A consumer has utility functionUq1 q2 q122q22 income Y 24 a

A consumer has utility functionU(q1, q2) = (q1)^(2)+2(q2)^(2), income Y= 24, and faces prices p1= 1 and p2= 3. Find all consumption bundles that satisfy the necessary condition fora utility maximizing choice. Then determine which of these is optimal.

Solution

The utility function is U(q1, q2) = (q1)^(2) + 2(q2)^(2). The income is Y = 24, and the level of prices are p1= 1 and p2 = 3.

The equation of budget is 24 = q1*1 + 3*q2 OR

24 = q1 + 3q2. At the optimum choice of bundle, MRS = price ratio

-2q1/4q2 = - 1/3

This gives a relation between q1 and q2 as q2 = 1.5q1. This gives the equation for all consumption bundles that satisfy the necessary condition fora utility maximizing choice.

Use this in the budget equation

24 = q1 + 3*1.5q1

24 = 5.5q1

q1* = 4.36 and q2* = 6.54

These is the consumption bundles at the optimum choice.