5.1 Introduction• In this chapter, we learn:

– How capital accumulates over time.– How diminishing MPK explains differences

in growth rates across countries. – The principle of transition dynamics.– The limitations of capital accumulation, and

how it leaves a significant part of economic growth unexplained.

• The Solow Growth Model:

– Builds on the production model by adding a theory of capital accumulation

– Was developed in the mid-1950s by Robert Solow of MIT

– Was the basis for the Nobel Prize he received in 1987

• Additions / differences with the model:

– Capital stock is no longer exogenous.– Capital stock is now “endogenized.”– The accumulation of capital is a possible

engine of long-run economic growth.

5.2 Setting Up the Model

• Start with the previous production model– Add an equation describing the accumulation

of capital over time.• The production function:

– Cobb-Douglas– Constant returns to scale in capital and labor– Exponent of one-third on K

• Variables are time subscripted (t).

Production

• Output can be used for consumption or investment.

• This is called a resource constraint.– Assuming no imports or exports

Consumption Investment Output

• Goods invested for the future determines the accumulation of capital.

• Capital accumulation equation:

Capital Accumulation

Next year’s capital

This year’s capital

Investment Depreciation rate

• Depreciation rate

– The amount of capital that wears out each period

– Mathematically must be between 0 and 1 in this setting

– Often viewed as approximately 10 percent

• Stock– A quantity that survives from period to

period.• tractor, house, factory

• Flow– A quantity that lasts a single period

• meals consumed, withdrawal from ATM

• A change in stock is a flow of investment.

• Change in capital stock defined as

• Thus:

• The change in the stock of capital is investment subtracted by the capital that depreciates in production.

• To understand capital accumulation, we must assume the economy begins with a certain amount of capital, K0.

• Suppose:– The initial amount of capital is 1,000

bushels of corn.– The depreciation rate is 0.10.

Case Study: An Example of Capital Accumulation

• Saving

– The difference between income and consumption

– Is equal to investment

• Farmers eat a fraction of output and invest the rest.

• Therefore:

– Consumption is the share of output we don’t invest.

Investment

FractionInvested

• To keep things simple, labor demand and supply not included

• The amount of labor in the economy is given exogenously at a constant level.

Labor

• Differences between Solow model and production model in previous chapter:– Dynamics of capital accumulation added– Left out capital and labor markets, along

with their prices• Why include the investment share but

not the consumption share?– No need to—it would be redundant– Preserve five equations and five

unknowns

Case Study: Some Questions about the Solow Model

5.3 Prices and the Real Interest Rate

• If we added equations for the wage and rental price, the following would occur:

– The MPL and the MPK would pin them.– Omitting them changes nothing.

• The real interest rate

- The amount a person can earn by saving one unit of output for a year

- Or, the amount a person must pay to borrow one unit of output for a year

- Measured in constant dollars, not in nominal dollars

• A unit of investment becomes a unit of capital

- The return on saving must equal the rental price of capital.

• Thus:

- The real interest rate equals the rental price of capital which equals the MPK.

• The model needs to be solved at every point in time, which cannot be done algebraically.

• Two ways to make progress– Show a graphical solution– Solve the model in the long run

• We can start by combining equations to go as far as we can with algebra.

5.4 Solving the Solow Model

• Combine the investment allocation and capital accumulation equation.

Depreciation

Investment

• Substitute the fixed amount of labor into the production function.

• We have reduced the system into two equations and two unknowns (Yt, Kt).

• The Solow Diagram– Plots the two terms that govern the change

in the capital stock

– New investment looks like the production functions previously graphed but scaled down by the investment rate.

Using the Solow Diagram• If the amount of investment is greater

than the amount of depreciation:– The capital stock will increase until

investment equals depreciation.• here, the change in capital is equal to 0• the capital stock will stay at this value of capital

forever• this is called the steady state

• If depreciation is greater than investment, the economy converges to the same steady state as above.

• Notes about the dynamics of the model:–When not in the steady state, the

economy exhibits a movement of capital toward the steady state.

–At the rest point of the economy, all endogenous variables are steady.

–Transition dynamics take the economy from its initial level of capital to the steady state.

• As K moves to its steady state by transition dynamics, output will also move to its steady state.

• Consumption can also be seen in the diagram since it is the difference between output and investment.

Output and Consumption in the Solow Diagram

• In the steady state, investment equals depreciation.

• Sub into the production function

Solving Mathematically for the Steady State

• Solve for K*

• The steady-state level of capital is– Positively related with the

• investment rate• the size of the workforce• the productivity of the economy

– Negatively correlated with• the depreciation rate

• Plug K* into the production function to get Y*.

• Plug in our solved value of K*.

• Higher steady-state production– Caused by higher productivity and

investment rate• Lower steady-state production

– Caused by faster depreciation

• Finally, divide both sides of the last equation by labor to get output per person (y) in the steady state.

• Note the exponent on productivity is different here (3/2) than in the production model (1).– Higher productivity has additional effects

in the Solow model by leading the economy to accumulate more capital.

• Recall the steady state.

• The capital to output ratio is the ratio of the investment rate to the depreciation rate:

• Investment rates vary across countries.• It is assumed that the depreciation rate is

relatively constant.

5.5 Looking at Data through the Lens of the Solow Model

The Capital-Output Ratio

Differences in Y/L• The Solow model gives more weight to TFP in

explaining per capita output than the production model.

• We can use this formula to understand why some countries are so much richer.

• Take the ratio of y* for two countries and assume the depreciation rate is the same:

From Chapter 4 See figure 5.3 (next slide)

• We find that the factor of 66 that separates rich and poor countries’ income per capita is decomposable:

– TFP differences– Investment differences

• The economy reachs a steady state because investment has diminishing returns.– The rate at which production and investment

rise is smaller as the capital stock is larger.• Also, a constant fraction of the capital stock

depreciates every period.– Depreciation is not diminishing as capital

increases.• Eventually, net investment is zero.

– The economy rests in steady state.

5.6 Understanding the Steady State

• Important result: there is no long-run economic growth in the Solow model.

• In the steady state, growth stops, and all of the following are constant:– Output– Capital– Output per person– Consumption per person

5.7 Economic Growth in the Solow Model

• Empirically, however, economies appear to continue to grow over time.– Thus, we see a drawback of the model.

• According to the model:– Capital accumulation is not the engine of

long-run economic growth.– After we reach the steady state, there is no

long-run growth in output.– Saving and investment

• are beneficial in the short-run• do not sustain long-run growth due to

diminishing returns

• Harvest starts with a small stock of seed.– Grows larger each year, for a time– Settles down to a constant level

• Diminishing returns– A fixed number of farmers cannot harvest

huge amounts of corn.– Growth eventually stops.

Meanwhile, Back on the Family Farm

• Can growth in the labor force lead to overall economic growth?– It can in the aggregate.– It can’t in output per person.

• The presence of diminishing returns leads capital per person and output per person to approach the steady state.– This occurs even with more workers.

Case Study: Population Growth in the Solow Model

• The Solow model:

– Does not explain long-run economic growth

– Does help to explain some differences across countries

• Economists can experiment with the model by changing parameter values.

5.8 Some Economic Experiments

• Suppose the investment rate increases permanently for exogenous reasons.– The investment curve

• rotates upward– The depreciation curve

• remains unchanged– The capital stock

• increases by transition dynamics to reach the new steady state

• this happens because investment exceeds depreciation

– The new steady state• is located to the right• investment exceeds depreciation

An Increase in the Investment Rate

– The capital stock• increases by transition dynamics to

reach the new steady state• this happens because investment

exceeds depreciation

– The new steady state• is located to the right• investment exceeds depreciation

An Increase in the Investment Rate

• What happens to output in response to this increase in the investment rate?

– The rise in investment leads capital to accumulate over time.

– This higher capital causes output to rise as well.

– Output increases from its initial steady-state level Y* to the new steady state Y**.

A Rise in the Depreciation Rate• Suppose the depreciation rate is

exogenously shocked to a higher rate.– The depreciation curve

• rotates upward– The investment curve

• remains unchanged– The capital stock

• declines by transition dynamics until it reaches the new steady state

• this happens because depreciation exceeds investment

– The new steady state• is located to the left

• What happens to output in response to this increase in the depreciation rate?– The decline in capital reduces output.– Output declines rapidly at first, and then

gradually settles down at its new, lower steady-state level Y**.

Experiments on Your Own

• Try experimenting with all the parameters in the model:– Figure out which curve (if either) shifts.– Follow the transition dynamics of the Solow

model.– Analyze steady-state values of capital (K*),

output (Y*), and output per person (y*).

• Hiroshima and Nagasaki– Returned close to their original economic

position in just a few decades• Vietnam

– In both villages that were bombed or left untouched, poverty, literacy, and consumption were similar 30 years after the war.

• Implications of Solow growth model?

Case Study: Wars and Economic Recovery

5.9 The Principle of Transition Dynamics

• If an economy is below steady state– It will grow.

• If an economy is above steady state.– Its growth rate will be negative.

• When graphing this, a ratio scale is used.– Allows us to see that output changes more

rapidly if we are further from the steady state– As the steady state is approached, growth

shrinks to zero.

• The principle of transition dynamics–The farther below its steady state an

economy is, (in percentage terms)• the faster the economy will grow

–The farther above its steady state• the slower the economy will grow

–Allows us to understand why economies grow at different rates

Understanding Differences in Growth Rates

• Empirically, for OECD countries, transition dynamics holds:– Countries that were poor in 1960 grew quickly.– Countries that were relatively rich grew slower.

• Looking at the world as whole, on average, rich and poor countries grow at the same rate.– Two implications of this:

• most countries have already reached their steady states

• countries are poor not because of a bad shock, but because they have parameters that yield a lower steady state

• South Korea– 6 percent per year– Increased from 15 percent of U.S. income

to 75 percent• Philippines

– 1.7 percent per year– Stayed at 15 percent of U.S. income

• Transition dynamics predicts– South Korea must have been far below its

steady state.– Philippines is already at steady state.

Case Study: South Korea and the Philippines

• Assuming equal depreciation rates

• The long-run ratio of per capita incomes depends on– The ratio of productivities (TFP levels) – The ratio of investment rates

5.10 Strengths and Weaknesses of the Solow Model

• The strengths of the Solow Model:– It provides a theory that determines how

rich a country is in the long run.• long run = steady state

– The principle of transition dynamics• allows for an understanding of

differences in growth rates across countries

• a country further from the steady state will grow faster

• The weaknesses of the Solow Model:– It focuses on investment and capital

• the much more important factor of TFP is still unexplained

– It does not explain why different countries have different investment and productivity rates.• a more complicated model could

endogenize the investment rate – The model does not provide a theory of

sustained long-run economic growth.

Summary• The starting point for the Solow model is

the production model.• The Solow model

– Adds a theory of capital accumulation. – Makes the capital stock an endogenous

variable• The capital stock today

– Is the sum of past investments– Consists of machines and buildings that

were bought over the last several decades

• The goal of the Solow model is to deepen our understanding of economic growth, but in this it’s only partially successful.

• The fact that capital runs into diminishing returns means that the model does not lead to sustained economic growth.

• As the economy accumulates more capital – Depreciation rises one-for-one– Output and therefore investment rise less

than one-for-one• because of the diminishing marginal product of

capital

• Eventually, the new investment is only just sufficient to offset depreciation.– The capital stock ceases to grow.– Out capital stock ceases to grow.– The economy settles down to a steady

state.

• The first major accomplishment of the Solow model is that it provides a successful theory of the determination of capital.– Predicts that the capital-output ratio is

equal to the investment-depreciation ratio• Countries with high investment rates

– Should thus have high capital-output ratios – This prediction holds up well in the data.

• The second major accomplishment of the Solow model is the principle of transition dynamics.– The farther below its steady state an

economy is, the faster it will grow.• Transition dynamics

– Cannot explain long-run growth– Provide a nice theory of differences in

growth rates across countries.• Increases in the investment rate or TFP

– Increase a country’s steady-state position and growth for a number of years

• In general, most poor countries have– Low TFP levels– Low investment rates,

• the two key determinants of steady-state incomes

• If a country maintained good fundamentals but was poor because it had received a bad shock– It would grow rapidly– This is due to the principle of transition

dynamics.

Additional Figures for Worked Exercises

Macroeconomics

This concludes the LectureSlide Set for Chapter 5

byCharles I. Jones

Second Edition

W. W. Norton & CompanyIndependent Publishers Since 1923

Additional Solow graph examples from previous edition of slides

Investment, Depreciation

Capital, Kt

At this point, dKt = sYt, so

The Solow Diagram graphs these two pieces together, with Kt on the x-axis:

Suppose the economy starts at this K0:

Investment, Depreciation

Capital, Kt

K0

• We see that the red line is above the green at K0

• Saving = investment is greater than depreciation

• So ∆Kt > 0 because

• Then since ∆Kt > 0, Kt increases from K0 to K1 >

K0

K1

Now imagine if we start at a K0 here:

Investment, Depreciation

Capital, Kt

K0

• At K0, the green line is above the red line

• Saving = investment is now less than depreciation

• So ∆Kt < 0 because

• Then since ∆Kt < 0, Kt decreases from K0 to K1 < K0

K1

We call this the process of transition dynamics: Transitioning from any Kt toward the economy’s

steady-state K*, where ∆Kt = 0

Investment, Depreciation

Capital, Kt

At this value of K, dKt = sYt, so

K*

No matter where we start, we’ll transition to K*!

We can see what happens to output, Y, and thus to growth if we rescale the vertical axis:

Investment, Depreciation, Income

Capital, KtK*

•Saving = investment and depreciation now appear here

•Now output can be graphed in the space above in the graph

•We still have transition dynamics toward K*

•So we also have dynamics toward a steady-state level of income, Y*

Y*