1. Lecture 1 A 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. Kuhn-Tucker
Example: Max lnx + lny
s.t. 2x+y m

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) is assumed to be: - f(0) = 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.