Question

Question 8: Consider a utility function u(xi , X2)- + X2. 1. What is the optimal bundle with p P,and income m? 2. (optional) Would the corner solution where all income is spent on single good be possible?

Answer 1

u = (x1)^0.5 + (x2)^0.5

(1)

Budget line: m = p1.x1 + p2.x2

Utility is maximized when MU1/MU2 = p1/p2

MU1 = $\partial$ u/$\partial$x1 = 0.5 / (x1)^0.5

MU2 = $\partial$ u/$\partial$x2 = 0.5 / (x2)^0.5

MU1/MU2 = [0.5 / (x1)^0.5] / [0.5 / (x2)^0.5] = (x2/x1)^0.5 = p1/p2

Squaring,

x2/x1 = p1² / p2²

x2 = x1.(p1² / p2²)

Substituting in budget line,

m = p1.x1 + p2.[x1.(p1² / p2²)]

m = p1.x1 + [x1.(p1² / p2)]

m = x1.[p1 + (p1² / p2)]

m = x1.[(p1.p2 + p1²) / p2)]

x1 = (m.p2) / (p1.p2 + p1²)

x2 = [(p1² / p2²).(m.p2)] / (p1.p2 + p1²) = (m.p1) / (p1.p2 + p2²)

(2)

If all income is spent on one good, either x1 is zero (everything is spent on x2) or x2 is zero (everything is spent on x1). Since demand function of both goods are independent of the amount of the goods, a corner solution is possible.