Question

A consumer has utility for protein bars and vitamin water summarized by the Cobb-Douglas utility function U(qB,qW) = qBqW.

a. Show that the consumer’s MRS at a generic bundle (qB,qW) is MRS = - MUB/MUW = - qW/qB.

b. Show that the consumer’s MRS would equally be - qW/qB. if the consumer’s utility function was V(qB,qW) = qB0.5qW0.5.

c. Find the consumer’s Marshallian demand for protein bars when M = 100 and Pw = 1.

d. Is the demand function you found in part c) elastic, inelastic, unit-elastic? Does the price elasticity of the demand function you found in part c) depend on the current combination of price and quantity?

e. Find the consumer’s Engel curve for vitamin water when PB = PW = 1.

f. What is the consumer’s optimal bundle when M = 100 and PB = PW = 1?

Suppose the price of protein bars increases to P’B = 2.

g. Find the new optimal bundle.

h. Find the substitution effect of the price increase on purchases of protein bars using the Slutsky method.

i. Find the substitution effect of the price increase on purchases of protein bars using the Hicks method.

Answer 1

U = qb.qw

a)

MUq_b =

MUq_w =

MRS = -MU_qb/MU_qw = -qw/qb

B)

U = qb^0.5.qw^0.5

MU_qb =

MU_{qw =}

MRS = -MU_qb/MU_qw = -0.5qb^-0.5qw^0.5/0.5qb^0.5qw^-0.5 = -qw/qb

C)

At consumer equilibrium

MU_qb/MU_qw = -Pb/Pw

-qw/qb = -pb/1, qw = qb*Pb

The budget constraint of the consumer can be written as:

qb*Pb + qw*Pw = m, where m is income

qb*Pb + qw = 100

Put qw = qb*Pb

qb*Pb +qb*Pb = 100

2qb*Pb = 100

qb*Pb = 50

qb* = 50/Pb

D)

Elasticity of demand =

Elasticity = -50/Pb²*(Pb/50/Pb) (Put qb = 50/Pb)

= (-50/Pb²)(Pb²/50)

= -1. Hence the price elasticity is unit elastic. The elasticity has absolute value of 1 and thus does not depends on price or quantity.