Class Slides for EC 204Spring 2006

To Accompany Chapters 7-8

Y =F(K, L) (constant returns to scale)

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

y =f(k) where y=Y/L and k=K/L

MPK =f(k+1)- f(k)

The Supply of Goods and the Production Function:

The Solow Growth Model

y =c+i

c=(1-s)y

y =(1-s)y+i

i =sy

The Demand for Goods and the Consumption Function:

i =sf(k)

Change in Capital Stock=Investment-Depreciation

Δk = i - δk

Δk =sf(k)-δk

At the Steady State: Δk =0

This will happen at a particular value of k=k*

Growth in the Capital Stock and the Steady State:

Golden Rule Maximizes the Level of Consumption per Worker

We compare different Steady States to decide which

one achieves this.

y =c+i

c=y- i

c*=f(k*)-δk*

Take derivative w.r.t. k*: f'(k*)-δ =0

Implies that we choose the k* where MPK=δ

The Golden Rule Steady State

Economy has Too Much Capital

Economy has Too Little Capital

Growth Rate of Population: n=ΔL/L

Roughly 0.01 for the United States (i.e., 1% a year)

Determining Steady State:

Δk =i - (δ+n)k

Think of (δ+n)k as the "break-even" level of investment

Thus,

Δk =sf(k)- (δ +n)k

Δk =0 when k=k*

Allowing for Population Growth in the Solow Model

What is the Golden Rule Steady State?

c=y- i

c*=f(k*)- (δ +n)k*

MPK =(δ +n) or MPK-δ =n

Population Growth versus Labor Force Growth

• Really should have growth of labor force in model rather than population growth

• But if the labor force participation rate is stable over time, then

• Population growth equals labor force growth

• Take a look at the data:

Labor Force Participation

20

30

40

50

60

70

80

90

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

Percent of Population

Male

Female

Total

Labor Force Participation

• Overall rate has increased steadily over the past half century

• Men’s participation rate has dropped sharply

• Women’s participation rate has increased

• What about older workers?

Labor Force Participation Age 65 and Over

0

5

10

15

20

25

30

35

40

45

50

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

Percent of Population

Male

Female

Per Capita Personal Income as a Percentage of U.S. Average By Region

50

70

90

110

130

150

170

1929 1934 1939 1944 1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999

SoutheastPlains Great Lakes

Mideast

SouthwestRocky Mt.

NewEngland

Far West

Production Function: Y =F(K, L ×E)

where E is the "efficiency of labor."

We can view E as capturing the improvement in

labor productivity over time.

(L ×E) measures the number of "effective workers."

Allowing for Technological Progress in the Solow Model

Need to rewrite the production function in terms

of effective labor units:

Y/LE = F(K/LE, 1)

y = f(k) where y=Y/LE and k=K/LE

Assume that technological progress causes E to grow

at a constant rate per year: ΔE/E = g

We call "g" the rate of labor-augmenting

technological progress.

Because the labor force (L) is growing at rate "n,"

the number of "effective workers" (L×E)

is growing at rate n+g.

Determining Steady State:

Δk =i - (δ+n +g)k

Think of (δ+n+g)k as

the "break-even" level of investment.

Thus,

Δk =sf(k)- (δ +n+g)k

Δk =0 when k=k*

What is the Golden Rule Steady State?

c=y- i

c*=f(k*)- (δ +n+g)k*

MPK =(δ +n+g) or MPK-δ =n+g

Does the U.S. Have Too Muchor Too Little Capital?

Too Much Capital Implies: MPK-δ < n+g

Too Little Capital Implies: MPK-δ > n+g

Four Facts for U.S:

1. Real GDP grows an average of 3% per year

2. k=2.5y

3. δk =0.1y

4. MPK×k =0.3y

€

Thus,

(n +g) = 0.03

δ = δk/k = (0.1y)/(2.5y) = 0.04

MPK = (MPK × k)/k = (.03y)/(2.5y) = 0.12

Plug in to show that MPK -δ = 8 percent per year

which is greater than (n +g) = 3 percent per year.

So, U.S. has too little capital and should save more to reach Golden-Rule steady state

Policies to Promote Growth

According to Golden Rule the U.S. Capital Stock is too small.

1. Increase Saving Rate: Public and Private Saving

2. Allocate More Efficiently Economy’s Investment

3. Encourage Technological Progress

Worldwide Slowdown in Economic Growth

Growth rate fell sharply in early 1970s worldwide and remained low.

Slowdown in growth was due to a slowdown in total factor productivity growth--closely related to the efficiency of laborin Solow Model.

Real income in the United States today is about 20 percent lowerthe it would have been if the slowdown had not occurred.

Recently, some economist believe that productivity growth has picked up and that the long-run growth of the economy is now againclose to what it was before the slowdown.

Reasons for the Slowdown

1. Measurement Problems--but would have to have gotten worse over time.

2. Oil Prices--timing is correct, yet magnitude and 1986 drop?

3. Worker Quality--demographics and social norms lead to less experienced workforce. Also, educational attainment not increasing as fast as in past and quality of education may be lower.

4. Depletion of Ideas--1950s and 1960s were unusual, had a backlog of ideas that hadn’t been implemented fully due to depression and war. Growth from 1870-1950 not much different!

Information Technology and the “New Economy”

Took some time for computers to be used effectively

Similar to electric motor in late 19th-early 20th centuries

Three channels:

1. Direct productivity gains in computer sector

2. Accumulation of “info-technology” capital

3. Indirect productivity gains in other sectors

Testing the Solow Model’s Predictions

• Balanced growth: Y/L and K/L have grown at about 2 percent per year over last half century, so Y/K ratio roughly constant; real wages grow at about 2 percent rate while real rental approximately constant

• Convergence: Across regions of U.S.; Conditional convergence across countries

• Factor Accumulation versus Production Efficiency: Both matter for growth

Accounting for the Sources of Growth

Y =AF(K, L)

ΔY =ΔA ×F(K, L) + MPK ×ΔK +MPL ×ΔL

ΔY =ΔAA

×AF(K, L) + MPK ×KΔKK

+MPL ×LΔLL

ΔY =ΔAA

×Y + MPK ×KΔKK

+MPL ×LΔLL

ΔYY

=ΔAA

+ MPK ×K

Y×

ΔKK

+MPL ×L

Y×

ΔLL

ΔYY

=ΔAA

+αΔKK

+(1−α)ΔLL

Table 1 Contributions to Growth of Real Output in Nonfarm Business Sector, 1974-1999 (annual percent change)

1974-90 1991-95 1996-99Growth Rate of Output 3.06 2.75 4.82Contribution from:Capital 1.35 1.01 1.85

Information Technology Capital 0.49 0.57 1.10Other Capital 0.86 0.44 0.75

Labor Hours and Quality 1.38 1.26 1.81Multifactor Productivity 0.33 0.48 1.16

Multifactor Productivity in Computer Sector plusComputer-related Semiconductor Sector

0.17 0.23 0.49

Multifactor Productivity in Other Sectors 0.16 0.25 0.67

Source: Tables 1 and 4 in Stephen D. Oliner and Daniel E. Sichel, “The Resurgence ofGrowth in the Late 1990s: Is Information Technology the Story?” Journal of EconomicPerspectives, Vo lume 14, Number 4, Fal 2000.

Endogenous Growth Theory

Y =AK

ΔK =sY-δK

ΔY/Y = ΔK/K =sA-δ

If sA is greater than δ, then economy

grows forever even without assuming

exogenous technological progress.