9781118041581

Continuous Variables

In the earlier procurement example, identifying an efficient agreement was

made easier by the limited number of distinct contract alternatives. In the next

example, the two issues at stake can be varied continuously. Nonetheless, the

same principles apply in finding an efficient agreement.

A QUANTITY-PRICE CONTRACT A buyer and seller are negotiating the terms

of a delivery contract specifying price and output quantity (Q). The buyer’s

total value from purchasing Q units is B 3Q Q^2 /20. The seller’s cost of pro-

ducing Q units is C Q^2 /40. The parties seek an agreement as to the quantity,

Q, and the total payment from buyer to seller (call this R). What order quan-

tity is part of an efficient agreement?

A direct way to characterize an efficient agreement is to find the value-

maximizing order quantity. The sum of buyer and seller profits is

Total net benefit (B C) is maximized by setting marginal benefit equal to

marginal cost: MB dB/dQ  3 Q/10 and MC dC/dQ Q/20. Setting

these equal to each other gives Q 20. At this quantity, the buyer’s benefit is

B 3(20)  202 /20 40, and the seller’s cost is C  202 /40 10. The rele-

vant negotiation region for the payment, R, is the range between 10 and 40, and

the maximum total profit is 40  10 30. (This assumes each party faces a

zero profit from a disagreement; i.e., each has no other profitable alternative.)

A graphical analysis provides additional insight into the meaning of effi-

ciency when continuous variables are the object of negotiation. In Figure 15.2,

the axes list the variables, Q and R. Thus, any point on the graph represents

possible terms of an agreement. The next step is to show the profit implica-

tions of any agreement. This is done by means of profit contours, the series of

curves in the figure.^6 The black curves show the seller’s profit contours; the

colored curves are the buyer’s. For instance, the lowest seller contour (marked

S0) shows all combinations of Q and R that provide exactly a zero profit.

This is identical to the firm’s cost curve: R C Q^2 /40. The curve is upward

sloping; to maintain a zero profit, the firm must receive a higher R for pro-

ducing a larger Q. The next highest contour (S7, only part of which is

shown) shows Q and R combinations yielding a profit of 7. In general, higher

profit contours are simply vertical displacements of lower ones. The seller’s

(BR)(RC)BC.

646 Chapter 15 Bargaining and Negotiation

(^6) In a great many economic settings, a slightly different terminology is used. Figure 15.2 often is called
an Edgeworth box,and the contours are called indifference curves.For instance, we examined an indi-
vidual’s indifference curves in the appendix to Chapter 3. When the individual gains from an increase
in either variable, the indifference curves will be downward sloping. (To leave the individual indif-
ferent, a reduction in one variable must be compensated by an appropriate increase in the other.)
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