i 0 4 � represents the elasticity of the price function Until now I assumed η

I 0 4 ? represents the elasticity of the price

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i > 0 (4) η represents the elasticity of the price function. Until now I assumed η always being 0,5. However, 0,5 is an estimated average value (Dolan, 1987) and in practice η may vary. For values of η between -1 and 1 the following function applies (see also Figure 2): 1 1 0 FFR FFR (5) Working Paper 981
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U NFAIR ALLOCATION OF GAINS UNDER EQUAL PRICE IN COOPERATIVE PURCHASING Figure 2: Dependency of FFR to η (for all η the maximum added value is reached by q=100%) With an elasticity of 1,0 and a corresponding FFR of 0%, all organisations increasing their volume through the consortium will receive fewer gains. The smallest organisation will always receive the largest part of the gains. The largest organisation will always receive the smallest part of the gains. The 25% rule The 38% rule applies to organisations increasing or decreasing their cooperative volume. Now I consider the situation where the total volume of a consortium is fixed. Figure 3 illustrates this scenario for different quantities of organisation 2 . Figure 3: Type of gains for organisation 2 with different quantities of q 2 while N is constant 14 th Annual IPSERA 2005 Conference, Archamps, France 982 equalprice i equalprice i + n i = M i
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S CHOTANUS F. At the point where q 2 becomes 50% of the total volume, the added value of this organisation reaches its maximum value. At the point where q 2 becomes 25% of the total volume, the EP- outcome for organisation 2 already reaches its maximum. With theorem and proof 3 I prove that this is always the case in the model. Again, this percentage is independent of p o , c 1 and c 2 in the price structure, the number of organisations, and the division of the volumes of these organisations. To conclude, when using EP and assuming a continuous price function, organisations purchasing 25% of the total volume will receive the maximum allocation of gains. Larger or smaller members will receive a smaller amount of gains. I define 25% as the Second Fairness Ratio (SFR) of EP with an average price function. Theorem 3 While using the EP concept and given the price function (2), consortium members purchasing 25% of the total volume will receive the maximum allocation of gains. Proof 0 2 0 2 · ( ( ) - ( )) = · - i i i i i equalprice v q price q price N p c q p c q N Here, N fixed and, 0 2 0 2 ' - 0 2 1 1 - 0 2 4 25% i i i i i p c p c equalprice v q N q N N q if q SFR N then SFR Once more, the dependent variable in this proof is η . If η = -1, SFR = 50%. This is a fair situation as SFR equals the point where the added value reaches its maximum (50%). When η > -1, SFR < 50%, this could lead to an unfair situation as SFR reaches its maximum before the added value does. For SFR the following function applies (see also Figure 4): 1 1 SFR FFR (6) Working Paper 983
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U NFAIR ALLOCATION OF GAINS UNDER EQUAL PRICE IN COOPERATIVE PURCHASING Figure 4: D ependency of SFR to η (for all η the maximum added value is reached by q=50%) Limitations and further research
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