Question

Consider a representative consumer with the utility function ?(?, ?) = ?? + ? and the budget constraint ??? + ??? ≤ ?.

Assume throughout that all prices and quantities are positive and infinitely divisible.

1.a.Derive the consumer’s indirect utility function ?(∙).

b.Then, derive the consumer’s expenditure function, e(∙), directly from ?(∙).

c.Finally, derive the consumer’s Hicksian/compensated demand functions (denoted ? and ? , respectively) from e(∙). ??

2.Assume initially that ?? = ?? = 1 and ? = 10. Define and calculate the compensating variation (??) associated with a doubling of the price of good ?.

3. Derive the consumers equilibrium cross-price elasticity between goods ? and ? and evaluate the value of this elasticity at the initial parameter values given in part (2)

Answer 1

1.(a)

Utility function: U(x,y) = xy + y

Budget constraint: p_xx + P_yy ≤ M

For marshallian demands:

MRS = Slope of Budget constraint

put the value of xP_x into budget constraint,

put y* into (1+y)P_y = xP_x

we get,

Indirect utility function :

; where (1+y) = xP_x/ P_y has been used.

The above expression represents the consumer's utility function.

1.(b)

Duality between Indirect utility and expenditure,

To find out the the expenditure function, put e(.) in place of M in the indirect utility function and equate it to 1. and solve for e(.).

The above expression represents the consumer's expenditure function.

1.(c)

To derive Hicksian demand functions directly from expenditure function:

we use Shephard's lemma:

Hicksian demand for x :

Hicksian demand for y:

(2) Compensating Variation:

Due to the change in the price of a good, original utility level of the consumer also changes. The adjustment in income needed to return the consumer to its original utility level. In other words, Compensating variation is the change in income needed to compensate the consumer against the price change.

at P_x = P_y = 1 and M = 10,

Use Marshallian Demands,

and

Original Utility :

Now P_x =1, P_y = 2 and

To find out optimal consumption bundle for this setting, use Hicksian demands,

   

and

Total income needed to afford new consumption bundle(x',y')

Compensating Variation(CV) = M' - M

CV = 13.29 - 10

CV = 3.29