Joseph has the utility function U(F,H) = 10F2H, where F is the quantity of food he consumes per year and H is the quantity of housing per week. Suppose the price of food is $10 and the price of housing is $5, while Joseph has an income of $150/week

a. Calculate Joseph's MRS as a function of the quantities F and H.
b. Write out Joseph's constrained optimization problem with the information provide
d.
c. Using the substitution method, solve for Joseph's optimal consumption bundle of food and housing.
d. Show that at the optimum, Joseph consumes the bundle along the budget constraint where MRS = MRT.

a. MRS = -2 ∗ 10FH/10F2 = -2H/F.
b. max 10F2H subject to 10F + 5H = 150
c. Solve the BC for F = 15 - .5H. Substitute this function of F into the utility function to get the unconstrained maximization problem:
max 10(15-.5H)2H
The derivative is
10(15 - .5H)2 + 10 ∗ 2(15 - .5H)(-.5)H = 0
Solving for H = 10. Plug H = 10 into the BC to get F = 10.
d. The MRS = -2H/F. At the optimal bundle from (c), MRS = -2. The MRT = -10/5 = -2. Hence the slope of the BC equals the slope of the IC at the optimal bundle.