Technological Progress and the Production Function

1. xY = F(xK,xAN) • Y/AN = f(K/AN) Technological Progress and the Production Function AN = Effective Labor = Labor in Efficiency Units Assuming: • Constant returns to scale • Given state of technology • 2Y = F(2K,2AN)

2. f(K/AN) Output per effective worker, Y/AN Capital per effective worker, K/AN Technological Progress and the Production Function Decreasing returns to Kapital per Effective Worker

3. Production f(K/AN) Investment sf(K/AN) Investment, Capital, & Output per Effective Worker Output per effective worker, Y/AN Capital per effective worker, K/AN

4. Determining the needed to maintain a given Investment per effective worker to keep capital per effective worker steady Assume: • A population growth rate/yr (gN) • N grows at same rate as gN • Rate of technological progress gA Then: Growth rate of effective labor (AN) = gA + gN If: gA = 2% & gN = 1%, then AN growth = 3%

5. The level of investment needed to maintain : Determining the needed to maintain a given • 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 =

6. Required investment ( + gA + gN)K/AN Production f(K/AN) * B Investment sf(K/AN) C Observe (K/AN)0: AC > AD D A (K/AN)o (K/AN)* Dynamics of Capital & Output Output per effective worker, Y/AN Capital per effective worker, K/AN

7. Dynamics of Capital & Output Observations about the Steady State: • Growth rate of Y = growth rate of AN = gY • gY = (gA + gN) • Outputgrowth 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

8. Growth: rate of 1. 2. 3. 4. 5. 6. 7. Capital per effective worker 0 Output per effective worker 0 Capital per worker gA Output per worker gA Labor gN Capital gA+gN Output gA+gN Dynamics of Capital & Output The Characteristics of Balanced Growth

9. f(K/AN) B ( + gA + gN)K/AN 0 A Savings = s0 s1f(K/AN) Output per effective worker, Y/AN s0f(K/AN) 0 1 1 Savings increase to s1 S1f(K/AN) Steady-state = & Steady-state = & (K/AN)0 (K/AN)1 0 1 Capital per effective worker, K/AN The Effects of the Savings Rate

10. Associated with s1 > s0 Associated with s1 > s0 B B Output, Y (log scale) Capital, K (log scale) B B A A slope (gN + gA) Associated with s0 slope (gA + gN) Associated with s0 A A t t Time Time The Effects of an Increase in the Savings Rate

11. Technological 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 further behind have been growing faster

12. The Facts of Growth Revisited Understanding These Trends Capital Accumulation vs. Technological Progress • Determinants of Fast Growth: • Higher rate of technological progress (gA) • Higher level of capital/effective worker (K/AN)

13. Inferring rate of technological progress, gA 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 gY = αgK + (1- α)(gN + gA) For Y = F(K,AN) 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)

14. Technological Progress and Growth Capital Accumulation vs. Technological Progress The Findings • 1950-1973 high growth of output per capita due to 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 technological progress