lecture 1 a simple representative model: two period kornkarun cheewatrakoolpong, ph.d....
TRANSCRIPT
Lecture 1A Simple Representative Model:
Two PeriodKornkarun Cheewatrakoolpong, Ph.D.
MacroeconomicsPh.D. Program in Economics
Chulalongkorn University, 1/2008
Reading List
• Manuelli’s notes chapter 1
• Romer chapter 1
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)
0)( xgi
m
iii xgxfxL
1
)()(),(
m
ii xDgxDf
1
0)()( 0)( xgi
0,0)( iii xf
Kuhn-Tucker
Example: Max lnx + lny
s.t. 2x+y m
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
Solow Model
f(k) is assumed to be:
- f(0) = 0
- f’(k) >0
- f’’(k) <0
- satisfy inada condition
f(k)
k
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 =
( ) ( ( ))L t n L t
( ) ( ( ))A t g A t
( ) ( ) ( )K t sY t K t
Solow Model
• Dynamics of the model
2
( ) ( )( ) [ ( ) ( ) ( ) ( )]
( ) ( ) [ ( ) ( )]
( ) ( )( ) [ ( ) / ( ) ( ) / ( )]
( ) ( ) [ ( ) ( )]
( ) ( )( ) ( )[ ]
( ) ( )
( ) ( ( )) ( )[ ]
K t K tk t A t L t L t A t
A t L t A t L t
K t K tk t L t L t A t A t
A t L t A t L t
sY t K tk t k t n g
A t L t
k t sf k t k t n g
Solow Model
(n+g+ )k
sf(k)
Investment/AL
kk*
Solow Model
kk*
k
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
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.
)()(),( 2121 cucuccu
A Two Period Model
• Social Planner’s Problem is
s.t.
)()(max 21 cucu 01 kce
2)( ckf
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)
)()(max 21 cucu
0)( 2211 cpcep
A Two Period Model
• Competitive equilibrium means the price (p1,p2) and consumption (c1,c2,k) such that:
1. k solves firm’s profit maximization problem.
2. c1,c2 solves consumer’s utility maximization problem.
3. Market clearing condition
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?
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?
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:
ehc
azhehh
haec
'
)1()1('
2
1
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:
i) Assume that all individuals have the same h, find the solution to the planner’s problem.
ii) Decentralize the solution in i) as a competitive equilibrium.
10
)()(),( 2121 cucuccu