Econ 2, Foldvary

Chapter 12, Production and Growth

The mathematics of growth:

How long is half of forever?

How long it takes for growth to occur: F=Peit

F: future amount; P: present amount; i: rate of growth; t: time

F/P = eit

If F/P is 2, then 2 = eit

The natural log of (2) = .69 = i*t

Growth and Productivity

The ranking of countries has changed as rich economies decline and poor ones advance.

The production function:

Y = A*f(L,K,H,N) (p. 253)

Output is a function of the factors.

L: quantity of labor; H human capital; N: natural resources; K capital goods

The natural resources of an economy do not limit growth, as they can be imported.

 The physical relationship between output and inputs,

A: scale due to technology, returns to scale, government policy:

Productivity: measured by the quantity of goods produced per worker.

Technology is embedded in capital goods and human capital.

The amount and productivity of the factors and the technology and production methods determine productivity.

The large differences in wealth among countries is mostly due to differing productivity.

Productivity is affected by government policy:

            taxes and restrictions can reduce productivity.

Robert Solow’s model of economic growth.

Based on savings, population growth, and technological progress.

Simplify components of GDP to Y=A*f(K, L)

K: capital goods. L: labor.

Hold technology, land, fixed.

Assume constant returns to scale: zY = A*f(zK, zL)

Output per worker depends on the amount of K per worker.

Y/L = A*f(K/L, L/L)

Y/L = f(K/L, 1) = A*f(K/L)

The Cobb-Douglas function is:

Y = A*KαLβ

Alpha and Beta are elasticities between zero and one.

With constant returns to scale, alpha + beta = 1.

Simplify: set A equal to 1.

Express all quantities relative to the size of the labor force:

Use lower-case letters for quantities per worker: y=Y/L, k= K/L

y = f(k)

There is eventually diminishing marginal product, with other factors held constant, in accord with the law of diminishing returns, or law of diminishing marginal product.

(Fig. 1)

Graphed, y increases with k, but at a decreasing rate.

The slope is the marginal product of capital goods.

MPK = f(k+1) - f(k)

The demand side: spending is either for consumption or for investment.

Economic meanings: using up; creation of; economic value.

y = c + i

income = consumption plus investment

The savings rate is s, between zero and one.

The consumption function:

c = (1-s)y.

y = (1-s)y + i

i = sy

Investment = savings = fraction of income not spent on consumption.

How increases in K affect growth:

Investment per worker is capital goods per worker:

i = s*f(k)

K increases with investment, and decreases with depreciation.

The depreciation rate: δ

The fraction of capital goods that wears out per year.

If a machine lasts 20 years, what is the depreciation rate?

The amount of capital goods that depreciates each year is δk

Δk = i - δk

Δk = s*f(k)- δk

The steady state is the long-run equilibrium.

The steady-state of growth is where investment equals depreciation.

If there is little K, K grows. If there is much K, K shrinks as depreciation > investment.

It’s because investment has diminishing returns, while depreciation is constant rate.

Suppose Y = K.5L.5

Y/L = (K.5L.5)/L

y = k.5 Output per worker equals the square root of K/L

Assume s = .3 and δ = .1 and k=4

How much is output per worker? y = 2 units

How much is consumption? c = (1-s)y = 1.4

How much is investment? i = .6

How much is depreciation? δk = .4

With i = .6 and δk = .4, Δk = .2

Next year we start with k = 4.2

But with diminishing marginal product of capital goods, eventually growth stops.

Δk = s*f(k)- δk = 0 in steady state.
k/f(k) = s/δ
k/k.5 = .3/.1
k = 9
In the steady state, k = 9, and i = δk

The Solow residual is a number explaining productivity growth in an economy aside from an increase in inputs.
It is a "residual" because it is the part of growth that cannot be explained through capital accumulation or an increase in labor or a new discovery of natural resources.
The Solow Residual is sometimes called the rate of growth of total factor productivity.
The Solow Residual is caused by better technology, greater economies of scale, better education, and more economic freedom (less deadweight loss due to lower taxes on labor and enterprise or fewer restrictive regulations).

An economy can develop rapidly and become wealthy with few natural resources if the economic policy allows entrepreneurs to produce, i.e. if there is a large degree of economic freedom.

Some countries with a lot of natural resources have declined because of poor policies.

Examples: Nigeria, Venezuela, Nauru.

Nauru, a small island country in the Pacific Ocean, had one major natural resource: phosphates, used in detergents and other cleaning chemicals.

While that was being mined, the country was wealthy, but it spent its wealth lavishly on airplanes, travel, and other luxuries. Now the phosphate has been depleted and the country is poorer and will be even poorer in the future. Also, the mining has destroyed the natural environment. 80% of the land is uninhabitable. Some of the income from mining was put in a trust, but the trust has lost millions of dollars through failed investments, speculation in the Tokyo stock market, and international financial scams.

In contrast, Taiwan developed rapidly since 1950 with few natural resources, but with the right economic policies.

The key to economic growth is economic freedom,

including the protection of property rights.


Economic freedom includes property rights and free trade.

Especially important is economic opportunity for women, which is lacking in poor countries.