Question

1. David consumes only ice cream and Dr. Pepper.  At his current consumption bundle his marginal utility from a scoop of ice cream is 8 and his marginal utility from a can of Dr. Pepper is 4. The current price of a scoop of ice cream is 50 cents while the cost of a can of Dr. Pepper is 35 cents.  Is David maximizing his utility? Explain why or why not and, if he is not, explain how he could increase his utility while keeping his total expenditures constant.
2. Sam has $405 each year to spend on Pizza and Tacos.  Pizza costs $3 per slice and Tacos cost $2 per slice.  Sam also has a utility function of U = PT², where P is slices of pizza per year and T is Tacos per year.  Draw Sam’s Budget constraint.  Solve for Sam’s optimal consumption bundle for the year.

Answer 1

Let the ice cream consumption and pepper consumption be denoted by x and y respectively. We have the following information:

MUx = 8, MUy = 4, Px = .50, Py = .35

Assuming both x and y are normal goods, according to the law of equil\-marginal utility, David will maximize his utility when MUx/Px = MUy/Py

Here, MUx/Px = 8/.50 = 16

MUy/Py = 4/.35 = 11.42

Thus, he is not maximizing his utility.

He can maximize his utility by increasing the consumption of Dr. Pepper and decreasing the consumption of ice creams.

2) M = 405, Pp = 3 and Pt = 2

Sam's budget constraint is 3p + 2t = 405

The problem of the consumer is to Max U = pt^2

subject to 3p + 2t = 405

Using Lagrange and solving we get,

Z = pt^2 + L(405 - 3p - 2t)

dZ/dp = t^2 - 3L = 0

dZ/dt = 2pt - 2L = 0

dZ/dL = 405 - 3p - 2t = 0

t/2p = 3/2

or, 2t = 6p

or, t = 3p

Putting this in the last equation,

6p + 3p = 405

or, p = 45 amd t = 135