Question

Ron consumes two goods, X and Y. His utility function is given by U(X,Y) = 44XY. The price of X is $11 a unit; the price of Y is $8 a unit; and Ron has $352 to spend on X and Y.

a. Provide the equation for Ron’s budget line. (Your answer for the budget line should be in the form Y = a – bX, with specific numerical values given for a and b.)

b. Provide the numerical value of Ron’s marginal rate of substitution of X for Y when utility is maximized.

c. Provide the numerical value of Ron’s utility-maximizing amount of X.

d. Provide the numerical value of Ron’s utility-maximizing amount of Y.

e. Provide the numerical value of Ron’s marginal utility of income when utility is maximized.

Answer 1

Given Utility function :- U(x, y) = 44xy

Px = $11; Py = $8. And M(income) = $352

a). Budget line : xPx + yPy = M

x11 + y8 = 352

8y = 352 - 11x

y = 352/8 - 11x/8 = 44 + 11x/8.

b). Marginal Rate of Substitution (MRS) = Marginal Utility of x(MUx)/Marginal Utility of y(MUy)

= 44y/44x = y/x.

We know that In Cobb douglas utility is maximized when MRS = Px/Py So,

MUx/MUy = y/x = Px/ Py

MRS = 11/8

d). From question b). We can conclude that

y/x = 11/8

x = 8y/11

And from budget line 11x + 8y = 352

Now, from both above equations,  

11(8y/11) + 8y = 352

8y + 8y = 352

16y = 352 So, y = 22

c).. And x = 8*22/11 = 16.

So, the optimal bundle(x*, y*) = (16, 22).