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Hans has $27 which he decides to spend on x and y. Commodity x costs $16 per unit and commodity y costs $10 per unit. He has the utility function U(x, y) = 5x^2 + 2y^2 and he can purchase fractional units of x and y. Hans will choose A. only x. only y. B. some of each commodity but more y than x. C. some of each commodity but more x than y. D. equal amounts of the two commodities.

Respuesta :

Hans has $27 which he decides to spend on x and y. Commodity x costs $16 per unit and commodity y costs $10 per unit. He has the utility function U(x, y) = 5x^2 + 2y^2 and he can purchase fractional units of x and y. Hans will choose (A)will only buy 1 unit of  x.

Explanation:

Given  the utility function U(x, y) = 5x^2 + 2y^2 a

By Applying  Lagrange  we get

L(x,y,µ)=5x²+2y²+µ(16x+10y-27)

dL/dx=10x+µ16=0

dL/dy=4y+µ10=0

dL/µ=16x+10y-27=0 ---------->(iii)

From one and two equations we get

25x=16y

x=0.64y

Substituting the value of x in  the third equation  we get

16( 0.64y)+10y-27=0

Further simplifying we get y

20.24 y=27

y=27/20.24=1,3 units

16x+10y=27

16 x+13-27=0

16 x=14

x=14/16=0.8 units that do not reach one unit

Hans will choose (A)will only buy 1 unit of  x.

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