Question

A. In the consumption/leisure model, let the consumer’s utility function be U(C,l)=C.25+l.25. Suppose Pc=$1 and w1=$10 determine the optimal amount of C and L and the equation for the labor supply.

B. Suppose w2=$12 determine the new optimal C and L and determine the substitution and income effects of the price change of leisure.

Answer 1

Solution :-

(A) :-

let the consumer’s utility function be U(C,l)=C^0.25 + l^0.25

Suppose Pc = $1 and w1= $10

L = labour supply

Constraint C = w1 L

I + L = T

Let T = 1

I + L = 1

I = 1 - L

Therefore,

U(C,l)= C^0.25 + l^0.25

= C^0.25 + ( 1 - L)^0.25

Subject to,

C = w1 L

U( L) = ( w1 L)^0.25 + ( 1- L) ^0.25

U/L =( 0.25 x w1)/( w1 L)^0.75 + (0.25 x -1)/( 1-L)^0.75 = 0

(w1^0.25)/(L^0.75) = 1/ ( 1 - L) ^ 0.75

( 1 - L)^0.75 = (L^0.75)/(w1^0.25)

1 - L = L/(w1)^1/3

L =[ (w1)^1/3]/( 1 + w1^1/3)........ labour supply equation.

So, optimal amount of C and L

L = ( 10^1/3)/( 1 + 10^1/3)

C = ( 10^4/3)/( 1 + 10^1/3)

(B) :-

Suppose w2 = $12

Then, new optimal C and L are

L = ( 12^1/3)/( 1 + 12^1/3)

C = ( 12^4/3)/( 1 + 12^1/3)

Now,

C = w2 L

C = w2 ( 1 - l )

C + w2 l = w2 ......New constraint

C + w1 l = w1......old constraint

U1 = C^0.25 + l^0.25

= ( 6.83)^0.25 + ( 0.68)^0.25

= 1.616 + 0.908

= 2.52

At point C

U = 2.52

And constraint is :-

C + w1 l = w2

C = 8.35 .....does not change

So,

I = [ 2.52 - ( 8.35)^0.25]^4

= [ 2.52 - 1.699]^4

= [ 0.821]^4

I = 0.45

* Point A to C is substitution effect and

Point C to B is income effect which is 0 for consumtion.

* Leisure has substitution effect which is negative and income effect which is positive ( C to B) and final figure is 0.695.