Question

Consider now Solow-Swan model with a general production functionY=AF(K, N). Derive the steady state level of capital. Show that the steady state level of capital increasing in A.

Answer 1

The production function model was applied to the study of growth problems by Robert Solow (American economist, Massachusetts Institute of Technology, Nobel prize 1990).

Solow began with a production function of the Cobb-Douglas type:

Q = A K^a L ^b

where A is multifactor productivity , a and b are less than one, indicating diminishing returns to a single factor, and a + b = 1 , indicating constant returns to scale.

Solow noted that any increase in Q could come from one of three sources:

1. an increase in L . However, due to diminishing returns to scale, this would imply a reduction in Q / L or output per worker.
2. an increase in K . An increase in the stock of capital would increase both output and Q / L

an increase in A or in multifactor productivity could also increase Q / L or output per worker.

To concentrate attention on what happens to Q / L or output per worker (and hence, unless the employment ratio changes, output per capita), Solow rewrote the Cobb-Douglas production function in what we shall refer to as per capita form:

Q / L = A K ^a L ^{b - 1} = A K ^a / L ^{1 - b}

since multiplying by L ^{b - 1} is the same as dividing by L ^{1 - b} . Also, since we assumed that a + b = 1, a = 1-b.

Q = A K ^a / L ^a = A ( K / L ) ^a

Defining q = Q / L and k = K / L, that is, letting small letters equal per capita variables , we have

q = A k a

which is the key formula we will work with. We will examine how the model works when growth comes through capital accumulation, and how it works when growth is due to innovation.

Growth by capital accumulation

1. In addition to the production function, we need two other pieces of information:

2. the savings function -- how much of output do people in our model economy save? The simplest assumption (which we will examine in more detail later in the course, and will conclude can be a fairly good representation of people's behavior) is that people save a given fraction of output. For the sake of having a specific example, we assume that people save 1 / 4 of output, or what comes to the same thing, 25 cents for every dollar of income. The savings function is therefore:

   s = 0.25 q

   the equilibrium condition . We shall find that if capital accumulation is the only source of growth, the economy will approach an equilibrium or steady state . It will reach the steady state when savings is just sufficient to replace the depreciated capital stock. If we assume that in each time period capital depreciates totally, the equilibrium condition is simply

s = k

Note that if depreciation were only 10 percent of capital stock, the equilibrium condition would be s = 0.10 k . Although this is a more realistic figure for yearly depreciation, we assume 100 percent depreciation for simplicity -- and if you are troubled by the lack of realism, you may think of our time periods as decades rather than years.

Let A = 100 and a = 0.5 in the Solow per capita production function. Note that a = 0.5 means "take the square root of k" and A = 100 means "then multiply it by 100" to get the ouput per worker.
That is, let our production function be:

q = 100 k ^0.5

Consider what happens if we begin with 100 units of capital per worker. We can use the production function to calculate that q = 1000.

The next step is to use the savings function to calculate how much of this output is saved. If s = 0.25 q then 250 units per capita of output are saved -- and the savings of one period become the capital of the next period.

Note that this means in the next period the capital stock will have increased from 100 to 250 .
Since the production function is unchanged, the output next period will be q = 100 (250) ^0.5 = 1581

We again note that savings is 0.25 of output; and .25 x 1581 = 395.3, so that savings next period will be 395.3.

Therefore capital in the third period will be 395.3, and output in the third period will be:

q = 100 (395.3) ^0.5 = 1988

This procedure can be continued as long as you can punch a calculator; the results for the first 7 periods are:

Note that output grows throughout, but that the change in output slows down -- since the production function exhibits diminishing returns, this is not surprising.

Will the growth stop? That is, will output converge to a steady state? The answer is yes . We can find steady state equilibrium by making use of the equilibrium condition:

s = k

Substitute for s the savings function to obtain:

0.25 q = k

Substitute for output the production function to obtain:

0.25 ( 100 k ^0.5 ) = k

Finally, divide through by k ^0.5 to obtain:

k ^0.5 = 25

and square both sides to get the equilibrium capital stock

k = 625

If the equilibrium capital stock is 625, equilibrium output (found using the production function q = 100 k ^0.5 ) will be:

q = 2500

Note that if savings is 1 / 4 of output, this means that equilibrium savings is 625 -- just enough to replace the capital stock next period, and to keep the economy in a steady-state with output at 2500 and capital stock of 625 ever after.

Predictions of the model

If the Solow model is correct, and if growth is due to capital accumulation , we should expect to find

1. Growth will be very strong when countries first begin to accumulate capital, and will slow down as the process of accumulation continues. Japanese growth was stronger in the 1950s and 1960s than it is now.
2. Countries will tend to converge in output per capita and in standard of living. As Hong Kong, Singapore, Taiwan (etc) accumulate capital, their standard of living will catch up with the initially more developed countries. When all countries have reached a steady state, all countries will have the same standard of living (at least if they have the same production function, which for most industrial goods is a reasonable assumption).

Certainly there is some evidence favoring these predictions. However, there are some problems as well:

1. The US growth rate was lower , at least on a per capita basis, in the 19th century than in the twentieth century.
2. The Soviet Union under Stalin saved a higher percentage of national income than the US. Because of the higher savings rate and because it started from a lower level of capital, it should have caught up very rapidly. It did not.
3. Less developed countries, with some exceptions -- such as Taiwan, Korea, Singapore and Hong Kong -- are not in general catching up to the developed countries. Indeed, in many cases, the gap is increasing .

Do these facts mean that the Solow model is wrong? Not necessarily, since increase in output per capita can be due to an increase in multifactor productivity as well as an increase in capital per worker.