2. Physical Capital

The physical objects that make workers more productive in producing output

Includes:

-machines in factories

-buildings in which we work

-infrastructures (roads, ports, bridges, etc.)

-vehicles

-computers

1. Capital is productive: enables workers to produce more output

2. Capital has to be produced/accumulated through investment: requires the sacrifice of some consumption

3. Capital is rival in its use: only a limited number of people can use a given piece of capital at one time

4. Capital can earn a return

5. Capital depreciates with usage and due to passage of time

Workers with more capital can produce more output

Differences in the quantity of capital are a natural explanation to consider for differences in income among countries

Basis of the Solow model

-Created in 1956 independently by both Robert Solow and Trevor Swan

-Production function tells us how labor and capital are transformed into output

-Solow model illustrates the importance of physical capital in explaining differences among countries in their levels of income per capita

-Theory based around capital accumulation

-Useful framework to explain differences in income-per-worker across countries

-Cannot explain all of the phenomenon we observe but still instructive

1. Constant returns to scale, if we increase the amount of each input by z 1

2. Diminishing marginal product

Alpha is the relative importance of K & L for the Output produced

It represents the share of K & L in total output. Alpha = 0,3 implies that 30% of the output is attributed to capital. It doesn’t directly indicate absolute quantities of K & L.

• α is the output elasticity of labor, indicating the responsiveness of output to changes in the amount of labor input.

• The value of α determines the share of total output attributable to labor. If α=0.6, for instance, it implies that 60% of the total output is due to the labor input.

If we increase all inputs by a given proportion z > 1, and it:

1. increases output more than proportionally, then the technology is said to exhibit increasing returns to scale in production or economies of scale. The average cost per unit decreases.

   F(zK,zL) > zF(K,L)

2. increases output less than proportionally, then the technology exhibits decreasing returns to scale in production or diseconomies of scale. The average cost per unit increases.

   F(zK,zL) < zF(K,L)

3. increases output proportionally, then the technology exhibits constant returns to scale in production. The average cost per unit is constant.

   F(zK,zL) = zF(K,L)

   
   

The marginal product of capital (MPK) is the first derivative of output with respect to capital

The first derivative is the slope of the production function

To check for diminishing marginal product, we compute the second derivative with respect to capital

MPK is smaller at higher levels of K than at lower levels of K

If the second derivative for K is:

negative -> Diminishing marginal product

positive -> Increasing marginal product

0 -> Constant marginal product

We assume that firms are perfectly competitive

-There are many firms, all producing the same homogenous output Firms enter and exit freely

-They all produce using a similar Cobb-Douglas function

-They are all price-takers for the use of labor and capital

Firms maximize profit taking prices as given:

To solve the maximization problem take the derivatives with respect to K and L and equate them to zero.

First order Conditions:

From the first-oder conditions:

Notice that total payments to factors are equal to total output:

This means that firms all have zero profits:

This is consistent with our assumption that firms are perfectly competitive with each other, and firms enter and exit freely. If there were profits, more firms would enter and compete them away.

Factor shares are the fraction of total output that is paid to each factor

Factor shares of output are thus constant when we use the Cobb-Douglas, regardless of the amount of K or L, consistent with the stylized facts on factor shares

Those facts suggest that α = 1/3 and (1 − α) = 2/3, roughly

-Assume that the quantity of labor (L) is constant over time (i.e. no population growth)

-Assume that the production function itself does not change over time (i.e. no improvement in productivity)

-> Thus, all of the action in the Solow model comes from changes in capital

-> Depreciation (the wearing out of old capital)

-> Investment (the building of new capital)

-We have a capital stock K this year

-If we make zero new investment, the capital stock next year will be:

Where δ is the depreciation rate of capital

If we save s share of today’s income (Yt) and invest this into capital, then the capital stock next year will be:

The change in capital from today to next year is then:

By dividing by L and expressing it in per-worker terms, we obtain:

Example:

If investment is equal to depreciation, then the capital stock will not change over time (i.e. ∆kt+1 = 0)

-The point at which this happens is called the steady state

-1:1 relation between k and y (recall: y=f(k) )

-In the steady state: No chance in income per worker

First, what do we mean by “solve”? We mean that we want to be able to express the endogenous variables in terms of only exogenous ones

Endogenous variables - things we are trying to explain

1. Output Y and/or output per worker y

2. Capital K and/or capital per worker k

Exogenous variables - things we take as given

1. α, capital’s share in output

2. s, the savings rate

3. δ, the depreciation rate

4. k0, the initial level of capital per worker

We can solve the Solow model for:

1. The steady state level of capital per worker

2. The steady state level of output per worker

Model equations in per-worker specification:

Steady state equilibrium condition:

Solution:

-Consider two countries i and j at their steady-state levels of income per worker

-Assume that they have the same levels of productivity (A) and the same rates of depreciation (δ)

-The only difference is in their investment rates (s)

At the steady-state:

Example:

Country i’s investment rate is 20%, country j’s investment rate is 5%. Use α = 1/3. What is the ratio of country i’s income to that of country j?

-In this simple version of the Solow model (with no growth in productivity), once a country reaches its steady state, there is no longer growth (i.e. income per worker will remain constant)

-Therefore, all of the growth we observe in the Solow model will be transitional - it will occur during the transition to a steady state

-Still, we can say something about relative growth rates, i.e. why some countries grow faster than others

-Think about countries that are not in the steady state

-Growth rate of capital per worker is defined as:

Using our model of capital accumulation:

Let us graph the two terms on the right-hand side of this equation (Speed of convergence):

Consider two countries with the same productivity level:

-If two countries have the same rate of investment but different levels of income, the country with lower income will have higher growth

-If two countries have the same level of income but different rates of investment, the country with a higher rate of investment will have higher growth

-A country that raises its level of investment will experience an increase in its rate of income growth

-Suppose all countries have the same parameters underlying the Solow model

-Under that assumption, prediction of the Solow model is that all countries will eventually reach the same steady state level of income per worker

-Implies countries that are initially poorer should grow faster than countries

that are initially richer. This is known as β convergence

-There seems to be lack of absolute convergence of countries to the same steady state

-However, the assumption that all countries have the same technology and tastes is very strong

-If one instead focuses on countries with similar characteristics, such as OECD economies you get a different picture

-Barro and Sala-i Martin estimated that when one controls for differences in characteristics, countries (with same human capital and life expectancy) converge to each other by about 2 percent per year

Convergence may be found only if we hold constant (i.e. “condition”) on certain fundamental differences between countries

Saving rate by decile of income per capita

There is a strong relationship between saving rates and income per capita

Suppose that poor countries have a naturally lower saving rate than rich countries

Assume no flows of investment among countries: in every country (saving = investment)

The saving rate depends on the level of income:

Two countries identical in terms of the underlying determinants of their incomes could end up with different steady-state levels of income per capita

A country at the lower steady state can be “trapped” there

Example of multiple steady states: a country’s initial position determines which of several possible steady states it will move toward

-Examine the role of capital in economic growth in the Solow model

-Limitation of this simple version of the Solow model

-> Take savings/investment rates as exogenous: unable to say anything about the source of differences in investment rates

-> No growth in the long run: once reaching the steady-state countries do not grow at all!

-> Next lectures: expand the Solow model to accommodate other factors of production, differences in productivity, and technological change over time

-We can express the model in continuous time using calculus

-So far, time was divided into discrete periods

-It is often more convenient to shrink the length of periods to 0

-Difference equations then become differential equations

Math reminders:

-Chain rule

-Product rule

-Growth rates