5.16. (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 MUx ! y and MUy ! x $ 10. The price of x is Px and the price of y is Py, 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!(2Px) % 5.

b) Suppose now that I ! 100. Since x must never be negative, what is the maximum value of Px for which this consumer would ever purchase any x?

c) Suppose Py ! 20 and Px ! 20. On a graph illustrating the optimal consumption bundle of x and y, show that since Px 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!Py ! 5 units of y and no units of x.)

d) Compare the marginal rate of substitution of x for y with the ratio (Px !Py) 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 Px. Does its location depend on the value of Py?