Question

(12 Points) Suppose Paul's utility depends on the amount of time spent playing on the Internet (x) and the amount of time playing video games (y), and his utility function is U(x, y)-3x°2y°8. He has 8 hours of free time to spend on these two activities each week, and his goal is to maximize his utility. The price he pays for each activity is one. Draw the graphical constrained maximization problem if Paul considers Internet and Video games as imperfect substitutes. Set up the Lagrangian for this constrained maximization problem What are the necessary conditions for the optimum from the Lagrangian? What is the optimal amount of time spent surfing the Internet and playing video games each week? a. b. c. d.

Answer 1

The utility is given as $\dpi{80} U = 3x^{0.2}y^{0.8}$ . As he have 8 hours of time, the constraint would be $\dpi{80} x+y=8$ , since the price he pays is 1.

(a) The graph is as below (Pauls utility is $\dpi{80} U = 3x^{0.2}y^{0.8}$ , and this means he consider x and y as imperfect substitutes).

(b) The Lagrangian function would be $\dpi{80} L = 3x^{0.2}y^{0.8} + \lambda (8-x-y)$ .

(c) The FOCs are as below.

$\dpi{80} \frac{\partial L}{\partial x} = 0$ or $\dpi{80} \frac{\partial }{\partial x}(3x^{0.2}y^{0.8} + \lambda (8-x-y)) = 0$ or $\dpi{80} 3*0.2x^{-0.8}y^{0.8} + \lambda (-1) = 0$ or $\dpi{80} 0.6x^{-0.8}y^{0.8} = \lambda$ .

$\dpi{80} \frac{\partial L}{\partial y} = 0$ or $\dpi{80} \frac{\partial }{\partial y}(3x^{0.2}y^{0.8} + \lambda (8-x-y)) = 0$ or $\dpi{80} 3*0.8x^{0.2}y^{-0.2} + \lambda (-1) = 0$ or $\dpi{80} 2.4x^{0.2}y^{-0.2} = \lambda$ .

$\dpi{80} \frac{\partial L}{\partial \lambda} = 0$ or $\dpi{80} \frac{\partial }{\partial \lambda}(3x^{0.2}y^{0.8} + \lambda (8-x-y)) = 0$ or $\dpi{80} (8-x-y) = 0$ or $\dpi{80} x+y=8$ .

Comparing first two FOCs, we have $\dpi{80} 0.6x^{-0.8}y^{0.8} = 2.4x^{0.2}y^{-0.2}$ or $\dpi{80} y = 4x$ , which is the necessary condition, or the utility maximizing combination of goods.

(d) Putting $\dpi{80} y = 4x$ in the constraint, we have $\dpi{80} x+4x=8$ or $\dpi{80} x^* = 8/5 = 1.6$ hours. As $\dpi{80} y^* = 4x^*$ , we have $\dpi{80} y^* = 4*1.6 = 6.4$ . Hence, optimal amount of internet time would be 1.6 hours or 1 hour 36 minutes, while optimal amount of video game time would be 6.4 hours or 6 hours 24 minutes.