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