Question

Suppose the price of good 1 is ?1 = $6, the price of good 2 is ?2 = $12. Alice has ? = $120 available to spend on these two goods. Alice has utility function ?(?1, ?2) = √?1√?2. a) Write down the optimality condition that must hold at the optimal solution to Alice’s utility maximization problem. b) Find Alice’s marginal utilities for good 1 and for good 2 and find her marginal rate of substitution. c) Find Alice’s demand functions for her optimal quantities to consume of goods 1 and 2 (i.e., find formulas for her optimal values of ?1 and ?2 as functions of only prices and income). d) Given the prices and the money she has available, what is Alice’s optimal consumption of goods 1 and 2?

Answer 1

u = x1^1/2x2^1/2

a) Optimality condition is MRS = price ratio where MRS = MU1/MU2.
So, MU1/MU2 = p1/p2

b) MU1 = ∂u/∂x1 = (1/2)x1^(1/2)-1x2^1/2 = (1/2)x1^-1/2x2^1/2
MU2 = ∂u/∂x2 = (1/2)x2^(1/2)-1x1^1/2 = (1/2)x2^-1/2x1^1/2

MRS = MU1/MU2 = (1/2)x1^-1/2x2^1/2/(1/2)x2^-1/2x1^1/2 = x2^(1/2)+(1/2)/x1^(1/2)+(1/2) = x2/x1

c) Now, x2/x1 = p1/p2
So, x2 = (p1/p2)x1

Budget constraint: p1x1 + p2x2 = m
So, p1x1 + p2(p1/p2)x1 = m
So, p1x1 + p1x1 = 2p1x1 = m
So, x1 = m/2p1

x2 = (p1/p2)x1 = (p1/p2)(m/2p1)
So, x2 = m/2p2

d) x1 = m/2p1 = 120/(2*6) = 120/12 = 10
x2 = m/2p2 = 120/(2*12) = 5