Econ 112, Foldvary

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)

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.

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

spending = 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.

Economic Logic chapter 17 emphasizes frugality,

which implies spending wisely and a high amount of savings.

High savings leads to high investment and economic growth.

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.

Economic Logic says on p. 375 at the top,

capital is the key to prosperity, and the origin is in savings.

Savings creates a demand for future consumption.

Investments are made today in order to supply goods for future consumption.

Higher savings shifts some production from current to future consumption.

The decrease in consumption is offset by greater investment.

But there also needs to be investment in human capital - education and training.

But while investment increases capital goods, depreciation reduces capital goods.

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

▵k = i - δk

▵k = 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, ▵k = .2

Next year we start with k = 4.2

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

▵k = 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

So why have economies kept growing rather than getting stuck in the steady state?

The answer is technological progress, also referred to as the Solow residual.

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.

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.

In many less developed countries, governments have imposed barriers to development,

in the form of high taxes, restrictions, permit requirements, and bureaucratic requirements,

as discussed on page 107 of Science of Economics, chapter 14 on economic growth,

The land tenure system also keeps people poor, as the ownership of the best land is concentrated in a few hands.

Also, corruption, crime, and violence prevent growth and cause misery.

The key to economic growth is economic freedom, including the protection of property rights.


Economic freedom includes property rights and free trade.

Economic logic, bottom of page 374, quotes Adam Smith in saying that

for a nation to increase its wealth, little else is needed than peace, easy taxes, and justice.

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

But at the end of the chapter, Science of Economics points out that some persons may want to preserve their way of life, and avoid economic development. It should not be forced on people.

Governments try to stimulate growth by increasing demand or by increasing supply.

The increase demand by spending money, as with the recent federal stimulus bills.

Supply-side policy seeks to increase supply be reducing the costs of taxes and regulations.

Supply-side policy is based on Say’s Law, put forth by the French economist

Jean Baptiste Say, 1803.

Say’s law states that production provides income to the factors, which then enables the factors to buy goods. Therefore greater supply results in greater demand.

So there is no such thing as a general oversupply of goods.\

However, without the demand to consume, there would be no production.

So consumption and production co-determine each other.