Question

[Utility Maximization] Mary spends her income on housing (H) and food (F). Her utility function is given by: U(H, F) = 3HF − H + F Suppose the price of food is $1 per unit and the price of housing is $2 per unit. Assume her income is $9.

a) Write down Mary’s budget constraint and find the expression for her marginal rate of substitution (MRS(HF)).

b) Assume the optimal choice of (H*,F*) is not a corner solution. Write the equation that represents the tangency between Mary’s budget constraint and her indifference curve.

c) Find the optimal quantity of housing (H*) and all other goods (F*).

Answer 1

(a)

Budget line is given by :

HP_H + FP_F = M where P_H and P_F are price of Housing(H) and Food(F) respectively and M = Income = 9

=> 2H + F = 9--------------Mary's Budget Constraint

(b)

Tangency condition :

In order to maximize utility a consumer chooses that combination where MRS = absolute value of Slope of Budget line.

Absolute value of Slope of Budget line = P_H/P_F = 2/1 = 2

MRS = MU_H/MU_F where MU_H and MU_F are Marginal utility of housing and Food respectively.

Mathematically :

Thus we have :

MRS = absolute value of Slope of Budget line => (3F - 1)/(3H + 1) = 2/1 => 3F - 1 = 6H + 2 => F - 2H = 1 ---------(1)

(c)

So From budget constraint we get :

2H + F = 9 and F - 2H = 1

=> 2H + 2H + 1 = 9 => H = 2 => F = 2*2 + 1 = 5

Hence, Optimal Quantities are : H* = 2 and F* = 5