Imagine a single consumer with the following very strange pattern of demand for a particular good. This good, which we will call a griffle, can come in any number of varieties. Specifically, any griffle is of a type t for some t E [O, 1]. Our consumer is interested in purchasing at most one and precisely one griffle; he will do so if there is a griffle whose price and type meet a hurdle that he sets. His preferences and demand behavior for griffles are characterized by a function T : [O, 1] -+ (0, oo) as follows. Suppose N -types of griffles are for sale, types t1, t2 , • . • , tN . Suppose that p(tn) is the price of a griffle of type tn. Then if for each n, p(tn) > T(tn), our consumer will not purchase any griffle. On the other hand, if for some tn, p(tn) :::; T(tn), then our consumer will purchase precisely one unit of some type n that maximizes the difference T(tn)-p(tn). Note well: If more than one type n maximizes this difference, our consumer will purchase precisely one unit of some one of the maximands; it is indeterminate which he will purchase.
(a) Suppose that our consumer's demand behavior is characterized by the function T(t) = T0 + kit -t0 1 for some constants To, k , and t0 E [O, 1]. Being very specific, suppose that To = 5, k = 1, and t0 = .6. Suppose three types of griffle are being sold: types .3, .5, and .7 at prices 3, 3.3, and 4, respectively. What will our consumer do (in terms of purchasing griffle) in these circumstances?
(b) Fix the prices of types .3 and .7 at 3 and 4, respectively. Draw this consumer's demand curve for griffie of type .5 as a function of the price of this type of griffie.