Answer to Question #222883 in Microeconomics for Ariel

Question

Suppose a consumer’s utility function is given a U = 100X0.25Y0.75.The prices of the two commodities X and Y are Birr 2 and Birr 5 per unit respectively. If the consumer’s income is Birr 280, how many units of each commodity should the consumer buy to maximize his/her utility? 

Answer 1

U = 100X^0.25Y^0.75

Budget constraint is

$2x + 5y = 280$

Maximize utility function

subject to the budget constraint

Max U = 100X^0.25Y^0.75

$2x + 5y ≤ 280$

$x≥0 , y≥0$

We introduce a lagrangian function to our function

$L =$ 100X^0.25Y^0.75 + ƛ ($280- 2x - 5y$ )

FIRST ORDER CONDITIONS

ɚL/ɚX = 25X^-0.75Y^0.75 - ƛ2 = 0 .......EQUATION 1

ɚL/ɚY=75X^0.25Y^-0.25- ƛ5 = 0 ......... EQUATION 2

ɚL/ɚ$ƛ =$ ƛ = 280- 2x - 5y = 0 ..............EQUATION 3

WE COMBINE EQUATION 1 AND 2

 ƛ=(25X^-0.75Y^0.75 ) ÷ 2 = (75X^0.25Y^-0.25 ) ÷ 5

SOLVING THIS WE GET;

5 ÷6 =X ÷ Y

Thus x= 0.833y

y= 1.2x

Using equation three we substitute one of the above rquations

280- 2x - 5y = 0

2x- 5y=280 but x= 0.833y

0.833y x 2 - 5y=280

Solving this we get

y= 168 units

From y= 1.2x we substitute y = 168 and solve to find x

168 = 1.2x

x = 140 units

Therefore the consumer maximizes utility at y= 168 units and x = 140 units