Saving, Capital Accumulation, and Output The Long Run Observation: The savings rate since 1950 U.S. 18.6% Germany 24.6% Japan 33.7% What do you think… Would increasing the U.S. savings rate lead to sustained higher U.S. growth in the future? …NO!

Interactions between Output and Capital Two long-run relations between output and capital The amount of capital determines the amount of output being produced The amount of output determines the amount of savings and investment and, thus, the amount of capital

Interactions between Output and Capital Capital, Output, and Saving/Investment The amount of capital (K)  amount of output (Y) The amount of output (Y)  the amount of savings (S) & investment (I = S when G-T=0)  amount of capital (K)

Interactions between Output and Capital When employment (N) is constant and there is no technological progress (f is constant) I = S in closed economy where G – T = 0 S = sY: Private saving is proportional to income s = Savings rate (between 0 & 1) Investment and Capital Accumulation: Kt+1 = (1-d) Kt + It Where d = depreciation rate

Interactions between Output and Capital Investment and Capital Accumulation Per worker output and capital accumulation Capital/worker in t+1 = Capital/Worker in t, adjusted for depreciation and investment Investment/worker = Savings rate x Output/worker in t

Interactions between Output and Capital The Effects of Output on Capital Accumulation Investment and Capital Accumulation The relation between output and capital accumulation Reorganizing: Savings/worker - depreciation Change in

Implications of Alternative Saving Rates Dynamics of Capital and Output: The change in Kapital/worker from t to t+1 depends on Investment/Worker – Depreciation/Worker - Change in capital from year t to year t+1 = Invest- ment during year t depreciation during year t

Implications of Alternative Saving Rates Dynamics of Capital and Output Graphically Output per worker f(Kt/N) Depreciation per worker Kt/N Investment per worker sf(Kt/N) Y*/N A B Output per worker, Y/N C AB = Output/worker AC = Investment/worker D AD = Depreciation AC > AD (Ko/N) K*/N Capital per worker, K/N

Implications of Alternative Saving Rates Steady-State Capital and Output Steady-State Value of Capital/Worker: Investment just offsets depreciation Steady-State Value of Output/Worker

Implications of Alternative Saving Rates The Saving Rate and Output What are the effects of the saving rate on the rate of output per worker? - The saving rate has NO effect on the long run growth rate of output/worker Growth rate = ZERO in long run! - An increase in the saving rate leads to a higher growth of output/worker for some time, but not forever. - The saving rate determines the level of output/worker in the long run

Implications of Alternative Saving Rates The Effects of Different Saving Rate Depreciation per worker Kt/N Output per worker, Y/N Capital per worker, K/N Investment s1f(Kt/N) K1/N Investment s0f(Kt/N) (K0/N)

Implications of Alternative Saving Rates The Effects of Different Saving Rate Investment s0f(Kt/N) s1f(Kt/N) Output per worker, Y/N Capital per worker, K/N Depreciation per worker Kt/N Output per worker f(Kt/N) D Y1/N K1/N B A (K0/N) Y0/N C I >

Implications of Alternative Saving Rates The Effects of Different Saving Rate Output per worker, Y/N Time (No technological progress) Associated with saving rate s1 > s0 Y1/N Y0/N Associated with saving rate s0 t

Implications of Alternative Saving Rates The Effects of Different Saving Rate (Technological progress) Output per worker, Y/N (log scale) Time Associated with saving rate s1 > s0 Associated with saving rate s0 t

Implications of Alternative Saving Rates The Savings Rate and the Golden Rule Does an increase in saving lead to an increase in consumption in the long run? Two Scenarios: Saving Rate = 0 Capital = 0 Output = 0 Consumption = 0 Saving Rate = 1 Consumption = 0 Output replaces depreciation

Implications of Alternative Saving Rates Maximum steady state Consumption per worker: At Golden Rule Level of Capital Consumption per worker, C/N Saving rate, s sG 1

Getting a Sense of Magnitudes Assume: (Constant return to scale and decreasing returns to either capital or labor)

Getting a Sense of Magnitudes Then In steady-state is constant and the left side = 0 and:

Getting a Sense of Magnitudes The Effects of the Saving Rate on Steady-State Output Steady-State Output/Worker: Higher saving rate and lower depreciation both lead to higher and in the long run. Double s  Quadruple K/N and double Y/N

Getting a Sense of Magnitudes The Dynamic Effects of an Increase in the Saving Rate

Getting a Sense of Magnitudes The Dynamic Effects of an Increase in the Saving Rate

Getting a Sense of Magnitudes The U.S. Saving Rate and the Golden Rule What saving rate that would maximize steady-state consumption? In Steady-State:

Getting a Sense of Magnitudes The Saving Rate and the Steady-State Levels of Capital, Output, and Consumption per Worker Capital Output Consumption Saving Rate, per Worker, per Worker, per Worker, s K/N Y/N C/N 0.0 0.0 0.0 0.0 0.1 1.0 1.0 0.9 0.2 4.0 2.0 1.6 0.3 9.0 3.0 2.1 0.4 16.0 4.0 2.1 0.5 25.0 5.0 2.5 0.6 36.0 6.0 2.4 .. .. .. .. 1.0 100.0 10.0 0.0

Getting a Sense of Magnitudes The U.S. Saving Rate and the Golden Rule Observation If s < .50: increasing s will increase long-run consumption In the U.S., s < 20%

Physical Versus Human Capital Extending the Production Function Measuring the Impact of Human Capital: Education and On-the-Job Training