Mankiw 6e PowerPoints
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Economic Growth I:
Capital Accumulation and Population Growth
8
CHAPTER
CHAPTER 8 Economic Growth I
Chapters 8 and 9 cover one of the most important topics in macroeconomics. The material in these chapters is more challenging than average for the book, yet Mankiw explains it especially clearly.
This PowerPoint presentation provides an introduction with data to motivate the study of economic growth. If your classroom computer has live internet access, a better alternative might be to display a few dynamic cross-country graphs from Gapminder, an exciting, dynamic graphical database I describe in the notes accompanying a few of the following slides. I have included links to a few ready-made Gapminder graphs. I strongly recommend spending a few minutes checking out Gapminder – you and your students will find it very interesting, useful, and easy to use. See the following slides for more info and help getting started. This is worth it, trust me!
IN THIS CHAPTER, YOU WILL LEARN:
the closed economy Solow model
how a country’s standard of living depends on its saving and population growth rates
how to use the “Golden Rule” to find the optimal saving rate and capital stock
CHAPTER 8 Economic Growth I
Why growth matters
Data on infant mortality rates:
20% in the poorest 1/5 of all countries
0.4% in the richest 1/5
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.
Poverty is associated with oppression of women and minorities.
Economic growth raises living standards and reduces poverty….
CHAPTER 8 Economic Growth I
* * * INSTRUCTOR PLEASE READ!!! * * *
Before you teach this chapter, please check out the links I’ve included to Gapminder. This amazing website creates dynamic cross-country scatter-type graphs using data from reputable sources (such as the World Bank’s World Development Indicators). The graphs are really quite amazing and will surely spark student interest in the topic and, if you wish, class discussion.
I prepared some introductory slides on economic growth a few editions ago, and they remain. But the Gapminder graphs are better; please check them out and consider using one or more of them in class.
If you’re learning about Gapminder for the first time from my slides, let me say in advance:
You’re welcome!
Income and poverty in the world
selected countries, 2010
Indonesia
Uruguay
Poland
Senegal
Kyrgyz Republic
Nigeria
Zambia
Panama
Mexico
Georgia
Peru
CHAPTER 8 Economic Growth I
* * * INSTRUCTOR PLEASE READ!!! * * *
I created this graph for previous editions of these slides, and I have not deleted it in case you are not able to use the following, far superior alternative:
PLEASE! Do yourself and your students a favor and spend a few minutes (before class) checking out this amazingly useful resource, here at:
http://gapminder.org or http://tools.google.com/gapminder
You can download “Gapminder Desktop” for later use without an internet connection, or if you have internet access in your classroom, just click on the links I provide in a later slide.
Trust me – Gapminder will impress you and your students, and will definitely motivate them about the importance of economic growth!
On the following slide, I have included links to a few Gapminder graphs which might be useful here.
—
Information about the data on THIS slide:
Source: World Bank Databank
http://databank.worldbank.org/data/home.aspx
This chart includes data on every country for which the poverty data were available. These countries include:
Argentina, Armenia, Bangladesh, Belarus, Colombia, Dominican Republic, Ecuador, Ethiopia*, Georgia, India, Indonesia, Jordan, Kyrgyz Republic, Macedonia FYR, Madagascar, Malawi, Mali, Mexico, Moldova, Nepal, Nigeria, Panama, Paraguay, Peru, Poland, Romania, Rwanda*, Senegal*, Serbia, Sierra*, Sri Lanka, Swaziland, Thailand, Togo*, Tunisia, Turkey, Ukraine, Uruguay, Zambia
* poverty rate missing in 2010 but present in 2011; the 2011 figure was used
2010 POV 341.077218457632 359.579526836732 419.223813294044 447.753323774006 503.161788464637 519.022354470948 594.341875583768 664.064231684515 673.694661381466 880.038514144186 992.643834910162 1224.95386293374 1419.11267396222 1431.60123914182 1631.53319453364 2399.92005365438 2613.68950836498 2946.65606133866 2973.98170874555 3093.53952962037 3100.50163991722 3124.7842960129 4206.77992157625 4370.72103318115 4442.29595377534 4508.11222024666 4802.66275766213 5073.07494484382 5156.7285201281 5249.58602241539 5818.8548592158 6179.75887163099 7355.10020314682 7670.30376269901 8780.24202219057 9132.95725866257 10135.4209159865 11520.2725838834 12302.0299191861 66 82.31 92.62 79.56 52.65 82.37 57.25 76.54 78.66 22.9 55.22 86.64 68.76 84.49 4.35 23.85 35.6 46.12 0.08 60.4 13.22 19.91 4.25 1.59 9.13 10.59 4.05 0.62 9.88 12.74 0.09 15.82 13.8 1.3 4.54 1.87 4.71 1.18 0.19 Income per capita in U.S. dollars
% of population living on $2/day or less
links to prepared graphs @ Gapminder.org
notes: circle size is proportional to population size,
color of circle indicates continent, press “play” on bottom to see the cross section graph evolve over time
Income per capita and
Life expectancy
Infant mortality
Malaria deaths per 100,000
Cell phone users per 100 people
CHAPTER 8 Economic Growth I
This slide provides links to some dynamic graphs at Gapminder.org. I strongly encourage you explore the site for a few minutes and then share it with your students.
You can start by checking out some or all of the graphs I’ve linked to here. For class, I suggest you show the life expectancy graph and one or two of the other ones.
Disclaimers/warnings: Gapminder is an external site not under the control of Worth Publishers. It could be taken down at any time (though this is very unlikely). I had to use “TinyURL” to create links small enough for PowerPoint to recognize; TinyURL is an external service not under the control of Worth Publishers, and it is possible that the TinyURL service could be interrupted or discontinued (though this is unlikely).
Also, please note that Gapminder can now be downloaded to your Mac or Windows computer and run without an internet connection.
Why growth matters
Anything that effects the long-run rate of economic growth – even by a tiny amount – will have huge effects on living standards in the long run.
1,081.4%
243.7%
85.4%
624.5%
169.2%
64.0%
2.5%
2.0%
…100 years
…50 years
…25 years
increase in standard
of living after…
annual growth rate of income per capita
CHAPTER 8 Economic Growth I
Why growth matters
If the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher from 2000–2010, the average person would have earned $2,782 more during the decade.
CHAPTER 8 Economic Growth I
The $2,782 figure is in 2005 prices. In current prices, the figure would be a bit higher.
Also, the result would have been a little higher if we divided GDP by working aged adults rather than the entire population.
For these reasons, we can consider $2,782 to be a lower bound on the extra income someone would have earned if the growth rate were just 0.1% higher.
How I did this calculation:
Created an annual real GDP per capita series with data on real GDP (in 2005 dollars) and population from FRED.
Computed the annual growth rate of real GDP per capita from 2000 to 2010.
Added 0.1% to each of these growth rates.
Computed what real GDP per capita would have been each year using these 0.1% higher growth rates.
Computed the difference between these hypothetical income per capita values and the actual ones.
Added up these differences for the decade and got the total: $2,782.
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
CHAPTER 8 Economic Growth I
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
CHAPTER 8 Economic Growth I
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
CHAPTER 8 Economic Growth I
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.
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
CHAPTER 8 Economic Growth I
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.
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)
CHAPTER 8 Economic Growth I
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.
The production function
Output per
worker, y
Capital per
worker, k
f(k)
Note: this production function exhibits diminishing MPK.
1
MPK = f(k +1) – f(k)
CHAPTER 8 Economic Growth I
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
CHAPTER 8 Economic Growth I
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)
CHAPTER 8 Economic Growth I
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)
CHAPTER 8 Economic Growth I
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.
Output, consumption, and investment
Output per
worker, y
Capital per
worker, k
f(k)
sf(k)
k1
y1
i1
c1
CHAPTER 8 Economic Growth I
Depreciation
Depreciation per worker, δk
Capital per
worker, k
δk
δ = the rate of depreciation
= the fraction of the capital stock that wears out each period
1
δ
CHAPTER 8 Economic Growth I
Capital accumulation
Change in capital stock = investment – depreciation
Δk = i – δk
Since i = sf(k) , this becomes:
Δk = s f(k) – δk
The basic idea: Investment increases the capital stock, depreciation reduces it.
CHAPTER 8 Economic Growth I
The equation of motion for 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)
consumption per person: c = (1 – s) f(k)
Δk = s f(k) – δk
CHAPTER 8 Economic Growth I
The steady state
If investment is just enough to cover depreciation
[sf(k) = δk ],
then capital per worker will remain constant:
Δk = 0.
This occurs at one value of k, denoted k*,
called the steady state capital stock.
Δk = s f(k) – δk
CHAPTER 8 Economic Growth I
The steady state
Investment and depreciation
Capital per
worker, k
sf(k)
δk
k*
CHAPTER 8 Economic Growth I
Moving toward the steady state
Investment and depreciation
Capital per
worker, k
sf(k)
δk
k*
Δk = sf(k) − δk
depreciation
Δk
k1
investment
CHAPTER 8 Economic Growth I
Moving toward the steady state
Investment and depreciation
Capital per
worker, k
sf(k)
k*
k1
Δk
k2
Δk = sf(k) − δk
δk
CHAPTER 8 Economic Growth I
Moving toward the steady state
Investment and depreciation
Capital per
worker, k
sf(k)
k*
k2
investment
depreciation
Δk
Δk = sf(k) − δk
δk
CHAPTER 8 Economic Growth I
Moving toward the steady state
Investment and depreciation
Capital per
worker, k
sf(k)
k*
Δk
k2
Δk = sf(k) − δk
δk
CHAPTER 8 Economic Growth I
Moving toward the steady state
Investment and depreciation
Capital per
worker, k
sf(k)
k*
k2
Δk
k3
Δk = sf(k) − δk
δk
CHAPTER 8 Economic Growth I
Moving toward the steady state
Investment and depreciation
Capital per
worker, k
sf(k)
k*
k3
Summary:
As long as k < k*, investment will exceed depreciation,
and k will continue to grow toward k*.
Δk = sf(k) − δk
δk
CHAPTER 8 Economic Growth I
NOW YOU TRY
Approaching k* from above
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?
CHAPTER 8 Economic Growth I
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
CHAPTER 8 Economic Growth I
A numerical example, cont.
Assume:
s = 0.3
δ = 0.1
initial value of k = 4.0
CHAPTER 8 Economic Growth I
As each assumption appears on the screen, explain its 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.”
Approaching the steady state:
A numerical example
Year k y c i δk Δk
1 4.000 2.000 1.400 0.600 0.400 0.200
2 4.200 2.049 1.435 0.615 0.420 0.195
3 4.395 2.096 1.467 0.629 0.440 0.189
4 4.584 2.141 1.499 0.642 0.458 0.184
…
10 5.602 2.367 1.657 0.710 0.560 0.150
…
25 7.351 2.706 1.894 0.812 0.732 0.080
…
100 8.962 2.994 2.096 0.898 0.896 0.002
…
∞ 9.000 3.000 2.100 0.900 0.900 0.000
CHAPTER 8 Economic Growth I
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.
NOW YOU TRY
Solve for the steady state
Continue to assume
s = 0.3, δ = 0.1, and y = k 1/2
Use the equation of motion
Δk = s f(k) − δk
to solve for the steady-state values of k, y, and c.
CHAPTER 8 Economic Growth I
Suggestion: give your students 3-5 minutes to work on this exercise in pairs. Working alone, a few students might not know that they need to start by setting the change in capital per worker equal to zero. 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 students need a hint, remind them that the steady state is defined by the condition that the change in capital per worker equals zero. A further hint is that the 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.
ANSWERS
Solve for the steady state
CHAPTER 8 Economic Growth I
The first few lines of this slide show the calculations and intermediate steps necessary to arrive at the correct answers, which are given in the last 2 lines of the slide.
An increase in the saving rate
Investment
and
depreciation
k
δk
s1 f(k)
An increase in the saving rate raises investment…
…causing k to grow toward a new steady state:
s2 f(k)
CHAPTER 8 Economic Growth I
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.
Prediction:
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.
Are the data consistent with this prediction?
CHAPTER 8 Economic Growth I
After showing this slide, you might also note that 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.)
This, of course, is relevant because actual U.S. public saving has fallen sharply since 2001.
International evidence on investment rates and income per person
Income per person in 2010
(log scale)
Investment as percentage of output
(average 1960-2010)
CHAPTER 8 Economic Growth I
Figure 8.6, p.223.
This scatterplot shows the experience of 100 countries, each represented by a single point. The horizontal axis shows the country’s rate of investment, and the vertical axis shows the country’s income per person. High investment is associated with high income per person, as the Solow model predicts.
Source:
Alan Heston, Robert Summers and Bettina Aten, Penn World Table Version 7.1, Center for International Comparisons of Production, Income and Prices at the University of Pennsylvania, July 2012.
41.08590317 22.17685318 28.41178322 25.52403259 12.59492207 23.9267005899999 26.17610359 19.87400627 12.51677704 23.13699913 19.48469734 12.53583336 16.37870789 20.9617195099999 53.19124985 11.83430195 23.83876419 36.94510269 19.4893493699999 28.94630814 35.53129196 18.98803139 11.50132847 22.80346107 17.87476158 28.97130013 13.86082268 14.85276222 19.2807273899999 15.60844612 28.3821315799999 22.35596657 37.64845276 7.39858865699998 21.57948112 28.91446877 17.95537567 14.56109715 29.2307682 21.9295311 35.66128159 25.27108192 22.00703239 23.99117088 27.66852188 25.44742203 26.73413277 24.61987114 30.76105499 34.01845551 14.41149426 30.98752022 28.81638718 25.42226028 11.34215641 32.40745926 31.11654091 18.43187904 30.39206314 23.37963295 30.58334732 12.45245171 17.05015564 22.05856133 19.32754707 23.66590691 18.73665619 14.07253838 31.18475151 19.11530495 23.19507408 18.00296211 23.35935783 23.15378952 27.08708 7.679425716 13.90959644 18.04723549 22.55971336 27.09989166 25.4724368999999 18.87962723 28.27882195 17.74635887 24.96299934 32.36548233 18.33903694 28.15144157 15.86350155 10.67754173 17.80090904 20.32007217 21.16180801 27.37434578 14.76590824 3.973996162 7220.849431 14512.06175 49727.91563 42488.6822 1502.087644 29697.32074 39759.47574 1290.724114 4432.779106 9754.691878 1149.928906 451.1570776 2055.871851 42678.59603 4126.969072 1768.584474 15960.80033 8124.918354 8975.414416 976.927163599999 2628.452247 12983.01742 1448.664554 38746.50592 11600.34756 7345.691681 5265.55153 6827.965692 11126.31029 813.2109171 36374.6233 35222.51278 15026.03725 1429.990886 2288.379944 28435.59031 7071.22547 982.546609999999 905.9784027 3803.382624 41930.24704 40099.3694 3995.606345 4312.092746 38163.02596 29325.26777 31779.18833 9681.04033499999 34655.03836 5789.815045 1467.754394 28768.22326 1675.678181 93496.5642799999 815.6960461 811.0727548 13992.5397 1151.591705 10388.24469 13430.03238 4133.720642 862.841802299999 1321.292551 42546.43482 31969.70844 2592.882558 552.340345899999 1629.502422 58958.65392 2489.887143 11495.74442 4851.179775 9009.563851 3595.869194 22583.45345 1225.829758 1657.729016 34736.04031 8907.69685 30816.09764 4611.932751 40890.70522 45368.07991 4058.636446 1320.950164 9212.03212399999 867.7367709 25525.53349 11589.95653 1292.806442 38463.11369 46568.56743 13671.16122 11777.99098 2267.444053 369.1481259
The Golden Rule: Introduction
Different values of s lead to different steady states.
How do we know which is the “best” steady state?
The “best” steady state has the highest possible
consumption per person: c* = (1–s) f(k*).
An increase in s
leads to higher k* and y*, which raises c*
reduces consumption’s share of income (1–s),
which lowers c*.
So, how do we find the s and k* that maximize c*?
CHAPTER 8 Economic Growth I
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*) − δk*
In the steady state: i* = δk* because Δk = 0.
CHAPTER 8 Economic Growth I
Then, graph
f(k*) and δk*,
look for the
point where
the gap between them is biggest.
The Golden Rule capital stock
steady state output and depreciation
steady-state capital per
worker, k*
f(k*)
δ k*
CHAPTER 8 Economic Growth I
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.
The Golden Rule capital stock
c* = f(k*) − δk*
is biggest where the slope of the production function
equals
the slope of the depreciation line:
steady-state capital per
worker, k*
f(k*)
MPK = δ
δ k*
CHAPTER 8 Economic Growth I
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.
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?
CHAPTER 8 Economic Growth I
Remember: policy makers 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 (e.g., a revenue-neutral shift from the income tax to a consumption tax)
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
t0
c
i
y
CHAPTER 8 Economic Growth 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.
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.
time
t0
c
i
y
CHAPTER 8 Economic Growth I
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.
Population growth
Assume the population and labor force grow
at rate n (exogenous):
EX: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02).
Then ΔL = n L = 0.02 ×1,000 = 20,
so L = 1,020 in year 2.
CHAPTER 8 Economic Growth I
Break-even investment
(δ + n)k = break-even investment,
the amount of investment necessary
to keep k constant.
Break-even investment includes:
δ 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 is spread more thinly over a larger population of workers.)
CHAPTER 8 Economic Growth I
The equation of motion for k
With population growth,
the equation of motion for k is:
break-even investment
actual investment
Δk = s f(k) − (δ + n) k
CHAPTER 8 Economic Growth I
Of course, actual investment and break-even investment here are in per-worker magnitudes.
The Solow model diagram
Investment, break-even investment
Capital per
worker, k
sf(k)
(δ + n ) k
k*
Δk = s f(k) − (δ+n)k
CHAPTER 8 Economic Growth I
The impact of population growth
Investment, break-even investment
Capital per
worker, k
sf(k)
(δ +n1) k
k1*
(δ +n2) k
k2*
An increase in n causes an increase in break-even investment,
leading to a lower steady-state level of k.
CHAPTER 8 Economic Growth I
Prediction:
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.
Are the data consistent with this prediction?
CHAPTER 8 Economic Growth I
This and the preceding slide establish an implication of the model.
The following slide confronts this implication with data.
International evidence on population growth and income per person
Income per person in 2010
(log scale)
Population growth
(percent per year, average 1961-2010)
CHAPTER 8 Economic Growth I
Figure 8-13, p.235.
This figure is a scatterplot of data from 100 countries. It shows that countries with high rates of population growth tend to have low levels of income per person, as the Solow model predicts.
So far, we’ve now learned two things a poor country can do to raise its standard of living: increase national saving (perhaps by reducing its budget deficit) and reduce population growth.
Source:
Alan Heston, Robert Summers and Bettina Aten, Penn World Table Version 7.1, Center for International Comparisons of Production, Income and Prices at the University of Pennsylvania, July 2012.
2.3383586 1.4017294 1.4728454 0.30749 2.127048 0.414577 0.2680454 3.0116331 2.1503955 2.0860415 2.4453459 2.5506819 2.5028899 1.2365773 1.9208657 2.5218625 1.5973305 1.4441582 2.0612856 2.929 2.8714115 2.6078269 3.6143322 0.3723145 2.2504261 2.4486655 2.2214213 1.7256336 2.0786038 2.6211256 0.3425583 0.6617476 2.5269555 3.2677621 2.5424266 0.5127122 2.4212914 2.490712 1.8921647 2.8788211 1.6900161 1.1338755 1.9561049 1.7888812 0.9876893 2.5010135 0.3826443 1.121188 0.5995807 4.15108199999999 3.2348592 1.3607067 1.6268712 0.9260687 2.7507082 3.0509705 2.4509631 2.2688774 1.3475309 2.165577 1.8875647 2.2317242 2.1430215 0.7618232 1.1763564 2.6853826 2.8431196 2.6316063 0.5351756 2.5784828 2.2034846 2.4417227 2.1647951 2.53884549999999 0.3470569 2.7371854 2.6893195 1.516911 2.0980366 0.8389559 1.5222485 0.387379 0.7085062 3.2303914 2.8546168 1.7794091 3.0700423 0.7648954 2.0500969 3.0999675 0.3496063 1.0950952 0.5332462 2.5995651 2.8809257 2.1666933 7220.849431 14512.06175 49727.91563 42488.6822 1502.087644 29697.32074 39759.47574 1290.724114 4432.779106 9754.691878 1149.928906 451.1570776 2055.871851 42678.59603 4126.969072 1768.584474 15960.80033 8124.918354 8975.414416 976.927163599999 2628.452247 12983.01742 1448.664554 38746.50592 11600.34756 7345.691681 5265.55153 6827.965692 11126.31029 813.2109171 36374.6233 35222.51278 15026.03725 1429.990886 2288.379944 28435.59031 7071.22547 982.546609999999 905.9784027 3803.382624 41930.24704 40099.3694 3995.606345 4312.092746 38163.02596 29325.26777 31779.18833 9681.04033499999 34655.03836 5789.815045 1467.754394 28768.22326 1675.678181 93496.5642799999 815.6960461 811.0727548 13992.5397 1151.591705 10388.24469 13430.03238 4133.720642 862.841802299999 1321.292551 42546.43482 31969.70844 2592.882558 552.340345899999 1629.502422 58958.65392 2489.887143 11495.74442 4851.179775 9009.563851 3595.869194 22583.45345 1225.829758 1657.729016 34736.04031 8907.69685 30816.09764 4611.932751 40890.70522 45368.07991 4058.636446 1320.950164 9212.03212399999 867.7367709 25525.53349 11589.95653 1292.806442 38463.11369 46568.56743 13671.16122 11777.99098 2267.444053 369.1481259
The Golden Rule with population growth
To find the Golden Rule capital stock,
express c* in terms of k*:
c* = y* − i*
= f (k* ) − (δ + n) k*
c* is maximized when
MPK = δ + n
or equivalently,
MPK − δ = n
In the Golden
Rule steady state,
the marginal product
of capital net of depreciation equals the population
growth rate.
CHAPTER 8 Economic Growth I
Alternative perspectives on population growth
The Malthusian Model (1798)
Predicts population growth will outstrip the
Earth’s ability to produce food, leading to the impoverishment of humanity.
Since Malthus, world population has increased sixfold, yet living standards are higher than ever.
Malthus neglected the effects of technological progress.
CHAPTER 8 Economic Growth I
Alternative perspectives on population growth
The Kremerian Model (1993)
Posits that population growth contributes to economic growth.
More people = more geniuses, scientists & engineers, so faster technological progress.
Evidence, from very long historical periods:
As world pop. growth rate increased, so did rate of growth in living standards
Historically, regions with larger populations have enjoyed faster growth.
CHAPTER 8 Economic Growth I
Michael Kremer, “Population Growth and Technological Change: One Million B.S. to 1990,” Quarterly Journal of Economics 108 (August 1993): 681-716.
CHAPTER SUMMARY
1. 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
2. An increase in the saving rate leads to:
higher output in the long run
faster growth temporarily
but not faster steady-state growth
CHAPTER 8 Economic Growth I
CHAPTER SUMMARY
3. 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.
CHAPTER 8 Economic Growth I
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