Econ 208
Marek Kapicka
Lecture 2
Basic Intertemporal Model
Where are we?

1) A Basic Intertemporal Model

A) Consumer Optimization

B) Market Clearing

C) Adding capital stock

D) Welfare Theorems

E) Infinite horizon
Consumer’s optimization

Consumers maximize utility subject to
budget constraints
max U (c1 )  U (c2 ) s.t c1  b1  y1
c1 ,c2 ,b1
c2  y2  b1 (1  r )

Lagrangean
L(c1 , c2 , b1 , 1 , 2 )  max U (c1 )  U (c2 )
c1 ,c2 ,b1
 1 ( y1  c1  b1 )
 2 ( y2  b1 (1  r )  c2 )
Consumer’s optimization


First order conditions
U (c1 )  1
U (c2 )  2
1  2 (1  r )
Euler Equation
U (c1 )  (1  r )U (c2 )
A) Consumer’s optimization

Log utility:
c2
 (1  r ) 
c1

Solution:
y2
y1 
1 r
c1* 
1 
b1*  y1  c1*
Where are we?

A Basic Intertemporal Model

A) Consumer Optimization

B) Market Equilibrium

C) Adding capital stock

D) Welfare Theorems

E) Infinite horizon
B) Market Equilibrium

Suppose that there is N identical agents

Market clearing condition is
Nb (r )  0
*
1

Log utility:
*
y2
y1 
1 r
y1 
1 
1 y2
*
r 
 y1
Where are we?

A Basic Intertemporal Model

A) Consumer Optimization

B) Market Clearing

C) Adding capital stock

D) Welfare Theorems

E) Infinite horizon
C) Adding Capital Stock

Shortcomings of the previous model


Production is not determined within the
model
Solution: Introduce production


There is a firm producing output using
capital stock it owns
Consumers own the firm, get the profits
C) Adding Capital Stock
Firm’s Problem

Production function
y1  F ( K1 )
y2  F ( K 2 )

Capital changes according to
K 2  (1   ) K1  I1
K 3  (1   ) K 2


Initial capital stock K1 given
Capital stock K3 can be sold at the end of period 2
C) Adding Capital Stock
Firm’s Problem

Profits
1  Y1  I1
 2  Y2  K 3

Maximize the present value of profits
max 1 
I

2
1 r
In the optimum:
FK ( K2 )  r  
C) Adding Capital Stock
Consumer’s problem revisited

Budget Constraints:
C1  B1  1 (r )
C2   2 (r )  B1 (1  r )


B1 are savings from period 1 to period 2
r is the interest rate
C) Adding Capital Stock
Market Equilibrium

Market Clearing
C1  I1  F ( K1 )
C2  F ( K 2 )  K 3

Properties of Equilibrium:
U (c1 )   (1  r )U (c2 )
  [ FK ( K 2 )  1   ]U (c2 )
Where are we?

A Basic Intertemporal Model

A) Consumer Optimization

B) Market Clearing

C) Adding capital stock

D) Welfare Theorems

E) Infinite horizon
D) Efficiency of Equilibrium
Pareto Efficiency




Thought experiment: How to choose
consumption and investment if one doesn’t
need to obey the markets
The only constraints are the resource
constraints
This is the best one can possibly do!
Will the solution coincide with the market
solution?
D) Efficiency of Equilibrium
Pareto Efficiency

Pareto Efficient Allocation satisfies
max U (C1 )  U (C2 ) s.t C1  I1  F ( K1 )
C1 ,C2 , I1
C2  F ( K 2 )  K 3

Properties of Pareto Optimum:
U (c1 )   [ FK ( K2 )  1   ]U (c2 )
D) Efficiency of Equilibrium
Welfare Theorems



The allocation is the same as in the
competitive equilibrium
The equilibrium allocation is (Pareto)
efficient
Practical advantages of this result:


Solving for Pareto Optimum is easier
How to figure out what the prices must
be?
Where are we?

A Basic Intertemporal Model

A) Consumer Optimization

B) Market Clearing

C) Adding capital stock

D) Welfare Theorems

E) Infinite horizon
E) Infinite Horizon

Shortcomings of the previous model:



2 periods are arbitrary
Solution: Infinite number of periods
Solve the Pareto Problem

max
{ct , k t 1 }
t

 U (Ct )
t 0
s.t.
Ct  K t 1  F ( K t )  (1   ) K t
K 0 given
E) Infinite Horizon
Euler Equation again and Steady State

Consumption satisfies:
U (ct )   [ FK ( Kt 1 )  1   ]U (ct 1 )

Steady State:
U (C ss )   [ FK ( K ss )  1   ]U (C ss )
1   [ FK ( K ss )  1   ]
FK ( K ss ) 
1

  1