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MACROECONOMICS MACROECONOMICS

Topic 6

Economic Growth:

Capital Accumulation and Population Growth

(Chapter 7) ICU, Winter Term 2011

Principles of Macroeconomics

Learning objectives

Learning objectives

In this chapter, we will learn :

ƒ

the Solow model, a framework to explain the growth of an economy

ƒ

how a country’s standard of living depends on its saving and population growth rates

ƒ

how to use the “Golden Rule” to find the optimal saving rate and capital stock

Why grow th matters

ƒ

Data on infant mortality rates:

ƒ

20% in the poorest 1/5 of all countries

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0.4% in the richest 1/5

ƒ

In Pakistan, 85% of people live on less than $2/day.

ƒ

One-fourth of the poorest countries have had famines during the past 3 decades.

ƒ

Poverty is associated with oppression of women and minorities.

I ncome and poverty in the w orld

selected countries, 2000

Madagascar

India

Bangladesh Nepal

Botswana

Mexico

Chile

S. Korea Brazil

Russian Federation Thailand

Peru China

Kenya

Why grow th matters

ƒ

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.

624.5% 169.2%

64.0% 2.0%

…100 years

…50 years

…25 years

percentage increase in standard of living after… annual growth

rate of income per capita

Why grow th matters

ƒ

If the annual growth rate of U.S. real GDP per capita had been just one-tenth of 1% higher during the 1990s, the U.S. would have

generated an additional $496 billion of income during that decade.

The lessons of grow th theory

…can make a positive difference in the lives of hundreds of millions of people.

These lessons help us

ƒ understand why poor countries are poor

ƒ design policies that can help them grow

ƒ learn how our own

growth rate is affected

Things you need to keep in mind before

going into details

ƒ

We focus on the determinants of economic growth and the standard of living.

ƒ

How do we measure these concepts?

…by GDP per capita and the growth rate of GDP.

ƒ

What determines GDP?

…things that affect the aggregate production function ⁽Æ^{Topic 3]}: capital, labor, and technology.

Y = F(K,L)

ƒ

The time span that is relevant here is the very long run _Æ see the next slide for a review.

The concept aggregate production function is central here!

So GDP is central here!

REVI EW: Time span in macroeconomics

prices

factors of production and

technology

time length in reality

studied in

very long run flexible varying a decade

~ decades ^{Topic 6}

long run flexible fixed many years

~ a decade

Topics 3~5

short run rigid fixed ^{months ~}

years

Topics 7~11

ƒ Of course, in reality prices, factors of production and technology are

Note:

The Solow model

ƒ

due to Robert Solow, won Nobel Prize for

contributions to the study of economic growth

ƒ

a major paradigm:

ƒ

widely used in policy making

ƒ

benchmark against which most

recent growth theories are compared

ƒ

looks at the determinants of economic growth and the standard of living in the very long run

How Solow model is different from

Chapter 3’s model

1. capital K is no longer fixed: investment causes it to grow, depreciation causes it to shrink

2. labor L is no longer fixed:

population growth causes it to grow

3. the consumption function is simpler

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

On the other hand, being similar to Chapter 3, we continue to assume a closed economy, so there is no trade (no net exports, NX) with foreigners.

The production function

ƒ

In aggregate terms: Y = F (K, L)

ƒ

Define: y = Y/L = output per worker k = K/L = capital per worker

ƒ

Assume constant returns to scale: zY = F (zK, zL ) for any z > 0

ƒ

Pick z = 1/L. Then Y/L = F (K/L, 1)

L is a proxy for population

Æ “per worker” and “per capita” are the same.

The production function

Output per worker, y

Capital per worker, k

f(k)

Note: this production function exhibits diminishing marginal product of capital (MPK). Note: this production function exhibits diminishing marginal product of capital (MPK).

1

MPK = f(k +1) – f(k)

The national income identity

ƒ

^{Y = C + I} (remember, no G and no NX )

ƒ

Divide both sides by L to change to “per worker” terms:

y = c + i

where c = C/L and i = I /L

The consumption function

ƒ

s = the saving rate (=saving/Y),

or the fraction of income that is saved

(s is an exogenous parameter, 0<s<1) Thus saving = sY.

ƒ

Y = C + saving _Æ consumption is also a

constant fraction of income: C = (1–s)Y. Divide both sides by L to change to per

worker terms: c = (1–s)y

Thus by assumption, the consumption function is simpler than that in Ch. 3.

Saving and investment

ƒ

Per-worker saving

= saving/L = (sY)/L = s(Y/L) = sy

ƒ

National income identity is y = c + i Rearrange to get: i = y – c = sy

(investment = saving, like in chap. 3!)

ƒ

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

s<1 _Æ this line must line below the f(k) line

During the production process, capital wears out, and we call this depreciation.

Depreciation

Depreciation per worker, ^δk

δ_k δ = the rate of depreciation

= the fraction of the capital stock that wears out each period

1

δ

Capital accumulation

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

Change in capital stock = investment – depreciation

Δ_{k = i –} δ_k

Since i = sf(k) , this becomes:

Δ k = s f(k) – ^δ k

The equation of motion for k

Δ k = s f(k) – ^δ k

ƒ

The Solow model’s central equation

ƒ

Determines behavior of capital over time…

ƒ

…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)

The steady state

Δ k = s f(k) – ^δ k

If investment is just enough to cover depreciation [sf(k) = ^δk ],

then capital per worker will remain constant: Δ_{k = 0.}

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

As we have seen previously, the steady state is defined as a situation in which all variables (i.e. k, y, c, i ) remain unchanged.

The steady state

Investment and

depreciation

sf( k) δ_k

This point

corresponds to the steady state.

Moving tow ard the steady state

Investment and

depreciation

Capital per worker, k

sf( k) δ_k

k^*

Δk = sf( k) ^{− δ}k

depreciation Δ_k

k₁ investment

Moving tow ard the steady state

Investment and

depreciation

sf( k) δ_k

k

Δk = sf( k) ^{− δ}k

Moving tow ard the steady state

Investment and

depreciation

Capital per worker, k

sf( k) δ_k

k^*

Δk = sf( k) ^{− δ}k

k₂ investment

depreciation Δ_k

k₁

Moving tow ard the steady state

Investment and

depreciation

sf( k) δ_k

k

Δk = sf( k) ^{− δ}k

Moving tow ard the steady state

Investment and

depreciation

Capital per worker, k

sf( k) δ_k

k^* k₃

Summary: As long as k < k^*, investment will exceed

depreciation, and k will continue to grow toward k^*. Summary: As long as k < k^*, investment will exceed

depreciation, and k will continue to grow toward k^*.

Δk = sf( k) ^{− δ}k

k_{1 k2}

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?

Investment and

depreciation

Capital per worker, k

sf( k) δ_k

k^*

Δk = sf( k) ^{− δ}k

k₁ depreciation

−^Δk

investment

investment < depreciation

Æ Æ

ANSWERS:

Approaching k* from above

Investment and

depreciation

sf( k) δ_k

Δk = sf( k) ^{− δ}k

− ^k ANSWERS:

Approaching k* from above

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

A numerical example , cont.

Assume:

ƒ

^{s = 0.3}

ƒ

^{δ = 0.1}

ƒ

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

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}

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 Δ_{k = s f(k)} − δ_k

to solve for the steady-state values of k, y, and c.

ANSWERS:

S o lv e f o r th e S te a d y S ta te

An increase in the saving rate

An increase in the saving rate raises investment…

…causing k to grow toward a new steady state:

Investment and

depreciation

δ_k

s₁ f(k) s₂ f(k)

Prediction:

ƒ

^{Higher s ⇒ higher k*.}

ƒ

And since y = f(k) ,

higher k^{* ⇒} higher y^* .

ƒ

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.

I nternational evidence on investment rates and income per person

Income per person in 2003 (log scale)

The line showing the average relationship btw i and y

The Golden Rule: I ntroduction

ƒ

Different values of s lead to different

steady states. How do we know which is the “best” steady state?

ƒ

^{The best} steady state is defined as the one that has the highest possible consumption per person

Basic idea: Here “best” = “highest well- being”, and after all, what determines economic well-being is not income (or output) but consumption.

The Golden Rule: I ntroduction

ƒ

Denote this consumption c* and note that c* = (1–s)f(k*).

ƒ

An increase in s has to opposite effects on c* :

ƒ

^{s↑ Æ k*↑ Æ y*=f(k*)↑ Æ c*↑.}

ƒ

^{s↑ Æ (1–s)↓ Æ c*↓.}

ƒ

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

c^* = y^{* −} i^*

= f (k^*) ⁻ i^*

= f (k^*) ^{− δ}k^*

In the steady state: i^* = ^δk^*

because ^Δk = 0.

The Golden Rule capital stock

Then, graph f(k^*) and ^δk^*, look for the point where

the gap between them is biggest.

steady state output and depreciation

f( k^* ) δ_k^*

The Golden Rule capital stock

c^* = f(k^*) ^{− δ}k^*

is biggest where the slope of the

production function equals

the slope of the depreciation line: c^* = f(k^*) ^{− δ}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^* ) δ_k^*

MPK = ^δ

The transition to the

Golden Rule steady state

ƒ

The economy does NOT have a tendency to move toward the Golden Rule steady state.

ƒ

Achieving the Golden Rule requires that policymakers adjust s.

ƒ

This adjustment leads to a new steady state with higher consumption.

ƒ

But what happens to consumption

Starting w ith 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.

then increasing c^* requires a fall in s. In the transition to the Golden Rule, consumption is

higher at all points in time.

t₀ time c

i y

Starting w ith too little capital

then increasing c^* requires an

increase in s.

Future generations enjoy higher

consumption, but the current one experiences then increasing c^* requires an

increase in s.

Future generations enjoy higher

consumption, but the current one experiences

c

i y

Population grow th

ƒ

So far we have assumed a constant population. To make the analysis more realistic, here we

consider population growth.

ƒ

Assume the population and labor force grow at rate n (exogenous):

ƒ EX: Suppose L = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02).

ƒ ^{Then Δ}L = n L = 0.02 ^× 1,000 = 20 _Æ L = 1,020 in year 2.

Break- even investment

Defining this concept makes the incorporation of population growth into the framework built so far straightforward.

ƒ ^{( δ + n)k}

can be interpreted as break-even investment, i.e. the amount of investment necessary to keep k constant.

ƒ

Break-even investment includes:

ƒ

^δk to replace capital as it wears out

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^{n k} to equip new workers with capital

(Otherwise, k would fall as the existing capital stock

The equation of motion for k

ƒ

With population growth,

the equation of motion for k is:

Δ k = s f(k) ⁻ ( ^δ + n) k

break-even investment actual

investment

The Solow model diagram

Investment, break-even

investment

sf( k) (^δ + n ) k Δ_{k = s f(k)} − ₍δ_{+ n)k}

The impact of population grow th

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.

Prediction:

ƒ

^{Higher n ⇒} lower k*.

ƒ

And since y = f(k) , lower k* ^⇒ lower y*.

ƒ

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.

I nternational evidence on population grow th and income per person

Population growth

The line showing the average relationship btw n and y is downward sloping _Æ consistent with the Solow model.

Income per person in 2003 (log scale)

The Golden Rule w ith population

grow th

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,

In the Golden Rule steady state, the marginal product of capital net of depreciation

Alternative perspectives on population

grow th

The Malthusian Model (1798)

ƒ

Predicts population growth will outstrip the Earth’s ability to produce food, leading to the impoverishment of humanity.

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Since Malthus, world population has increased sixfold, yet living standards are higher than ever.

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Malthus neglected the effects of technological progress.

Alternative perspectives on population

grow th

The Kremerian Model (1993)

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Posits that population growth contributes to economic growth.

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More people = more geniuses, scientists & engineers, so faster technological progress.

ƒ

Evidence, from very long historical periods:

ƒ As world pop. growth rate increased, so did rate of growth in living standards

Chapter Summary

Chapter Summary

1. The Solow growth model shows that,

in the long run, a country’s standard of living depends:

ƒ

positively on its saving rate

ƒ

negatively on its population growth rate

2. An increase in the saving rate leads to:

ƒ

higher output in the long run

ƒ

faster growth temporarily

ƒ

but not faster steady state growth

Chapter Summary

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

Chapter Summary

Chapter Summary

3.

A question remains:

The versions of the Solow model we have

studied so far predict that the growth rate of the economy decreases toward the steady

state, and then finally be zero (i.e. no growth!) there. But in reality, many economies do grow over a long period of time.

Æ How should we reconcile the Solow model with this fact?

Æ Next chapter.