Lecture 1 A Simple Representative Model: Two Period

1. Lecture 1A Simple Representative Model: Two Period Kornkarun Cheewatrakoolpong, Ph.D. Macroeconomics Ph.D. Program in Economics Chulalongkorn University, 1/2008

2. Reading List • Manuelli’s notes chapter 1 • Romer chapter 1

3. Kuhn-Tucker Consider the following maximization problem: Max f(x) s.t. For i = {1,…,m} Then we can define a saddle function L s.t. FOC: (1) (2) (3)

4. Example: Max lnx + lny s.t. 2x+y m Kuhn-Tucker

5. Solow Model • The production function is taken in the form of: Y(t) = F(K(t),A(t)L(t)) • Assumptions concerning the productions • CRS in capital and effective labor F(cK,cAL) = cF(K,AL) • We can write down the production function in this form: F(K/AL,1) =(1/AL)F(K,AL) • Given k=K/AL, y= Y/AL, f(k) = F(k,1), then y = f(k) output per effective labor

6. Solow Model f(k) f(k) is assumed to be: - f(0) = 0 - f’(k) >0 - f’’(k) <0 - satisfy inada condition k

7. Solow Model • The evolution of the inputs into Production • Continuous time model with n,g are exogeneously given • Fraction of output for investment = s • Depreciation rate =

8. Solow Model • Dynamics of the model

9. Solow Model Investment/AL (n+g+ )k sf(k) k k*

10. Solow Model k k*

11. Solow Model • The Balanced growth path (steady state) When k converges to k* - labor grows at rate n - knowledge grows at rate g - k grows at rate n+g - AL grows at rate n+g

12. A Two Period Model • Discrete time model • A large number of identical households • Each lives for two periods • The utility is given by: • The technology is represented by f(k), using k units of the first period consumption then you get f(k) units of the second period consumption.

13. A Two Period Model • Social Planner’s Problem is s.t.

14. A Two Period Model • Competitive equilibrium Firm’s problem: max p2f(k) – p1k Consumer’s problem: s.t. (Here we assume that a consumer owns firm)

15. A Two Period Model • Competitive equilibrium means the price (p1,p2) and consumption (c1,c2,k) such that: • k solves firm’s profit maximization problem. • c1,c2 solves consumer’s utility maximization problem. • Market clearing condition

16. A Two Period Model • The first welfare theorem If the vector price p and the allocation (c1,c2,k) constitute a competitive equilibrium, then this allocation is the solution of the planner problem. Question: Does the first welfare theorem hold in our setting?

17. A Two Period Model • The Second Welfare Theorem For every Pareto optimal allocation (c1,c2,k), there is a price vector p such that (c1,c2,k,p) is a competitive equilibrium. Question: Does the first welfare theorem hold in our setting?

18. A Two Period Model Example: Human Capital Accumulation Consider a two period economy in which an individual who has initial human capital has to decide what fraction a of his endowment e to allocate to producing goods in the first period. The fraction 1-a is used to accumulate human capital. The first period consumption and the end of period human capital h’ can be written as:

19. A Two Period Model Example: Human Capital Accumulation (cont’) Given that z is the productivity of current human capital. is the depreciation rate of human capital. Each individual has a utility function given by: • Assume that all individuals have the same h, find the solution to the planner’s problem. • Decentralize the solution in i) as a competitive equilibrium.