Answers
The utility function is 
, while the budget constraint would be 
. The Lagrangian function to maximize the utility subjected to the constraint would be as 
. The FOCs would be as below.
or 
or 
or 
or 
or 
.
or 
or 
or 
or 
or 
.
or 
or 
or .
Comparing the first two FOCs, we have 
or 
or 
, and since x and y can not be negative, we have 
as the required condition for utility maximizing combination of x and y.
Putting this in the constraint, we have 
or 
or 
, and since , we have 
or 
.
Hence, the optimal amount if time spent will be 10 hours of internet surfing and 10 hours of gaming.
h + -= LU
r+y = 20
L= 19 1 + y + (20-r-y)
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V 9 + (20-r-y)) = 0 2 + y
19 r+ y a +w21(-1) = 0
y(1+y) (x + y)2 ( ry + y)2
ry + y (x + y)2 ry (+y)
(もっ
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a ry +(20-1-y)) = 0
LU r+y (r + y)2 12 + (-1)=0
r(r + y) (x + y)2 ry + y)2 (
r+ rg ( + y )2 ᎢᏴ {r +1 )2-
(r - 2 =A
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0 0 + 2x (20-r-y)) = 0 + y
20-r- y) = 0
r+y = 20
(r + y)2 (r + y)2
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r+ (r) = 20
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.
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