0. The first 11 slides contain introductory information to motivate the study of economic growth. Most instructors likely will not test students on the details of this material. Therefore, and because the rest of the chapter is challenging, I encourage you to consider going through the introductory material a bit faster than the remainder of the material.
The central equation of the Solow model first appears on slide 24. The textbook does not assign a special name to this equation, but the equation is referred to very often, so giving it a name makes life easier for students and for the instructor. In these slides, therefore, it’s referred to as “the equation of motion for k.”
Because this name is not used in the textbook, one could reasonably argue that it should not be used here. If you take this position, please let me know in an I am doing annual updates to these slides (which will publish in summer 2003 and summer 2004), so I will have the chance to incorporate feedback from users like yourself into the revisions.
One final note before we begin: I strongly encourage you to take a look at William Easterly’s brilliant new book The Elusive Quest for Growth (MIT Press, 2001). It has lots of compelling real world examples (some of which I’ve included in the PowerPoint presentation of the two economic growth chapters). It also explains economic theory in ways that students (and college-educated non-econ majors) find easy to grasp. After reading it, I find I do a better job teaching Mankiw’s growth chapters to my students.

1. Chapter 7 learning objectives
Learn the closed economy Solow model
See how a country’s standard of living depends on its saving and population growth rates
Learn how to use the “Golden Rule” to find the optimal savings rate and capital stock

2. The importance of economic growth
…for poor countries
for poor countries:
The following slides present selected poverty statistics and estimates of the reduction in poverty caused by economic growth.
The point: If we can learn how to help poor countries rise out of poverty, we will be making a huge difference in the lives of billions of people, as well as creating new markets for our exports.
…for rich countries:

3. selected poverty statistics
In the poorest one-fifth of all countries,
daily caloric intake is 1/3 lower than in the richest fifth
the infant mortality rate is 200 per 1000 births, compared to 4 per 1000 births in the richest fifth.
source: The Elusive Quest for Growth, by William Easterly. (MIT Press, 2001)

4. selected poverty statistics
In Pakistan, 85% of people live on less than $2/day
One-fourth of the poorest countries have had famines during the past 3 decades. (none of the richest countries had famines)
Poverty is associated with the oppression of women and minorities
source: The Elusive Quest for Growth, by William Easterly. (MIT Press, 2001)

5. Estimated effects of economic growth
A 10% increase in income is associated with a 6% decrease in infant mortality
Income growth also reduces poverty. Example:
Growth and Poverty in Indonesia
change in # of persons living below poverty line
change in income per capita
source: The Elusive Quest for Growth, by William Easterly. (MIT Press, 2001)
This is just one of many concrete examples of the negative relationship between economic growth and the poverty rate.
+76%
-25%
-12%
+65%

6. Income and poverty in the world selected countries, 2000
source: The Elusive Quest for Growth, by William Easterly. (MIT Press, 2001)
Again, the point here is this:
Learning about economic growth

7. The importance of economic growth
…for poor countries
…for rich countries
for poor countries:
The following slides present selected poverty statistics and estimates of the reduction in poverty caused by economic growth.
The point: If we can learn how to help poor countries rise out of poverty, we will be making a huge difference in the lives of billions of people, as well as creating new markets for our exports.
…for rich countries:

8. Huge effects from tiny differences
In rich countries like the U.S., if government policies or “shocks” have even a small impact on the long-run growth rate, they will have a huge impact on our standard of living in the long run…

9. Huge effects from tiny differences
percentage increase in standard of living after…
annual growth rate of income per capita
…25 years
…50 years
…100 years
2.0%
64.0%
169.2%
624.5%
2.5%
85.4%
243.7%
1,081.4%
These calculations show that a one-half point increase in the growth rate has, in the long run, a HUGE impact on the standard of living.

10. Huge effects from tiny differences
If the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher during the 1990s,
the U.S. would have generated an additional $449 billion of income during that decade
The $449 billion is in “today’s dollars” (i.e. measured in the prices that prevailed in the first quarter of 2002).
In case you’re wondering how I did this calculation:
Computed actual quarterly growth rate of real income per capita from 1989:4 through 1999:4.
Added one-fourth of one-tenth of one percent to each quarter’s actual growth rate.
Computed what real income per capita would have been with the new growth rates.
Multiplied this hypothetical real income per capita by the population to get hypothetical real GDP.
Computed the difference between hypothetical and actual real GDP.
Cumulated these differences over the period 1990:1-1999:4.
Like the original real GDP data, the cumulative difference was in 1996 dollars. I multiplied this amount by 10%, the amount by which the GDP deflator rose between 1996 and 2002:1, so the final result ($449 billion) is expressed in 2002:1 dollars, which I simply call “today’s dollars.”
DATA SOURCE:
Real GDP, GDP deflator - Dept of Commerce, Bureau of Economic Analysis.
Population - Dept of Commerce, Census Bureau.
All obtained from “FRED” - the St. Louis Fed’s database, on the web at

11. The lessons of growth theory
…can make a positive difference in the lives of hundreds of millions of people.
These lessons help us
understand why poor countries are poor
design policies that can help them grow
learn how our own growth rate is affected by shocks and our government’s policies

12. The Solow Model
due to Robert Solow, won Nobel Prize for contributions to the study of economic growth
a major paradigm:
widely used in policy making
benchmark against which most recent growth theories are compared
looks at the determinants of economic growth and the standard of living in the long run

13. How Solow model is different from Chapter 3’s model
1. K is no longer fixed: investment causes it to grow, depreciation causes it to shrink.
2. L is no longer fixed: population growth causes it to grow.
3. The consumption function is simpler.
It’s easier for students to learn the Solow model if they see that it’s just an extension of something they already know, the classical model from Chapter 3. So, this slide and the next point out the differences.

14. How Solow model is different from Chapter 3’s model
4. No G or T (only to simplify presentation; we can still do fiscal policy experiments)
5. Cosmetic differences.
The cosmetic differences include things like the notation (lowercase letters for per-worker magnitudes instead of uppercase letters for aggregate magnitudes) and the variables that are measured on the axes of the main graph.

15. The production function
In aggregate terms: Y = F (K, L )
Define: y = Y/L = output per worker
k = K/L = capital per worker
Assume constant returns to scale: zY = F (zK, zL ) for any z > 0
Pick z = 1/L. Then
Y/L = F (K/L , 1)
y = F (k, 1)
y = f(k) where f(k) = F (k, 1)
When everything on the slide is showing on the screen, explain to students how to interpret f(k):
f(k) is the “per worker production function,” it shows how much output one worker could produce using k units of capital.
You might want to point out that this is the same production function we worked with in chapter 3. We’re just expressing it differently.

16. The production function
Output per worker, y
Capital per worker, k
f(k) 1
MPK =f(k +1) – f(k)
Note: this production function exhibits diminishing MPK.

17. The national income identity
Y = C + I (remember, no G )
In “per worker” terms: y = c + i where c = C/L and i = I/L

18. The consumption function
s = the saving rate, the fraction of income that is saved
(s is an exogenous parameter)
Note: s is the only lowercase variable that is not equal to its uppercase version divided by L
Consumption function: c = (1–s)y (per worker)

19. Saving and investment saving (per worker) = y – c = y – (1–s)y = sy
National income identity is y = c + i
Rearrange to get: i = y – c = sy (investment = saving, like in chap. 3!)
Using the results above, i = sy = sf(k)
The real interest rate r does not appear explicitly in any of the Solow model’s equations. This is to simplify the presentation. You can tell your students that investment still depends on r, which adjusts behind the scenes to keep investment = saving at all times.

20. Output, consumption, and investment
Output per worker, y
Capital per worker, k
f(k) y1 k1 c1
sf(k) i1

21. Depreciation  = the rate of depreciation
= the fraction of the capital stock that wears out each period
Depreciation per worker, k
Capital per worker, k k  1

22. Capital accumulation The basic idea:
Investment makes the capital stock bigger,
depreciation makes it smaller.

23. Capital accumulation k = s f(k) – k
Change in capital stock = investment – depreciation
k = i – k
Since i = sf(k) , this becomes:
k = s f(k) – k

24. The equation of motion for k
k = s f(k) – k
the Solow model’s central equation
Determines behavior of capital over time…
…which, in turn, determines behavior of all of the other endogenous variables because they all depend on k. e.g.,
income per person: y = f(k)
consump. per person: c = (1–s) f(k)

25. The steady state k = s f(k) – k
If investment is just enough to cover depreciation [sf(k) = k ],
then capital per worker will remain constant: k = 0.
This constant value, denoted k*, is called the steady state capital stock.

26. Investment and depreciation
The steady state
Investment and depreciation
Capital per worker, k k
sf(k) k*

27. Moving toward the steady state
k = sf(k)  k
Investment and depreciation
Capital per worker, k k
sf(k) k*
investment k1 k
depreciation

28. Moving toward the steady state
k = sf(k)  k
Investment and depreciation
Capital per worker, k k
sf(k) k* k1 k

29. Moving toward the steady state
k = sf(k)  k
Investment and depreciation
Capital per worker, k k
sf(k) k* k1 k k2

30. Moving toward the steady state
k = sf(k)  k
Investment and depreciation
Capital per worker, k k
sf(k) k*
investment k
depreciation k2

31. Moving toward the steady state
k = sf(k)  k
Investment and depreciation
Capital per worker, k k
sf(k) k* k2 k

32. Moving toward the steady state
k = sf(k)  k
Investment and depreciation
Capital per worker, k k
sf(k) k* k k2 k3

33. Moving toward the steady state
k = sf(k)  k
Investment and depreciation
Capital per worker, k k
Summary: As long as k < k*, investment will exceed depreciation, and k will continue to grow toward k*.
sf(k) k* k3

34. Now you try:
Draw the Solow model diagram, labeling the steady state k*.
On the horizontal axis, pick a value greater than k* for the economy’s initial capital stock. Label it k1.
Show what happens to k over time. Does k move toward the steady state or away from it?

35. A numerical example Production function (aggregate):
To derive the per-worker production function, divide through by L:
Then substitute y = Y/L and k = K/L to get

36. A numerical example, cont.
Assume:
s = 0.3
 = 0.1
initial value of k = 4.0
As each assumption appears on the screen, explain it’s interpretation.
I.e.,
“The economy saves three-tenths of income,”
“every year, 10% of the capital stock wears out,” and
“suppose the economy starts out with four units of capital for every worker.”

37. Approaching the Steady State: A Numerical Example
Year k y c i k k
Before revealing the numbers in the first row, ask your students to determine them and write them in their notes. Give them a moment, then reveal the first row and make sure everyone understands where each number comes from.
Then, ask them to determine the numbers for the second row and write them in their notes.
After the second round of this, it’s probably fine to just show them the rest of the table.

38. Approaching the Steady State: A Numerical Example
Year k y c i k k … 

39. Exercise: solve for the steady state
Continue to assume s = 0.3,  = 0.1, and y = k 1/2
Use the equation of motion k = s f(k)  k to solve for the steady-state values of k, y, and c.
Suggestion: give your students 3-5 minutes to work on this exercise in pairs. Working alone, a few students might not know to start by setting k = 0. But working in pairs, they are more likely to figure it out.
Also, this gives students a little psychological momentum to make it easier for them to start on the end-of-chapter exercises (if you assign them as homework).
(If any need a hint, remind them that the steady state is defined by k = 0. A further hint is that they answers they get should be the same as the last row of the big table on the preceding slide, since we are still using all the same parameter values.)

40. Solution to exercise:

41. An increase in the saving rate
An increase in the saving rate raises investment…
…causing the capital stock to grow toward a new steady state:
Investment and depreciation k k
s2 f(k)
s1 f(k)
Next, we see what the model says about the relationship between a country’s saving rate and its standard of living (income per capita) in the long run (or steady state).
An earlier slide said that the model’s omission of G and T was only to simplify the presentation. We can still do policy analysis. We know from Chapter 3 that changes in G and/or T affect national saving. In the Solow model as presented here, we can simply change the exogenous saving rate to analyze the impact of fiscal policy changes.

42. Prediction: Higher s  higher k*.
And since y = f(k) , higher k*  higher y* .
Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run.
Of course, the converse is true, as well: a fall in s (caused, for example, by tax cuts or government spending increases) leads ultimately to a lower standard of living.
In the static model of Chapter 3, we learned that a fiscal expansion crowds out investment. The Solow model allows us to see the long-run dynamic effects: the fiscal expansion, by reducing the saving rate, reduces investment. If we were initially in a steady state (in which investment just covers depreciation), then the fall in investment will cause capital per worker, labor productivity, and income per capita to fall toward a new, lower steady state. (If we were initially below a steady state, then the fiscal expansion causes capital per worker and productivity to grow more slowly, and reduces their steady-state values.)

43. International Evidence on Investment Rates and Income per Person
Graph to be updated for final version of this PowerPoint presentation.

44. The Golden Rule: introduction
Different values of s lead to different steady states. How do we know which is the “best” steady state?
Economic well-being depends on consumption, so the “best” steady state has the highest possible value of consumption per person: c* = (1–s) f(k*)
An increase in s
leads to higher k* and y*, which may raise c*
reduces consumption’s share of income (1–s), which may lower c*
So, how do we find the s and k* that maximize c* ?

45. The Golden Rule Capital Stock
the Golden Rule level of capital, the steady state value of k that maximizes consumption.
To find it, first express c* in terms of k*:
c* = y*  i*
= f (k*)  i*
= f (k*)  k*
In general: i = k + k
In the steady state: i* = k* because k = 0.

46. The Golden Rule Capital Stock
steady state output and depreciation
steady-state capital per worker, k*
 k*
Then, graph f(k*) and k*, and look for the point where the gap between them is biggest.
f(k*)
Students sometimes confuse this graph with the other Solow model diagram, as the curves look similar.
Be sure to clarify the differences:
On this graph, the horizontal axis measures k*, not k. Thus, once we have found k* using the other graph, we plot that k* on this graph to see where the economy’s steady state is in relation to the golden rule capital stock.
On this graph, the curve measures f(k*), not sf(k).
On the other diagram, the intersection of the two curves determines k*. On this graph, the only thing determined by the intersection of the two curves is the level of capital where c*=0, and we certainly wouldn’t want to be there.
There are no dynamics in this graph, as we are in a steady state. In the other graph, the gap between the two curves determines the change in capital.

47. The Golden Rule Capital Stock
c* = f(k*)  k* is biggest where the slope of the production func. equals the slope of the depreciation line:
 k*
f(k*)
If your students have had a semester of calculus, you can show them that deriving the condition MPK =  is straight-forward:
The problem is to find the value of k* that maximizes c* = f(k*)  k*.
Just take the first derivative of that expression and set equal to zero:
f(k*)   = 0
where f(k*) = MPK = slope of production function and  = slope of steady-state investment line.
MPK = 
steady-state capital per worker, k*

48. The transition to the Golden Rule Steady State
The economy does NOT have a tendency to move toward the Golden Rule steady state.
Achieving the Golden Rule requires that policymakers adjust s.
This adjustment leads to a new steady state with higher consumption.
But what happens to consumption during the transition to the Golden Rule?
Remember: policymakers can affect the national saving rate:
- changing G or T affects national saving
- holding T constant overall, but changing the structure of the tax system to provide more incentives for private saving (i.e., shifting from income tax to consumption tax in such a way that leaves total revenue unchanged)

49. Starting with too much capital
then increasing c* requires a fall in s.
In the transition to the Golden Rule, consumption is higher at all points in time.
time y c i
t0 is the time period in which the saving rate is reduced. It would be helpful if you explained the behavior of each variable before t0, at t0 , and in the transition period (after t0 ).
Before t0: in a steady state, where k, y, c, and i are all constant.
At t0:
The change in the saving rate doesn’t immediately change k, so y doesn’t change immediately.
But the fall in s causes a fall in investment [because saving equals investment] and a rise in consumption [because c = (1-s)y, s has fallen but y has not yet changed.]. Note that c = -i, because y = c + i and y has not changed.
After t0:
In the previous steady state, saving and investment were just enough to cover depreciation. Then saving and investment were reduced, so depreciation is greater than investment, which causes k to fall toward a new, lower steady state value.
As k falls and settles on its new, lower steady state value, so will y, c, and i (because each of them is a function of k).
Even though c is falling, it doesn’t fall all the way back to its initial value.
Policymakers would be happy to make this change, as it produces higher consumption at all points in time (relative to what consumption would have been if the saving rate had not been reduced. t0

50. Starting with too little capital
then increasing c* requires an increase in s.
Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption. y c
Before t0: in a steady state, where k, y, c, and i are all constant.
At t0:
The increase in s doesn’t immediately change k, so y doesn’t change immediately.
But the increase in s causes investment to rise [because higher saving means higher investment] and consumption to fall [because we are saving more of our income, and consuming less of it].
After t0:
Now, saving and investment exceed depreciation, so k starts rising toward a new, higher steady state value. The behavior of k causes the same behavior in y, c, and i (qualitatively the same, that is).
Ultimately, consumption ends up at a higher steady state level. But initially consumption falls. Therefore, if policymakers value the current generation’s well-being more than that of future generations, they might be reluctant to adjust the saving rate to achieve the Golden Rule. Notice, though, that if they did increase s, an infinite number of future generations would benefit, which makes the sacrifice of the current generation seem more acceptable. i t0
time

51. Population Growth
Assume that the population--and labor force-- grow at rate n. (n is exogenous)
EX: Suppose L = 1000 in year 1 and the population is growing at 2%/year (n = 0.02).
Then L = n L = 0.02  1000 = 20, so L = 1020 in year 2.

52. Break-even investment
( + n)k = break-even investment, the amount of investment necessary to keep k constant.
Break-even investment includes:
 k to replace capital as it wears out
n k to equip new workers with capital (otherwise, k would fall as the existing capital stock would be spread more thinly over a larger population of workers)

53. The equation of motion for k
With population growth, the equation of motion for k is
k = s f(k)  ( + n) k
actual investment
break-even investment
Of course, “actual investment” and “break-even investment” here are in “per worker” magnitudes.

54. The Solow Model diagram
k = s f(k)  ( +n)k
Investment, break-even investment
Capital per worker, k
( + n ) k
sf(k) k*

55. The impact of population growth
Investment, break-even investment
( +n2) k
( +n1) k
An increase in n causes an increase in break-even investment,
sf(k)
k2*
leading to a lower steady-state level of k.
k1*
Capital per worker, k

56. Prediction: Higher n  lower k*.
And since y = f(k) , lower k*  lower y* .
Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.
Of course, the converse is true, as well: a fall in s (caused, for example, by tax cuts or government spending increases) leads ultimately to a lower standard of living.
In the static model of Chapter 3, we learned that a fiscal expansion crowds out investment. The Solow model allows us to see the long-run dynamic effects: the fiscal expansion, by reducing the saving rate, reduces investment. If we were initially in a steady state (in which investment just covers depreciation), then the fall in investment will cause capital per worker, labor productivity, and income per capita to fall toward a new, lower steady state. (If we were initially below a steady state, then the fiscal expansion causes capital per worker and productivity to grow more slowly, and reduces their steady-state values.)

57. International Evidence on Population Growth and Income per Person

58. The Golden Rule with Population Growth
To find the Golden Rule capital stock, we again express c* in terms of k*:
c* = y*  i*
= f (k* )  ( + n) k*
c* is maximized when MPK =  + n
or equivalently, MPK   = n
In the Golden Rule Steady State, the marginal product of capital net of depreciation equals the population growth rate.

59. Chapter Summary
The Solow growth model shows that, in the long run, a country’s standard of living depends
positively on its saving rate.
negatively on its population growth rate.
An increase in the saving rate leads to
higher output in the long run
faster growth temporarily
but not faster steady state growth.
Before leaving this chapter, you should emphasize that we have not yet answered an important question: What causes the kind of sustained growth in living standards that we’ve experienced in the U.S. and elsewhere over the very long run? The Solow model as described in Chapter 7 has a steady state in which income per capita remains constant.
Chapter 8 addresses this issue by introducing technological progress into the Solow model.

60. Chapter Summary
If the economy has more capital than the Golden Rule level, then reducing saving will increase consumption at all points in time, making all generations better off.
If the economy has less capital than the Golden Rule level, then increasing saving will increase consumption for future generations, but reduce consumption for the present generation.

61. Thanks for your attention!!
Dr. Weng