A consumer\'s utility function is U = In(xy^4) (a) Find the values of x and y wh

Question

A consumer's utility function is U = In(xy^4) (a) Find the values of x and y which maximise utility subject to the budgetary constraint 10x + 5y = 150 Use the method of substitution to solve this problem. (b) Show that the ratio of marginal utility to price is the same for x and y. (a) x = and y = (Simplify your answers.) (b) The values of the marginal utilities at the optimum are U/x = and U/y = (Give your answers to three decimal places as needed) The ratio of the marginal utilities and the ratio of the prices are both equal to u/y/U/x = p_y/p_x

Explanation / Answer

U =ln (xy^4) ---------------------------(1)

Differentiation of equation 1 w.r.t. X

MUx = dU/dX = (1/XY^4)*(Y^4) = 1/X ---------------(2)

Differentiation of equation 1 w.r.t. Y

MUY = dU/dY = (1/XY^4)*(X*4Y^3) = 4/Y -------------(3)

MUX/MUY = Px/Py

(1/X)/(4/Y) = 10/5 = 2

Y/4X = 2

Y = 8X

150 = 10X + 5Y = 10X+5*8X = 50X

X = 3

Y = 8X

Y= 24

A.

X = 3                      Y = 24

B.

dU/dX = 1/X

dU/dY = 4/Y

C.

Py / Px = 5/10 = 1/2 = (dU/dY) / (dU/dX)

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