Question

Malachi only consumes 2 goods: DVD rentals and coffee. His utility function is

U(R. C) = R0.75 0.25

where is the number of rentals and is cups of coffee. Malachi has $16 in his pocket right now, and he plans to spend all of the $16 on DVD rentals and coffee today.

b) Suppose that Malachi signs up for a membership that reduces the price of a rental to $2, while leaving his income unchanged.

Find the substitution effect, income effect and total effect of the decrease in the price of video rentals on Malachi's consumption of rentals and coffee.

Answer 1

Solution: To determine the substitution effect, we will solve the problem of Malachi's expenditure-minimization using the lower price of rentals with the original utility as the constraint. Malachi's original utility is Ū= 30.7520.25 2.71 Thus, Malachi's expenditure-minimization problem is min 2R+ 2C s.t. 2.71 = R0.75C0.25 & as a Lagrangian: min L(R,C,X) = 2R+ 2C + X(2.71 – R0.75 0.25) FOC: 04 = 2 – 10.75R-0,25 0.25 = 0 R.C R.C. or, c = 2 – 10.25R0.75C-0.75 = 0 or, OL = 2.71 – Rº.75(0.25 = 0 From the the first two conditions we will solve R(C) (0.5)0.75R-0.25 0.25 = (0.5)0.25R0.75 -0.75 0.375C0.25 0.75 = 0.125R0.75 R0.25 R=30 Plugging into the utility constraint and determing C* and R*: 2.71 = R0.75C0.25 = (3C)0.75 0.25 30.75C = 2.71 C* 1.2 R* = 3C* 3(1.2) 3.6 Thus we can conclude that the substitution effect of the fall in the price of rentals would be to consume 0.6 more rentals 0.6 =3.6 - 3) and 0.8 (2 - 1.2 =-0.8) fewer cups of coffee To know the income effect, we have to determine the optimal bundle at the new price of rentals and the original income. We can apply again the generic solution to the Cobb-Douglas utility function with the rentals price equal to $2 per gallon:

(a)l (0.75)16 12 R = Pe="2"= 2 = 6 (1 - a) (0.25)16 4 P = =-= 2 Thus, the income effect of the fall in the rental prices is to consume 2.4 more rentals (= 6 - 3.6 = 2.4) & 0.8 +2 -1.2 = 0.8) more cups of coffee. The total effect would be consuming 3= 6 - 3 = 3) more rentals and the same number of coffees (as there would be an expectation from the Cobb-Douglas utility function).