Mathematics 1 Deneme Sınavı Sorusu #1233374
The utility function of a consumer is defined by U(x,y)=x0.5y0.5 who is constrained by the budget 120=2x+4y (He has 120 liras to spend on x and y, whose prices are 2 and 4 liras respectively). What are the amounts of (x,y) that maximizes his utility given that he spends all his budget?
|
(30,20) |
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(30,15) |
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(60,0) |
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(0,30) |
|
(40,10) |
We can decrease the number of unknown variables to one by substitution.
Since 120=2x+4y, y=(120-2x)/4=30-0.5x
Then we can rewrite his utility function as U(x,y)=x0.5y0.5=U(x)=x0.5(30-0.5x)0.5=(x(30-0.5x))0.5=(30x-0.5x2)0.5
For the maximum utility the first derivative of utility with respect to x must be equal to zero. Thus:
Ux=0.5(30-x)(30x-0.5x2)-0.5=0
30-x=0, which means x=30. Substituting this in y=30-0.5x=15. Thus the utility maximizing combination of (x,y)=(30,15)
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