This week Perlie Mong gave a talk on the Solow Growth Model.
The Solow growth model is designed to show how growth in the capital stock, growth in the labor force, and advances in technology interact in an economy as well as how they affect a nation’s total output of goods and services. Due to time constraint, this week’s presentation focused on how to build the Solow model and how population growth affects economic growth.
The supply of goods in the Solow model is based on the production function, which states that output depends on the capital stock and the labor force: Y = F(K, L). Because the Solow model assumes constant returns to scale, allow us to analyze all quantities in the economy relative to the size of the labor force. To see that this is true, set z = 1/L in the preceding equation to obtain Y/L = F(K/L, 1).
This equation shows that the amount of output per worker Y/L is a function of the amount of capital per worker K/L. (The number 1 is constant and thus can be ignored.) The assumption of constant returns to scale implies that the size of the economy—as measured by the number of workers—does not affect the relationship between output per worker and capital per worker. Because the size of the economy does not matter, it will prove convenient to denote all quantities in per worker terms. We designate quantities per worker with lowercase letters, so y = Y/L is output per worker, and k = K/L is capital per worker. We can then write the production function as y = f (k), where we define f(k) = F(k, 1). The slope of this production function shows how much extra output a worker produces when given an extra unit of capital. This amount is the marginal product of capital MPK. Mathematically, we write MPK = f(k + 1) − f (k). Graphically, it looks like this:
The demand for goods in the Solow model comes from consumption and investment. In other words, output per worker y is divided between consumption per worker c and investment per worker i: y = c + i. This equation is the per-worker version of the national income accounts identity for an economy. Notice that it omits government purchases (which for present purposes we can ignore) and net exports (because we are assuming a closed economy).
The Solow model assumes that each year people save a fraction s of their income and consume a fraction (1 – s). We can express this idea with the following consumption function:
c = (1 − s)y, where s, the saving rate, is a number between zero and one. To see what this consumption function implies for investment, substitute (1 – s)y for c in the national income accounts identity: y = (1 − s)y + i. Rearrange the terms to obtain i = sy. This equation shows that investment equals saving. Thus, the rate of saving s is also the fraction of output devoted to investment.
In short, for any given capital stock k, the production function y = f(k) determines how much output the economy produces, and the saving rate s determines the allocation of that output between consumption and investment.
At any moment, the capital stock is a key determinant of the economy’s output, but the capital stock can change over time, and those changes can lead to economic growth. In particular, two forces influence the capital stock: investment and depreciation. Investment is expenditure on new plant and equipment, and it causes the capital stock to rise. Depreciation is the wearing out of old capital, and it causes the capital stock to fall. Let’s consider each of these forces in turn. As we have already noted, investment per worker i equals sy. By substituting the production function for y, we can express investment per worker as a function of the capital stock per worker: i = sf(k). This equation relates the existing stock of capital k to the accumulation of new capital i. This relationship can be illustrated graphically:
To incorporate depreciation into the model, we assume that a certain fraction δ of the capital stock wears out each year. Here δ is called the depreciation rate. For example, if capital lasts an average of 25 years, then the depreciation rate is 4 percent per year (δ = 0.04). The amount of capital that depreciates each year is δk. We can express the impact of investment and depreciation on the capital stock with this equation: Change in Capital Stock = Investment – Depreciation (Δ k = i − δ k), where Δ k is the change in the capital stock between one year and the next. Because investment i equals sf(k), we can write this as
Δ k = sf (k) − δ k.
There is a single capital stock k* at which the amount of investment equals the amount of depreciation. If the economy finds itself at this level of the capital stock, the capital stock will not change because the two forces acting on it—investment and depreciation—just balance. That is, at k*, Δ k = 0, so the capital stock k and output f(k) are steady over time (rather than growing or shrinking). We therefore call k* the steady-state level of capital. The steady state is significant for two reasons. As we have just seen, an economy at the steady state will stay there. In addition, and just as important, an economy not at the steady state will go there. That is, regardless of the level of capital with which the economy begins, it ends up with the steady-state level of capital. In this sense, the steady state represents the long-run equilibrium of the economy. It can be illustrated graphically:
To see why an economy always ends up at the steady state, suppose that the economy starts with less than the steady-state level of capital, such as level k1 in Figure 7-4. In this case, the level of investment exceeds the amount of depreciation. Over time, the capital stock will rise and will continue to rise—along with output f(k)—until it approaches the steady state k*.
Similarly, suppose that the economy starts with more than the steady-state level of capital, such as level k2. In this case, investment is less than depreciation: capital is wearing out faster than it is being replaced. The capital stock will fall, again approaching the steady-state level. Once the capital stock reaches the steady state, investment equals depreciation, and there is no pressure for the capital stock to either increase or decrease.
In short, the Solow model shows that the saving rate is a key determinant of the steady-state capital stock. If the saving rate is high, the economy will have a large capital stock and a high level of output in the steady state. If the saving rate is low, the economy will have a small capital stock and a low level of output in the steady state. This conclusion sheds light on many discussions of fiscal policy. A government budget deficit can reduce national saving and crowd out investment. Now we can see that the long-run consequences of a reduced saving rate are a lower capital stock and lower national income. This is why many economists are critical of persistent budget deficits. It is worth noting that the Solow model says that higher saving leads to faster growth in the Solow model, but only temporarily. An increase in the rate of saving raises growth only until the economy reaches the new steady state. If the economy maintains a high saving rate, it will maintain a large capital stock and a high level of output, but it will not maintain a high rate of growth forever.
International evidence does support the Solow model, as demonstrated by the graph below:
The basic Solow model shows that capital accumulation, by itself, cannot explain sustained economic growth: high rates of saving lead to high growth temporarily, but the economy eventually approaches a steady state in which capital and output are constant. To explain the sustained economic growth that we observe in most parts of the world, we must expand the
Solow model to incorporate the other two sources of economic growth—population growth and technological progress. We start with population growth. Instead of assuming that the population is fixed, as we did in Sections 7-1 and 7-2, we now suppose that the population and the labor force grow at a constant rate n. As we noted before, investment raises the capital stock, and depreciation reduces it. But now there is a third force acting to change the amount of capital per worker: the growth in the number of workers causes capital per worker to fall.
We continue to let lowercase letters stand for quantities per worker. Thus, k = K/L is capital per worker, and y = Y/L is output per worker. Keep in mind, however, that the number of workers is growing over time. The change in the capital stock per worker is Δ k = i − (δ + n)k. We can think of the term (δ + n)k as defining break-even investment—the amount of investment necessary to keep the capital stock per worker constant. Break-even investment includes the depreciation of existing capital, which equals δ k. It also includes the amount of investment necessary to provide new workers with capital. The amount of investment necessary for this purpose is nk, because there are n new workers for each existing worker and because k is the amount of capital for each worker. The equation shows that population growth reduces the accumulation of capital per worker much the way depreciation does. Depreciation reduces k by wearing out the capital stock, whereas population growth reduces k by spreading the capital stock more thinly among a larger population of workers. Our analysis with population growth now proceeds much as it did previously. First, we substitute sf(k) for i. The equation can then be written as Δ k = sf(k) − (δ + n)k. The impact of population growth on the steady state can be demonstrated graphically like this:
An economy is in a steady state if capital per worker k is unchanging. As before, we designate the steady-state value of k as k*. If k is less than k*, investment is greater than break-even investment, so k rises. If k is greater than k*, investment is less than break-even investment, so k falls. In the steady state, the positive effect of investment on the capital stock per worker exactly balances the negative effects of depreciation and population growth. That is, at k*, Δ k = 0 and i*= δ k*+ nk*. Once the economy is in the steady state, investment has two purposes. Some of it (δ k*) replaces the depreciated capital, and the rest (nk*) provides the new workers with the steady-state amount of capital.
Population growth alters the basic Solow model in two ways. First, it brings us closer to explaining sustained economic growth. In the steady state with population growth, capital per worker and output per worker are constant. Because the number of workers is growing at rate n, however, total capital and total output must also be growing at rate n. Hence, although population growth cannot explain sustained growth in the standard of living (because output per worker is constant in the steady state), it can help explain sustained growth in total output. Second, population growth gives us another explanation for why some countries are rich and others are poor. Consider the effects of an increase in population growth. Figure 7-12 shows that an increase in the rate of population growth from n1 to n2 reduces the steady-state level of capital per worker from k*1 to k*2. Because k*is lower and because y*= f(k*), the level of output per worker y*is also lower. Thus, the Solow model predicts that countries with higher population growth will have lower levels of GDP per person. Notice that a change in the population growth rate, like a change in the saving rate, has a level effect on income per person but does not affect the steady-state growth rate of income per person.