An improvement in technology shifts The Sources of Growth An improvement in technology shifts the production function up Output per worker, Y/N Capital per worker, K/N F(K/N, 1) F(K/N, 1) B´ A A´ Growth comes from capital accumulation and technological progress. Because of decreasing returns to capital, capital accumulation by itself cannot sustain growth.

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) ΔK = I – Depreciation = sY - δK

Interactions between Output and Capital 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 - Change in capital from year t to year t+1 = Invest- ment during year t depreciation during year t

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

Steady-State Capital and Output Steady-State Value of Capital/Worker: Investment just offsets depreciation Steady-State Value of Output/Worker

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 >

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

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

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

In steady-state is constant and the left side = 0 and: Assume: (Constant return to scale and decreasing returns to either capital or labor) Then In steady-state is constant and the left side = 0 and: Double s  Quadruple K/N and double Y/N

What saving rate that would maximize steady-state consumption? The U.S. Saving Rate and the Golden Rule What saving rate that would maximize steady-state consumption? In Steady-State: If s < .50: increasing s will increase long-run consumption In the U.S., s < 20%