Respuesta :
Answer:
x = 250
y = 125
u(x,y) = 3125500
Step-by-step explanation:
As given,
The utility function u(x, y) = 100xy + x + 2y
[tex]P_{x}[/tex] = 2 , [tex]P_{y}[/tex] = 4
Now,
Budget constraint -
[tex]P_{x}[/tex] x + [tex]P_{y}[/tex] y = 1000
⇒2x + 4y = 1000
So,
Let v(x, y) = 2x + 4y - 1000
Now,
By Lagrange Multiplier
Δu = Δv
⇒< 100y + 1, 100x + 2 > = < 2, 4 >
By comparing, e get
100y + 1 = 2 ........(1)
100x + 2 = 4 .........(2)
Divide equation (2) to equation (1) , we get
[tex]\frac{100y + 1}{100x + 2} = \frac{1}{2}[/tex]
⇒2(100y+1) = 1(100x+2)
⇒200y + 2 = 100x + 2
⇒200y = 100x
⇒2y = x
Now,
As 2x + 4y = 1000
⇒2x + 2(2y) = 1000
⇒2x + 2x = 1000
⇒4x = 1000
⇒x = 250
Now,
As 2y = x
⇒2y = 250
⇒y = [tex]\frac{250}{2}[/tex] = 125
∴ we get
x = 250
y = 125
Now,
u(250, 125) = 100(250)(125) + 250 + 2(125)
= 3125000 + 250 + 250
= 3125000 + 500
= 3125500
⇒u(250, 125) = 3125500
The maximized value of the utility function is $3125500
The utility function is given as:
[tex]U(x,y) = 100xy + x + 2y[/tex]
The prices per unit are also given as:
[tex]P_x = 2[/tex]
[tex]P_y = 4[/tex]
When a person receives $1000, then the budget function is:
[tex]2x + 4y = 1000[/tex]
Divide through by 2
[tex]x + 2y = 500[/tex]
Differentiate the utility function with respect to x and y
[tex]U'(x) = 100y + 1[/tex]
[tex]U'(y) = 100x + 2[/tex]
So, we have:
[tex]U'(x) = P_x[/tex] and [tex]U'(y) = P_y[/tex]
The above equations become
[tex]100y + 1 = 2[/tex] and [tex]100x + 2 = 4[/tex]
Divide both equations
[tex]\frac{100y + 1}{100x + 2} = \frac 24[/tex]
Reduce the fractions
[tex]\frac{100y + 1}{100x + 2} = \frac 12[/tex]
Cross multiply
[tex]100x + 2 = 200y + 2[/tex]
Subtract 2 from both sides
[tex]100x = 200y[/tex]
Divide both sides by 100
[tex]x = 2y[/tex]
Recall that:
[tex]x + 2y = 500[/tex]
So, we have:
[tex]2y + 2y = 500[/tex]
[tex]4y=500[/tex]
Divide through by 4
[tex]y=125[/tex]
Recall that:
[tex]x = 2y[/tex]
So, we have:
[tex]x = 2 \times 125[/tex]
[tex]x = 250[/tex]
Also, we have:
[tex]U(x,y) = 100xy + x + 2y[/tex]
Substitute the values of x and y, in the above function
[tex]U(250,125) = 100 \times 250 \times 125 + 250 + 2 \times 125[/tex]
[tex]U(250,125) = 3125500[/tex]
Hence, the maximized value of the utility function is 3125500
Read more about utility functions at:
https://brainly.com/question/24922430
