© 2020 EMANUEL GOMES CONFIDENTIAL AND PROPRIETARYMiguel Lebre de Freitas M.C, Escher, Day and Night, 1938

The endowment

economy with

uncertainty

0 EMANUEL GOMES CONFIDENTIAL AND PROPRIETARY

Miguel Lebre de Freitas

M.C, Escher, Day and Night, 1938

Miguel Lebre de Freitas

The endowment economy with uncertainty

Questions to be addressed

• How households decide how much of their income to consume today and how much to save for an uncertain future?

• Do savings increase with uncertainty? In which circumstances?

• How is the macroeconomic equilibrium affected by consumer uncertainty?

Intro

• When people take consumption decisions, they assess whether changes in their incomes are temporary or permanent.

• So far we assumed that the consumer is fully informed about his future incomes

• In real life, people do not exactly how the future will look like.

• Agents form expectations about the future but mistakes are possible

• In this chapter we analyse the impact of uncertainty on consumer decisions

The endowment economy

with uncertainty

•

•

•

The endowment economy

with uncertainty

•

•

•

The endowment economy

with uncertainty

Keep previous assumptions

• Single homogenous good

• Production is exogenous

(no investment, no labour market)

• Two period economy

• Flexible prices

Optimal consumption

revisited

Keep previous assumptions

• Single homogenous good

• Production is exogenous

(no investment, no labour market)

• Two period economy

• Flexible prices

Novelty:

 Uncertainty regarding period 2

• In past exercises, we have assumed that

information regarding future income is revealed beforeconsumers decide consumption in period 1.

• Now, we assume that information regarding future income is revealed afterconsumers decide

consumption in period 1.

• And the question is whether consumption smoothing will still hold (ex ante) or not.

Optimal consumption

revisited

The household maximizes his life-time utility function

Subject to the budget constraint

Utility maximization implies:

•

Euler:

•

Optimal consumption:

Revisit Consumer problem (deterministic case)

Optimal consumption

The endowment economy

   

1 U u C u C

  



r Q Q

r C C

 

 

 1 1

1 2 1 2

 

^'  

1

' 1 r u C

C

u  

 

2

1 1

1

1

2 1

C Q Q

r





 

        

Utility maximization implies:

•

Euler:

•

Optimal savings:

The household maximizes his life-time utility function

Subject to the budget constraint

Utility maximization implies:

•

Euler:

•

Optimal consumption:

Revisit Consumer problem (deterministic case)

Optimal consumption

The endowment economy

• Savings are zero when the income path is horizontal

   

1 U u C u C

  



r Q Q

r C C

 

 

 1 1

1 2 1 2

 

^'  

1

' 1 r u C

C

u  

 

2

1 1

1

1

2 1

C Q Q

r





 

        

Simple case:

r    0

 

^'  

' C u C

u  C

 C

 

1 1 2

1 2

S

 Q  Q

Two scenarios for future output

• Two states of nature

2 2

2

1

H L

Q p

Q Q p

  

 

The endowment economy with uncertainty

Optimal consumption

revisited

•

Once C1 is decided, the realization of C2 will depend on the materialization of the shock

•

“ex post” there will be no consumption smoothing

Two scenarios for future output

• Two states of nature

• How much will be future consumption?

…it depends

2 2

2

1

H L

Q p

Q Q p

  

 

  

2^H

1

C   r Q  C  Q

  

2^L

1

C   r Q  C  Q

The endowment economy with uncertainty

Optimal consumption

revisited

The household maximizes expected utility

Subject to

•

Euler equation (stochastic):

Consumer problem under uncertainty

      ^{ }  

1 2 2

1 2 2

, ,

1

H L

H L

C C

Max

 U  u C    pu C   p u C  

  

2^L

1

C   r Q  C  Q

  

2^H

1

C   r Q C   Q

 

¹

 

^{ }  

' ' 1 '

1

H L

u C r pu C p u C

   

     

The endowment economy with uncertainty

Optimal consumption

revisited

The marginal utility of current consumption equals the discount adjusted expected

marginal utility of future consumption

The household maximizes expected utility

Subject to

•

Euler equation (stochastic):

Consumer problem under uncertainty

      ^{ }  

1 2 2

1 2 2

, ,

1

H L

H L

C C

Max

 U  u C    pu C   p u C  

  

2^L

1

C   r Q  C  Q

  

2^H

1

C   r Q C   Q

 

¹

 

^{ }  

' ' 1 '

1

H L

u C r pu C p u C

   

      ^{u C '}  

^ ₁ ¹ _ ^ _ ^r

^{E u C } _ ^'  

^ _

The endowment economy with uncertainty

Optimal consumption

revisited

How to solve this?

The marginal utility of current consumption

equals the discount adjusted expected value of the marginal utility of future consumption

 

¹

 

' '

1

u C r E u C



  

   

The endowment economy with uncertainty

 

'

u C ^{E u C } _ ^'  

^ _

 

'

u C ^{u E C '}   



The trick is to replace by

In the Euler equation

 

'

E u C     ^{u E C '}   



• We cannot compare with

• But we could compare with

Optimal consumption

revisited

Three cases shall be distinguished

• We cannot compare with

• But we could compare with

How to solve this?

The marginal utility of current consumption

equals the discount adjusted expected value of the marginal utility of future consumption

 

¹

 

' '

1

u C r E u C



  

    ^{u E C '}   

 _{ } ^{E u C  '}  

^ _

  

  

' '

u E C   E u C   

  

  

' '

u E C   E u C   

Random walk

Precautionary savings Imprudent

The endowment economy with uncertainty

 

'

u C ^{E u C } _ ^'  

^ _

 

'

u C ^{u E C '}   

 Optimal consumption

revisited

2

CL ^E[C₂^] C₂^H

 

' u C

C

The endowment economy with uncertainty

Graphical illustration

•

Solution’ type depends on the convexity of

(u’’’)

•

Distinct from Risk Aversion, that is implied by the concavity of the Utility Function (u’’<0)

  



'

u E C

The log utility function belongs to the second category .

Marginal Utility

 

^ln

u C  C

 

'

1

u C  C

  

  

' '

u E C   E u C   

  

  

' '

u E C   E u C   

Random walk

Precautionary savings

 

'''

2

0

u C  C 

Optimal consumption

revisited