Solow ’ s Model (Modeling economic growth)

Solow model I: Constant productivity Assumptions of the model Population grows at rate n L ’ = (1 + n)L Population equals labor force No productivity growth Capital depreciates at rate 

1. Per-capital Income Production function: Y = F(K, L) In “ per worker ” terms: y = f(k) Relationship between variables:

From the above we can get: Per-person or per-capita income level (y) depends on each worker ’ s capital equipment(k). Per-person or per-capita income level (y) depends on each worker ’ s capital equipment(k). y=f(k) shows DMR. Can you draw the graph with y and k? Can you draw the graph with y and k?

–Growth rate is measured by the slope of the tangent line of the y or f(k) curve. –Growth rate decreases as the per-capita capital stock rises. It is true for all countries- “ Convergence ” –Countries that start further away from the steady state grow faster

2. Actual Supply of Capital Assume FIXED SAVINGS RATE or APS: s =S/N/Y/N = savings /income Given an income of y –Actual savings= s · y = s f(k)

EXAMPLE Savings rate of 40% – s =.4 (you save a fraction of your income) Can you draw the actual savings curve in the previous graph you have drawn?

Minimum Capital Requirement to just keep up for each work is proportional to population growth rate(n) and capital depreciation rate(  *if you do not replenish the economy with the minimum requirement of capital, then the level of capital and thus the level of production or income fall. 3. Required Capital for Just Keep-Up

Example) Y = 100; L = 20; K = 10 y = Y/L = 5 y = Y/L = 5 k = K/L = 10/20 = 0.5 k = K/L = 10/20 = 0.5 n = 3% ;  n = 3% ;   Then you need 8% of capital every year to keep constant each worker’s capital equipment.

4. Equilibrium or Not The Change in capital per worker is the actual supply of capital over the minimum required capital We may call this net investment.

Thus: –If  k > 0: economy accumulates capital per worker –If  k < 0: economy reduces capital per worker –If  k = 0: constant capital per worker: steady state

Graphically k (  +n)k s f(k) k* k0k0  k > 0  k < 0 f(k)f(k)

Steady-state Per-capita Income or y* = Y/N is determined where s f(k*) = (  +n)k*.

Implications of the model The economy converges, over time, to its steady state. –If the economy starts BELOW the steady state, it accumulates capital until it reaches the steady state. –If the economy starts ABOVE the steady state, it reduces capital until it reaches the steady state.

Growth rates –Capital per worker grows at rate 0 –Output per worker grows at rate 0 –Total capital: K = k · L grows at rate n –Total output: Y = y · L grows at rate n

Comparative statistics Parameters of the model: s, n,  Predictions of the model: In steady state: –Higher savings rate implies higher income per worker –Higher population growth implies lower income per worker

Savings rate and growth k (+n)k(+n)k s1f(k)s1f(k) kss s2f(k)s2f(k) kss 2

Note that an increase in savings rate do increase the level of income, but not the rate of growth of income.

Population growth rate and growth k (+n1)k(+n1)k sf(k) kss (+n2)k(+n2)k kss 2

Technical Innovations How is this different for the y curve from an increase in savings rate?