Question

_{Question 1. Suppose that Bruce cares only about barbecued chicken (B) and fish (F). His utility function is u = 30.470.6. The price of barbecued chicken is $5, and the price of fish is $10. Bruce has a budget of $100. (a) Write down the consumers maximization problem. (b) What are the consumption quantities of B and F that will maximize Bruce's utility?}

Answer 1

a)

Given U=B^0.4F^0.6

Budget constraint is given by

5*B+10*F=100

or 100-5B-10F=0

So,

Maximize U=B^0.4F^0.6

Subject to

100-5B-10F=0

Lagrange is given by

L=B^0.4F^0.6+$\lambda$*(100-5B-10F)

b)

First order conditions give us

dL/dB=0.4B^-0.6F^0.6-5$\lambda$=0

or 0.4B^-0.6F^0.6=5$\lambda$ -----(1)

dL/dF=0.6B^0.4F^-0.4-10$\lambda$=0

or 0.6B^0.4F^-0.4=10$\lambda$ -----(2)

(100-5B-10F)=0 -----(3)

Divide equation (1) by equation (2), we get

(0.4B^-0.6F^0.6)/(0.6B^0.4F^-0.4)=5/10

(2/3)*(F/B)=1/2

or F=0.75*B

Set F=0.75B in equation 3, we get

100-5B-10*0.75B=0

100-12.5B=0

B=8 -----(optimal consumption of B)

F=0.75*B=0.75*8=6 -----(optimal consumption of F)