Solow began with a production function of the Cobb-Douglas type:
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:
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:
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.
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:
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:
This procedure can be continued as long as you can punch a calculator; the results for the first 7 periods are:
Period | Capital | Output | Savings | Change in Output |
---|---|---|---|---|
1 | 100 | 1000 | 250 | ---- |
2 | 250 | 1581 | 395.3 | 581 |
3 | 395.3 | 1988 | 497 | 407 |
4 | 497 | 2229 | 557 | 241 |
5 | 557 | 2360 | 590 | 131 |
6 | 590 | 2429 | 607 | 69 |
7 | 607 | 2464 | 616 | 35 |
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:
Substitute for s the savings function to obtain:
Substitute for output the production function to obtain:
Finally, divide through by k ^{ 0.5 } to obtain:
and square both sides to get the equilibrium capital stock
If the equilibrium capital stock is 625, equilibrium output (found using the production function q = 100 k ^{ 0.5 }) will be:
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.