1. 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.

2. 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
The Solow growth model is the starting point to determine why growth differs across similar countries.
In 1960, South Korea and the Philippines were similar in many respects. Both were relatively poor countries: per capita GDP was about $1,800 in Korea and $2,200 in the Philippines, less than 15 percent of the U.S. level. Both had populations on the order of 25 million, about half of whom were of working age. Similar fractions of the population in both countries worked in industry and agriculture. About 5 percent of Koreans in their early twenties attended college, versus 13 percent of Filipinos. Between 1960 and 2007, however, the paths of these two countries diverged dramatically. In the Philippines, per capita GDP grew at a relatively modest rate of about 1.7 percent per year. In contrast, South Korea became one of the world’s fastest-growing economies, with growth just under 6 percent per year. By 2007, per capita GDP in Korea had risen to nearly $24,000, more than half the U.S. level. In contrast, per capita GDP in the Philippines was only $5,000. How do we understand this astounding difference in economic performance between two countries that on the surface looked relatively similar? Why was growth in South Korea so much faster? The starting point in economics for thinking about these questions is what’s known as the Solow growth model. This model was developed in the mid-1950s by Robert Solow of MIT and was the basis for the Nobel Prize he received in Since the 1950s, the model has been extended in a number of important directions and is now probably the most widely used in all of macroeconomics.

3. 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.
“endogenized” means that it is converted from an exogenous to an endogenous variable.
The Solow framework builds on the production model we developed in Chapter 4, but introduces a new element: a theory of capital accumulation. Rather than the capital stock being given at some exogenous level, agents in the economy can accumulate tools, machines, computers, and buildings over time. This accumulation of capital is endogenized in the Solow model—it is converted from an exogenous variable into an endogenous variable. Other than this one change, the Solow model is the production model, so the insights we developed in Chapter 4 will serve us well here. The Solow model allows us to consider the accumulation of capital as a possible engine of long-run economic growth. Perhaps standards of living have increased by a factor of 10 in the last century because we have increased the amount of capital available for each worker. Perhaps some countries are richer than others because they invest more in accumulating capital.

4. 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).
Variables are time subscripted as they may potentially change over time.
Before diving into equations, let’s consider a simple representation of the Solow model. Think of the economy as a large family farm that grows a single crop, corn. To begin, the farm has a silo containing several bags of seed corn. The family farmers plant some kernels in the spring, tend their crop over the summer, and then harvest it as autumn draws near. They reserve three-fourths of the corn for eating over the course of the year and store the remaining fourth in their silo, to plant the next year. Key to this process is that one kernel of seed corn produces ten ears of corn, each containing hundreds of kernels that can be eaten or planted the following year. So as the years pass, the size of the harvest grows larger and larger, as does the quantity of corn kernels stored in the silo. That in a nutshell—or a corn kernel—is the Solow model we will develop. We start with our familiar production model and add an equation describing the accumulation of capital over time. This capital accumulation will proceed much like the collection of seed corn on the family farm. As in the production model, the farm produces a final output good Y, using the capital stock K and labor L. We assume this production function is Cobb-Douglas and exhibits constant returns to scale in K and L. Moreover, the exponent on capital is again 1/3—recall that this exponent reflects the fact that one-third of GDP is paid to capital. Notice that here we add the time subscript t to our production function. Output, capital, and labor can all potentially change over time.

5. Output can be used for consumption or investment.
This is called a resource constraint.
Assuming no imports or exports
Consumption
Investment
Output
Such an equation is called a resource constraint: it describes a fundamental constraint on how the economy can use its resources. We are assuming the farm is a closed economy: there are no imports or exports in this equation.

6. Capital Accumulation
Goods invested for the future determines the accumulation of capital.
Capital accumulation equation:
Depreciation rate
Next year’s capital
This year’s capital
Investment
The seed corn invested for the future determines the accumulation of capital, as can be seen in the equation.

7. 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
Typically, economists think of this depreciation rate as taking on a value of 0.07 or 0.10, so that 7 percent or 10 percent of the capital stock is used up in the production process each period. On the family farm, you might think of depreciation as the fraction of the seed corn in the silo that gets eaten by rats before it can be planted. The capital accumulation equation, then, says that the amount of corn in the silo next spring will be equal to the amount in the silo this year, plus the new additions of seed corn from this year’s harvest, less the amount the rats eat. One way in which the farm analogy is not perfect is that in our model, capital survives from one period to the next. It’s more like a tractor that remains available in the next period than like seed corn that gets planted and used up every year. However, if you think of the seed corn as being recovered at the end of each harvest (since one kernel of seed corn leads to a bucket of kernels at harvest), the analogy works.

8. A change in stock is a flow of investment.
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.
You’ll have noticed that we refer to Kt as the capital stock. Economists often find it helpful to distinguish between stocks and flows.
A stock is a quantity that survives from period to period—a tractor, house, or semiconductor factory, for example. We also speak of the stock of government debt, a debt that persists over time unless it is paid off.
In contrast, a flow is a quantity that lasts for a single period—the breakfast you ate this morning, or a withdrawal you make from your bank’s ATM. Stocks and flows are intimately related. Stocks satisfy an accumulation equation, like the capital accumulation equation in this model. The change in a stock is a flow: the change in the capital stock, for example, is the flow of investment, and the change in your bank balance (a stock) is the flow of deposits and withdrawals you make.

9. Change in capital stock defined as
Thus:
The change in the stock of capital is investment subtracted by the capital that depreciates in production.
Note that Δ is the "change over time" operator. The change in the capital stock is equal to new investment I less the amount of capital that depreciates in production. Notice that today’s capital stock is the result of investments undertaken in the past. This works fine for all periods in the model other than the first. To get the model started, we simply assume that the economy is endowed with some initial capital and the model begins at date t = 0.

10. Case Study: An Example of Capital Accumulation
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.

11. To read this table, start with the first row and go across, from left to right. The farm economy begins with 1,000 bushels of corn and invests 200 more. The depreciation rate is 10 percent, so during the course of production in year 0, 100 (= 0.10 x 1,000) bushels get eaten by rats. Investment net of (minus) depreciation is therefore = 100 bushels. The capital stock rises by this amount, so the amount of corn in the silo at the start of period 1 is 1,100 bushels. In year 1, 10 percent of this higher stock is lost again to rats, so investment net of depreciation in this year is = 90, and so on. A key insight from this table is that the capital stock is simply the sum of past investments: the capital stock in an economy today consists of machines and buildings that were purchased in previous decades. Economists often compute the capital stock in an economy with this basic approach: we make an educated guess about an initial stock of capital at some date long ago and then add up the amount of investment that occurs each year, net of depreciation. Because the stock from long ago will mostly have depreciated by today, this approach is not particularly sensitive to (does not vary much with) the initial educated guess. While Table 5.1 is just an example, it reveals something fundamental about the Solow model: the amount by which the capital stock increases each period is smaller and smaller each year.

12. Saving The difference between income and consumption
Is equal to investment
On the family farm, our farmers forgo consuming some corn (they save the corn) and place it in the silo to be used as next year’s seed (investment). Now consider the national income identity for our Solow model. The left side of this expression is saving: the difference between income and consumption. The right side is investment. So the equation tells us that saving equals investment in this economy. We’ve already defined the real interest rate as the amount a person can earn by saving one unit of output. We now see that the unit of saving gets used as a unit of investment, and a unit of investment becomes a unit of capital. Therefore, the return on saving is equal to the price at which the unit of capital can be rented. Finally, we saw in Chapter 4 that the rental price of capital is given by the marginal product of capital in equilibrium. Therefore, we have the following result: the real interest rate equals the rental price of capital that clears the capital market, which in turn is equal to the marginal product of capital.

13. Investment Farmers eat a fraction of output and invest the rest.
Therefore:
Consumption is the share of output we don’t invest.
Fraction
Invested
The family farm uses the output Y for consumption or investment, but how does the family choose how much to consume and how much to invest? We’ll assume the family farmers eat a constant fraction of the output each period and invest the remainder. s bar can be any number between 0 and 1.

14. 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.

15. That completes our setup of the Solow growth model, summarized in Table 5.2: there are essentially five equations and five unknowns. Here, though, we have a dynamic model, so these five equations hold at each point in time, as long as we’d like the model to run.

16. That completes our setup of the Solow growth model, summarized in Table 5.2: there are essentially five equations and five unknowns. Here, though, we have a dynamic model, so these five equations hold at each point in time, as long as we’d like the model to run.

17. Case Study: Some Questions about the Solow Model
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
If you compare the Solow model with the model of production presented in Chapter 4, you’ll notice several differences. The main one is that we have added dynamics to the model in the form of capital accumulation. But we have also left out the markets for capital and labor and the corresponding prices—the rental price of capital and the wage rate. This is a simplification that makes the model easier to work with. The consumption equation is part of the model, but we do not need to add it to the list in Table 5.2 because it would be redundant.

18. 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.
We’ve left prices the rental price of capital and the wage rate for labor—out of our Solow model to keep the model as simple as possible. If we added those elements back in, we would end up with two more endogenous variables and two more equations. These would pin down the wage as the marginal product of labor and the rental price of capital as the marginal product of capital, just as in the production model of Chapter 4. Otherwise, though, nothing would change. You might think about these additional elements as lying in the background of our Solow model. They are available in case we ever need them, but it is simpler for most of this chapter to leave them there. For now, however, there is one other important price to consider, called the real interest rate. The real interest rate in an economy is equal to the rental price of capital, which in turn is given by the marginal product of capital. Let’s take this statement in pieces. First, 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. We say that it is real because it is measured in units of output (or constant dollars) rather than in nominal dollars. Why is this rate equal to the rental price of capital? Consider the source of the supply of capital in an economy. In Chapter 4, we simply assumed the existence of a fixed amount of capital. In this chapter, we understand that capital comes from the decision by households to forgo consumption and save instead. This saving is equal to investment, which becomes the capital supplied in the economy.

19. 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
We’ve left prices—the rental price of capital and the wage rate for labor—out of our Solow model to keep the model as simple as possible. If we added those elements back in, we would end up with two more endogenous variables and two more equations. These would pin down the wage as the marginal product of labor and the rental price of capital as the marginal product of capital, just as in the production model of Chapter 4. Otherwise, though, nothing would change. You might think about these additional elements as lying in the background of our Solow model. They are available in case we ever need them, but it is simpler for most of this chapter to leave them there. For now, however, there is one other important price to consider, called the real interest rate. The real interest rate in an economy is equal to the rental price of capital, which in turn is given by the marginal product of capital. Let’s take this statement in pieces. First, 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. We say that it is real because it is measured in units of output (or constant dollars) rather than in nominal dollars. Why is this rate equal to the rental price of capital? Consider the source of the supply of capital in an economy. In Chapter 4, we simply assumed the existence of a fixed amount of capital. In this chapter, we understand that capital comes from the decision by households to forgo consumption and save instead. This saving is equal to investment, which becomes the capital supplied in the economy.

20. 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.
On the family farm, our farmers forgo consuming some corn (they save the corn) and place it in the silo to be used as next year’s seed (investment). Now consider the national income identity for our Solow model. The left side of this expression is saving: the difference between income and consumption. The right side is investment. So the equation tells us that saving equals investment in this economy. We’ve already defined the real interest rate as the amount a person can earn by saving one unit of output. We now see that the unit of saving gets used as a unit of investment, and a unit of investment becomes a unit of capital. Therefore, the return on saving is equal to the price at which the unit of capital can be rented. Finally, we saw in Chapter 4 that the rental price of capital is given by the marginal product of capital in equilibrium. Therefore, we have the following result: the real interest rate equals the rental price of capital that clears the capital market, which in turn is equal to the marginal product of capital.

21. 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.
To solve the Solow model completely, we need to write the endogenous variables as functions of the parameters of the model. Furthermore, we need to do this not just for a single time period, but for every point in time. It turns out that the model cannot be solved algebraically in this fashion. However, we can make some progress in two complementary ways. First, we can show graphically what the solution looks like. Second, we can solve the model “in the long run,” a phrase that will take on more meaning once we’ve studied the model more carefully. Keep a pencil and some paper handy as you read through the solution of the Solow model below; working through the equations yourself as we go along is one of the most effective ways of learning the model. The first step is to combine equations in a way that will take us as far as we can go algebraically.

22. Combine the investment allocation and capital accumulation equation.
Depreciation
Investment

23. Substitute the fixed amount of labor into the production function.
We have reduced the system into two equations and two unknowns (Yt, Kt).
At this point, we have reduced our system of five equations and five unknowns to two equations and two unknowns (Kt and Yt). Moreover, it should be clear that we could simply plug in the production function for output the "change in capital" equation (5.5) if we wished. Then we would have a single dynamic equation describing the evolution of the capital stock. While this equation cannot be solved algebraically, we can learn a lot about it by analyzing it graphically.

24. 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.

25. The Solow diagram plots two curves, which are both functions of capital K. The first curve is new investment s bar Y. Because the production function is increasing in K but with diminishing returns, investment shows this same curvature. The second curve is depreciation d bar K, which is just a linear function of K. The change in the capital stock—net investment—is the vertical difference between these two curves. Arrows on the horizontal axis indicate how the capital stock changes.

26. 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.
In other words, the amount of seed corn we add to the silo exceeds the amount that gets eaten by rats, so the total amount of seed corn in the silo rises. Net investment is positive, and the capital stock increases. This process continues—and the economy moves in the direction of the arrows—until the economy reaches a capital level K *. At this point, the two curves in the Solow diagram intersect. This means that the amount of investment being undertaken is exactly equal to the amount of capital that wears out through depreciation. And since investment equals depreciation, the change in the capital stock is equal to zero, and the capital stock remains constant. In the absence of any shocks, the capital stock will remain at K * forever, as each period’s investment is just enough to offset the depreciation that occurs during production. This rest point is called the steady state of the Solow model.

27. 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.
We call the behavior of the economy away from its steady state the transition dynamics of the economy. The arrows in the previous Figure 5.1 illustrate the transition dynamics in the basic Solow model—that is, as the economy “transits” to its steady state. An important thing to notice about transition dynamics is that they always move the economy toward the steady state. No matter how we choose the initial level of capital, K(0), if we wait long enough, the economy will converge to the steady state K *. Try moving K0 around in the Solow diagram to test this out. Remarkably, the economy will also converge to the steady state if we start it off with a level of capital that is larger than K *. In this case, the amount of capital that wears out in production exceeds the amount of investment. Net investment is therefore negative, and the capital stock declines. This process continues until the economy settles down at K *.

28. Output and Consumption in the Solow Diagram
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.
We can plot this production function on the Solow diagram as well, and it allows us to see how output evolves over time (see Figure 5.2 on next slide). As transition dynamics take the economy from K0 to K *, output rises over time from Y0 to its steady-state level, Y *.

29. Here, the production function is added to the Solow diagram, plotting Y as a function of K. Output rises as the economy transits from K0 to the steady state K *. Consumption is the difference between output and investment.

30. Solving Mathematically for the Steady State
In the steady state, investment equals depreciation.
Sub into the production function
As mentioned earlier, we can’t solve the Solow model mathematically for the level of capital at each point in time. The Solow diagram helps us understand what is going on in the absence of an exact mathematical expression. But we can solve mathematically for the steady-state level of capital, and this is what we do now.

31. The steady-state level of capital is
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
This equation pins down the steady-state level of capital K* as a function of the underlying parameters of the model, and it’s worth pausing to make sure you understand the result. A higher investment rate s bar leads to a higher steady-state capital stock: if we invest 20 percent of our harvest as seed corn rather than 10 percent, more seed will accumulate in the silo. The steady-state level of capital also increases if the underlying level of productivity A bar is higher. This might seem surprising at first—why does the level of capital depend on how productive the economy is? The answer is that if the farm is more productive, the harvest will be larger, and the larger harvest will translate into more seed corn in the silo. Finally, the steady-state capital stock also depends on the depreciation rate and the size of the workforce. A higher rate of depreciation reduces the capital stock, as the rats gobble more of the seed corn. A larger workforce produces more output, which leads to more investment and hence more capital in the steady state.

32. 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
As with the steady-state capital stock, a higher rate of investment and higher productivity lead to a higher steady-state level of production, but faster depreciation lowers it. The constant returns to scale of the underlying production function show up in that doubling labor leads in the long run to a doubling of steady-state production.

33. 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.
Why the difference in the exponents on productivity A? The answer is that the level of the capital stock itself depends on productivity. In the Solow model, a higher productivity parameter raises output directly just as in the production model. But there is an additional effect in the Solow model. The higher productivity level leads the economy to accumulate more capital as well. This explains the larger exponent in the Solow framework.

34. 5.5 Looking at Data through the Lens of the Solow Model
The Capital-Output Ratio
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.
In reality, different countries have different investment rates. Over the last 30 years, the investment rate in Japan has averaged more than 35 percent of GDP, while the rate in the United States has been about 24 percent. In the poorest countries of the world, investment rates are only 5 percent or so. It is conventional to assume that the depreciation rate is relatively similar across countries; a typical value would be about 7 percent. According to the equation, countries with high investment rates should have high capital-output ratios.

35. 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:
Recall that in Chapter 4’s production model, we found that most of the differences in per capita GDP across countries were explained by differences in total factor productivity (TFP), with a modest contribution from differences in capital per person. What the Solow model does is to endogenize these differences in capital per person. Since it adds nothing new regarding TFP differences, we would expect that they remain as a key explanatory factor. In fact, the Solow model leads to an even more important role for TFP differences. Why? In the Solow model, notice that differences in capital are explained in part by differences in investment rate and in part by differences in the productivity parameter. That is, some of the differences we observe in capital per person across countries are themselves due to differences in productivity. This means that the Solow model gives an even larger role to TFP than our production model did. One way to see this is to look back at the equation for output per person in steady state. As was pointed out earlier, the exponent on the productivity parameter A is now larger than 1, suggesting a more important role for productivity. A second way to see the larger role of TFP in the Solow model, though, is to calculate how important differences in investment rates are for explaining differences in y *. In Chapter 4, we found that the ratio of per capita GDP in the five richest countries to GDP in the five poorest countries was a factor of 66. With the production model, a factor of 11 of this difference was attributed to productivity and a factor of 6 to capital.
From Chapter 4
See figure 5.3 (next slide)

36. Just as the Solow model predicts, a key determinant of a country’s capital-output ratio is its investment rate.
It turns out that the Solow prediction holds up remarkably well. Countries like Japan and Norway experience high average investment rates and high capital-output ratios. The United States lies somewhere in the center, with an investment rate of 24 percent and a capital-output ratio of 3. Uganda and Egypt have low investment rates and low capital-output ratios.

37. We find that the factor of 66 that separates rich and poor countries’ income per capita is decomposable:
TFP differences
Investment differences
How do we arrive at the figures 26 and 2.5? Looking back at Figure 5.3, we see that investment rates in the richest countries average about 25 or 30 percent, while investment rates in the poorest countries are about 5 percent. The ratio of investment rates between rich and poor countries is thus no more than 30/5 = 6, suggesting that differences in investment rates explain a factor of square root of 6, which is approximately 2.5, of the differences in income, and leaving an even larger factor of 66/2.5 = 26 for productivity differences. While the Solow model is successful in helping us understand why countries have different amounts of capital, it deepens the puzzle of why some countries are so much richer than others. Since it assigns an even larger role to TFP differences than our production model did, we are led to ask again, Why is it that some countries use their inputs so much more efficiently than others?

38. 5.6 Understanding the Steady State
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.
The steady-state result—that the Solow model eventually approaches a constant level of capital K * and a constant level of production Y * no matter where it begins—is quite remarkable. It is therefore worth spending some time thinking about where this result comes from and what it means. The fact that production exhibits diminishing returns to capital accumulation means that each addition to the capital stock increases production—and therefore investment—by less and less. But it increases depreciation by the same amount. Eventually, the amount of investment the economy generates is equal to the amount of capital that depreciates. Net investment is zero, and the economy stabilizes at the steady state.

39. 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
One of the most important implications of the steady-state result is that there is no long-run growth in the Solow model. In the long run, the economy settles down to a constant level of production Y * and a constant amount of capital K *. The economy grows from K0 to K *—but eventually growth stops as the capital stock and production converge to constant levels.

40. 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
Let’s contrast this theoretical prediction with the facts of economic growth that we observed in Chapter 3. There we saw that economic growth was a widespread phenomenon, both across countries and over time. Per capita GDP in the United States, for example, has grown at an average annual rate of 2 percent per year for more than a century. Moreover, this rate of growth is higher than the growth rate in previous centuries. Economic growth shows no signs of disappearing, but in the Solow model, this is exactly what happens. One of the central lessons of the Solow model therefore comes as something of a surprise: capital accumulation cannot serve as the engine of long-run economic growth. The fact that we save and invest in additional factories, machine tools, computers, and bulldozers does lead output to grow in the medium run. But in the long run, the diminishing returns to capital accumulation cause the return to these investments to fall. Eventually depreciation and new investment offset each other, and the economy settles down to a constant level of output per person. Although we can’t help but feel disappointed by this result, it is a testimony to the importance of other results we will derive in the Solow model that it still remains one of the most important models in macroeconomics. The capital accumulation hypothesis is an important one to explore, but in the end, the Solow model teaches us that it is not the answer to the question of what causes long-run growth.

41. Meanwhile, Back on the Family Farm
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.
In our farm example, the family of corn farmers eat a constant fraction of their harvest and plant the remainder as seed for next year’s crop. Our model suggests that starting from a small stock of seed corn, the harvest grows larger and larger each year, at least for a while. Eventually, though, the harvest settles down to some constant level: the farm produces 1,500 bushels of corn every year. Why does this occur? Why doesn’t the process of planting seed corn year after year, each kernel of which can produce ten ears of corn, lead to an ever-growing harvest? The amount of corn that can be successfully harvested is limited by the amount of labor the farmers can provide. As we add more and more corn plants to the farm, the amount that a family of six can harvest may rise, but it runs into diminishing returns: six farmers can effectively harvest 6 acres or 600 acres, but what about 600,000 acres? Eventually a fixed number of farmers simply cannot harvest all the corn that’s planted, and growth stops. Diminishing returns to the amount of corn a given number of farmers can cultivate lead to a situation where the new investment—which is proportional to each year’s harvest—is exactly offset by depreciation, just as in the Solow diagram.

42. Case Study: Population Growth in the Solow Model
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.
What would happen if the number of farmers we could employ on the family farm were to grow? Can growth in the labor force lead to overall economic growth? The short answer is that it can in the aggregate, but not in output per person. That is, total capital and total production can grow as the labor force grows. But the presence of diminishing returns leads capital per person and output per person to settle down in a steady state; fundamentally, the amount of corn each farmer can produce runs into diminishing returns as well. At first, one farmer can tend to 100 acres, 1,000 acres, and maybe even 10,000 acres of corn. But as the number of acres per farmer grows, eventually she has no time to cultivate the additional crop. Investment per farmer falls to equal depreciation per farmer, and growth in output per farmer stops. An increase in labor by itself doesn’t really change the lessons of the Solow model: even with more workers, the economy eventually settles down to a steady state with no growth in output per person. An intuitive way to understand this result is to recall that the Solow economy exhibits constant returns to scale. Adding more family farms won’t affect the analysis of a single farm (recall the standard replication argument). Therefore each farm ends up in a steady state with constant output per farm, just as in the Solow model.

43. 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.
While the Solow model leads us to look elsewhere for a source of long-run economic growth, it nevertheless helps us understand some of the other facts of growth documented in Chapter 3. For example, it does help us understand differences in growth rates between countries like South Korea and the Philippines. Developing these results will occupy the remainder of the chapter; we return to the question of the source of long-run growth in the next chapter. One of the best ways to understand how an economic model works is to experiment by changing some of the parameter values. After building a toy world populated by little robots, the economist-god is free to watch the world for a while and then change it suddenly to see how it responds. Such experiments can reveal useful insights.

44. An Increase in the Investment Rate
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

45. 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

46. Starting from K *, new investment exceeds depreciation when the investment rate rises to s bar prime. This causes the capital stock to increase, until the economy reaches the new steady state at K **.

47. 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**.
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 key point to appreciate from this example is that the increase in the investment rate causes the economy to grow over time, at least until the new steady state is reached. In the long run, both steady-state capital and steady-state production are higher. Because labor is constant, this also means the level of output per person is permanently higher as well.

48. What happens to output in response to this increase in the investment rate? The answer is shown in Figure 5.5(a). 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 **. Part (b) shows the hypothetical behavior of output over time, assuming that the increase in the investment rate occurs in the year Notice that output grows fastest immediately following the change in the investment rate. Then over time, the growth rate of output falls, and the economy converges smoothly to its new steady state. We will explore these transition dynamics in more detail later in the chapter.

49. 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
As a second experiment, let the economy again start from its steady state, but now suppose the depreciation rate increases permanently. In a real economy, we might imagine this could happen if changes in the climate, for example, led to more severe weather so that buildings and vehicles depreciated more quickly. How would the economy evolve in this case?

50. Starting from K *, depreciation exceeds new investment when the depreciation rate rises to d bar prime. This causes the capital stock to decline, until the economy reaches the new steady state at K **.

51. 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**.

52. A permanent increase in the depreciation rate causes output to decline over time until it reaches its new steadystate level Y **.

53. 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*).

54. Case Study: Wars and Economic Recovery
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?
Does war have a long-lasting detrimental effect on economic performance that persists for decades after the war ends? Most of us would guess that the answer is yes. But in fact, some recent research calls that instinct into question, offering some intriguing evidence. Donald Davis and David Weinstein of Columbia University have considered the effect of the nuclear bombs that were dropped on the Japanese cities of Hiroshima and Nagasaki near the end of World War II. The bombing of Hiroshima leveled over two-thirds of the city and in short order killed perhaps more than 20 percent of the population, with more casualties following over time from radiation poisoning. The explosion over Nagasaki was twice as powerful, but hilly geography and a missed target lessened the destruction. Perhaps 40 percent of the city was destroyed and about 8.5 percent of the population killed. Despite this devastating loss of life and economic capability, however, Hiroshima and Nagasaki recovered relatively quickly. In just a few decades, these cities had returned to close to their original economic position relative to other Japanese cities—by 1960 for Nagasaki and by 1975 for Hiroshima. Edward Miguel and Gérard Roland of the University of California at Berkeley have studied the effect on subsequent economic development of the most intense bombing campaign in military history—the U.S. bombing of Vietnam from 1964 to The heaviest bombing occurred in the central region of the country in Quang Tri province, and the infrastructure of this region was virtually destroyed: of 3,500 villages in the region, only 11 were left untouched by bombing. Despite the enormous humanitarian and economic losses, however, Miguel and Roland find that by 2002—about 30 years after the war—poverty rates, consumption levels, literacy, and population density in this region look much like those in other areas of Vietnam that were not bombed. The research suggests that economies (and people) are surprisingly robust. Once wars are completely ended, economies can at least sometimes recover from massive destruction over the course of a single generation. Can you relate these studies to what you’ve learned in the Solow model?

55. 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 experiments we have just conducted with the Solow model suggest yet another possible explanation for differences in growth rates across countries. Notice that if an economy is “below” its steady state, it grows rapidly. On the other hand, if it is “above” its steady state, its growth rate is negative. Perhaps, then, differences in growth rates reflect how near or far countries are from their steady states. Take a look back at Figures 5.5(b) and 5.7(b), showing how output evolves over time. In these graphs, output is drawn on a ratio scale. This means that the growth rate of output can be inferred from the slope of the output path. In Figure 5.5, immediately following the change in the investment rate, output grows rapidly, and then gradually declines as the economy approaches its new steady state. Similarly, when the depreciation rate increases in Figure 5.7, the growth rate is initially a large negative number. Once again, though, as the economy approaches its steady state, the growth rate moves smoothly back to zero.

56. 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
These patterns apply more generally. The overall result is important enough that we give it a name, the principle of transition dynamics: the farther below its steady state an economy is (in percentage terms), the faster the economy will grow; similarly, the farther above its steady state, the slower it will grow. Consider two economies, one just a little below its steady state, say 5 percent below, and another far below its steady state, say 50 percent. The principle of transition dynamics says that not only will both economies grow, but the economy that is 50 percent below will grow faster. One of the most important traits of the principle of transition dynamics is that it gives us a coherent way of understanding why countries may grow at different rates—why China is growing rapidly, for example, while Argentina is growing slowly. We now look at this phenomenon in more detail.

57. 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
With the principle of transition dynamics in mind, let’s take a look at the evidence on growth rates across countries. The OECD group is made up of relatively rich countries, mostly from Western Europe, but it also includes Australia, Japan, New Zealand, and the United States. OECD countries that were relatively poor in 1960, including Japan, Ireland, Portugal, and Spain, grew quickly. In contrast, countries that were relatively rich in 1960—including New Zealand, Switzerland, and the United States—were among the slowest-growing countries in the OECD. How do we understand these facts? The principle of transition dynamics suggests a possible explanation. Suppose all the countries of the OECD will have similar incomes in the long run. In this case, countries that are poor in 1960 would be far below their steady state, while countries that are rich would be closer to (or even above) their steady state. The principle of transition dynamics predicts that the poorest countries should grow quickly while the richest should grow slowly. Around the world, however, the picture is very different: there is essentially no correlation between how rich a country is and how fast it grows. The fact that, on average, rich and poor countries grow at the same rate suggests that most countries—both rich and poor—have already reached their steady states. Most poor countries are not poor because they received a bad shock and have fallen below their steady state; if this were the case, we would expect them to grow faster than rich countries—as relatively poor OECD countries did, for example, in Figure 5.8. Instead, the poor countries of the world are typically poor because, according to the Solow model, the determinants of their steady states—investment rates and total factor productivity levels—are substantially lower than in rich countries. Without some change in these determinants, we can’t expect most poor countries to grow rapidly.

58. Among OECD countries, those that were poor in 1960 grew the fastest, and those that were rich in 1960 grew the slowest.

59. For the world as a whole, growth rates are largely unrelated to initial per capita GDP. This suggests that most poor countries are poor because of a low steady-state income, not because some shock has pushed them below their steady state.

60. Case Study: South Korea and the Philippines
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.
We began this chapter by asking: Why has South Korea grown so much faster (6 percent per year) than the Philippines (1.7 percent per year) in the last half century? We can now apply the principle of transition dynamics to suggest an answer. Perhaps starting in the 1960s, changes in the investment rate and productivity took place in Korea that moved the country’s steady-state income to a much higher level. Starting from around 15 percent of U.S. income in 1960, maybe Korea’s steady-state level moved up to 50 or 75 percent while the Philippines’ level remained at 15 percent. This scenario means that Korea in 1960 would have been far below its steady state, so we would expect its economy to grow quickly. And the Philippines, in contrast, would have been roughly at its steady state, and we would expect that economy to grow at about the same rate as the United States, preserving its ratio of 15 percent.

61. 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
What does the Solow model tell us about the determinants of the steady-state ratio of income in South Korea to income in the United States? Assuming these economies have the same rate of depreciation, the ratio of their steady-state incomes is given by the equation shown. That is, the long-run ratio of per capita incomes depends on the ratio of productivities (TFP levels) and the ratio of investment rates. While the U.S. rate is relatively stable, the rate in Korea rises dramatically, from about 12 percent in the 1950s to more than 40 percent in the early 1990s. According to the Solow model, this rise should significantly increase Korea’s steady-state income. The principle of transition dynamics then tells us that the Korean economy will grow rapidly as it makes the transition to its new, higher level. In contrast, the relatively stable investment rate in the Philippines suggests that the steady state in this economy is not changing, and the principle of transition dynamics then implies that the Philippines will not grow rapidly over this period.

62. The investment rate in South Korea rose sharply between 1960 and 1990, while it remained relatively stable in the United States and the Philippines.

63. 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 key elements of the Solow framework lie at the heart of virtually every model in modern macroeconomics. These key elements are a production function that depends on capital and labor and an accumulation equation that shows how forgoing consumption today leads to a higher capital stock tomorrow. There are two main strengths of the Solow model. First, it provides a theory that determines how rich a country is in the long run—that is, in the steady state. Countries will be rich to the extent that they have a high rate of investment, a high TFP level, and a low rate of depreciation. If we extended the Solow model, we might add a high rate of investment in human capital—including education and on-the-job training—to this list. Second, through the principle of transition dynamics, the Solow model helps us understand differences in growth rates across countries. The farther a country is below its steady state, the faster it will grow. Countries like China, Ireland, and South Korea have grown rapidly between 1960 and 2007 because the key determinants of their steady-state income (investment rates, TFP levels) have risen.

64. 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.
Along with these strengths, though, come three basic shortcomings of the Solow model. First, the main mechanism studied in the model is investment in physical capital, but our quantitative analysis showed that differences in investment rates explain only a small fraction of the differences in income across countries. Instead, TFP differences—which remain something of a mystery—are even more important here in explaining income differences than they were in the production model. Second, why is it that countries have different productivity levels and different investment rates? If one of the key reasons for increased growth is a rise in the investment rate, the Solow model doesn’t explain why, for example, the investment rate rose in South Korea but not in the Philippines. On this question, the economics literature provides some insight. Economists extending the Solow model have endogenized the investment rate—that is, they have made s an endogenous variable. In these models, the long-run investment rate depends on how patient people are and on the taxes and subsidies that a government levies on investment, among other things. It is possible, then, that the Korean government removed substantial barriers to investment, while the Philippines maintained these barriers. Such a story may go part of the way toward understanding the Korean growth miracle. The final and perhaps most important shortcoming of the Solow model is that it does not provide a theory of long-run growth. We might have thought that the mechanism by which saving leads to investment in computers, factories, and machine tools could serve as an engine of long-run growth. However, what we have seen in this chapter is that the diminishing returns to capital accumulation in the production function mean that capital accumulation by itself cannot sustain growth. As an economy accumulates capital, the marginal product of capital declines. Each additional unit of capital produces a smaller and smaller gain in output, so that eventually the additional output produced by investment is only just enough to offset the wear and tear associated with depreciation. At this point, growth stops in the Solow model.

65. 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

66. 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.

67. 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.

68. Countries with high investment rates
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.

69. Increases in the investment rate or TFP
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

70. 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.

71. Additional Figures for Worked Exercises

74. Macroeconomics Second Edition This concludes the Lecture
Slide Set for Chapter 5
Macroeconomics
Second Edition by
Charles I. Jones
W. W. Norton & Company
Independent Publishers Since 1923

75. Additional Solow graph examples from previous edition of slides

76. The Solow Diagram graphs these two pieces together, with Kt on the x-axis:
Investment, Depreciation
Capital, Kt
At this point,
dKt = sYt, so

77. Suppose the economy starts at this K0:
Investment, Depreciation
Capital, Kt
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 K0 K1

78. Now imagine if we start at a K0 here:
Investment, Depreciation
Capital, Kt
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 K0

79. 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 K*
No matter where we start, we’ll transition to K*!
At this value of K, dKt = sYt, so

80. We can see what happens to output, Y, and thus to growth if we rescale the vertical axis:
Investment, Depreciation, Income
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* K* Y*
Capital, Kt