1. Growth Facts Solow Growth Model Optimal Growth Endogenous Growth
Economic Growth
Growth Facts
Solow Growth Model
Optimal Growth
Endogenous Growth

2. Readings Williamson, Ch 6, p 200-222 (Solow Growth)
Williamson, Ch 7 (Endogenous Growth)

3. Table 6.1 Economic Growth in Eight Major Countries, 1870–2005

4. Growth Facts
Before Industrial Revolution (1800) standards of living was very similar across countries.
Large differences in per-capita incomes across countries appeared after IR.
Positive correlation between per capita output and per capital investment.
Negative correlation between per capita output and population growth.
Rich countries: Convergence
Poorest countries: No Convergence

5. Figure 6.2 Output per Worker vs. Investment Rate

6. Figure 6.3 Output per Worker vs. the Population Growth Rate

7. Figure 6.5 Convergence Among the Richest Countries

8. Figure 6.6 No Convergence Among the Poorest Countries

9. Questions:
(1) What explains differences in economic growth and standard of living among countries?
(2) How do factors like saving, population, and technology affect economic growth and productivity?
(3) What does CE model say about optimal growth?
(4) How does the accumulation of “human capital” contribute to economic growth?
(4) Policy implications?

10. Two key variables of interest:
(1) yt = Yt/Nt = output per person or average labor productivity. Also can measure standard of living.
(2) ΔY/Y = growth rate of Y.

11. Solow Growth Model F(xK,xN) = xF(K,N)
Robert Solow (MIT) – 1989 Nobel Winner
A constant returns to scale production function says
F(xK,xN) = xF(K,N)
Notice that from CRTS production function we can write
where kt = Kt/Nt.

12. Per-capita GDP (y) depends only on the capital–labor ratio (k).
Competitive Market-Clearing:
Yt = Ct + It
Assume proportional aggregate saving- investment rate:
(Yt - Ct)= It = sYt
Population grows at rate n:
Nt+1 = (1+n)Nt
Population = Labor Force = Nt

13. Definition of (Net) Investment:
A steady state equilibrium is where k and y are constant (no pressure to change):

14. In the steady state k = K/N and y = Y/N are constant. Therefore
* Dy = Dk = 0 *

15. Saving Rate (s)
Increase s  increases k* and y*
(i) Permanent increase in standard of living (y*)
(ii) Temporary increase in economic growth.
Productivity/Technology (z)
Increase z  increases k* and y*
(i) Perm increase in standard of living
(ii) One time inc. in z  temp increase in economic growth.
(iii) Sustained inc. in z  perm inc. in growth

16. Figure Effect of an Increase in the Savings Rate on the Steady State Quantity of Capital per Worker

17. Figure 6.16 Effect of an Increase in the Savings Rate at Time T

18. Figure 6.23 Natural Log of the Solow Residual, 1948–2001

19. Population growth (n)
increase n  decreases k* and y*
(i) Permanent decrease in standard of living
(ii) Permanent increase in economic growth
Application – The convergence hypothesis: Economic growth and output per person in countries with similar levels of technology, savings and population growth should converge.

20. Application/Evidence
* Growth convergence among developed countries.
* Japanese/German “miracle” after WWII
* Non-convergence among poor countries
Explanation: Barriers to Technological Adoption.
(i) Trade Restrictions
(ii) Intellectual Property Rights / Monopolies

21. Figure 7.2 Convergence in Income per Worker Across Countries in the Solow Growth Model

22. Figure 7.3 Convergence in Aggregate Output Across Countries in the Solow Growth Model

23. Productivity Levels in Selected Countries

24. Figure 6.4 No Convergence Among All Countries

25. Figure 7.4 Differences in Total Factor Productivity

26. Optimal Growth How much should a growing economy consume and save?
(i) What’s the “best” s?
(ii) What’s the “best” k*?
Two Responses:
(1) Maximize Steady State Consumption
(Solow Model)
(2) Maximize PDV of discounted household utility. (Optimal Growth Model)

27. Solow - The Golden Rule Steady-state per capita consumption: OR
Golden-Rule: The optimal kGR* maximizes c*
“treat others (future generations) as you would like to be treated”:
FOC 

28. Figure 6.17 Steady State Consumption per Worker

29. Policy Public policies to increase saving rate (s)
(1) Reduce budget deficit (government spending)
(2) Encourage private saving (IRA accounts, Social Security reform)
Public policies to increase productivity (z)
(1) Infrastructure
(2) Human capital (R&D)
(3) Industrial policies

30. The Optimal Growth Model – Modified Golden Rule
Optimal Growth Model  Basic CE model w/
* exogenous/inelastic labor supply
* representative consumer so all variables are already in per-capita terms.
The saving rate (s) is endogenous in Optimal Growth Model.
Recall that CE Model  Pareto Optimal

31. Social Planner’s Objective: subject to
State Variable:
Control Variable:
Bellman Equation:

32. First Order Condition:
Steady State (Modified Golden Rule):
where b = 1/(1+r) and r is rate of time preference.
Is this level of capital k*MGR accumulation socially optimal?

33. Social Efficiency (Solow Growth):
Individual Efficiency (Optimal Growth):
as steady state implies r* = rate of time preference = r.
(i) n = r = r  kGR= kMGG  Dynamic Efficiency
(ii) n > r = r  kGR < kMGR  over accumulation
(iii) n < r = r  kGR > kMGR  under accumulation

34. Endogenous Growth
Solow Model can only explain non-convergence via barriers to technological adoption.
In Solow Model sustained growth is due to exogenous forces. Does not explain where growth comes from.
Endogenous growth models attempt to explain economic growth from society’s choice of human capital accumulation:
Human Capital = Stock of Knowledge and Skills

35. A Simple Example Definitions: Labor Supply = ut Education = (1-ut)
Human Capital Stock = Ht
Efficiency Units of Labor = utHt
Real Wage per Efficiency Unit = wt
Budget Constraint for t = 1,2:
(BC)

36. Investment in Human Capital:
(HC)
Production Function:
Representative Household: Given H1 and wt,
max U(c1,c2, …) subject to (BC) and (HC).

37. Representative Firm: Maximize Pt = Yt – wtutHt.
Market-Clearing: Yt = ct
General Case: In a steady state equilibrium where ut = ut+1 = u,
Output Growth = Consumption Growth
= Human Capital Growth = b(1-u) – 1 > 0 if b(1-u) > 1.

38. Implications
Sustained economic growth can be explained by endogenously by human capital accumulation.
Higher rates of growth can be attained by
(i) Greater time allocation to education (1-u)
(ii) More efficient educational systems (b)
Countries with identical u and b will experience same growth rate but levels of output and standard of living may never converge.

39. Figure 7.9 Growth and Education

40. Figure 7.10 Income per Worker and Education

41. Figure 7.8 No Convergence in the Endogenous Growth Model

42. Policy Implications:
(i) There is a potential role for governments to subsidize education/worker retraining or encourage efficiency (increasing b / decreasing u).
(ii) There are externalities to human capital accumulation  too little private investment.
Private Marginal benefit = MC
< Society Marginal Benefit

43. (iii) Optimal Human Capital Accumulation:
* Social MB = MC
* Short-Run Marginal Cost of Lost Consumption = Long Run MB of increased future consumption.

44. Figure 7.7 Effect of a Decrease in u on the Consumption Path in the Endogenous Growth Model