Technological Progress and the Production Function ) , ( AN K F Y 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)
Mar 15, 2016
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