This interpretation o f endogenous bargaining power is consistent with Beghin and Karp’s explanation endogenous bargaining power See Beghin and Karp (1991), footnote 4.

Appendix-6A: Derivative Properties of the Fear Functions

Let us consider Aumann and Kurz's fear of ruin at price Px. Player l's fear of ruin is defined by

(6A.1) _{/ i W =}

n ^ - n f

dnjdP,

’

and player 2's fear of ruin is defined by (6A.2) _{f 2W} n 2(Px) - n d2

dll 2 / dPx

Differentiating equation (6A.1) with respect to the relative price yields

(6A.3)

(noMrij-nf)

V

<j2n,

dP2 \ / Therefore, the sufficient condition for > 0 is that d2

n,

dP2< 0.

Satisfaction of the condition means that player l's fear of ruin increases with Pv Given the differentiability of the rental function, the necessary and sufficient condition for a m

dP, > 0 is that the term in the parentheses on the right be positive. Similarly, differentiating (6A.2) with respect to Px yields

(6 A.4)

dA(P>)

1

dPt

<n')2

i2rr \

(n’2)2-(n 2- n d2)— ±

O/j 4T2(/j) d 2

n.

A sufficient condition for — < 0 is that----t2- < 0. If the condition is satisfied then

dPx dP2

an increase in Px implies a fall in player 2's fear of ruin. That is, at a higher relative price of commodity 1, player 1 becomes more fearful of ruin and player 2 becomes bolder. At a lower relative price of commodity 1, player 1 becomes bolder and player 2 becomes more fearful of ruin.

If the second order derivative of the payoff function with respect to Px is nonpositive, we are done. If this is not the case, that is if d 2^ / dP2 > 0 , then the information is not sufficient to determine the sign of the derivative of the fear of ruin as relative price changes.

The purpose o f this A ppendix is to obtain conditions under w hich >0

and < o when the second order derivatives o f the payoff functions are positive.

If the underlying production functions are continuously differentiable and the elasticities o f factor substitution are finite then the sectoral payoff functions are continuously differentiable in Px. Therefore, under these standard conditions, we can use T aylor series to approxim ate the sectoral rental functions. The series can, then, be used to evaluate the slope o f fear o f ruin functions o f the players.

First, consider the case o f player 1. Since Taylor series can be constructed around any arbitrary value o f P{ provided the derivatives exist, we can choose P* such that IIj (P*) =

nf

. Then, for all P{ such that

n

e SR we have P{ > P*. The second order Taylor series expansion o f the payoff function around P* can be w ritten as

(6A.5) n , (/>) =

nf

+ a(/> - P‘) + ft(f> - />* f .

w here, a = > 0; b = (1 / 2)Yl"(P") > 0 . and the derivative o f the payoff function is given by

(6A.6) dTll / dPx = a + 2b(P{ - P*)

Therefore, the fear function can be w ritten as (6A.7)

_{A W}

- P j ) + b(Pt - P ‘)2

a + 2b(,Pi - P ‘)

D ifferentiating the fear function with respect to Px yields after sim plification

(6 A. 8) 4 f i W _ a2 + 2[b{Px - P /) ] 2 + 2ab(P - / Q . Q

Sim ilarly, we can obtain a T aylor series approxim ation o f the rental income function fo r player 2 by expanding around P* w here n 2( / f ) = I I 2. Then for all Pxsuch that n e /* ) e SR we m ust have Px < P*.

For player 2, we can write

(6A .9)

n2(/>)

= Ud2+ c(/> - P?)+ d(Px - P‘ f

w here, c = YV2{Pxa)< 0 and d = r i ''( />/*)/ 2 If d<0 , then we are done. So suppose that d>0.

Following similar steps we can obtain (6A.10) where, d A W ._ c1 + 2{d(P,-P;)]2 + 2cd(Pl - P : ) dP > [c + 2rf(/> - / > “)]2 [c + 2d(Pt - c t f - P n + d t f - P ' )1 Sl( 1 c + 2 d ( / > - / f )

These results remain valid if we use Taylor series of third order, provided that the third order derivative of the rental function remain nonnegative for player 1 and nonpositive for player 2 at the point of expansion. As we increase the order of expansion, the signs of the derivatives of the fear functions become ambiguous.

Therefore, the sufficient conditions for fear functions to be well behaved are that (i) the elasticities of factor substitution be fmite; and

(ii) either of the following two conditions holds:

(a) the second order derivatives of the rental functions with respective to Px are negative everywhere;

(b) all third and higher order derivatives of the rental functions with respect to the relative price vanish at the point of expansion.

The first condition assures that there are no comer solutions. This condition together with the continuous differentiablity of the output supply functions with respect to price, then, guarantees the differentiablity of the rental functions21 at each price.

The second condition is related to the higher order derivatives of the rental functions. This condition is satisfied if the rental functions, in the relevant range for bargaining, are approximately quadratic (concave or convex) in relative price of either commodity. The presence of nonlinearities makes it is difficult to assess whether the condition (ii) is satisfied or not.

However, the rental functions are bounded and strictly increasing in relative price of own commodity, eventually at higher prices, the second order derivatives of the

21 Recall that rental income, measured in units o f own output, at each {rice is equal to output supply less the cost o f mobile factor.

rental functions with respect to the relative price of own commodity have to be nonpositive.

Condition (i) is normally satisfied. We do not normally expect factors of production to be perfectly substitutable. Condition (ii) may be violated in some cases. Particularly this condition may be violated if the rental functions are wavy, though strictly increasing in relative price of own commodity. If this is so, then the bargaining solution that equalizes generalized fear may not be stable. Player l ’s generalized fear or ruin may fall as the relative price of commodity 1 rises and player 2's generalized fear of ruin may fall as the relative price of commodity 2 rises. Thus, each player may like to put demands that are likely to be incompatible. Hence, we regard condition (ii) as a condition for stability of the bargaining solution. However, in this study, we assume that the conditions (i) and (ii) are satisfied.

Appendix -6B: Proof of Roth's Theorem

We will prove Roth's theorem by way of the following lemma. The proof basically follows Binmore (1987a).

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