Solow’s Growth Model The mainline Classical Theory of Economic Growth

Solow’s Economic Growth Model ‘The’ representative Neo-Classical Growth Model: focusing on savings and investment, and eventually on technology. It explains the long-run evolution of economy quite well with all being held Constant – Dynamic Model

Features Focusing on capacity of Savings to meet the demand for Investment as Capital Requirements and, beyond that, as Capital Accumulation for expansion of Production capacity; Convergence is usual but possibility of Technical Innovation for Sustained Economic Growth

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. Math

1) Per-capital Income Production function: Y = F(K, L: fixed T) 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). y=f(k) shows DMR. 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 –Supply of K is from 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

Investment above and beyond this requirement will lead to Accumulation of Capital and Expansion of Production and an Increase in per-capita Income. (but still the growth rate continues to fall; per-capita income rises in decelerating manners)

Example) Y = 100; L = 20; K = 10 y = Y/L = 5 k = K/L = 10/20 = 0.5 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

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

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

3. 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

4. Comparative statistics 1) Parameters of the model: s, n,  2) Once-and-for-all increase in K 3) Technical Innovations

(1) 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 does increase the level of income, but not the rate of growth of income.

*Is there the optimal savings rate? Yes, there is the Dynamically Optimal savings rate: The savings rate that maximizes consumption(= Income– Investment or savings in the steady state. It is the Golden Rule Savings Rates

*Golden Rule The Golden Rule of Savings Rate is such that MPk = n +  This is the condition that the economy is dynamically efficient.

C 1 is consumption for Golden Rule where MPk = n + , which is lager than C2 or any others. Note: This graph is for the case where there is no depreciation; add  to n everywhere. If  for simplicity, then

What does this mean? The Golden Rule Point can be reached with a lower savings rate than the current equilibirum. The economy has intrinsic incentive to over-save and under-consume.

(2) Population growth rate and growth; A lower rate of n raises the growth rate of Y. k (+n1)k(+n1)k sf(k) kss (+n2)k(+n2)k kss 2 : Population control raises the growth rate of national income

How is this relevant to Japan? 1) In the s: Worked positive. 2) In the 1970s- Worked negative.

(3) An Injection of Capital It really does not do anything for the economy and income in the long-run; It does not lead to sustainable economic growth in the long-run. Try the llustration.

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

*Solow Model carves up technical innovation in the long run for sustainable economic growth; Technical Innovation fights against Convergence. MP = f (K/L, T)

The bigger questions are: What is technical innovation? How does it happen? How can we promote it? These are not easy questions at all.

Practice Click here for Solow’s Model simulation for Economic GrowthClick here for Solow’s Model simulation for Economic Growth Note: The production function used in the above illustration is not the original Solow model, but is an revised version with the later developed theory of Human Capital. Thus it is a hybrid of Solow model and Endogenous Growth Model.

Shortcoming When it comes to Capital, the Solow Model emphasizes the importance of an endogenous factor as opposed to an exogenous factor; People’s savings rate matters, but Injected Capital from outside does not lead to sustained economic growth. However, when it comes to Technology, he just presents it as an important factor like an exogenous factor or an accident, and does not explain how it endogenously evolves from inside of the growing economy.