Question

Suppose there are two goods, X and Y.  The price of good X is $2 per unit and the price of good Y is $3 per unit.  A given consumer with an income of $300 has the following utility function:

U(X,Y) = X^0.8Y^0.2

        which yields marginal utilities of:

MU_X= 0.8X^-0.2Y^0.2

MU_Y= 0.2X^0.8Y^-0.8

        a.     What is the equation for this consumer’s budget constraint in terms of X and Y?

        b.    What is the equation for this consumer’s marginal rate of substitution (MRS_XY)?  Simplifyso you only have one X term and one Y term.

        c.     Suppose this consumer is exhausting his income by buying X and Y in such a way that his MRS_XY= 2.6. Given the prices above, should this consumer buy more X and less Y or more Y and less X?  Explain briefly.

        d.    Solve for the utility maximizing combination of X and Y for this consumer.

               Remember that you have to first set the slope of the budget constraint equal to the slope of the indifference curve, then solve for X or Y in terms of the other.  Substitute that into the budget constraint and solve for the one variable you have. Then use that value in your first equation to solve for the other.

Answer 1

a. Budget constraint: Px*X + Py*Y = Income
So, 2X + 3Y = 300

b. MRSXY = MUX/MUY = (0.8X-0.2Y0.2)/(0.2X0.8Y-0.8) = 4Y0.2+0.8/X0.8+0.2 = 4Y/X

c. MRSXY = 2.6
Px/Py = 2/3 = 0.67
So, MRS > Px/Py. Thus, he should buy more X and less Y because utility is maximized when MRSXY = Px/Py. When he buys more of X and less of Y, his MUX will decrease and MUY will increase which will decrease MRSXY until it equals PX/PY.

d. Utility is maximized when MRSXY = Px/Py
So, 4Y/X = 2/3
So, X = 4Y*(3/2) = 6Y

2X + 3Y = 300
So, 2(6Y) + 3Y = 300
So, 12Y + 3Y = 15Y = 300
So, Y = 300/15
Thus, Y = 20

X = 6Y = 6*20 = 120
So, X = 120