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This problem shows that an optimal consumption choice need not be interior and may be at a corner point.) Suppose that a consumer’s utility function is U(x, y) = xy + 10y. The marginal utilities for this utility function are MU_{x}= y and MU_{y}= x + 10. The price of x is P_{x}and the price of y is P_{y}, with both prices positive. The consumer has income I.
a) Assume first that we are at an interior optimum. Show that the demand schedule for x can be written as x = I/(2P_{x}) − 5.
b) Suppose now that I = 100. Since x must never be negative, what is the maximum value of P_{x}for which this consumer would ever purchase any x?
c) Suppose P_{y}= 20 and P_{x}= 20. On a graph illustrating the optimal consumption bundle of x and y, show that since P_{x}exceeds the value you calculated in part (b), this corresponds to a corner point at which the consumer purchases only y. (In fact, the consumer would purchase y = I/P_{y}= 5 units of y and no units of x.)
d) Compare the marginal rate of substitution of x for y with the ratio (P_{x}/P_{y}) at the optimum in part (c). Does this verify that the consumer would reduce utility if she purchased a positive amount of x?
e) Assuming income remains at 100, draw the demand schedule for x for all values of P_{x}. Does its location depend on the value of P_{y}?
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