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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, 8thedition, by N. Gregory Mankiw

ECO62UdayanRoy

- 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

- There are two productive resources:
- Capital, K
- Labor, L

- These two productive resources are used to produce one
- final good, Y

- 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

- 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

- Note:

The Solow-Swan model assumes that production functions obey 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.

- 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) .

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

- 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)

- 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.

- 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

- 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)

- 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

- 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

- 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!

- 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

- 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:

Now we are ready for dynamics!

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.

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.

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

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

Investment and break-even investment

(δ + n)kt

sf(kt)

k1

k*

Capital per worker, k

Steady state

Investment and depreciation

(δ + n)kt

sf(kt)

k1

investment

Break-even investment

k1

k*

Capital per worker, k

Steady state

Investment and depreciation

(δ + n)kt

sf(kt)

k1

k1

k*

Capital per worker, k

k2

Steady state

Investment and depreciation

(δ + n)kt

sf(kt)

k2

investment

Break-even investment

k1

k*

Capital per worker, k

k2

Steady state

Investment and depreciation

(δ + n)kt

sf(kt)

k2

k1

k*

Capital per worker, k

k2

k3

Steady state

Investment and depreciation

(δ + n)kt

sf(kt)

k1

k*

Capital per worker, k

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

k2

k3

Steady state

- 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.

- 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

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

- 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

- 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

- 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

- 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?

Investment and depreciation

(δ + n)kt

sf(kt)

k1

k*

Capital per worker, k

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

Investment and break-even investment

(δ + n)kt

s2f(k)

s1f(k)

k

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*

Investment and break-even investment

(δ + n)kt

s2f(k)

s1f(k)

k

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*

(δ + n)kt

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*

(δ + n)kt

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

(δ + n)kt

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*

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

- 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.

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

- 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.

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.

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.