Economic growth i capital accumulation and population growth
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PART III Growth Theory: The Economy in the Very Long Run. Economic Growth I: Capital Accumulation and Population Growth. Chapter 8 of Macroeconomics , 8 th edition, by N. Gregory Mankiw ECO62 Udayan Roy. The Solow-Swan Model. This is a theory of macroeconomic dynamics

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Economic growth i capital accumulation and population growth

PART III Growth Theory: The Economy in the Very Long Run

Economic Growth I: Capital Accumulation and Population Growth

Chapter 8 of Macroeconomics, 8thedition, by N. Gregory Mankiw

ECO62UdayanRoy


The solow swan model

The Solow-Swan Model

  • This is a theory of macroeconomic dynamics

  • Using this theory,

    • you can predict where an economy will be tomorrow, the day after, and so on, if you know where it is today

    • you can predict the dynamic effects of changes in

      • the saving rate

      • the rate of population growth

      • and other factors


Two productive resources and one produced good

Two productive resources and one produced good

  • There are two productive resources:

    • Capital, K

    • Labor, L

  • These two productive resources are used to produce one

    • final good, Y


The production function

The Production Function

  • The production function is an equation that tells us how much of the final good is produced with specified amounts of capital and labor

  • Y = F(K, L)

    • Example: Y = 5K0.3L0.7


Constant returns to scale

Constant returns to scale

  • Y = F(K, L) = 5K0.3L0.7

    • Note:

      • if you double both K and L, Y will also double

      • if you triple both K and L, Y will also triple

      • … and so on

    • This feature of the Y = 5K0.3L0.7 production function is called constant returns to scale

The Solow-Swan model assumes that production functions obey constant returns to scale


Constant returns to scale1

Constant returns to scale

  • Definition: The production function F(K, L) obeys constant returns to scale if and only if

    • for any positive number z (that is, z > 0)

    • F(zK, zL) = zF(K, L)

  • Example: Suppose F(K, L) = 5K0.3L0.7.

    • Then, for any z > 0, F(zK, zL) = 5(zK)0.3(zL)0.7 = 5z0.3K0.3z0.7L0.7 = 5z0.3 + 0.7K0.3L0.7 = z5K0.3L0.7 =zF(K, L)

At this point, you should be able to do problem 1 (a) on page 232 of the textbook.


Constant returns to scale2

Constant returns to scale

  • CRS requires F(zK, zL) = zF(K, L) for any z > 0

  • Let z be 1/L.

  • Then,

So, y = F(k, 1) .

From now on, F(k, 1) will be denoted f(k), the per worker production function.

k = K/Ldenotes per worker stock of capital

y = Y/Ldenotes per worker output

So, y= f(k) .


Per worker production function example

Per worker production function: example

At this point, you should be able to do problems 1 (b) and 3 (a) on pages 232 and 233 of the textbook.

This is what a typical per worker production function looks like: concave


The cobb douglas production function

The Cobb-Douglas Production Function

  • Y = F(K, L) = 5K0.3L0.7

    • This production function is itself an instance of a more general production function called the Cobb-Douglas Production Function

  • Y = AKαL1 − α ,

    • where A is any positive number (A > 0) and

    • α is any positive fraction (1 > α > 0)


Per worker cobb douglas production function

Per worker Cobb-Douglas production function

  • Y = AKαL1 − αimplies y = Akα = f(k)

In problem 1 on page 232 of the textbook, you get Y = K1/2L1/2, which is the Cobb-Douglas production with A = 1 and α = ½.

In problem 3 on page 233, you get Y = K0.3L0.7, which is the Cobb-Douglas production with A = 1 and α = 0.3.


Per worker production function graph

Per worker production function: graph


Income consumption saving investment

Income, consumption, saving, investment

  • Output = income

  • The Solow-Swan model assumes that each individual saves a constant fraction, s, of his or her income

  • Therefore, saving per worker = sy = sf(k)

  • This saving becomes an addition to the existing capital stock

  • Consumption per worker is denoted c = y – sy = (1– s)y


Income consumption saving investment graph

Income, consumption, saving, investment: graph


Depreciation

Depreciation

  • But part of the existing capital stock wears out

  • This is called depreciation

  • The Solow-Swan model assumes that a constant fraction, δ, of the existing capital stock wears out in every period

  • That is, an individual who currently has k units of capital will lose δk units of capital though depreciation (or, wear and tear)


Depreciation1

Depreciation

  • Although 0 < δ < 1 is the fraction of existing capital that wears out every period, in some cases—as in problem 1 (c) on page 219 of the textbook—depreciation is expressed as a percentage.

  • In such cases, care must be taken to convert the percentage value to a fraction

    • For example, if depreciation is given as 5 percent, you need to set δ = 5/100 = 0.05


Depreciation graph

Depreciation: graph


Dynamics

dynamics


Dynamics what time is it

Dynamics: what time is it?

  • We’ll attach a subscript to each variable to denote what date we’re talking about

  • For example, ktwill denote the economy’s per worker stock of capital on date tandkt+1 will denote the per worker stock of capital on date t + 1


How does per worker capital change

How does per worker capital change?

  • A worker haskt units of capital on date t

  • He or she adds syt units of capital through saving

  • and loses δkt units of capital through depreciation

  • So, each worker accumulates kt + syt−δktunits of capital on date t + 1

  • Does this mean kt+1 = kt + syt−δkt?

  • Not quite!


Population growth

Population Growth

  • The Solow-Swan model assumes that each individual has n kids in each period

  • The kids become adult workers in the period immediately after they are born

    • and, like every other worker, have n kids of their own

    • and so on


Population growth1

Population Growth

  • Let the growth rate of any variable x be denoted xg. It is calculated as follows:

  • Therefore, the growth rate of the number of workers, Lg, is:


How does per worker capital change1

How does per worker capital change?


Dynamics algebra

Dynamics: algebra


Dynamics algebra1

Dynamics: algebra

Now we are ready for dynamics!


Dynamics algebra2

Dynamics: algebra


Dynamics algebra3

Dynamics: algebra

At this point, you should be able to do problems 1 (d) and 3 (c) on pages 232 and 233 of the textbook. Please give them a try.


Dynamics algebra to graphs

Dynamics: algebra to graphs

Although this is the basic Solow-Swan dynamic equation, a simple modification will help us analyze the theory graphically.

Now we are ready for graphical analysis.


Dynamics algebra to graphs1

Dynamics: algebra to graphs

This version of the Solow-Swan equation will help us understand the model graphically.


Dynamics algebra to graphs2

Dynamics: algebra to graphs

This version of the Solow-Swan equation will help us understand the model graphically.

2. The economy shrinks if and only if per worker saving and investment [sf(kt)] is less than (δ + n)kt.

3. The economy is at a steady state if and only if per worker saving and investment [sf(kt)] is equal to (δ + n)kt.

1. The economy grows if and only if per worker saving and investment [sf(kt)] exceeds (δ + n)kt.

4. (δ + n)kt is called break-even investment


Dynamics graph

Dynamics: graph


Economic growth i capital accumulation and population growth

Investment and break-even investment

(δ + n)kt

sf(kt)

k1

k*

Capital per worker, k

Steady state


Moving toward the steady state

Investment and depreciation

(δ + n)kt

sf(kt)

k1

investment

Break-even investment

k1

k*

Capital per worker, k

Moving toward the steady state

Steady state


Moving toward the steady state1

Investment and depreciation

(δ + n)kt

sf(kt)

k1

k1

k*

Capital per worker, k

Moving toward the steady state

k2

Steady state


Moving toward the steady state2

Investment and depreciation

(δ + n)kt

sf(kt)

k2

investment

Break-even investment

k1

k*

Capital per worker, k

Moving toward the steady state

k2

Steady state


Moving toward the steady state3

Investment and depreciation

(δ + n)kt

sf(kt)

k2

k1

k*

Capital per worker, k

Moving toward the steady state

k2

k3

Steady state


Moving toward the steady state4

Investment and depreciation

(δ + n)kt

sf(kt)

k1

k*

Capital per worker, k

Moving toward the steady state

As long as k < k*, investment will exceed break-even investment, and k will continue to grow toward k*

k2

k3

Steady state


Steady state

Steady state


The steady state algebra

The steady state: algebra

  • The economy eventually reaches the steady state

  • This happens when per worker saving and investment [sf(kt)] is equal to break-even investment [(δ + n)kt].

  • For the Cobb-Douglas case, this condition is:

At this point, you should be able to do problems 1 (c) and 3 (b) on pages 232 and 233 of the textbook. Please give them a try.


Steady state and transitional dynamics algebra

Steady State and Transitional Dynamics: algebra


Solow swan predictions for the steady state

Solow-swan predictions for the steady state


There is no growth

There is no growth!

  • The Solow-Swan model predicts that

    • Every economy will end up at the steady-state; in the long run, the growth rate is zero!

      • That is, k = k* and y = f(k*) = y*

    • Growth is possible—temporarily!—only if the economy’s per worker stock of capital is less than the steady state per worker stock of capital (k < k*)

      • If k < k*, the smaller the value of k, the faster the growth of k and y


There is no growth1

There is no growth!


From per worker to total

From per worker to total

  • We have seen that per worker capital, k, is constant in the steady state

  • Now recall that k = K/L

  • and that L increases at the rate of n

  • Therefore,in the steady state, the total stock of capital, K, increases at the rate of n


From per worker to total1

From per worker to total

  • We have seen that per worker income y = f(k) is constant in the steady state

  • Now recall that y = Y/L

  • and that L increases at the rate of n

  • Therefore,in the steady state, the total income, Y, must also increase at the rate of n

  • Similarly,although per worker saving and investment, sy, is constant in the steady state, total saving and investment, sY, increases at the rate n


From per worker to per capita

From per worker to per capita

  • Recall that L = amount of labor employed

  • Suppose P = amount of labor available

    • This is the labor force

    • But in the Solow-Swan model everybody is capable of work, even new-born children

    • So P can also be considered the population

  • Suppose u = fraction of population that is not engaged in production of the final good.

    • u is assumed constant

  • Then L = (1 – u)P or L/P= 1 – u


Per worker to per capita

Per worker to per capita


Steady state summary

Steady state: summary


Solow swan predictions for changes to the steady state

Solow-swan predictions for changes to the steady state


A sudden fall in capital per worker

A sudden fall in capital per worker

  • A sudden decrease in k could be caused by:

    • Earthquake or war that destroys capital but not people

    • Immigration

    • A decrease in u

  • What does the Solow-Swan model say will be the result of this?


A sudden fall in capital per worker1

Investment and depreciation

(δ + n)kt

sf(kt)

k1

k*

Capital per worker, k

A sudden fall in capital per worker

At this point, you should be able to do problem 2 on pages 232 and 233 of the textbook. Please give it a try.

2. A gradual return to the steady state

1. A sudden decline


An increase in the saving rate

Investment and break-even investment

(δ + n)kt

s2f(k)

s1f(k)

k

An increase in the saving rate

An increase in the saving rate causes a temporary spurt in growth. The economy returns to a steady state. But at the new steady state, per worker capital, output, and saving are all higher. Per worker consumption is a bit trickier.

k1*

k2*


An increase in the saving rate1

Investment and break-even investment

(δ + n)kt

s2f(k)

s1f(k)

k

An increase in the saving rate

1. An increase in the saving rate raises investment …

2. … causing k to grow (toward a new steady state)

3. This raises steady-state per worker output y* = f(k*) and saving sy*.

4. The growth rate begins at zero, becomes positive for a while, and eventually returns to zero.

k1*

k2*


An increase in the saving rate2

(δ + n)kt

An increase in the saving rate

1. Recall that the saving rate is a fraction between 0 and 1.

2. What can we say about the steady state levels of k, y, and c when s = 0?

3. And when s = 1?

Investment and break-even investment

4. So, how is consumption per worker, c, affected by changes in s?

f(k)

sf(k)

c*

f(k*)

sf(k*)

k

k*


Being the grasshopper is not good

(δ + n)kt

Being the grasshopper is not good!

1. Recall that the saving rate is a fraction between 0 and 1.

2. What can we say about the steady state levels of k, y, and c when s = 0?

3. They are all zero!

Investment and break-even investment

f(k)

sf(k) when s = 0

k

k*= 0


Being a miserly any is not good either

(δ + n)kt

Being a miserly any is not good either!

1. Recall that the saving rate is a fraction between 0 and 1.

2. What can we say about the steady state levels of k, y, and c when s = 1?

Investment and break-even investment

f(k)

= sf(k)

c*= 0

f(k*)

sf(k*)

k

k*


Effect of saving on steady state consumption

Effect of saving on steady state consumption

For the Cobb-Douglas case, it can be shown that the Golden Rule saving rate is equal to capital’s share of all income, which is approximately 30% or 0.30.

Steady state consumption per worker, c* = (1 – s)f(k*)

The US saving rate is well below 0.30. So, according to the Solow-Swan model, if we save more we will, in the long run, consume more too!

Golden Rule consumption per worker

Golden Rule Saving rate

Saving rate, s

0

1


A decrease in the saving rate

A decrease in the saving rate


An increase in the saving rate3

An increase in the saving rate


What do we get for thrift

What do we get for thrift?

  • In the long run, a higher rate of saving and investment gives us

    • A higher per worker income

    • But not a faster rate of growth

  • And consumption per worker, c* = (1 – s)y*, may increase or decrease or stay unchanged when s increases.

At this point, you should be able to do problem 4 on page 233 of the textbook. Please try it.


Whole lecture in one slide

Whole lecture in one slide!


Effect of saving evidence

Effect of saving: evidence


Faster population growth

Faster population growth

3. This reduces steady-state per worker output y* = f(k*), saving sy*, and consumption (1 – s)y*.

4. The growth rate begins at zero, becomes negative for a while, and eventually returns to zero.

5. The effect is the same if depreciation increases


What do we get for having fewer kids

What do we get for having fewer kids?

  • In the long run, a lower rate of population growth gives us

    • A higher per worker income

    • But not a faster rate of growth

At this point, you should be able to do problem 6 on page 233 of the textbook. Please try it.


Faster population growth evidence

Faster population growth: evidence


Alternative perspectives on population growth

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.


Alternative perspectives on population growth1

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.


Alternative perspectives on population growth kremer

Alternative perspectives on population growth: Kremer


Robert solow and trevor swan

Robert Solow and Trevor Swan


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