# The Solow Growth Model

## Chapter Summary

#### Key Concepts

1. The starting point for the Solow model is the production model of Chapter 4. To that framework, the Solow model adds a theory of capital accumulation. That is, it makes the capital stock an endogenous variable.

2. The capital stock is the sum of past investments. The capital stock today consists of machines and buildings that were bought over the last several decades.

3. The goal of the Solow model is to deepen our understanding of economic growth, but 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, but 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, and the capital stock ceases to grow. Output stops growing as well, and the economy settles down to a steady state.

4. There are two major accomplishments of the Solow model. First, it provides a successful theory of the determination of capital by predicting that the capital-output ratio is equal to the investment-depreciation ratio. Countries with high investment rates should thus have high capital-output ratios, and this prediction holds up well in the data.

5. The second major accomplishment of the Solow model is the principle of transition dynamics, which states that the farther below its steady state an economy is, the faster it will grow. While the model cannot explain long-run growth, the principle of transition dynamics provides a nice theory of differences in growth rates across countries. Increases in the investment rate or total factor productivity can increase a country's steady-state position and therefore increases growth, at least for a number of years. These changes can be analyzed with the help of the Solow diagram.

6. In general, most poor countries have low TFP levels and 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, we would see it grow rapidly, according to the principle of transition dynamics.

**1. ****Introduction**

*The Solow growth model*is the starting point to determine why growth differs across similar countries, and it builds on the production model by adding a theory of capital accumulation.- The capital stock is no longer exogenous.
- Rather, it is "endogenized," because it is converted from an exogenous to an endogenous variable.

**2. ****Setting Up the Model**

- We start with the production model from Chapter 4 and add an equation describing the accumulation of capital over time.

**a. ****Production**

- The production function is Cobb-Douglas, has constant returns to scale in capital and labor, and has an exponent of 1/3 on capital.
- Variables in the production function are time subscripted, because they may change over time.
- Output can be used for either consumption or investment.
- A
*resource constraint*describes how an economy can use its resources.

- A

**b. ****Capital Accumulation**

- The
*capital accumulation equation*says that the capital stock next year equals the sum of the capital started with this year plus the amount of investment undertaken this year minus depreciation.*Depreciation*is the amount of capital that wears out each period and the depreciation rate is viewed as approximately 10 percent.- Thus, the change in the capital stock is investment less depreciation.

- The economy is endowed with an initial amount of capital in period 0.

**c. ****Case Study: An Example of Capital Accumulation**

- The capital stock is the sum of past investments.
- The amount by which the capital stock increases each period is smaller and smaller every year.

**d. ****Labor**

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

**e. ****Investment**

- The amount of investment in the economy is equal to a constant investment rate times total output.
- Therefore, consumption equals output times the quantity one minus the investment rate.

**f. ****The Model Summarized**

- The Solow model has five equations and five unknowns.
- The model is dynamic, which means that the five equations hold at all points in time—or, all endogenous variables are time subscripted.

**g. ****Case Study: Some Questions about the Solow Model**

** i. ****What about wages and the rental price of capital?**

- Leaving out the rental price of capital and the wage rate simplifies the model.

** ii. ****Why isn't part of Table 5.2?**

- The resource constraint and the allocation of investment equations imply the above equation; thus, including it would be redundant.

** iii. ****What are stocks and flows?**

- A
*stock*(for example, the capital stock) is a quantity that survives from period to period. - A
*flow*is a quantity that lasts for a single period. - Note that stocks satisfy accumulation equations and that the change in a stock is a flow.

**3. ****Prices and the Real Interest Rate**

- If we added equations for the wage and rental price, the MPL and the MPK would pin them down, respectively.
- Omitting them changes nothing.

- The
*real interest rate*is the amount a person can earn by saving one unit of output for a year, or equivalently the amount a person must pay to borrow one unit of output for a year.- It is measured in constant dollars, not in nominal dollars.

*Saving*is the difference between income and consumption.- Saving equals investment.

- A unit of saving is a unit of investment, which becomes a unit of capital and therefore the return on saving must equal the rental price of capital.
- The real interest rate in an economy is equal to the rental price of capital, which is equal to the marginal product of capital.

**4. ****Solving the Solow Model**

- To solve the model, write the endogenous variables as functions of the parameters of the model.
- Graphically show what the solution looks like and solve the model in the long run.
- Combine the investment allocation equation with the capital accumulation equation.
*Net investment*is investment minus depreciation.- Substitute the supply of labor into the production function.
*The Solow diagram*plots the two parts of the net investment equation.- The investment rate times output looks similar to the production function of Chapter 4.
- Depreciation is a line from the origin.

**a. ****Using the Solow Diagram**

- When the amount of investment is greater than the amount of depreciation, the capital stock will increase.
- The capital stock will rise until investment equals depreciation.
- At this point, the change in capital is equal to 0.
- Absent any shocks, the capital stock will stay at this value of capital forever, and the point where investment equals depreciation is called the
*steady state.*

- The capital stock will rise until investment equals depreciation.
- When not in steady state, the economy obeys
*transition dynamics,*or in other words, the process of capital moving toward a steady state. - Notice that when depreciation is greater than investment, the economy converges to the same steady state as above.
- At the rest point of the economy, all endogenous variables are steady.

**b. ****Output and Consumption in the Solow Diagram**

- Using the production function, it is evident that as capital moves to its steady state by transition dynamics, output will also move to its corresponding steady state by transition dynamics.
- Note that consumption is the difference between output and investment.

**c. ****Solving Mathematically for the Steady State**

- In steady state, investment equals depreciation. If we evaluate this equation at the steady-state level of capital, we can solve mathematically for it.
- The steady-state level of capital is positively related with the investment rate, the size of the workforce, and the productivity of the economy.
- The steady-state level of capital is negatively correlated with the depreciation rate.

- Substituting in the steady-state level of capital into the production function yields an expression for steady-state output.
- Notice the value of steady state output depends on the same parameters as steady-state capital but with different exponents.

- Dividing by the size of the labor force gives us a steady-state level of output per capita.
- Notice that the exponent on the productivity parameter is greater than in the Chapter 4 model.
- This is because a higher productivity parameter raises output as in the production model. However, higher productivity also implies the economy accumulates additional capital.

- Notice that the exponent on the productivity parameter is greater than in the Chapter 4 model.

**5. ****Looking at Data through the Lens of the Solow Model**

**a. ****The Capital-Output Ratio**

- The capital-to-output ratio is given by the ratio of the investment rate to the depreciation rate.
- While investment rates vary across countries, it is assumed that the depreciation rate is relatively constant.
- Empirically, countries with higher investment rates have higher capital-to-output ratios.

**b. ****Differences in Y/L**

- The Solow model gives more weight to TFP in explaining per capita output than the production model does.
- Some of the differences in capital per person are explained due to differences in productivity.

- Differences in output per person are explained by a factor of 26 for productivity differences, as opposed to only a factor of 11 in Chapter 4.

**6. ****Understanding the Steady State**

- The economy will settle in a steady state because the investment curve has diminishing returns.
- As capital increases, production and investment rise.
- However, the rate at which production and investment rise is smaller as the capital stock grows larger.

- As capital increases, production and investment rise.
- A constant fraction of the capital stock depreciates every period, which implies depreciation is not diminishing as capital increases.
- In summary, as capital increases, diminishing returns implies that production and investment increase by less and less, but depreciation increases by the same amount. Eventually, net investment is zero and the economy rests in steady state.

**7. ****Economic Growth in the Solow Model**

- There is no long-run economic growth in the Solow model.
- In steady state, output, capital, output per person, and consumption per person are all constant and growth stops.
- Empirically, economies appear to continue to grow over time.
- Thus, capital accumulation is not the engine of long-run economic growth.
- Saving and investment are beneficial in the short-run, but diminishing returns to capital do not sustain long-run growth.

- Thus, capital accumulation is not the engine of long-run economic growth.

**a. ****Meanwhile, Back on the Family Farm**

- Note that the number of farmers on the family farm is fixed.
- As the farm becomes larger and larger, a fixed number of farmers can harvest the crop less and less successfully—and growth will stop.

**b. ****Case Study: Population Growth in the Solow Model**

- Growth in the labor force can cause aggregate output to grow over time but will
*not*result in sustained growth in output per person. - Even with population growth, the economy will settle in a steady state without growth in output per person.

**8. ****Some Economic Experiments**

- While the Solow model does not explain long-run economic growth, it does help to explain some differences across countries.
- Economists can experiment with the model by changing parameter values.

**a. ****An Increase in the Investment Rate**

- Suppose the investment rate increases permanently for exogenous reasons.
- The investment curve rotates upward, but the deprecation line remains unchanged.
- The economy is now below its new steady state and the capital stock and output will increase over time by transition dynamics.
- In the long run, steady-state capital and steady-state output are higher.

**b. ****A Rise in the Depreciation Rate**

- Suppose the depreciation rate is exogenously shocked to a higher rate.
- The depreciation curve rotates upward and the investment curve remains unchanged.
- The new steady state is located to the left of the old one, which means that immediately after the shock, depreciation exceeds investment.
- The capital stock declines by transition dynamics until it reaches the new steady state.
- Note output declines rapidly at first but less rapidly as it converges to the new steady state.

**c. ****Experiments on Your Own**

- Try experimenting with all the parameters in the model.
- First, figure out which curve (if either) shifts.
- Second, follow the transition dynamics of the Solow model.
- Third, analyze the steady-state values of capital, output, and output per person.

**d. ****Case Study: Wars and Economic Recovery**

- Note that the dropping of the bombs on Hiroshima and Nagasaki destroyed large portions of the cities, yet within a few decades these cities had returned close to their former economic position relative to other cities.
- Heavy bombings in Vietnam devastated some regions more than others, yet about thirty years after the Vietnam War ended, these regions have similar characteristics as those regions that were not bombed.
- Some economies can recover from massive destruction within a generation.

**9. ****The Principle of Transition Dynamics**

- When the depreciation rate and the investment rate were shocked, output was plotted over time on a ratio scale.
- The ratio scale allows us to see that output changes more rapidly the further we are from the steady state. As the steady state is approached, growth shrinks to zero.

*The principle of transition dynamics*says that the farther below its steady state that an economy is, in percentage terms, the faster the economy will grow.- Similarly, the farther above its steady state, in percentage terms, the slower the economy will grow.

- This principle allows us to understand why economies may grow at different rates at the same time.

**a. ****Understanding Differences in Growth Rates**

- Empirically, OECD countries that were relatively poor in 1960 grew quickly, while countries that were relatively rich grew slower.
- If the OECD countries have the same steady state then the principle of transition dynamics predicts this.

- Looking at the world as a whole, on average, rich and poor countries grow at the same rate.
- Thus, this implies that most countries have already reached their steady states.
- This implies that countries are poor not because of a bad shock, but because they have parameters that yield a lower steady state.

**b. ****Case Study: South Korea and the Philippines**

- An increased investment rate raised South Korea's steady state between 1960 and 1990, but the Philippines remained in its steady state.
- Thus, we expect the South Korean economy to grow at a much higher rate because it is much further from its new steady state in relative terms.

**10. ****Strengths and Weaknesses of the Solow Model**

- The strengths of the Solow model are:
- It provides a theory that determines how rich a country is in the long run.
- The principle of transition dynamics allows for an understanding of differences in growth rates across countries.

- The weaknesses of the Solow model are:
- It focuses on investment and capital, while the much more important factor of TFP is still unexplained.
- It does not explain why different countries have different investment and productivity rates.
- It does not provide a theory of sustained long-run economic growth.

**11. ****Chapter Review**

**Summary****Key Concepts****Review Questions****Exercises****Worked Exercises**