A college student is considering whether to operate a lawn-mowing business for the summer or work in a business owned by his family. His time is worth $w1 per our and he can work as many hours as he chooses in the family business at this rate. If he starts his own business, he will have to buy gasoline for his lawn mower at a price of $w2 per gallon. He can rent a small mower for $w3 per hour. The mower uses 1/3 gallon of gasoline per hour. With this mower, he can cut 1 unit of lawn in an hour. The college student can rent a large mower for $w4 per hour. This mower uses 1 gallon of gasoline per hour. He can cut 3 units of lawn per hour with the large mower.

(a) Let X1 denote the hours of labor, X2 gallons of gasoline, and let X3 and X4 denote the hours of the small mower and the large mower, respectively. Show that the production functions with the two mowers are as follows: f(X1, X2, X3) = min{X1, 3*X2, X3) f(X1, X2, X4) = 3*min{X1, X2, X4}

(b) Derive the cost functons.

(c) Show that using the small mower is a cheaper way to cut grass if 2*w1 < w4 - 3*w3

(d) How high a price must the college student receive for cutting a unit of lawn, in order for setting up his own lawn-mowing firm to be better than working in the family business for him?