**Chapter 4: The Theory of
Economic Growth**

J. Bradford DeLong

--Draft 2.0--

**1999-08-28: 7,290 words**

**The Importance of Long-Run
Growth**

*Long-run growth and macroeconomics.*

Macroeconomists pay more attention to long-run growth than they did two decades ago. Most of this is the result of increasing awareness of the importance of the subject: not only is long-run economic growth

theultimately most important aspect of how the economy performs, but economic policies can have powerful effects on the rate of long-run growth.

*Sources of long-run growth.*

Macroeconomists tend to break the study of long-run growth into two parts.

The first part is the determination of an economy's steady-state capital-output ratio (and the speed with which it will converge to that steady-state capital-output ratio). The second part is the determination of the rate of invention and innovation. An economy with a higher capital-output ratio will be a richer economy (if it has access to the same inventions and innovations). But an economy with higher rates of invention and innovation--faster total factor productivity growth--tend to become richer faster.

One way to increase real GDP per worker is to increase the capital stock per worker. The capital stock per worker can be increased in many ways--more investment, slower depreciation, or slower population growth. As the capital stock per worker rises, the value of the machines and workspace available to the average worker rises. With more assistance from capital, the average worker is more productive.

*Diminishing returns.*

But boosting productivity by raising capital per worker is subject to diminishing returns: each successive increase in capital per worker generates less of an increase in production than did the one before. Eventually the boost to total productivity given by further increases in the capital stock does not even amount to enough to make up for the wear-and-tear on the extra capital employed. The bulk of increases in productivity and material standards of living over the course of decades or generations has to come from a different source than simply "deepening" the amount of capital that the economy possesses.

*The key importance of technology.*

This additional, extra source of increases in productivity and material standards of living is

technology.Improvements in technology--and economists use "technology" in the broadest possible sense to include improvements in the so-called technologies of organization, government, and education--are the greatest amplifiers of productivity. Yet economists have relatively little to say about what governs technological progress. Why did we see better technology raise living standards by 2 percent annually a generation ago, but by less than 1 percent annually today? Why was the rate of technological progress only 0.25 percent per year in the early 1800s?

Economists point to expanding literacy, better communications, the institutionalization of research and development as causes of faster technological progress now than in the distant past. But they have too little to say about the causes of recent changes in the rate of productivity growth.

Thus discussions of economic growth by economists often end on an unsatisfying note. Economists have a lot to say about the causes and effects of capital deepening, the sources of large differences in productivity levels and material standards of living between countries, and the relationship between economic policies and relative rates of economic growth. But they nevertheless largely duck some of the big questions.

**The Production Function**

*The production function.*

__Writing down an equation.__

Economists summarize the relationship between the level of

potential output(real GDP) that the economy can produce in any year and thefactor inputs--capital, labor, the level of technology broadly defined--available in that year using a mathematical tool called theproduction function:Where:

F() -- a placeholder that stands for the idea that there is a

production function--a rule that relates how large potential output is to how much of the factor inputs are available--itself.E -- the efficiency of labor, taken as an indicator of the level of technology broadly defined (not just engineering techniques and scientific knowledge, but a whole range of other factors like management practices and social conventions that also affect the aggregate productivity of the economy).

Y -- the level of

output(real GDP).L -- the economy's

labor supply(the number of workers).K -- the economy's

capital stock(machines, bridges, buildings, and so on).Thus:

Y/L -- output per worker

K/L -- the economy's capital-labor ratio: how much capital the average worker has at his or her disposal to amplify his or her potential productivity.

__The production function is an abstraction.__

By even writing down such an aggregate production function we have already made a number of breathtakingly-large leaps of abstraction. We have assumed that it is useful to talk about a single measure of total output--that resources could be switched from one sector to another without losing much of their productive value or causing big changes in relative prices that create insuperable index-number problems. We have assumed that it doesn't matter (or doesn't matter much within appropriate limits) what kinds of investments have been made in the past. We have assumed that it useful to talk about a single measure of the overall level of technology E, and that this level of technology can be thought of as directly amplifying the productivity of the average worker, the average member of the economy's labor force L.

All of these are big assumptions. (Indeed, fierce intellectual struggles were waged during the 1950s and 1960s over whether writing down such a function was useful at all, or just a sterile waste of time; at one point Robert Solow--who won his Nobel Prize primarily for developing much of the theory of economic growth outlined in this chapter--declared that he was on "both sides.")

__Further simplifications.__

Moreover, we want to make further assumptions. Calculations are made much, much simpler (and little flexibility is lost) if we write this production function--this relationship between this year's level of potential output per worker and the factors of production available to the economy--in one particular form:

The economy's total output is proportional to the economy's stock of capital K raised to the power a, times the product of the size of the labor force L and the efficiency of labor E, that product raised to the power 1- a.

If we divide through by the number of workers in the economy so that our left-hand side is output per worker: the productivity of the economy.

We hope that these abstractions and the simplifications involved in writing down an aggregate production function and then choosing this one particular algebraic form will not, later on, significantly mislead us in some way. We analyze this production function; but we expect that the conclusions we reach for

thisproduction function will be more general, will hold for other production functions--including the economy'srealproduction function, to the extent that such a thing (approximately) exists--as well.

- First, this production function exhibits
*constant returns to scale:*double the amount of capital K in the economy, and double either the efficiency of labor E or the size of the labor force L (or maneuver both in any other way that doubles their product E x L), and you have doubled total output Y. Such constant-returns-to-scale mean that we can analyze the economy in per-worker terms, so that there is a sense in which the absolute size of the economy doesn't matter for most of the analysis. - Second, this production function is simple in the sense that how much output is produced for each possible level of the economy's labor force and capital stock depends on only two things: the parameter a and the level of the technology parameter, the measure of the efficiency of labor E.
- Third, this production is a simple power function--a variable or two raised to a power. Thus we can use a large amount of the tools of algebra developed three and four hundred years ago to help us analyze the properties of this production function.
- If some variable y=k
^{a}, then {the percentage change in y} will (for small percentage changes) be equal to a x {the percentage change in k}. - If y
^{1-a}= z, then y = z^{(1/(1-a))}. - If z = k/y, if k is growing at r percent per year, and if y is growing at s percent per year, then z will be growing at r - s percent per year. (We already covered this property in chapter 3.)
- The general property that if x is small, then (1+x)
^{a}is approximately equal to 1 + ax.

**[Box: Tools] Advantages of this
production function.**

Three features ofthisparticular production function make using it especially easy.

**[Box: Tools] Power functions.**

[To be written]

**[Box: Tools] Algebraic tools.**

We can use a large amount of the tools of algebra developed three and four hundred years ago to help us use this production function to analyze real-world questions.

Four of these algebraic tools turn out to be the most useful:

These algebraic rules prove very useful to us, but they are available for use only if the production function takes a simple form--like:

__Advantages of simplicity.__

This functional form is simple enough to keep analysis from becoming too convoluted too quickly. Potential output per worker (Y/L) depends on only three things: the level of capital per worker (K/L), the level of technology (E), and a parameter that is somewhere between zero and one labeled a, the exponent, the power to which K/L is raised in this algebraic expression for the production function. Yet this functional form is also sufficiently flexible to allow for a wide range of different cases. And it provides for a natural classification of increases in output per worker. For increases in total product per worker have two sources.

The first is an increased capital stock of machines, buildings, and infrastructure produced by investment. The second is an improvement in the level of technology: not more machines but better machines, and better organizations.

Think of increases in production per worker from an increased capital stock per worker as shifts along a production function that tells you how much the average worker can produce with the existing capital stock. Increases from better technology are then upward shifts in the production function.

*Potential output-per-worker as a
function of the capital stock.*

__Capital accumulation and higher output.__

Potential output per worker (Y/L) depends on the capital stock per worker (K/L). The higher the capital stock per worker--the more in the way of machines, buildings, infrastructure, and so forth available to amplify worker productivity--the higher is potential output per worker. Whenever net investment is more than enough to provide new entrants into the labor force with the capital they need, the capital stock per worker rises: The value of the machines and work space available to the average worker rises. With more assistance, the average worker is more productive.

__Diminishing returns.__

However, because the a in the production function is smaller than 1, boosting productivity by raising capital per worker is subject to diminishing returns: Each successive increase in capital generates less of an increase in production than did the one before. How fast potential output per worker increases with capital per worker depends on the value of the parameter a in the production function: a relatively low value of a (a value near zero) means that diminishing returns to capital set in quickly, and that the point at which further increases in capital per worker do little to raise potential output per worker arrives quickly; a relatively high value of a (a value near one) means that diminishing returns to capital set in only slowly, and that there is a large range within which increases in capital per worker generate large increases in potential output per worker.

__Flexibility through varying a.__

Do you want to analyze a situation in which boosting capital per worker raises output per worker at almost the same rate indefinitely? You can do that with the form of the production function 4.2.1.2 and a value of a near one. Do you want to analyze a situation in which boosting capital per worker beyond an initial, minimal level does little to raise potential output per worker? You can do that with the form of the production function 4.2.1.2 and a value of a near zero. Do you want to analyze an intermediate case? You can do that too with an intermediate value of the parameter a that tunes how quickly diminishing returns to capital set in.

Figure Legend:By changing the parameter a--the exponent of (K/L) in the algebraic form of the production function--you change the curvature of the production function, and thus the point at which diminishing returns to further increases in capital per worker set in.

*Production-per-worker as a function
of the level of technology.*

__Higher output through better technology.__

Potential output per worker (Y/L) depends also on the level of technology or efficiency of labor E. Better technology means a higher level of efficiency of labor parameter E, and thus a higher level of output per worker for any given level of capital per worker.

In the very long run, the fact that we have higher productivity levels and living standards than did our distant ancestors, and that we expect such economic progress to continue, is the result of invention, technological progress, and improvements in total factor productivity. Without such improvements in total factor productivity economic progress would grind to a halt: diminishing returns would diminish the benefits to investments that further raised the capital-labor ratio, and soon it would no longer be worthwhile to continue adding to the capital stock per worker.

Figure Legend:By changing the parameter E--the level of technology or the efficiency of labor in the algebraic form of the production function--you raise or lower how much potential output per worker is generated by each level of capital per worker.

*Production per worker as a function
of the capital-output ratio.*

__Finding a more convenient form for
the production function.__

The production function as we have written it so far expresses output per worker as a function of the capital stock per worker. But this is not the most convenient way of expressing it for our purposes: since we will be focusing on the capital-output ratio as our key variable in this chapter, we would prefer an expression telling us how output per worker depended on the capital-output ratio.

We can translate our production function into such an expression by noticing that the capital-labor ratio, K/L, is just equal to the capital-output ratio times output per worker:

Substituting this definition into our production function:

Dividing through to move all the output per worker terms to the left hand side:

And then raising both sides to the 1/(1-a) power to solve for the level of output per worker:

gives us what we wanted: output per worker as a function of the capital-output ratio (and the level of technology A, and the parameter a governing the curvature of the production function).

We can easily transform our diagram showing output per worker as a function of capital per worker into one showing output per worker as a function of the capital-output ratio by noticing that fixed values of the capital-output ratio are straight lines radiating from the bottom-left corner of the figure.

To figure out what level of output per worker (for a given, fixed level of the efficiency of labor E) correspond to a capital-output ratio of one, look for the point where the curve showing output per worker as a function of the capital stock per worker crosses the ray corresponding to a capital-output ratio of one.

**The steady-state capital-output
ratio**

*The steady-state capital-output ratio.*

__What kind of equilibrium do we look
for?__

No matter what particular situation is being analyzed or issue is being explored, the first instinct of an economist is to look for an

equilibrium: some economic quantity or group of quantities for which there are stable values. These equilibrium values need to be stable in two senses. First, if the economy is in a state in which these quantities are not at their equilibrium values, the economy will tend to change and these variables will approach their equilibrium values. Second, that if the economy is in a state in which these quantities are at their equilibrium values, then the economy will tend to remain in that state.But how are we to look for an equilibrium in the case examined in this chapter, the case of an economy engaged in long-run growth? After all, it seems that in a growing economy with improving technology and positive investment there is no stable equilibrium--there is no stable equilibrium level for technology, or for output per worker, or for the capital stock per worker. All these variables tend to grow over time.

M.I.T. economics professor Robert Solow won his Nobel Prize in large part for determining that there is a straightforward way to think about what an economic equilibrium is in a growing economy. The key is that even though there are no stable values for the economy's stock of capital, or its level of output, or its level of technology, there are stable relationships between these variables.

**[Box: Tools] Growth equilibrium.**

[How can it be an equilibrium if everything is changing? To be written]

__
__

__Focus on the steady-state capital-output ratio.__

For our purposes the most convenient economic variable to focus on is the capital-output ratio: the level of the capital stock per worker divided by potential output per worker. For in a growing economy there will be a stable equilibrium value for this capital-output ratio: And once we have determined this equilibrium value--what economists call the call the

steady-state capital-output ratio--it is then straightforward to calculate the path of economic growth that the economy will follow, a path that economists call thesteady-state growth path.

__Determining the steady-state capital-output
ratio.__

What is the steady-state capital-output ratio?

Consider an economy of which four things are true: First, potential output per worker Y/L--Y for potential output, / for "per", and L for the number of workers in the economy--is growing at some rate g: each year potential output per worker is g percent higher than it was last year. Second, the number of workers in the economy is growing at rate n: each year the number of workers in the economy is n percent higher than it was last year. Third, capital in the economy depreciates at rate d (a Greek letter: lower-case delta): each year d percent of the economy's capital stock wears out, breaks, becomes obsolete, must be replaced. Fourth, each year some s percent of total output Y is saved by the economy as a whole, and used to purchase investment goods to augment the economy's capital stock.

Since output per worker is growing at g percent per year, and the number of workers is growing at n percent per year, then the economy's total output is growing at n+g percent per year. Thus if the capital-output ratio in the economy is going to be unchanging--if the economy is going to be in its long-run growth equilibrium with the capital-output ratio at its steady state value and with output per worker growing along its long-run steady-state growth path--then the capital stock K must be growing at n+g percent per year too. This year's capital stock must be bigger than last year's by n+g percent of last year's capital stock:

(if the capital-output ratio is at its steady-state)

__Is the capital stock growing at the
appropriate rate?__

How can we tell whether the capital stock is in fact growing at the rate needed for the capital-output ratio to be stable? Remember that we also know that each year an amount equal to s x Y, to the savings rate s times this year's output Y is being added to the capital stock through new gross investment. And we also know a fraction d of the current capital stock is being lost due to depreciation--wearing out, falling apart, breaking down, or just becoming too obsolete to be worth using any more:

(by definition: always)

If the capital-output ratio is at its steady-state value, the amount that is being added to the capital stock by net investment--that is, gross investment minus depreciation--must be equal to the amount that is needed in order to match the proportional growth rate of capital to the proportional growth rate of output. Thus:

Moving all of the terms with the capital stock in them to the left-hand side:

Dividing through by this year's level of output:

__The answer: the steady-state capital
output ratio.__

Then dividing through by (n+g+d) in order to get the capital-output ratio all by itself on the left-hand side generates:

(if the capital-output ratio is at its steady-state value)

This is the equation to remember.

This is the payoff from the preceding short march through simple algebra.

The economy's steady-state capital-output ratio is equal to (a) the share s of output saved and invested in new capital, divided by (b) the economy's

investment requirements--divided by the sum of the labor force growth rate n, the long-growth rate of potential growth in output per worker g, and the rate of depreciation of the capital stock d.

*The capital-output ratio grows or
shrinks if...*

__If the capital-output ratio is less
than its steady-state value.__

What if the economy is not at its steady-state capital output ratio? What if this year's value of K/Y is less than s/(n+g+d)? Then the proportional rate of growth of the capital stock will be greater than n+g. To see this, notice that we can divide both sides of (4.2.1.2) by this year's capital stock to determine the proportional rate of growth of the economy's capital stock:

4.2.2.1

If K/Y is smaller than its steady-state value, Y/K will be larger than its steady state value--and so the capital stock will be growing faster than the long-run growth rate of output n+g. Thus the capital-output ratio will tend to rise--unless we are in a very unusual situation (an unusual situation that we will rule out, because it could happen given our production function only if a were to be greater than one) in which the fact that the capital stock is growing faster than n+g boosts the economy's productive potential so much that output grows not only faster than its long-run steady-state proportional growth rate n+g, but faster than the capital stock.

__If the capital-output ratio is greater
than its steady-state value.__

Conversely, if K/Y is larger than its steady-state value, Y/K will be smaller than its steady state value. The higher the capital-output ratio, the more gross investment that must go simply to replace depreciated capital-machines and buildings that wear out or become obsolete. And the smaller the share of investment that is available to boost the capital stock. The capital stock will be growing more slowly than the long-run growth rate of output n+g (and the capital stock may even be shrinking). Thus the capital-output ratio will tend to fall.

__If the capital-output ratio is equal
to its steady-state value.__

And if K/Y is equal to its steady-state value, then--unless one of its determinants s, n, g, or d changes--then K/Y will stay at its steady-state value.

The steady-state value of the capital-output ratio given in (4.2.1.6) thus fulfills all the conditions economists want for an equilibrium. If the capital-output ratio is at its steady-state value, it stays there. If it is less than its steady-state value, it grows. If it is greater than its steady-state value, it shrinks.

**[Box: Tools] Getting to the steady-state
capital-output ratio via diagrams.**

[To be written]

One way to calculate the steady-state capital-output ratio is to plot gross investment and investment requirements as functions of the capital-output ratio.

When gross investment (the upper plot line) is greater than the investment required to keep the capital-output ratio stable (the lower plot line), the capital-output ratio increases. When gross investment is less than investment requirements, the capital-output ratio decreases. Where the curves cross, the capital-output ratio is stable: National product and capital stock are growing at the same rate. This is the steady-state capital-output ratio.

*The steady-state growth path.*

__Determining output per worker on the
steady-state growth path.__

If the economy has reached its steady-state growth path, we can substitute our expression for the steady-state value of the capital-output ratio into the capital-output form of the production function:

to get:

This is a very useful expression.

It tells us how to find the steady-state growth path of output per worker of the economy--the path that it would be on if its capital-output ratio were at its equilibrium value.

**[Box: Examples] Numerical examples for
determining the steady-state growth path.**

[To be written]

__Using the steady-state growth path
as a tool to analyze an economy.__

The equation just above makes analyzing the growth of an economy relatively easily. First identify the economy's savings rate, labor force growth rate, depreciation rate, and long-run rate of growth of potential output. Next calculate the steady-state capital-output ratio--easy to find. And then you can calculate the steady-state growth path.

From the steady-state growth path, you can forecast the future of the economy, for the steady-state growth path is the path that the economy will be on in equilibrium. If the economy is on its steady-state growth path today, it will stay on its steady-state growth path in the future (as long as n, g, d, s, and a do not shift). If the economy is not on its steady-state growth path today, it will head for its steady-state growth path in the future (as long as n, g, d, s, and a do not shift). Thus forecasting becomes easy: we know that if the capital-output ratio is not at its equilibrium value, it is heading for its equilibrium value and will approach it soon.

**[Box: Details] How fast is the efficiency
of labor E growing?…**

One remaining loose end has to do with E, the efficiency of labor. We said that output per worker Y/L had the potential to grow at a proportional rate g in steady state. But we did not say how fast our measure of technology in the broadest sense, the variable E, the efficiency of labor, had to be growing in order for that to be the case.

We can easily determine how fast E must be growing if we remember the form of the production function:

and recall that the capital-output ratio K/Y is constant in steady-state. Then output per worker Y/L is proportional to the efficiency of labor E. So if E is growing at g percent per year, Y/L will be growing at g percent per year as well.

*How fast does the economy head for
its steady-state growth path?*

There is one last, final, loose end to be tied down. It has to do with how fast the economy approaches its steady-state growth path. Suppose that the capital-output ratio K/Y is not at its steady state? How fast does it approach its steady state?

__Calculating the out-of-steady-state
behavior of the capital-output ratio.__

We know that the proportional growth rate of the quotient K/Y is going to be equal to the proportional growth rate of the capital stock K minus the proportional growth rate of output Y. So let us calculate both.

The proportional growth rate of the capital stock K is equal to net investment--gross investment minus depreciation--divided by the current level of the capital stock:

Since we know that:

We know that the growth rate of output Y is equal to a times the growth rate of the capital stock K plus (1-a) times the growth rate of the product (E x L), the product of the efficiency of the labor force E and the size of the labor force L. This is one of the useful tools from algebra that can make our life easier:

__Solving the algebra.__

But we know what the growth rate of the capital stock K is--it is s(Y/K) - d.and We know that with E growing at rate g and L growing at rate n, their product is growing at rate n+g.

Thus we know that:

Subtracting the growth rate of GDP [Y] from the growth rate of the capital stock [K]:

Which simplifies to:

You can check that this is correct by asking what happens when K/Y is at its equilibrium value of s/(n+g+d). Then the terms inside the braces cancel, and the growth rate of the capital-output ratio is zero--as it must be if it is at equilibrium.

If we then pull the (n+g+d) term out to the front of the equation:

And notice that s/(n+g+d) is the steady-state capital-output ratio, we have:

__Interpreting the answer.__

How can we interpret this answer?

There is a straightforward interpretation. The right-hand side will be positive only when this year's Y/K times the steady-state K/Y is greater than one--only when the current capital-output ratio is lower than its steady state value. In such a year the capital output ratio will grow.

In a year an economy whose capital-output ratio is not at its steady-state value will reduce the gap between its current and steady-state values by a fraction of approximately [(1- a)(n+g+d)]. A high capital share (a) means that convergence is slower; with diminishing returns, additional investments yield smaller increases in production and make for more rapid convergence. A low capital share means sharply diminishing returns on investment, and thus rapid convergence.

A high growth rate (n+g) also means that convergence is faster; rapid growth signifies that past output was small, and so can have little importance for the present. A high depreciation rate (d) means that convergence is faster as well.

In an economy like that of the U.S., with a population-plus-productivity-growth rate (n+g) of about 3% per year, a depreciation rate (d) of about 3% per year, and a capital share (a) of about one-third, (1-a)(n+g+ d) is 4%.

If an economy closes four percent of the gap between its current and its steady-state capital-output ratio in a year, then in a 30-year generation it will close about 70% of the gap--slow closure from the perspective of a year or a business cycle or a presidential election cycle, but rapid closure from the perspective of generations or of history. No matter where its level of national product per worker starts, the level tends to converge to its steady-state growth path within several decades.

**[Box: WWW: Examples] The speed of convergence to steady-state.**

[To be written]

**[Box: Examples] West Germany's post-WWII
convergence to its steady state.**

Consider the example of West Germany after World War II. The end of World War II left Germany's cities and its economy in ruins. The allies had waged a truly total war against the Nazi tyranny. Wartime destruction had wrecked the German economy and pushed total output per worker far below its steady-state growth path. But within less than four decades, West Germany had converged again to its pre-World War II growth path. West Germany, at least, was once again among the richest, most productive economies on earth.

Figure: West German Convergence]

**Determinants of the steady-state
capital-output ratio**

*Labor force growth.*

The faster the growth rate of the labor force, the lower will be the economy's steady-state capital-output ratio. Why? Because each new worker who joins the labor force must be equipped with enough capital to be productive, and to on average match the productivity of his or her peers. The faster the rate of growth of the labor force, the larger the share of current investment that must go to equip new members of the labor force with the capital they need to be productive. Thus the lower will be the amount of investment that can be devoted to building up the average ratio of capital to output.

We can analyze the long-run effect on the economy of a change in the rate of labor force growth by using the equation for the steady-state level of output per worker:

A sudden and permanent increase in the rate of population growth n will lower the steady-state growth path level of output per worker because n appears only in the denominator of the right-hand side. The increase in population growth will have no immediate effect on output per worker--just after the instant it happens, the increased rate of population growth has had no time to increase the population. But over time the economy will converge--closing about 4% of the gap every year--to its new, lower, steady-state growth path corresponding to the higher rate of population growth.

[Figure: Changes in population growth rates, changes in the steady-state growth path, and transition to the new steady-state growth path]

We can determine the level of the new, post-change steady-state growth path relative to the level of the old, no-change steady-state growth path. Simply divide the equation for output per worker along the new steady state growth path by the equation for output per worker along the old steady state growth path:

Almost all of the terms on the right-hand side then cancel. All that is left is:

Divide the ratio of the old value of population growth (plus the growth rate g of the efficiency of labor, plus the depreciation rate d) by the new value of population growth (plus the growth rate g of the efficiency of labor, plus the depreciation rate d). Then raise the result to the (a/(1-a)) power. The answer is the level of output per worker on the new, post-change steady-state growth path relative to the value on the old, no-change steady-state growth path.

**[Box: Examples] Changes in labor force
growth and the steady state.**

For example, consider an economy in which the parameter a is 1/2--so that (a/(1-a)) is one--in which the underlying rate of productivity growth g is 1.5% per year and the depreciation rate d is 3.5% per year. Then a reduction in the population growth rate n from 3% per year to 1% per year will raise the level of output per worker on the steady state growth path by fully one-third:

*Depreciation.*

The higher the depreciation rate, the lower will be the economy's steady-state capital-output ratio. Why? Because a higher depreciation rate means that the existing capital stock wears out and must be replaced more quickly. The higher the depreciation rate, the larger the share of current investment that must go replace the capital that has become worn-out or obsolete. Thus the lower will be the amount of investment that can be devoted to building up the average ratio of capital to output.

You can calculate the long-run impact of a change in the rate of depreciation (arising from, say, a technology-driven shift toward either less durable or more durable kinds of capital) on the level of output per worker in the same way that you calculated the long-run impact of a change in the rate of population growth. This time the equation that will be left will be:

And the way you do the calculation is the same. Divide the ratio of the old value of depreciation d (plus the growth rate g of the efficiency of labor, plus the growth rate of the labor force n) by the new value of depreciation d (plus the growth rate g of the efficiency of labor, plus the growth rate of the labor force n). Then raise the result to the (a/(1-a)) power. The answer is the level of output per worker on the new, post-change steady-state growth path relative to the value on the old, no-change steady-state growth path.

**[Box: Examples] Changes in labor force
growth and the steady state.**

[To be written].

*The rate of technological progress.*

The faster the growth rate of productivity, the lower will be the economy's steady-state capital-output ratio. This is correct but counterintuitive. The faster is productivity growth, the higher is output now. But the capital stock depends on what investment was in the past. The faster is productivity growth, the smaller is past investment relative to current production, and the lower is the average ratio of capital to output.

The savings rate.

The higher the share of national product devoted to gross savings and investment investment, the higher will be the economy's steady-state capital-output ratio. Why? Because more investment increases the amount of new capital that can be devoted to building up the average ratio of capital to output. Double the share of national product spent on gross investment, and you will find that you have doubled the economy's capital intensity--doubled its average ratio of capital to output.

Increases in the rate of population growth or the depreciation rate lowered the level of output per worker along the steady-state growth path. Increases in the savings rate raise the level of output per worker along the steady-state growth path.

Once again you divide the new, post-change equation for the level of steady-state output per worker by the old, pre-change equation:

And once again you cancel almost all the terms on the right-hand side:

To be left with only the ratio of the savings rates, raised to the (a/(1-a)) power. That is the long-run effect on output per worker of a change in the savings rate.

**[Box: Examples] Changes in technological
progress and the steady state.**

[To be written].

For example, consider an economy in which the parameter a is 1/3--so that (a/(1-a)) is 1/2. Then an increase in the share of output devoted to gross savings and investment from 10% to 20% would raise the level of output per worker on the steady state growth path by more than two-fifths:

**The golden rule**

*Can the steady-state growth path
be "too high"?: the golden rule.*

__Focus on consumption per worker.__

Suppose that we have an economy on its steady-state growth path, and we are interested not in the level of output per worker but in the level of consumption spending per worker--where the share of total output Y devoted to consumption is simply one minus the share devoted to saving, so that consumption spending per worker is (1-s)Y.

Using the steady-state growth path version of our production function:

__Consumption per worker in steady-state
as a function of the savings rate.__

Now for the moment fix the growth rate of the labor force n, the efficiency of labor g, the depreciation rate d, the parameter a, and consider what the level of consumption per worker in the economy would be if one changed only the steady-state savings rate s. Examine low values of s near zero, high values of s near one, and intermediate values, looking in each case at the level of consumption per worker C/L relative to the efficiency of labor E along the economy's steady-state growth path.

When s is very low, near zero, consumption per worker will be a very low--near-zero--fraction of the efficiency of labor E. The very low numerator of the fraction on the right-hand side of (4.4.1.2) ensures that. When s is high, near one, consumption per worker will also be a very low--near zero--fraction of E. The very low value of the (1-s) in the initial parenthesis in (4.4.1.2) ensures that.

__Maximizing steady-state consumption
per worker.__

In between there is a value of the savings rate s at which consumption per worker along the steady-state growth path (measured relative to the current level of the technology parameter E, the efficiency of labor) reaches a maximum. This savings rate that sustains the maximum level (relative to E, the efficiency of labor) of steady-state consumption per worker is in a sense the "best" savings rate. Economists call it the

golden rulesavings rate, and they call the associated steady-state growth path thegolden rulesteady-state growth path.

__An economy can have "too much"
savings.__

So yes, there is a sense in which an economy can have "too much" savings: if an economy's savings rate is higher than the golden rule savings rate, economists call the economy

dynamically inefficient:it would be possible to raise everyone's level of consumption and thus of material well being if only the economy would save and invest less, would save an invest a smaller proportion of output.

*The marginal product of capital.*

__Building tools: determining the marginal
product of capital.__

In order to say anything else about the golden rule level of saving, we must first determine what the marginal product of capital in the economy is--what the boost to output provided by an extra unit of capital happens to be. To do this, begin with the production function (with all variables subscripted "o", for "old value):

and consider the effect on output of boosting the economy's capital stock by one (with Y now subscripted "+1", for the effect of adding one unit of capital):

We can rewrite the term inside the first parenthesis as:

Notice that the last two parentheses are simply our initial value of output Y

_{o}:And use the algebraic principle that (1+x)

^{}a is, if a is small, approximately equal to 1+ax:Thus the marginal product of capital--the amount by which output is increased by adding a unit to the economy's capital stock--is:

*Determining the golden rule savings
rate.*

__The effect of raising the capital
stock by one unit.__

With the marginal product of capital in hand, we can start to think about what the golden rule savings rate s* is. Suppose that we increase the savings rate by just enough to raise the capital stock by a single unit. If the economy is on its steady-state growth path increase in the capital stock will increase output by an amount equal to the marginal product of capital on the steady-state growth path:

__Does this increase in capital generate
enough extra output?__

Will this increase in output generated by the small increase in the capital stock allow the economy to increase consumption per worker?

The answer is maybe. Thereafter the economy must save an extra amount n+g in order to keep the capital-output ratio at its new, slightly higher level. Thereafter the economy must save an extra amount d to offset the extra depreciation on the new, slightly higher capital-output ratio. There is enough extra output left over to boost consumption if only if

the marginal product of capital(on the steady-state growth path) exceeds theinvestment requirementsn+g+d generated by having a single extra unit of capital.Thus an increase in the capital intensity of the economy--an increase in the savings rate--raises steady-state consumption per worker if and only if:

An increase in the capital intensity of the economy--an increase in the savings rate--lowers steady-state consumption per worker if and only if:

__The golden rule value.__

And if:

a small change in the savings rate neither raises nor lowers steady-state consumption per worker. Then we are at the peak of the figure.

We are at the golden-rule savings rate, which we can easily see to be:

**[Box: Examples] Determining the golden-rule
growth path.**

[To be written].

__Is the golden rule a guide for economic
policy?__

Suppose that the savings rate is not currently at the golden rule value. Would it be good economic policy to take steps to try to move the savings rate to its golden rule value?

If the savings rate is higher than the golden rule value, then the answer is yes. You could make everyone better off--now and in the future--by saving less and reducing the economy's capital-output ratio.

__Intertemporal and intergenerational
tradeoffs.__

If the savings rate is less than the golden rule value, then it is not so clear. Raising the savings rate will in the long run increase consumption per worker, because it will shift the level of consumption per worker (relative to the efficiency of labor) associated with the economy's steady-state growth path upwards as it shifts the steady-state growth path upwards. But in the short run it will decrease consumption per worker--because those alive today will have to cut back on their consumption to generate the increase in the savings rate.

Moreover, those alive today are poorer--have lower consumption per worker--than is likely for future generations. Why should a poorer group of people reduce their consumption to increase the consumption of a richer group?

But that's what we do--both individually and collectively--reduce our level of consumption in order for our descendants to have higher levels of consumption per worker. Few of us (few of us with children, at any rate) are comfortable with the idea that our children won't live any better than we will.

However, it is clear that the case for policies to raise the savings rate (and lower the consumption of the relatively poor current generation) gets weaker the closer s comes to a. When s is very close to a, it is hard to imagine that it could improve anyone's definition of what social welfare is by cutting present-day consumption by a (significant) amount in order to raise the consumption of future generations by (infinitesimal) magnitudes.