Question

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) = xy/(x+y). He has 20 hours of free time to spend on these two activities each week, and his goal is to maximize his utility. What is the optimal amount of time spent surfing the Internet and playing video games each week?

Answer 1

ucary) letne Internet . games . . a Time spent playing on y Time spent playing video If, Total free time = 20 houss 80, aty = 20 Hence, Max ux,y) = a hang subject to Jaty = 20] Setting up Lagrange : 1 DV Od (2.ty).og – ay (1) -= 0 0 ja "Taty) 2 yang = (x+y) •- By (1) - 1-09 on 20-4 = 0

from 6 and (ii x = (atyy - y = (aty) x - xy NE » (x+y) y = (x+y) x + G - 4G » Catyly = betyy x U Putting g = 2 in mi >> 20 - y -a = 0 » 20-2-2 = 0 » 20 = 2x >> F= 20 =10 -> 14 = 2 =10 8 x = 10 and y= 10
Hence, Paul will spend 10 hours in surfing the internet(x) and 10 hours playing video games(y) each week.