Economics 3307

Key Points of Chapter 4

The purpose of this chapter and Chapter 5 is to explain the following basic facts about economic growth:

1. Real output per person grows at a more or less constant rate over fairly long periods of time.

2. The stock of real capital grows at a more or less constant rate exceeding the growth of labor input.

3. The rates of growth of real output and the stock of real capital tend to be about the same, so that the ratio of real capital to real output (K/Y) has no pronounced long-term trend.

Point (1) is the main source of ongoing increases in the "standard of living" observed in the last 100-200 years in industrialized market economies. If we can develop an explanation of (1) that also implies (2) and (3), then we can have some confidence that the explanation is correct -- or at least that is captures some element of the "truth" about economic growth.

The Solow model with ongoing population growth and technological progress (which is covered in Chapter 5) can explain fact (1) in a way that is consistent with facts (2)-(3). We must have ongoing population growth and technological progress because:

- The Solow Model without population growth and without technological progress (which is covered in Chapter 4) predicts that real output is constant over time. In fact, real output rises about 3% per year in the U.S.

- The Solow Model with population growth and without technological progress (also in Chapter 4) predicts that real output should rise over time, but that real output per person is constant over time. In fact, real output per person rises about 2% per year in the U.S.

The Solow model begins with the assumption that total savings in the economy is a constant fraction of income. Recall from Chapter 2 that for all practical purposes, Real Output = Real Income. Thus the Solow model assumes that total saving in the economy is a constant fraction of output. We use the letter "s" to denote the savings rate, which is the fraction of income saved: s = (Saving/Income) = Saving/Output. If s = 0.2, then total saving is 20% of total income. If s = 0.45, then total saving is 45% of total income.

Recall from Chapter 3 the assumption of the neoclassical model that the aggregate supply of output is determined by the quantities of capital and labor in the economy. The real rental price of capital (R/P) and the real wage (W/P) adjust so that existing supplies of capital and labor are fully employed, and the aggregate production function Y = F(K,L) determines the amount of output actually produced in the economy.

Recall also from Chapter 3 that the total demand for output (we assume a closed economy) is equal to C + I + G. The neoclassical model assumes that the real interest rate (r) adjusts to ensure equality between supply (Y) and demand (C+I+G). Another way of saying that Y = C+I+G is to say that total saving = total investment.

When we combine (i) the assumption of the Solow model that total saving is a constant fraction of output so that S = sY, and (ii) the equilibrium condition that investment equals total saving so that I=S, we find that I = sY.

It turns out, for reasons that are not absolute necessary for you to understand, that when the production function exhibits constant returns to scale, we can work with the aggregate production function in a simpler way. The aggregate production function says that Y = F(K,L). That is, one output (Real GDP = Y) is a function of 2 inputs -- capital (K) and labor (L). With constant returns to scale, it is possible to represent the aggregate production technology as y = f(k), where y = Y/L and k = K/L.

Why do this? For two reasons. First, when we use y = f(k), we only have to keep up with two variables -- y and k -- rather than 3 variables -- Y, K, and L. Second, output per person (y) is a better measure of the standard of living than simple output (Y). If Y is rising only because of population growth, the rise in Y does not signal an improvement in the standard of living.

In steady state equilibrium, k and y are constant. The condition for steady-state equilibrium is that s·f(k) = (d+n)·k. Because s·f(k) = i = I/L, if s·f(k) < (d+n)·k, actual investment is less than break-even investment (the level of i needed to keep k constant) and k tends to fall. If s·f(k) > (d+n)·k, actual investment is greater than break-even investment and k tends to rise. Thus when k is not constant, it tends to move toward the level at which it is constant (the steady-state equilibrium level).

If k and y are constant, this means that K and Y grow at the same rate, which accords with fact #3 above. If y is constant, Y grows at the same rate as L, which grows at rate n, where n is the population growth rate. (Y/L) is constant. The fact that (Y/L) is constant means that we have NOT explained fact #1 above. The fact that k is constant means that (K/L) is constant, which means we have NOT explained fact #2 above. Thus we need to move on to Chapter 5 to explain these facts. The Chapter 5 model, however, is but a simple extension of the model with population growth in this chapter.

I will close with a few major points:

1. A rise in the savings rate (s) increases the steady-state level of output, but not the steady-state growth rate of output.

2. The Golden Rule of capital accumulation shows the one particular steady state capital stock that implies maximum per capita consumption. At the Golden Rule capital stock, MPK - d = n. This reduces to MPK = d when n = 0 (i.e., when there is no population growth).

3. To increase the steady state capital stock to the Golden Rule level, the savings rate should be increased. One way to do this is to reduce the government budget deficit.

4. If the savings rate is increased, then consumption per capita will be lower for a while during the transition to a new steady state -- even though consumption per capita will eventually be higher in the new steady state.

5. If an economy finds itself with a steady state capital stock above the Golden Rule level, a decrease in the savings rate will lead to higher steady state consumption per capita. In this case, the transition to the new steady state will also involve higher consumption.

6. The steady state equilibrium values of y and k will increase with a reduction in the rate of population growth and with a reduction in the rate of depreciation.