Technological Progress and the Production Function

),( ANKFY AN = Effective Labor = Labor in Efficiency Units Assuming:

•Constant returns to scale•Given state of technology

2Y = F(2K,2AN)

xY = F(xK,xAN)Y/AN = f(K/AN)

Technological Progress and the Production Function

f(K/AN)

Out

put p

er e

ffect

ive

wor

ker,

Y/A

N

Capital per effective worker, K/AN

Decreasing returns to Kapital per Effective Worker

Investment sf(K/AN)

Investment, Capital, & Output per Effective Worker

Production f(K/AN)

Out

put p

er e

ffect

ive

wor

ker,

Y/A

N

Capital per effective worker, K/AN

Determining the needed to maintain a givenANI

ANK

Assume:

Then:

•A population growth rate/yr (gN)

•N grows at same rate as gN

•Rate of technological progress gA

Growth rate of effective labor (AN) = gA + gN

If: gA = 2% & gN = 1%, then AN growth = 3%

Investment per effective worker to keep capital per effective worker steady

Determining the needed to maintain a givenANI

ANK

The level of investment needed to maintain :ANK

KKgg NA ofondepreciati)(

•Must offset depreciation, δK•Must outfit new workers with capital, gNK•Must give all workers additional capital to keep up, gAK

Amount of Investment Needed/Effective Worker to maintain a constant K/AN =

ANKgg NA )(

Dynamics of Capital & Output

Investment sf(K/AN)

Production f(K/AN)

Required investment ( + gA + gN)K/ANδ

Out

put p

er e

ffect

ive

wor

ker,

Y/A

N

Capital per effective worker, K/AN

A

B

(K/AN)o

C

D Observe (K/AN)0:AC > AD

(K/AN)*

ANY *

Dynamics of Capital & Output

Observations about the Steady State:

•Growth rate of Y = growth rate of AN = gY

gY = (gA + gN)

Output growth rate [= gA + gN] independent of s

•Capital growth rate gK = (gA + gN)

Capital keeps up with labor force and technology•Per worker output growth rate = gY – gN = gA

Dynamics of Capital & Output

The Characteristics of Balanced Growth

Growth: rate of1.2.3.4.5.6.7.

Capital per effective worker 0Output per effective worker 0Capital per worker gA

Output per worker gA

Labor gN

Capital gA+gN

Output gA+gN

The Effects of the Savings Rate

f(K/AN)

Out

put p

er e

ffect

ive

wor

ker,

Y/A

N

Capital per effective worker, K/AN

A

(K/AN)0

ANY

0

( + gA + gN)K/ANδ

s0f(K/AN)

Savings = s0

Steady-state = &

ANK

0

ANY

0

s1f(K/AN)

(K/AN)1

B

Savings increase to s1

S1f(K/AN)

Steady-state = &

ANK

1

ANY

1

ANY

1

The Effects of an Increase in the Savings Rate

Out

put,

Y (lo

g sc

ale)

Timet

Associated with s0

Associated with s1 > s0

B

slope (gA + gN)

B

A

A

Cap

ital,

K (l

og s

cale

)

Timet

Associated with s0

Associated with s1 > s0

B

slope (gN + gA)

B

A

A

Slide #11

Technological Progress and GrowthTechnological Progress and Growth

The Facts of Growth Revisited

A Review

Observations on growth in developed countries

since 1950:

•Sustained growth 1950-mid 1970s

•Slowdown in growth since the mid 1970s

•Convergence: countries that were furtherbehind have been growing faster

Slide #12

The Facts of Growth Revisited

Understanding These Trends

Determinants of Fast Growth:•Higher rate of technological progress (gA)•Higher level of capital/effective worker (K/AN)

Capital Accumulation vs. Technological Progress

Growth of Output per Capita, gY/N Rate of Technological Progress, gA

1950-73 1973-87 Change 1950-73 1973-87 Change(1) (2) (3) (4) (5) (6)

France 4.0 1.8 -2.2 4.9 2.3 -2.6

Germany 4.9 2.1 -2.8 5.6 1.9 -3.7

Japan 8.0 3.1 -4.9 6.4 1.7 -4.7

United Kingdom 2.5 1.8 -0.7 2.3 1.7 -0.6

United States 2.2 1.6 -0.6 2.6 0.6 -2.0

Average 4.3 2.1 -2.2 4.4 1.6 -2.8

Inferring rate of technological progress, gA

For Y = F(K,AN) gY = αgK + (1- α)(gN + gA)where α = capital share of national income (1 - α) = labor share of national income

Can measure Solow residual (total factor productivity) as gY not explained by capital growth and labor force growth

Residual = gY – {α gK + (1 – α) gN}Then (1-α) gA = Residual … or gA = Residual/ (1-α)

Technological Progress and GrowthTechnological Progress and Growth

The Findings

•1950-1973 high growth of output per capita dueto technological progress•Since 1973 slowdown in growth of output per

capita due to a decrease in the rate of technological progress

•Convergence is the result of technologicalprogress

Capital Accumulation vs. Technological Progress