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?