schooling, and social infrastructure are important determinants of level effects, {A(0)}, but they have no effect on the growth rate, {g}. Under plausible assumptions, theories of endogenous innovation will suggest that the growth rate, {g}, is identical across countries.

There are some facts, though, that this simple growth model cannot account for. Namely, there are notable cases where output per worker grows more quickly or more slowly than {g}. China, for example, over the last 30 years has grown much faster than the U.S. or Japan. South Korea had a similar growth miracle, starting in about 1960 and lasting until the 2000’s. Germany, from World War II until about 1980, grew at a very accelerated pace compared to the U.S. in the same period. How do we reconcile these facts with the assertion above that {g} is the same for all countries?

The key is noting that these growth accelerations were temporary. Germany grew very quickly, but after 1980 its growth rate fell back to a value nearly identical to the U.S. South Korea’s growth rate has diminshed as well in the 2000’s. What appears to be happening is that once output per worker approaches a frontier level, generally defined by the U.S., growth slows down. While China continues to grow quickly, it has not approached the U.S. level of output per worker.

Looking at these countries, what appears to be happening is that there is a level effect, or their {A(0)} has shifted up. However, it seems to take them a long time to move from their old level to the new, higher level. We call the temporary growth spurt that occurs when a country moves between levels transitional growth. Output per worker grows faster than {g} temporarily – although this could last a few decades – but then growth returns to the rate {g}.

Our simple model doesn’t offer a way of understanding this transitional growth. The first major extension we’ll make to this simple model is to add physical capital, which has to be slowly accumulated over time. Because of this slow accumulation, the economy will take an extended time to fully respond to a level effect.