Economic Growth I:Capital Accumulation and Population Growth

What I want you to 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

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….

Income and poverty in the world selected countries, 2000

0

10

20

30

40

50

60

70

80

90

100

$0 $5,000 $10,000 $15,000 $20,000

% o

f pop

ulat

ion

livin

g on

$2

per d

ay o

r les

s

Income per capita in dollars

Madagascar

India

BangladeshNepal

Botswana

Mexico

ChileS. Korea

Brazil Russian Federation

Thailand

Peru

ChinaKenya

links to prepared graphs @ Gapminder.orgnotes: circle size is proportional to population size,

color of circle indicates continent

Income per capita and Life expectancy Infant mortality Malaria deaths per 100,000 Adult literacy Cell phone users per 100,000

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

percentage increase in standard of living after…

annual growth rate of

income per capita

Why growth matters 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 $496 billion of income during that decade.

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

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

How Solow model is different from Chapter 3’s model1. 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

4. no G or T(only to simplify presentation; we can still do fiscal policy experiments)

5. cosmetic differences

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)

The production functionOutput per worker, y

Capital per worker, k

f(k)

Note: this production function exhibits diminishing MPK.

1MPK = f(k +1) – f(k)

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

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)

Saving and investment saving (per worker) = y – c

= y – (1–s)y = sy

National income identity is y = c + iRearrange to get: i = y – c = sy (investment = saving, like in chap. 3!)

Using the results above, i = sy = sf(k)

Output, consumption, and investment

Output per worker, y

Capital per worker, k

f(k)

sf(k)

k1

y1

i1

c1

DepreciationDepreciation per worker, k

Capital per worker, k

k

= the rate of depreciation = the fraction of the capital stock

that wears out each period

1

Capital accumulation

Change in capital stock = investment – depreciation

k = i – k

Since i = sf(k) , this becomes:

k = s f(k) – k

The basic idea: Investment increases the capital stock, depreciation reduces it.

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)

k = s f(k) – k

The steady state

If investment is just enough to cover depreciation [sf(k) = k ], then capital per worker will remain constant:

k = 0.

This occurs at one value of k, denoted k*, called the steady state capital stock.

k = s f(k) – k

The steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

Moving toward the steady stateInvestment

and depreciation

Capital per worker, k

sf(k)

k

k*

k = sf(k) k

depreciation

k

k1

investment

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k* k1

k = sf(k) k

k

k2

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

k = sf(k) k

k2

investment

depreciation

k

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

k = sf(k) k

k2

k

k3

Moving toward the steady state

Investment and

depreciation

Capital per worker, k

sf(k)

k

k*

k = sf(k) k

k3

Summary:As long as k < k*,

investment will exceed depreciation,

and k will continue to grow toward k*.

A numerical exampleProduction function (aggregate):

1/ 2 1/ 2( , )Y F K L K L K L

1/ 21/ 2 1/ 2Y K L KL L L

1/ 2( )y f k k

To derive the per-worker production function, divide through by L:

Then substitute y = Y/L and k = K/L to get

A numerical example, cont.

Assume:

s = 0.3

= 0.1

initial value of k = 4.0

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

Assumptions: ; 0.3; 0.1; initial 4.0y k s k

… 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

An increase in the saving rate

Investment and

depreciation

k

δk

s1 f(k)

*k1

An increase in the saving rate raises investment……causing k to grow toward a new steady state:

s2 f(k)

*k2

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.

International evidence on investment rates and income per person

0 5 10 15 20 25 30 35100

1,000

10,000

100,000Income per person in

2003 (log scale)

Investment as percentage of output (average 1960-2003)

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*?

The Golden Rule capital stock

the Golden Rule level of capital, the steady state value of k

that maximizes consumption.

*goldk

To find it, first express c* in terms of k*:

c* = y* i*

= f (k*) i*

= f (k*) k* In the steady state:

i* = k* because k = 0.

Then, graph f(k*) and k*, look for the point where the gap between them is biggest.

The Golden Rule capital stocksteady state output and

depreciation

steady-state capital per worker, k*

f(k*)

k*

*goldk

*goldc

* *gold goldi k

* *( )gold goldy f k

The Golden Rule capital stock

c* = f(k*) k*

is biggest where the slope of the production function equals the slope of the depreciation line:

steady-state capital per worker, k*

f(k*)

k*

*goldk

*goldc

MPK =

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?

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.

If goldk k* *

timet0

c

i

y

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.

If goldk k* *

timet0

c

i

y

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 L = n L = 0.02 1,000 = 20,so L = 1,020 in year 2.

L nL

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 is spread more thinly over a larger population of workers.)

The equation of motion for k

With population growth, the equation of motion for k is:

break-even investment

actual investment

k = s f(k) ( + n) k

The Solow model diagramInvestment, break-even investment

Capital per worker, k

sf(k)

( + n ) k

k*

k = s f(k) ( +n)k

The impact of population growth

Investment, break-even investment

Capital per worker, k

sf(k)

( +n1) k

k1*

( +n2) k

k2*

An increase in n causes an increase in break-even investment,leading to a lower steady-state level of k.

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.

International evidence on population growth and income per person

Income per person in

2003 (log scale)

Population growth (percent per year, average 1960-2003)

0 1 2 3 4 5100

1,000

10,000

100,000

The Golden Rule with population growth

To find the Golden Rule capital stock, 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.

Alternative perspectives on population growthThe 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. Thomas Malthus

Alternative perspectives on population growthThe 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.Michael Kremer

In the simple Solow model, the production technology is held constant. income per capita is constant in the steady

state.

Neither point is true in the real world: 1908-2008: U.S. real GDP per person grew by

a factor of 7.8, or 2.05% per year. examples of technological progress abound

Technological progress in the Solow model

Examples of technological progress From 1950 to 2000, U.S. farm sector productivity

nearly tripled. The real price of computer power has fallen an

average of 30% per year over the past three decades. Percentage of U.S. households with ≥ 1 computers:

8% in 1984, 62% in 2003 1981: 213 computers connected to the Internet

2000: 60 million computers connected to the Internet 2001: iPod capacity = 5gb, 1000 songs. Not capable

of playing episodes of True Blood. 2009: iPod capacity = 120gb, 30,000 songs. Can play episodes of True Blood.

Technological progress in the Solow model A new variable: E = labor efficiency Assume:

Technological progress is labor-augmenting: it increases labor efficiency at the exogenous rate g:

EgE

Technological progress in the Solow model We now write the production function as:

where L E = the number of effective workers. Increases in labor efficiency have the

same effect on output as increases in the labor force.

( , )Y F K L E

Technological progress in the Solow model Notation:

y = Y/LE = output per effective worker k = K/LE = capital per effective worker

Production function per effective worker:y = f(k)

Saving and investment per effective worker:s y = s f(k)

Technological progress in the Solow model

( + n + g)k = break-even investment: the amount of investment necessary to keep k constant.

Consists of: k to replace depreciating capital n k to provide capital for new workers g k to provide capital for the new “effective”

workers created by technological progress

Technological progress in the Solow model

Investment, break-even investment

Capital per worker, k

sf(k)

( +n +g ) k

k*

k = s f(k) ( +n +g)k

Steady-state growth rates in the Solow model with tech. progress

n + gY = yEL Total output

g(Y/ L) = yE Output per worker

0y = Y/(LE )Output per effective worker

0k = K/(LE )Capital per effective worker

Steady-state growth rateSymbolVariable

The Golden Rule with technological progressTo find the Golden Rule capital stock, express c* in terms of k*:

c* = y* i*

= f (k* ) ( + n + g) k*

c* is maximized when MPK = + n + g

or equivalently, MPK = n + g

In the Golden Rule steady state,

the marginal product of capital

net of depreciation equals the

pop. growth rate plus the rate of tech progress.

Growth empirics: Balanced growth Solow model’s steady state exhibits

balanced growth - many variables grow at the same rate. Solow model predicts Y/L and K/L grow at the

same rate (g), so K/Y should be constant. This is true in the real world.

Solow model predicts real wage grows at same rate as Y/L, while real rental price is constant. Also true in the real world.

Growth empirics: Convergence Solow model predicts that, other things equal,

“poor” countries (with lower Y/L and K/L) should grow faster than “rich” ones.

If true, then the income gap between rich & poor countries would shrink over time, causing living standards to “converge.”

In real world, many poor countries do NOT grow faster than rich ones. Does this mean the Solow model fails?

Growth empirics: Convergence Solow model predicts that, other things equal,

“poor” countries (with lower Y/L and K/L) should grow faster than “rich” ones.

No, because “other things” aren’t equal. In samples of countries with

similar savings & pop. growth rates, income gaps shrink about 2% per year.

In larger samples, after controlling for differences in saving, pop. growth, and human capital, incomes converge by about 2% per year.

Growth empirics: Convergence What the Solow model really predicts is

conditional convergence - countries converge to their own steady states, which are determined by saving, population growth, and education.

This prediction comes true in the real world.

Growth empirics: Factor accumulation vs. production efficiency Differences in income per capita among countries

can be due to differences in:1. capital – physical or human – per worker2. the efficiency of production

(the height of the production function) Studies:

Both factors are important. The two factors are correlated: countries with

higher physical or human capital per worker also tend to have higher production efficiency.

Growth empirics: Factor accumulation vs. production efficiency Possible explanations for the correlation

between capital per worker and production efficiency: Production efficiency encourages capital

accumulation. Capital accumulation has externalities that

raise efficiency. A third, unknown variable causes

capital accumulation and efficiency to be higher in some countries than others.

Average annual growth rates, 1970-89

closedopen

Growth empirics: Production efficiency and free trade Since Adam Smith, economists have argued that

free trade can increase production efficiency and living standards.

Research by Sachs & Warner:

0.7%4.5%developing nations

0.7%2.3%developed nations

Growth empirics: Production efficiency and free trade To determine causation, Frankel and Romer

exploit geographic differences among countries: Some nations trade less because they are farther

from other nations, or landlocked. Such geographical differences are correlated with

trade but not with other determinants of income. Hence, they can be used to isolate the impact of

trade on income.

Findings: increasing trade/GDP by 2% causes GDP per capita to rise 1%, other things equal.

Policy issues Are we saving enough? Too much? What policies might change the saving rate? How should we allocate our investment between

privately owned physical capital, public infrastructure, and “human capital”?

How do a country’s institutions affect production efficiency and capital accumulation?

What policies might encourage faster technological progress?

Policy issues: Evaluating the rate of saving Use the Golden Rule to determine whether

the U.S. saving rate and capital stock are too high, too low, or about right. If (MPK ) > (n + g ),

U.S. is below the Golden Rule steady state and should increase s.

If (MPK ) < (n + g ), U.S. economy is above the Golden Rule steady state and should reduce s.

Policy issues: Evaluating the rate of savingTo estimate (MPK ), use three facts about the U.S. economy:

1. k = 2.5 yThe capital stock is about 2.5 times one year’s GDP.

2. k = 0.1 yAbout 10% of GDP is used to replace depreciating capital.

3. MPK k = 0.3 yCapital income is about 30% of GDP.

Policy issues: Evaluating the rate of saving1. k = 2.5 y2. k = 0.1 y3. MPK k = 0.3 y

0.12.5

k yk y

0.1 0.042.5

To determine , divide 2 by 1:

Policy issues: Evaluating the rate of saving

MPK 0.32.5

k yk y

0.3MPK 0.122.5

To determine MPK, divide 3 by 1:

Hence, MPK = 0.12 0.04 = 0.08

1. k = 2.5 y2. k = 0.1 y3. MPK k = 0.3 y

Policy issues: Evaluating the rate of saving From the last slide: MPK = 0.08 U.S. real GDP grows an average of 3% per year,

so n + g = 0.03 Thus,

MPK = 0.08 > 0.03 = n + g Conclusion:

The U.S. is below the Golden Rule steady state: Increasing the U.S. saving rate would increase consumption per capita in the long run.

Policy issues: How to increase the saving rate Reduce the government budget deficit

(or increase the budget surplus). Increase incentives for private saving:

reduce capital gains tax, corporate income tax, estate tax as they discourage saving.

replace federal income tax with a consumption tax.

expand tax incentives for IRAs (individual retirement accounts) and other retirement savings accounts.

Policy issues: Allocating the economy’s investment In the Solow model, there’s one type of capital. In the real world, there are many types,

which we can divide into three categories: private capital stock public infrastructure human capital: the knowledge and skills that

workers acquire through education How should we allocate investment among these

types?

Policy issues: Allocating the economy’s investmentTwo viewpoints:

1. Equalize tax treatment of all types of capital in all industries, then let the market allocate investment to the type with the highest marginal product.

2. Industrial policy: Govt should actively encourage investment in capital of certain types or in certain industries, because they may have positive externalities that private investors don’t consider.

Possible problems with industrial policy The govt may not have the ability to “pick winners”

(choose industries with the highest return to capital or biggest externalities).

Politics (e.g., campaign contributions) rather than economics may influence which industries get preferential treatment.

Policy issues: Establishing the right institutions Creating the right institutions is important for

ensuring that resources are allocated to their best use. Examples: Legal institutions, to protect property rights. Capital markets, to help financial capital flow to

the best investment projects. A corruption-free government, to promote

competition, enforce contracts, etc.

Policy issues: Encouraging tech. progress Patent laws:

encourage innovation by granting temporary monopolies to inventors of new products.

Tax incentives for R&D Grants to fund basic research at universities Industrial policy:

encourages specific industries that are key for rapid tech. progress (subject to the preceding concerns).

CASE STUDY: The productivity slowdown

1.5

1.8

2.6

2.3

2.0

1.6

1.8

2.2

2.4

8.2

4.9

5.7

4.3

2.9

1972-951948-72

U.S.

U.K.

Japan

Italy

Germany

France

Canada

Growth in output per person(percent per year)

Possible explanations for the productivity slowdown Measurement problems:

Productivity increases not fully measured. But: Why would measurement problems

be worse after 1972 than before?

Oil prices:Oil shocks occurred about when productivity slowdown began. But: Then why didn’t productivity speed up

when oil prices fell in the mid-1980s?

Possible explanations for the productivity slowdown Worker quality:

1970s - large influx of new entrants into labor force (baby boomers, women).New workers tend to be less productive than experienced workers.

The depletion of ideas:Perhaps the slow growth of 1972-1995 is normal, and the rapid growth during 1948-1972 is the anomaly.

Which of these suspects is the culprit?

All of them are plausible, but it’s difficult to prove

that any one of them is guilty.

CASE STUDY: I.T. and the “New Economy”

2.0

2.6

1.2

1.2

1.5

1.7

2.2

1.5

1.8

2.6

2.3

2.0

1.6

1.8

2.2

2.4

8.2

4.9

5.7

4.3

2.9

1995-20071972-951948-72

U.S.

U.K.

Japan

Italy

Germany

France

Canada

Growth in output per person(percent per year)

CASE STUDY: I.T. and the “New Economy”Apparently, the computer revolution did not affect aggregate productivity until the mid-1990s.

Two reasons:1. Computer industry’s share of GDP much

bigger in late 1990s than earlier. 2. Takes time for firms to determine how to

utilize new technology most effectively.

The big, open question: How long will I.T. remain an engine of growth?

Endogenous growth theory

Solow model: sustained growth in living standards is due to

tech progress. the rate of tech progress is exogenous.

Endogenous growth theory: a set of models in which the growth rate of

productivity and living standards is endogenous.

A basic model Production function: Y = A K

where A is the amount of output for each unit of capital (A is exogenous & constant)

Key difference between this model & Solow: MPK is constant here, diminishes in Solow

Investment: s Y Depreciation: K Equation of motion for total capital:

K = s Y K

A basic model K = s Y K

Y K sAY K

If s A > , then income will grow forever, and investment is the “engine of growth.”

Here, the permanent growth rate depends on s. In Solow model, it does not.

Divide through by K and use Y = A K to get:

Does capital have diminishing returns or not? Depends on definition of “capital.” If “capital” is narrowly defined (only plant &

equipment), then yes. Advocates of endogenous growth theory

argue that knowledge is a type of capital. If so, then constant returns to capital is more

plausible, and this model may be a good description of economic growth.

A two-sector model Two sectors:

manufacturing firms produce goods. research universities produce knowledge that

increases labor efficiency in manufacturing.

u = fraction of labor in research (u is exogenous)

Mfg prod func: Y = F [K, (1-u )E L] Res prod func: E = g (u )E Cap accumulation: K = s Y K

A two-sector model In the steady state, mfg output per worker

and the standard of living grow at rate E/E = g (u ).

Key variables:s: affects the level of income, but not its

growth rate (same as in Solow model)u: affects level and growth rate of income

Facts about R&D1. Much research is done by firms seeking profits.

2. Firms profit from research: Patents create a stream of monopoly profits. Extra profit from being first on the market with a

new product.

3. Innovation produces externalities that reduce the cost of subsequent innovation.

Much of the new endogenous growth theory attempts to incorporate these facts into models to better understand technological progress.

Is the private sector doing enough R&D? The existence of positive externalities in the

creation of knowledge suggests that the private sector is not doing enough R&D.

But, there is much duplication of R&D effort among competing firms.

Estimates: Social return to R&D ≥ 40% per year.

Thus, many believe govt should encourage R&D.

Economic growth as “creative destruction” Schumpeter (1942) coined term “creative

destruction” to describe displacements resulting from technological progress: the introduction of a new product is good for

consumers, but often bad for incumbent producers, who may be forced out of the market.

Examples: Luddites (1811-12) destroyed machines that

displaced skilled knitting workers in England. Walmart displaces many “mom and pop” stores.

Chapter Summary1. Key results from Solow model with tech

progress steady state growth rate of income per person

depends solely on the exogenous rate of tech progress

the U.S. has much less capital than the Golden Rule steady state

2. Ways to increase the saving rate increase public saving (reduce budget deficit) tax incentives for private saving

Chapter Summary3. Productivity slowdown & “new economy”

Early 1970s: productivity growth fell in the U.S. and other countries.

Mid 1990s: productivity growth increased, probably because of advances in I.T.

4. Empirical studies Solow model explains balanced growth,

conditional convergence Cross-country variation in living standards is

due to differences in cap. accumulation and in production efficiency

Chapter Summary5. Endogenous growth theory: Models that

examine the determinants of the rate of tech. progress, which Solow takes as given.

explain decisions that determine the creation of knowledge through R&D.