Question

What is the money-metric utility function of a consumer with the utility function

U(q1, q2,) =?q1 +?q2 ?

Use prices p1=p2= 2 to determine the money-metric utility function.

Answer 1

U(q1, q2,) =?q1+?q2

p1 = p2 = 2

First we compute the optimal consumption bundle,

Marginal Utility of q1 =dU/dq1 = 1/2q1^-1/2

Marginal Utility of q2 =dU/dq2 = 1/2q2^-1/2

MRS = MU of q1/MUq2 = q2^1/2/q1^1/2

At equilibrium the consumer equtes MRS with the price ratio,

q2^1/2/q1^1/2 = 2/2

q2^1/2= q1^1/2

q2 = q1

Let M be the income of the consumer, hence the budget constraint of the consumer can be written as:

2q1 + 2q2 = M

substitute q1 = q2

2q1+2q1 = M

q1 = M/4

q2 = M/4

Substitute the value of q1 and q2 in the utility function we get,

U(q1, q2,) =?q1+?q2

U(q1, q2,) =?M/4+?M/4

U(q1, q2,) = ?M