#### Understanding the Growth Model and Growth of the Capital-Output Ratio

Understanding the Growth Model:

Economists' first instinct when analyzing any model is to look for a point of equilibrium. In the study of long-run growth, however, the key economic quantities are never stable. They are growing over time. The efficiency of labor is growing, the level of output per worker is growing, the capital stock is growing, the labor force is growing. How, then, can we talk about a point of equilibrium where things are stable if everything is growing?

The answer is to look for an equilibrium in which everything is growing together, at the same proportional rate. Such an equilibrium is one of steady-state balanced growth. If everything is growing together, then the relationships between key quantities in the economy are stable. And it makes this chapter easier if we focus on one key ratio: the capital-output ratio. Thus our point of equilibrium will be one in which the capital-output ratio is constant over time, and toward which the capital-output ratio will converge if it should find itself out of equilibrium.

How Fast Is the Economy Growing?

So how fast are the key quantities in the economy growing? Determining how fast the quantities in the economy are growing is straightforward if we remember our three mathematical rules:

a) The proportional growth rate of a product --P x Q, say--is equal to the sum of the proportional growth rates of the factors, is equal to the growth rate of P plus the growth rate of Q.

b) The proportional growth rate of a quotient--E/Q, say--is equal to the difference of the proportional growth rates of the dividend (E) and the divisor (Q).

c) The proportional growth rate of a quantity raised to a exponent--Qy, say--is equal to the exponent (y) times the growth rate of the quantity (Q).

The Growth of Capital per Worker:

Begin with capital per worker. To save on our breath and reduce the length of equations, let's use the expression g(kt) to stand for the proportional growth rate of capital per worker. The proportional growth rate is simply what output per worker will be next year minus what output per worker is this year, all divided by what output per worker is this year:

Capital-per-worker is a quotient: it is the capital stock divided by the labor force. Thus the proportional growth rate of capital-per-worker is the growth rate of the capital stock minus the growth rate of the labor force.

The growth rate of the labor force is simply the parameter n. That's what the parameter n is. The growth rate of the capital stock is a bit harder to calculate. We know that it is:

And we know that we can write next year's capital stock as equal to this year's capital stock, plus gross investment, minus depreciation:

If we substitute in for next year's capital stock, and rearrange:

Then we see that the proportional growth rate of capital per worker is:

To make our equations look simpler, let’s give the capital-output ratio K/Y a special symbol: k--a little k with a short stem (actually the Greek letter kappa)--and write that the proportional growth rate of capital per worker is:

This says that the growth rate of capital-per-worker is equal to the savings share of GDP (s) divided by the capital-output ratio (k, minus the depreciation rate (δ), minus the labor force growth rate (n). It goes through example calculations of what the growth rate of capital-per-worker is for sample parameter values. A higher rate of labor force growth will reduce the rate of growth of capital per worker: more workers means the available capital has to be divided up more ways. A higher rate of depreciation will reduce the rate of growth of capital per worker: more capital will rust away. And a higher capital-output ratio will reduce the proportional growth rate of capital per worker: the higher the capital-output ratio, the smaller is investment relative to the capital stock.

The Growth of Output per Worker:

Our Cobb-Douglas form of the production function tells us that the level of output per worker is:

Output per worker is the product of two terms, each of which is a quantity raised to a exponential power. So using our mathematical rules of thumb the proportional growth rate of output per worker--call it g(yt) to once again save on space:

α times the proportional growth rate of capital per worker,
plus (1 - α times the rate of growth of the efficiency of labor.

The rate of growth of the efficiency of labor is simply g. And the previous section calculated the growth rate of capital per worker g(k): s/kt - δ - n.

So simply plug these expressions in:

And simplify a bit by rearranging terms:

The Growth of the Capital-Output Ratio:

Now consider the capital-output ratio kt. It will be the key ratio that we will focus on--and our equilibrium will be when it is stable and constant. The capital-output ratio is equal to capital per worker divided by output per worker. So its proportional growth rate is the difference between their growth rates:

Which simplifies to:

Thus the growth rate of the capital-output ratio depends on the balance between the investment requirements—(n+g+δ)--and the investment effort—s--made in the economy.

From the growth rate of the capital-output ratio:

We can see that whenever the capital-output ratio kt is greater than s/(n+g+δ), the growth rate of the capital-output ratio will be negative. Output per worker will be growing faster than capital per worker. And the capital-output ratio will be shrinking. By contrast, we can also see that whenever the capital-output ratio kt is less than s/(n+g+δ), the capital-output ratio will be growing.

What happens when the capital-output ratio kt is equal to s/(n+g+δ)? Then the growth rate of the capital-output ratio will be zero. It will be stable, neither growing nor shrinking. If the capital-output ratio is at that value, it will stay there. If the capital-output ratio is away from that value, it will head toward there. No matter where the capital-output ratio kt starts, it will head for--converge to--home in on-- its steady-state balanced-growth value of s/(n+g+δ).

Thus the value s/(n+g+δ) is the equilibrium level of the capital-output ratio. It is a point at which the economy tends to balance, and to which the economy converges. The requirement that the capital-output ratio equal this equilibrium level becomes our equilibrium condition for balanced economic growth.

And to make our future equations even simpler, give the quantity s/(n+g+δ) that is the equilibrium value of the capital-output ratio a special symbol: k*:

Other Quantities:

When the capital-output ratio kt is at its steady state value of:

k* = s/(n+g+δ),

the proportional growth rates of capital per worker and output per worker are stable too. Output per worker is then growing at a proportional rate g:

g(yt) = g

The capital stock per worker is then growing at the same proportional rate g:

g(kt) = g

The total economy-wide capital stock is then growing at the proportional rate n+g: the growth rate of capital per worker plus the growth rate of the labor force. Real GDP is then also growing at rate n+g: the growth rate of output per worker plus the labor force growth rate.

The Level of Output per Worker On the Steady State Growth Path:

When the capital-output ratio is at its steady-state balanced-growth equilibrium value k*, we say that the economy is on its steady-state growth path. What is the level of output per worker if the economy is on its steady-state growth path?

If we define:

λ = α/(1-α)

and call λ the growth multiplier , then output per worker along the steady-state growth path is equal to the steady-state capital-output ratio raised to the growth multiplier, times the current level of the efficiency of labor:

Thus calculating output per worker when the economy is on its steady-state growth path is a simple three-step procedure:

a) First, calculate the steady-state capital-output ratio, k*=s/(n+g+δ), the savings rate divided by the sum of the population growth rate, the efficiency of labor growth rate, and the depreciation rate.

b) Second, amplify the steady-state capital-output ratio k* by the growth multiplier. Raise it to the λ = (α/(1-α)) power, where α is the diminishing-returns-to-capital parameter.

c) Third, multiply the result by the current value of the efficiency of labor Et, which can be easily calculated because the efficiency of labor is growing at the constant proportional rate g.

And the fact that an economy converges to its steady-state growth path makes analyzing the long-run growth of an economy relatively easily as well:

a) First calculate the steady-state growth path.

b) From the steady-state growth path, forecast the future of the economy: If the economy is on its steady-state growth path today, it will stay on its steady-state growth path in the future (unless some of the parameters--n, g, δ, s, and α shift).

c) If the economy is not on its steady-state growth path today, it is heading for its steady-state growth path and will get there soon.

Thus long-run economic forecasting becomes simple.

How Fast Does the Economy Head For Its Steady-State Growth Path?

The growth rate of the capital-output ratio will be approximately equal to a fraction (1-α) x (n+g+δ) of the gap between the steady-state and its current level.

For example, if (1-α) x (n+g+δ) is equal to 0.04, the capital-output ratio will close approximately 4 percent of the gap between its current level and its steady-state value in a year. If (1-α) x (n+g+δ) is equal to 0.07, the capital-output ratio closes 7 percent of the gap between its current level and its steady-state value in a year. A variable closing 4 percent of the gap each year between its current and its steady-state value will move halfway to its steady-state value in 18 years. A variable closing 7 percent of the gap each year between its current and its steady-state value will move halfway to its steady-state value in 10 years.

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