midterm

Chapter 5: The Solow Growth Model

Hikaru Saijo

University of California, Santa Cruz

5.2 Setting Up the Model: Production

• 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).

Yt = F (Kt , Lt) =

• Output can be used for consumption or investment.

= Yt

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

Capital Accumulation

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

• Capital accumulation equation:

Kt+1 =

• 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

d̄ = 0.10

• Change in capital stock defined as

∆Kt+1 ≡ Kt+1 − Kt

• Thus:

∆Kt+1 =

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

Labor

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

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

Lt = L̄

Investment

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

It = s̄Yt

• Therefore:

Ct =

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

• Unknowns/endogenous variables:

1 Production function:

2 Capital accumulation: ∆Kt+1 =

3 Labor force: Lt = L̄

4 Resource constraint: Yt =

5 Allocation of resources: It =

• Parameters/exogenous variables: Ā, s̄, d̄ , L̄, K̄0

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

• Saving • The difference between income and consumption • Is equal to investment

Yt − Ct︸ ︷︷ ︸ saving

= It︸︷︷︸ investment

• 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 .

5.4 Solving the Solow Model

• 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.

• Combine the investment allocation and capital accumulation equation.

∆Kt+1 =

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

Yt =

• 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

s̄Y d̄K

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

s̄Y = s̄ ĀK 1/3L̄2/3

Drawing the Solow diagram

Using the Solow Diagram

• If the amount of is greater than the amount of : • 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 is greater than , 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.

Output and Consumption in the Solow Diagram

• As capital 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.

Drawing the Solow diagram with output.

Solving Mathematically for the Steady State

• In the steady state, investment equals depreciation.

s̄Y ∗ = d̄K ∗

• Sub in the production function

= d̄K ∗

Solving for K ∗

The steady-state level of capital is

• Positively related with

•

•

•

• Negatively related with

•

Solving for Y ∗

The steady-state level of output is

• Positively related with

•

•

•

• Negatively related with

•

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

y ∗ ≡ Y ∗

L∗ =

• 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.

By what proportion does per capita output change in the long run in response to the following changes?

1 Saving rate decreases by 10%.

2 The productivity level falls by 20%.

3 The capital stock increases by 50% as a result of foreign investment.

5.5 Looking at Data through the Lens of the Solow

Model: The Capital-Output Ratio

• Recall the steady state.

s̄Y ∗ = d̄K ∗

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

K ∗

Y ∗ =

s̄

d̄

• Investment rates vary across countries.

• It is assumed that the depreciation rate is relatively constant.

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:

y ∗rich y ∗poor︸ ︷︷ ︸ 64

=

( Ā∗rich Ā∗poor

)3/2 ︸ ︷︷ ︸

32

× ( s̄∗rich s̄∗poor

)1/2 ︸ ︷︷ ︸

2

5.6 Understanding the Steady State

• The economy reaches 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.

5.7 Economic Growth in the Solow Model

• 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

• 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

5.8 Some Economic Experiments

• 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.

An Increase in the Investment Rate

• Suppose the investment rate increases permanently for exogenous reasons.

s̄ −→ s̄ ′

• What happens to the economy (capital and output) over time and in the long run?

• Use Solow diagram to answer the question.

A Rise in the Depreciation Rate

• Suppose the depreciation rate is exogenously shocked to a permanently higher rate.

d̄ −→ d̄ ′

• What happens to the economy (capital and output) over time and in the long run?

• Use Solow diagram to answer the question.

A Rise in the Productivity

• Suppose the productivity is exogenously shocked to a permanently higher rate.

Ā −→ Ā′

• What happens to the economy (capital and output) over time and in the long run?

• Use Solow diagram to answer the question.

A Destruction of Capital Stock

• Suppose the capital stock is exogenously shocked to a lower level (due to, for example, a natural disaster).

K −→ K ′

• What happens to the economy (capital and output) over time and in the long run?

• Use Solow diagram to answer the question.

5.9 The Principle of Transition Dynamics

• If an economy is below • It will grow.

• If an economy is above • 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 further below its steady state an economy is, (in percentage terms) • the the economy will grow

• The further above its steady state • the 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.

• For the world as a whole, on average, rich and poor countries grow at the same rate.

• Two implications of this: • Most countries (rich and poor) 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 (determinants of the steady state invest rates and A).

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 • 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

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