ECON222 - Lecture 12 (Week 12)

A Lagrange Method Application to the Choice of Saving and Consumption

in Two Life Periods Consider a person who:

1. lives two periods: present (0) and future (1);

2. has certain incomes: 0y at present and 1y in the future (in dollars);

3. can transfer financial resources from one period to another in a fixed interest rate )(r :

saving in the case of a transfer from period 0 to period 1, or borrowing in the case of a

transfer from period 1 to period 0;

4. leaves neither bequest nor debt at death;

5. who derives utility from spending on consumption at present and in the future in

accordance to a lifetime utility function ),( 10 CCu displaying positive, but

diminishing, marginal utilities from present and future consumptions (i.e., 2 2 2 2

0 1 0 1/ 0, / 0, / 0, / 0u C u C u C u C > > < < ) and, for simplicity, no effect of

consumption in one period on the utility from consumption in the other period (i.e.,

0 1 1 0( / ) / 0, ( / ) / 0u C C u C C = = ) ; and

6. who simultaneously chooses (spending on) consumption in period 0 ( 0C in dollars)

and consumption in period 1 ( 1C in dollars) at the beginning of period 0 so as to

maximize his/her utility subject to his/her inter-temporal budget constraint.

Formally, the decision problem of this person is:

0 1max ( , )u C C

subject to the intertemporal budget constraint

0

0011 )()1(S

CyryC ++= .

As displayed by the intertemporal budget constraint, the price of present consumption

comprises the purchasing price ($1) and the forgone interest on that dollar (i.e., 1 r+ ).

The corresponding Lagrange function is

0 1 1 0 0 1( , ) [ (1 )( ) ]= + + + u C C y r y C CL .

The Lagrange multiplier ( ) can be interpreted in the present case as the shadow value of the

budget available for the second period consumption. It is expressed in utility terms. It is the

marginal lifetime utility of funds available in the second period. Hence, it is positive.

The second order condition requires the determinant of the Hessian (H) of L (i.e., the matrix

of all the second derivatives of the Lagrange function as ordered and identified below by the

subscripts) to be positive at the critical point * * *0 1( , , )C C for that point to be constrained

minimum. That is, it is required that

1

0 0 0 1 0

1 0 1 1

0 1

2 20

* * * 2 20 1 1

2 2 2 2 20 1

( ) / 0 (1 )det ( , , ) 0 ( ) / 1

(1 ) 1 0

( ) / (1 ) ( ) / 0

C C C C C

C C C C C

C C

u C rH C C u C

r

u C r u C

+

* +

= *

+

= * + * >

L L L

L L L

L L L

This second order condition for constrained maximum lifetime utility is satisfied and an

interior solution, * * *0 1( , , )C C , exists.

The interior solution satisfies the following necessary conditions:

* * * *

* *0 1 0 1

0 0 0

( , ) ( , )(1 ) 0 (1 )u C C u C Cr rC C C

= + = = +

L (1)

* * * *

* *0 1 0 1

1 1 1

( , ) ( , )0u C C u C CC C C

= = =

L (2)

* *

1 0 0 1(1 )( ) 0

= + + =

y r y C CL . (3)

Dividing both sides of equation (1) by * and recalling (as shown in (2)) that

* * *0 1 1( , ) /u C C C = ], the marginal rate of substitution of future consumption for present

consumption is equal to the price ratio of present and future consumption:

* *0 1 0* *0 1 1

( , ) / 1( , ) /

u C C C ru C C C

= +

(4)

where 1 r+ dollars is the price of a unit of present consumption (one dollar plus the foregone interest on that dollar spent on present consumption) and 1 dollar is the price of a unit of future consumption. From equation (3),

))(1( *001*1 CyryC ++= . (5)

The system comprising equations (4) and (5) determines the values of *0C and *1C and

subsequently the individuals level of saving (when *0 0 0y C > ), or borrowing (when

0*00 ) Example Let,

5.01

5.00 9.0 CCu +=

0 1 $ 60,000y y= =

0.08r = .

1. Prove that the optimality condition leads to:

2* *

1 00.45 (1 )0.5

C r C = + .

From the optimality condition of equality between the MRS and 1+r:

1C

0y

1y

1 r+ 1 r+

0C

0.5* 0.5 *0.5 *0 0 1 1

* 0.5 *0.5 *1 1 0 0

/ 0.5 0.5 0.5 1/ 0.45 0.45 0.45

u c C C C ru c C C C

= = = = +

Hence,

0.5*1

*0

0.45 (1 )0.5

C rC

= +

Hence, 2

* *1 0

0.45 (1 )0.5

C r C = + .

2. Substitute the above expression into the budget constraint and show that

* 1 00 2

(1 )0.45 (1 ) (1 )0.5

y r yCr r

+ += + + +

.

By substituting 2

* *1 0

0.45 (1 )0.5

C r C = + into the budget constraint ))(1( *001

*1 CyryC ++= ,

2* *

0 1 0 00.45 (1 ) (1 )( )0.5

r C y r y C + = + + .

By rearrangement of terms, 2

*0 1 0

0.45 (1 ) (1 ) (1 )0.5

r r C y r y + + + = + +

and, consequently,

* 1 00 2

(1 )0.45 (1 ) (1 )0.5

y r yCr r

+ += + + +

.

3. Compute 0C , 1C and 0S ! Straightforward: Substitute the parameters values and calculate. Note that * *0 0 0S y C= .