Question

1. The Smith household allocates its monthly grocery budget between vegetables (V) and other food (F).  The Smiths’ preferences are represented by the utility function U(V,F) = V^0.3F^0.7.
   1. What is their marginal rate of substitution of vegetables for other food, MRSvf?
   2. Suppose the Smiths’ grocery budget is $600/month, the price of vegetables is $5 per pound, and the price of other food is $2 per pound. What quantities of F and V will the Smiths choose to consume?

Answer 1

We have a utility function U(V,F) = V^0.3F^0.7.

a) This gives MRS = -MUV/MUF = -0.3V^-0.7F^0.7 / 0.7V^0.3F^-0.3 = -3F/7V. Hence marginal rate of substitution of vegetables for other food, MRSvf = 3F/7V

b) Price ratio is PV/PF = 5/2. At the optimal choice we have MRS = -PV/PF or -3F/7V = 5/2. This gives F = (35/6)V

Use the budget equation to get 600 = VPV + FPF

600 = 5V + 2*(35/6)V

600 = 5V + 35V/3

V = 600*3/50 = 36 and F = 210

Hence optimal choices are F = 210 and V = 36.