1. ECO 402 Fall 2013 Prof. Erdinç Economic Growth The Solow Model

2. The Neoclassical Growth model Solow (1956) and Swan (1956) Simple dynamic general equilibrium model of growth

3. Output produced using aggregate production function Y = F (K, L ), satisfying: A1. positive, but diminishing returns F K >0, F KK 0, F LL <0 A2. constant returns to scale (CRS) Neoclassical Production Function

4. Production Function in Intensive Form Under CRS, can write production function Alternatively, can write in intensive form: y = f ( k ) - where per capita y = Y/L and k = K/L Exercise: Given that Y=L  f(k), show: F K = f’(k) and F KK = f’’(k)/L.

5. Competitive Economy Representative firm maximises profits and take price as given (perfect competition) Inputs paid by their marginal products : r = F K and w = F L –inputs (factor payments) exhaust all output: wL + rK = Y –general property of CRS functions (Euler’s THM)

6. A3: The Production Function F(K,L) satisfies the Inada Conditions Note: As f’(k)=F K have that Production Functions satisfying A1, A2 and A3 often called Neo-Classical Production Functions

7. Technological Progress = change in the production function F t Hicks-Neutral T.P. Labour augmenting (Harrod-Neutral) T.P. Capital augmenting (Solow-Neutral) T.P.

8. A4: Technical progress is labour augmenting Note: For Cobb-Douglas case three forms of technical progress equivalent:

9. Under CRS, can rewrite production function in intensive form in terms of effective labour units -note: drop time subscript to for notational ease - Exercise: Show that

10. A5: Labour force grows at a constant rate n A6: Dynamics of capital stock: net investment = gross investment - depreciation – capital depreciates at constant rate  Model Dynamics

11. National Income Identity Y = C + I + G + NX Assume no government (G = 0) and closed economy (NX = 0) Simplifying assumption: households save constant fraction of income with savings rate 0  s  1  I = S = sY Substitute in equation of motion of capital: … closing the model

12. Fundamental Equation of Solow-Swan model

13. Steady State Definition: Variables of interest grow at constant rate (balanced growth path or BGP) at steady state:

14. Solow Diagram: Steady State

15. Existence of Steady State From previous diagram, existence of a (non- zero) steady state can only be guaranteed for all values of n,g and  if - satisfied from Inada Conditions (A3).

16. Transitional Dynamics If, then savings/investment exceeds “depreciation”, thus If, then savings/investment lower than “depreciation”, thus By continuity, concavity, and given that f(k) satisfies the INADA conditions, there must exists an unique

17. Properties of Steady State 1. In steady state, per capita variables grow at the rate g, and aggregate variables grow at rate (g + n) Proof:

18. 2. Changes in s, n, or  will affect the levels of y* and k*, but not the growth rates of these variables. Prediction: In Steady State, GDP per worker will be higher in countries where the rate of investment is high and where the population growth rate is low - but neither factor should explain differences in the growth rate of GDP per worker. - Specifically, y* and k* will increase as s increases, and decrease as either n or  increase

19. Policies to Promote Growth 1.Are we saving enough? Too much or too little? 2.What policies may change the savings rate? 3.How should we allocate savings between physical and human capital? 4.What policies could generate faster technological progress?

20. Golden Rule Definition: (Golden Rule) It is the saving rate that maximises consumption in the steady-state. We can use the rule to evaluate if we are saving too much, too little or about right. Given we can use to find.

21. Golden Rule and Dynamic Inefficiency If our savings rate is given by then our savings rate is optimal and If then we must be under-saving If then we must be over-saving Check why this is the case!

22. Is Golden Rule attained in the US? Is it Dynamically Efficient? Let us check: Three Facts about the US Economy a) The capital stock is about 2.5 times the GDP b) About 10% of GDP is used to replace depreciating capital c) OR Capital income is 30% of GDP: Note alpha also measures the elasticity of output with respect to capital!

23. Is Golden Rule attained in the US? Is it Dynamically Efficient? Since US real GDP grows on average at 3% per year, i.e. Hence, US economy is under-saving because

24. Changes in the savings rate Suppose that initially the economy is in the steady state: If s increases, then Capital stock per efficiency unit of labour grows until it reaches a new steady-state Along the transition growth in output per capita is higher than g.