Exploring the Difference between Willingness to Pay and Willingness to Accept: Suppose you and I are identical in every way €” same exogenous income, same tastes over pizza and €œother goods€. The only difference between us is that I have a coupon that allows the owner of the coupon to buy as much pizza as he/she wants at 50% off.
A: Now suppose you approach me to see if there was any way we could make a deal under which I would sell you my coupon. Below you will explore under what conditions such a deal is possible.
(a) On a graph with pizza on the horizontal axis and €œother goods€ on the vertical, illustrate (as a vertical distance) the most you are willing to pay me for my coupon. Call this amount P.
(b) On a separate but similar graph, illustrate (as a vertical distance) the least I would be willing to accept in cash to give up my coupon. Call this amount R.
(d) Is P larger or smaller than R? What does your answer depend on? (Hint: By overlaying your lower graphs that illustrate P and R as areas along marginal willingness to pay curves, you should be able to tell whether one is bigger than the other or whether they are the same size depending on what kind of good pizza is.)
Graph 10.7: Trading Pizza Coupons

(e) True or False: You and I will be able to make a deal so long as pizza is not a normal good. Explain your answer intuitively.
B: Suppose your and my tastes can be represented by the Cobb-Douglas utility function u(x1, x2) = x10.5 x20.5, and suppose we both have income I = 100. Let pizza be denoted by x1 and €œother goods€ by x2, and let the price of pizza be denoted by p. (Since €œother goods€ are denominated in dollars, the price of x2 is implicitly set to 1.)
(a) Calculate our demand functions for pizza and other goods as a function of p.
(b) Calculate our compensated demand for pizza (x1) and other goods (x2) as a function of p (ignoring for now the existence of a coupon).
(c) Suppose p = 10 and the coupon reduces this price by half (to 5). Assume again that I have a coupon but you do not. How much utility do you and I get when we make optimal decisions?
(d) How much pizza will you consume if you pay me the most you are willing to pay for the coupon? How much will I consume if you pay me the least I am willing to accept?
(e) Calculate the expenditure function for me and you.
(f) Using your answers so far, determine R €” the least I am willing to accept to give up my coupon. Then determine P €” the most you are willing to pay to get a coupon. (Hint: Use your graphs from A (a) to determine the appropriate values to plug into the expenditure function to determine how much income I would have to have to give up my coupon. Once you have done this, you can subtract my actual income I = 100 to determine how much you have to give me to be willing to let go of the coupon. Then do the analogous to determine how much you€™d be willing to pay, this time using your graph from A (b).)
(g) Are we able to make a deal under which I sell you my coupon? Make sense of this given what you found intuitively in part A and given what you know about Cobb-Douglas tastes.
(h) Now suppose our tastes could instead be represented by the utility function u(x1, x2) = 50lnx1+ x2. Using steps similar to what you have just done, calculate again the least I am willing to accept and the most you are willing to pay for the coupon. Explain the intuition behind your answer given what you know about quasilinear tastes.
(i) Can you demonstrate, using the compensated demand functions you calculated for the two types of tastes, that the values for P and R are in fact areas under these functions (as you described in your answer to A(c)?

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