Macroeconomics

F I G U R E

7 - 3

Depreciation

per worker, 

dk

Depreciation,

dk

Capital

per worker, k

Depreciation

A constant frac-

tion

d

of the capital stock wears

out every year. Depreciation is

therefore proportional to the

capital stock.

---

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.

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 k

1

in

Figure 7-4. In this case, the level of investment exceeds the amount of deprecia-

tion. 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 k

2

. 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 cap-

ital stock to either increase or decrease.

Approaching the Steady State: A Numerical Example

Let’s use a numerical example to see how the Solow model works and how the

economy approaches the steady state. For this example, we assume that the pro-

duction function is

Y

= K

1/2

L

1/2

.

From Chapter 3, you will recognize this as the Cobb–Douglas production func-

tion with the capital-share parameter 

a

equal to 1/2. To derive the per-worker

C H A P T E R   7

Economic Growth I: Capital Accumulation and Population Growth

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