The Solow Growth Model

Romer01a.doc

The Solow Growth Model

Set-up

The Production Function

Assume an aggregate production function:

(1.1)

Notation:

Y output

K capital

L labor

A effectiveness of labor (productivity)

Technical change is labor-augmenting (also known as Harrod neutral).

The production function exhibits constant returns to scale:

for all . (1.2)

Setting in (1.2) yields the production function in intensive form:

(1.3)

Now define:

Then (1.3) becomes:

(1.4)

So output per effective unit of labor is a function of capital per effective unit of labor.

We further assume that:

.

We also note that is the marginal product of capital. By CRS:

Differentiating with respect to K:

We also assume that the Inada conditions hold:

.

Note: the Cobb-Douglas production function is one that satisfies the conditions assumed here.

Evolution of Inputs

Labor and capital grow at constant rates;

(1.8)

(1.9)

where the dot notation refers to a time derivative: .

The constant growth assumption also permits us to describe paths for L and A by:

.

By definition, output can be divided into consumption and investment. The fraction of output going to investment is s, which is assumed to be a constant. The constant rate of saving is a key feature of this model. Capital also depreciates at the rate . Thus the path of capital must satisfy the equation:

(1.10)

It is assumed that .

Analysis of the Model

Since

Using both quotient and product rules for derivatives:

(1.11)

(1.11a)[1]

Now recall (1.10):

(1.10)

Substituting (1.10) and the given growth rates for A and L into (1.11a):

(1.12)

(1.13)

Equation (1.13) is the key equation in the Solow model. It describes how the capital stock evolves over time. The first term on the RHS is the amount of investment per worker. The second term is the amount of investment that would be needed to keep k constant.

Two diagrams are useful here: Figure 1.2 and Figure 1.3 in Romer. (Note that these diagrams are drawn to satisfy the Inada conditions).

Balanced Growth Path

If , then and if , then . Therefore, over time k will approach .

When , . At such a point L and A are growing at constant rates n and g and K is growing at rate . Since both capital and effective labor are growing at rate , by CRS, output is also growing at rate . and are growing at rate g. We see that each variable in the model grows at a steady rate when .

At such a point, the economy is on a balanced growth path. On this path, the growth rate of output per worker depends only technical progress.

Some Stylized Facts

Growth rates of labor, capital, and output are constant.

Output and capital grow at about the same rate, so the capital-output ratio is roughly constant.

Output and capital grow faster than labor, so output per worker and capital per worker are rising.

These facts are compatible with the Solow growth model.

Comparative Dynamics

We will consider changes in the savings rate, the key model parameter. Policymakers might be able to influence this parameter by changing tax rates or the amount and/or composition of government spending.

Suppose the savings rate increases. This causes the line to shift upward. Investment exceeds its break-even level, so k begins to increase and does so until it reaches a new higher level of . During the transition, output per worker grows faster than A, because of the rise in k. So a permanent increase in the savings rate produces a temporary increase in the growth of output per worker.

The initial impact of the increase in s is to reduce c. Since output is not initially change, the added saving must reduce consumption. As k grows, y grows, and c grows. However, when the new balanced growth path is reached, it is questionable whether c is higher than before (illustrate via diagram).

The golden-rule level of occurs at the savings rate that maximizes c (illustrate via diagram).

Calibration Experiments

When the savings rate changes, the economy moves to a new balanced growth path. But how much does a change in s affect y, and how quickly?

With plausible functional forms and parameter values, Romer concludes that:

A significant (10%) increase in the saving rate has a modest (5%) effect on output along the balanced growth path.

Following a change in the savings rate, convergence to a new balanced growth path is slow. Half of the movement toward the new growth path is accomplished in 18 years.

These results seem to imply that it is difficult to increase an economy’s standard of living by way of higher saving.

Implications

The results noted above suggest that variations in s and k will probably account for little of the variation in growth and output across countries. Instead, variations in the productivity parameter, A, must account for most of the variation in output. In the Solow model, variations in A are not explained.

What is A?

The stock of knowledge?

Education of the labor force (human capital)?

Quality of infrastructure (public capital)?

Institutions regulating and enforcing property rights?

Solow on Growth Accounting

Also see the Solow Handout (in lieu of the discussion on pp. 26-27 in Romer).

Recall the production function:

(1.1)

Differentiate with respect to time:

, (1.28)

where:

and .

Divide on both sides by and rearrange to get:

(1.29a)

(1.29b)

Note that the ’s are elasticities of output with respect to the indicated inputs.

Subtracting from each side, and noting that , we obtain:

The rate of growth of the output/labor ratio depends on the rate of growth in the capital/labor ratio and a residual, the Solow residual. Everything in the equation above is fairly easily measured, except for . But that means that can be determined as a residual. One can use this equation to decompose growth into portions due to changes in capital per worker and changes due to technical change.

Empirical Results from Solow’s paper:

For the period 1909-29: Technical change accounts for 0.90 percentage points of per capita growth per year.

1930-1949: Technical change accounts for 2.25 percentage points of per capita growth per year.

Technical change accounts for 7/8 of per capita growth; increased capital per worker accounts for just 1/8.

Technical change appears to be highly variable from year to year.

Economic Convergence

Why should we expect “convergence” across economies (i.e. economies should be similar in terms of income, capital, etc.)?

Each country should approach its balanced growth path.

Countries with lower capital stocks will have a higher marginal product of capital, and will attract investment.

The ability to emulate best technology should allow convergence toward a common level for A.

Empirical evidence on the hypothesis of convergence is partly contradictory, but it seems clear that not all poor countries are in a process of catching up.

7

[1] This can also be written as:

.