MACROECONOMICS Chapter 7 Economic Growth I: Capital Accumulation and Population Growth.

MACROECONOMICSMACROECONOMICS

Chapter 7Chapter 7

Economic Growth I: Economic Growth I: Capital Accumulation and Capital Accumulation and

Population GrowthPopulation Growth

22

Solow Growth ModelSolow Growth Model

Real GDP in US is 5X its level 50 years Real GDP in US is 5X its level 50 years ago; per capita real GDP is 3X.ago; per capita real GDP is 3X.

In some poor countries, real GDP per In some poor countries, real GDP per person is only 2-5% of US.person is only 2-5% of US.

Using the production function with only K, Using the production function with only K, L, and L, and ΘΘ, Robert Solow developed a very , Robert Solow developed a very abstract theory to capture growth.abstract theory to capture growth.

33

Solow ModelSolow Model

To keep the analysis as simple as To keep the analysis as simple as possible, we will pretend that G=0, NX=0. possible, we will pretend that G=0, NX=0. After we develop the model, we can see how After we develop the model, we can see how

changing these parameters will affect the changing these parameters will affect the results.results.

First, the contribution of capital to growth First, the contribution of capital to growth and the importance of savings to capital and the importance of savings to capital accumulation will be discussed.accumulation will be discussed.

44

Solow ModelSolow Model

The impact of increasing available labor The impact of increasing available labor (dubbed population growth) will be (dubbed population growth) will be discussed after the basic model is discussed after the basic model is understood.understood.

The impact of technology change will be The impact of technology change will be the subject of the next chapter.the subject of the next chapter.

55

Accumulation of CapitalAccumulation of Capital

If L and If L and ΘΘ are fixed, the only factor of are fixed, the only factor of production that will bring about growth of Y production that will bring about growth of Y is K.is K.

We will use the same approach of supply We will use the same approach of supply of Y and demand for Y we used before, to of Y and demand for Y we used before, to determine how much K will increase each determine how much K will increase each period.period.

66

Supply of YSupply of Y

)(

1,

1

),(

),(

kfy

L

KF

L

YL

z

Let

zLzKFzY

LKFY

The production function is the familiar one with constant returns to scale. The little trick of defining z allows us to show the output as per worker real GDP and the input as capital per worker (also called capital-labor ratio).

The lower case depiction of the production function, therefore, says that per worker output depends on capital per worker.

77

Basic Rule of DerivativesBasic Rule of Derivatives

1

rs

sr

rxAydx

dZ

yAxZ

Negative exponent means reciprocal.

When x changes by one unit, by how many units will Z change?

88

Marginal Product of CapitalMarginal Product of Capital

4

3

14

1

4

3

5

5

K

L

dK

dY

KLdK

dY

dK

dYMPK

4

1

4

1

4

3

4

1

20

20

20

ky

L

K

L

Y

LKY

4

3

4

1

14

1

4

1

15

4

1120

KLdK

dy

K

LdK

dy

dK

dyMPK

Using an arbitrary Cobb-Douglas function, we can see how the production function can be presented in terms of GDP per worker.

The MPK will be decreasing as capital increases because as K goes up the denominator increases.

Likewise, when capital-labor ratio (k) increases, the marginal product of k decreases.

4

3

14

1

5

5

k

MPk

kMPk

The exercise here shows thatMPK=MPk and it doesn’t matter If one uses Y or y.

99

The Shape of Prod. Fn.The Shape of Prod. Fn.

Y

K

y

k

1

MPK

1

MPk

1010

Demand for YDemand for YThe total output (GDP) is divided between C, I, G, and NX. For simplicity, pretend that G=0 and NX=0. Then, Y = C + ILet’s show this equation as per worker:

IS

isy

iysy

ysc

YsCY

Ss

icyL

I

L

C

L

Y

)1(

)1(

)1(

Output per worker (y) is determined by capital per worker (k). Given k, we know what y will be. The output per worker (y) will be divided between consumption per worker and investment per worker according to the size of savings rate (s). The higher the savings rate, more of the output will be used for investment and less for consumption.

1111

Relationship of i and yRelationship of i and yf(k)

sf(k)

k

y

y=f(k) and i=sy which is the same as i=sf(k)

Consumption per worker, c

Invesment per worker, i

y=c+i

k1

y1

What happens if s rises?

1212

DepreciationDepreciation

Some capital stock is used up.Some capital stock is used up.Some capital stock becomes obsolete.Some capital stock becomes obsolete.Some capital stock is broken.Some capital stock is broken.Collectively, let’s say, in general a certain Collectively, let’s say, in general a certain

percentage of the capital will be lost per percentage of the capital will be lost per year: year: δδK.K.

1313

Capital AccumulationCapital Accumulation

Investments add to the capital stock.Investments add to the capital stock.Depreciation subtracts from the capital Depreciation subtracts from the capital

stock.stock.Net capital accumulation, Net capital accumulation, ΔΔK, K, then, then,

must be I – must be I – δδK.K.Per worker: Per worker: ΔΔk = i – k = i – δδkkAlternately, Alternately, ΔΔk = sf(k) – k = sf(k) – δδkk

1414

Capital Accumulation and Capital Accumulation and Steady StateSteady State

k (Capital per worker)

δk

sf(k)

δk, i

k*k1k2

δk2>i

Δk*=i

δk1<i

1515

Period k y c i Dep k acc1 40.00 50.30 35.21 15.09 8.00 7.092 47.09 52.39 31.43 20.96 18.84 2.123 49.21 52.97 31.78 21.19 19.68 1.504 50.71 53.37 32.02 21.35 20.29 1.065 51.78 53.65 32.19 21.46 20.71 0.756 52.53 53.84 32.31 21.54 21.01 0.537 53.05 53.98 32.39 21.59 21.22 0.378 53.42 54.07 32.44 21.63 21.37 0.269 53.68 54.14 32.48 21.65 21.47 0.18

10 53.86 54.18 32.51 21.67 21.55 0.1311 53.99 54.21 32.53 21.69 21.60 0.0912 54.08 54.24 32.54 21.69 21.63 0.0613 54.14 54.25 32.55 21.70 21.66 0.0414 54.19 54.26 32.56 21.71 21.67 0.0315 54.22 54.27 32.56 21.71 21.69 0.0216 54.24 54.28 32.57 21.71 21.70 0.0217 54.25 54.28 32.57 21.71 21.70 0.0118 54.26 54.28 32.57 21.71 21.71 0.0119 54.27 54.28 32.57 21.71 21.71 0.0120 54.28 54.29 32.57 21.71 21.71 0.00

1616

Calculating k*Calculating k*

At k*, sf(k)=At k*, sf(k)=δδk or sf(k*)= k or sf(k*)= δδk*k*Likewise, s/Likewise, s/δδ=k*/f(k*)=k*/f(k*)Suppose s=0.4, Suppose s=0.4, δδ=0.2, f(k)=10k=0.2, f(k)=10k0.33330.3333

2 = k/10k2 = k/10k0.33330.3333

8 = k8 = k22/10/1080 = k80 = k22

k* ≈ 9k* ≈ 9

1717

The Impact of War or The Impact of War or Natural DisasterNatural Disaster

Inv; dep.

k

δksf(k)

k*k

1818

Sudden Drop in the Sudden Drop in the Savings RateSavings Rate

Inv; dep.

k

δksf(k)

k*k

s’f(k)

1919

Drop in sDrop in s

6.31

10

10

10

15.0)10(15.0

)(

15.0

15.0

10

23

33

3

1

3

1

3

1

k

k

kk

kk

kk

kksf

s

ky

4.68

6667.16

3

50

15.

5.2

15.0)10(25.0

)(

15.0

25.0

10

2

3

3

1

3

1

3

1

3

1

k

k

kk

kk

kk

kksf

s

ky

2020

Test of Solow ModelTest of Solow Model

Solow model says, ceteris paribus, higher Solow model says, ceteris paribus, higher investment rates bring higher steady-state investment rates bring higher steady-state capital and higher income per worker.capital and higher income per worker.

How does one test this?How does one test this?What does Figure 7-6 show?What does Figure 7-6 show?

2121

Golden Rule of Capital Golden Rule of Capital

What steady state level of capital per What steady state level of capital per worker is optimal?worker is optimal?

Define optimal as maximum consumption Define optimal as maximum consumption per worker (well-being = consumption).per worker (well-being = consumption).

The higher the s, the higher the k.The higher the s, the higher the k.But which k is the best one?But which k is the best one?

2222

Which k Maximizes c?Which k Maximizes c?f(k)

s’f(k)

s’’f(k)

s’’’f(k)

y, i

k

δk s’>s’’>s’’’

2323

Golden Rule kGolden Rule k

Consumption per worker is at maximum Consumption per worker is at maximum when the slope of when the slope of δδk is exactly equal to k is exactly equal to the slope of f(k).the slope of f(k).Slope of Slope of δδk is k is δδk/k = k/k = δδ..Slope of f(k) is dy/dk. But dy/dk = dY/dK (see Slope of f(k) is dy/dk. But dy/dk = dY/dK (see

slide #7)slide #7)When When δδ = MPK, consumption per worker is = MPK, consumption per worker is

maximized.maximized.

2424

Which k Maximizes c?Which k Maximizes c?f(k)y, i

k

δk

c

i

sf(k)

2525

ExampleExample

If y = 10k0.25, and δ = 0.15, what is the golden rule k?

MPK = 2.5k-0.75

MPK = δ

2.5k-0.75 = 0.150.15

k0.75 = 16.6667

k = 16.66671.333

k ≈ 42.6

Compare with slide # 15!

2626


Population growth rate is given as n.Population growth rate is given as n. If the population growth is equal to labor If the population growth is equal to labor

force growth, next year’s L will be (1+n)L.force growth, next year’s L will be (1+n)L.To distinguish this year’s L from next To distinguish this year’s L from next

year’s, let’s say Lyear’s, let’s say Lt+1t+1 = (1+n)L = (1+n)Ltt..For K/L to be constant, the growth rate of For K/L to be constant, the growth rate of

K should also be n.K should also be n. If y*=f(k*), then at equilibrium Y, K, and L all If y*=f(k*), then at equilibrium Y, K, and L all

grow at the rate of n.grow at the rate of n.

2727


At equilibrium, i had to just match At equilibrium, i had to just match depreciation. Now that K has to also grow depreciation. Now that K has to also grow at rate n to keep k constant, i has to at rate n to keep k constant, i has to compensate for both depreciation and compensate for both depreciation and required capital growth:required capital growth:

Δk = i – δk – nk

Δk = i – (δ + n)k

It has to be nk because growth rate of L has to match growth rate of K.

2828


The impact of n on the model is to make The impact of n on the model is to make the the δδk line steeper.k line steeper.

The steady state will now be The steady state will now be

0 = sf(k) - (0 = sf(k) - (δδ+n)k+n)k

i = (i = (δδ+n)k+n)k

2929

Slowing Population GrowthSlowing Population Growth

k

i (δ+n1)k sf(k)

k1

kk2

(δ+n2)k

n2 < n1

3030


In the steady state consumption per In the steady state consumption per worker is not increasing but GDP (Y) is worker is not increasing but GDP (Y) is increasing at rate n.increasing at rate n.

A higher n implies a lower k and a lower y. A higher n implies a lower k and a lower y. Do higher n countries have lower per Do higher n countries have lower per capita incomes? Figure 7-13.capita incomes? Figure 7-13.

Golden Rule capital is now MPK = Golden Rule capital is now MPK = δδ+n+n

3131

Is Population Growth a Is Population Growth a Curse or a Blessing?Curse or a Blessing?

Malthus: resource constraintMalthus: resource constraintKremer: innovation and technology.Kremer: innovation and technology.

MACROECONOMICS Chapter 7 Economic Growth I: Capital Accumulation and Population Growth.

Documents