The Generalized Nash Bargaining Solution for Transfer Price Negotiations Under Incomplete Information

Abstract

In our model two divisions negotiate over type-dependent contracts to determine an intrafirm transfer price for an intermediate product. Since the upstream division’s (seller’s) costs and downstream division’s (buyer’s) revenues are supposed to be private information, we formally consider cooperative bargaining problems under incomplete information. This means that the two divisions consider allocations of expected utility generated by mechanisms that satisfy (interim) individual rationality, incentive compatibility and/or ex post efficiency. Assuming two possible types for buyer and seller each, we first establish that the bargaining problem is regular, regardless whether or not incentive and/or efficiency constraints are imposed. This allows us to apply the generalized Nash bargaining solution to determine fair transfer payments and transfer quantities. In particular, the generalized Nash bargaining solution tries to balance divisional profits, while incentive constraints are still in place. In that sense a fair profit division is generated. Furthermore, by means of illustrative examples we derive general properties of this solution for the transfer pricing problem and compare the model developed here with the models existing in the literature. We demonstrate that there is a tradeoff between ex post efficiency and fairness.

Appendices

Appendix

A Proofs of Propositions

Proof

(Proof of Proposition 2) The proof is analogous to the proof of Proposition 1 by Matsuo (1989). $\square$

Proof

(Proof of Proposition 3) We show regularity in the three cases by defining a mechanism that satisfies the IR constraints with strict inequality and obey the remaining restrictions.

Case 1 In the presence of IR and IC constraints, consider the mechanism $\mu ^{(Y,Q)}$ with

$$\begin{aligned} (Y_{HH},Q_{HH})&=\left( \frac{3(1-\varepsilon ) Y_{LL}+\delta R_H-(1-\varepsilon )(1+\delta )C_L+\delta (1-\varepsilon )C_H}{3\delta },\frac{1}{3}\right) ,\\ (Y_{LH},Q_{LH})&=\left( \frac{3(\delta -\varepsilon ) Y_{LL}+\delta R_H+\varepsilon (1+\delta )C_L+\delta (1-\varepsilon )C_H}{3\delta },1\right) ,\\ (Y_{HL},Q_{HL})&=\left( 0,0\right) ,\\ (Y_{LL},Q_{LL})&=\left( \frac{R_L+C_L-2\delta \left( R_H-C_H\right) -2\delta \varepsilon \left( C_H-C_L\right) }{6},\frac{1}{3}\right) . \end{aligned}$$

Case 2.1 In the presence of IR, IC, and EPE constraints and additionally inequality (1) is strict, implying $\varepsilon \delta R_H +(1-\varepsilon )R_L > \delta C_H + (1-\delta )(1-\varepsilon )C_L$, use the following mechanism $\mu ^{(Y,Q)}$ with

$$\begin{aligned} (Y_{HH},Q_{HH})&=\left( \frac{(1-\varepsilon )Y_{LL}+\varepsilon \delta R_H-(1-\delta )(1-\varepsilon )C_L}{\delta },1\right) ,\\ (Y_{LH},Q_{LH})&=\left( \frac{(\delta -\varepsilon )Y_{LL}+\varepsilon \delta R_H+(1-\delta )\varepsilon C_L}{\delta },1\right) ,\\ (Y_{HL},Q_{HL})&=\left( 0,0\right) ,\\ (Y_{LL},Q_{LL})&=\left( \frac{(1-\delta )(1-\varepsilon ) C_L+\delta C_H-\varepsilon \delta R_H+(1-\varepsilon ) R_L}{2(1-\varepsilon )},1\right) . \end{aligned}$$

Case 2.2 In the presence of IR, IC, and EPE constraints and additionally inequality (1) is not strict, implying $\varepsilon \delta R_H +(1-\varepsilon )R_L \le \delta C_H + (1-\delta )(1-\varepsilon )C_L$, we show that no strictly individually rational mechanism exists. To see this we first add constraint (IC2) multiplied by $(1-\varepsilon )$ to constraint (IC3) multiplied by $\delta$ and obtain:

$$\begin{aligned}&(1-\delta )(1-\varepsilon ) Y_{LL}-(1-\varepsilon )C_L+\delta R_H -\varepsilon \delta Y_{HH}\\&\quad \ge (1-\varepsilon )\delta Y_{HH}-(1-\varepsilon ) \delta Y_{LL}- (1-\varepsilon )\delta C_L+(1-\varepsilon ) \delta R_H. \end{aligned}$$

Rearranging yields

$$\begin{aligned} (1-\varepsilon ) Y_{LL}-(1-\varepsilon )(1-\delta )C_L+\varepsilon \delta R_H -\delta Y_{HH}\ge 0. \end{aligned}$$

If constraint (IR1) holds with strict inequality, we have $Y_{HH}>C_H$. Hence, we have

$$\begin{aligned} 0&\le (1-\varepsilon ) Y_{LL}-(1-\varepsilon )(1-\delta )C_L+\varepsilon \delta R_H -\delta Y_{HH}\\&<(1-\varepsilon ) Y_{LL}-(1-\varepsilon )(1-\delta )C_L+\varepsilon \delta R_H -\delta C_H\\&\le (1-\varepsilon ) (Y_{LL}-R_L). \end{aligned}$$

Therefore, $Y_{LL}>R_L$, which contradicts the strict inequality of constraint (IR4).

Case 3 In the presence of IR and EPE constraints, consider the mechanism defined in Proposition 4, which we will not repeat here.

In the three cases, 1, 2.1 and 3, tedious, yet straightforward calculations show that the given mechanism is strictly individually rational.¹⁴ Therefore, Theorem 3 in Myerson (1979) for regular bargaining problems can be applied showing the existence and uniqueness of the generalized Nash bargaining solution in terms of the agents’ expected utilities. $\square$

Proof

(Proof of Proposition 4) To prove Proposition 4, we make use of a technical lemma that can be found Appendix C. We may apply Lemma 1 to the transfer pricing game f being the generalized Nash product, where the $x_i$’s are the agents’ conditional utilities resulting from the mechanism $\mu ^{(Y,Q)}$ and the $l_i$’s are the type probabilities. The constraint in the lemma is satisfied with

$$\begin{aligned} c:=\delta R_H+(1-\delta ) (1-\varepsilon )R_L- \varepsilon \delta C_H-(1-\varepsilon ) C_L, \end{aligned}$$

(3)

since

$$\begin{aligned}&\varepsilon (U_1(\mu ^{(Y,Q)}|H))+ (1-\varepsilon ) (U_1(\mu ^{(Y,Q)}|L)) +\delta (U_2(\mu ^{(Y,Q)}|H))\nonumber \\&\qquad + (1-\delta ) (U_2(\mu ^{(Y,Q)}|L)) \nonumber \\&\quad = \varepsilon \delta (Y_{HH}-Q_{HH}C_H)\nonumber \\&\qquad + (1-\varepsilon ) \left( \delta (Y_{LH}-Q_{LH}C_L) + (1-\delta ) (Y_{LL}-Q_{LL}C_L) \right) \nonumber \\&\qquad + \delta \left( \varepsilon (Q_{HH}R_H-Y_{HH})+(1-\varepsilon )(Q_{LH}R_H-Y_{LH})\right) \nonumber \\&\qquad + (1-\delta ) (1-\varepsilon ) (Q_{LL}R_L-Y_{LL}) \nonumber \\&\quad = \varepsilon \delta Q_{HH}(R_H-C_H)+(1-\varepsilon )\delta Q_{LH}(R_H-C_L) \nonumber \\&\qquad +(1-\varepsilon )(1-\delta ) Q_{LL}(R_L-C_L) \nonumber \\&\quad \le \delta R_H+(1-\delta ) (1-\varepsilon )R_L- \varepsilon \delta C_H-(1-\varepsilon ) C_L \quad = \quad c \end{aligned}$$

(4)

is bounded by c. Actually, (4) holds because all transfer quantities are no greater than 1. Therefore, it holds with equality for EPE mechanisms. Moreover, the variables (expected utilities) are assumed to be nonnegative. It follows that the domain of the maximization problem includes all EPE and IR mechanisms. From Lemma 1, we know that the optimal solution exhibits the same coordinates, meaning that all conditional utilities are equal. This constitutes a system of linear equations. Straightforward calculations show that $\mu ^{*(Y,Q)}$ is a solution to that system. Now, as $\mu ^{*(Y,Q)}$ is IR and EPE, and is a maximizer of the maximization problem, it must be the generalized Nash bargaining solution when IR and EPE constraints are active. $\square$

Proof

(Proof of Proposition 5) To show (i), we demonstrate that $Q_{LH}<1$ leaves some room for a Pareto improvement. Since for any transfer payment between $R_H$ and $C_L$ both the selling and the buying division are always willing to trade, we further specify that $Q_{LH}$ needs to be 1.

Consider a mechanism $\mu ^{(Y,Q)}$ with $Q_{LH}<1$ and define a new mechanism ${\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})}$ by

$$\begin{aligned} ({\tilde{Y}}_{HH},{\tilde{Q}}_{HH})=(Y_{HH},Q_{HH}),\qquad&({\tilde{Y}}_{LH},{\tilde{Q}}_{LH})=(Y_{LH}+\gamma C_H ,Q_{LH}+ \gamma ), \\ ({\tilde{Y}}_{HL},{\tilde{Q}}_{HL})=(Y_{HL},Q_{HL}),\qquad&({\tilde{Y}}_{LL},{\tilde{Q}}_{LL})=(Y_{LL},Q_{LL}), \end{aligned}$$

where $\gamma$ is chosen such that ${\tilde{Q}}_{LH}=1$. This mechanism ${\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})}$ still satisfies the IR and IC constraints and gives both divisions at least the same expected utility for both their types and at least one division is strictly better off. The expected utilities from the mechanism ${\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})}$ are

$$\begin{aligned} U_1({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |H)&=U_1(\mu ^{(Y,Q)} |H),\\ U_1({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |L)&=U_1(\mu ^{(Y,Q)} |L)+\delta \gamma (C_H-C_L)> U_1(\mu ^{(Y,Q)} |L), \\ U_2({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |H)&=U_2(\mu ^{(Y,Q)} |H)+ (1-\varepsilon ) \gamma (R_H-C_H) > U_2(\mu ^{(Y,Q)} |H), \\ U_2({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |L)&=U_2(\mu ^{(Y,Q)} |L). \end{aligned}$$

We obtain for the IR constraints

$$\begin{aligned}&U_1({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |H) \ge 0, \qquad U_1({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |L)> 0, \\&U_2({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |H) > 0, \qquad U_2({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |L) \ge 0 \end{aligned}$$

and for the IC constraints

$$\begin{aligned} U_1({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |H)&=U_1(\mu ^{(Y,Q)} |H) \\&\ge \delta (Y_{LH}-Q_{LH}C_H) + (1-\delta ) (Y_{LL}-Q_{LL}C_H) \\&= \delta ({\tilde{Y}}_{LH}-{\tilde{Q}}_{LH}C_H) + (1-\delta ) ({\tilde{Y}}_{LL}-{\tilde{Q}}_{LL}C_H), \\ U_1({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |L)&=U_1(\mu ^{(Y,Q)} |L)+\delta \gamma (C_H-C_L)\\&\ge \delta (Y_{HH}-Q_{HH}C_L) \\&=\delta ({\tilde{Y}}_{HH}-{\tilde{Q}}_{HH}C_L),\\ U_2({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |H)&=U_2(\mu ^{(Y,Q)} |H)+ (1-\varepsilon ) \gamma (R_H-C_H)\\&\ge (1-\varepsilon ) (Q_{LL}R_H-Y_{LL})\\&= (1-\varepsilon ) ({\tilde{Q}}_{LL}R_H-{\tilde{Y}}_{LL}),\\ U_2({\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})} |L)&=U_2(\mu ^{(Y,Q)} |L) \\&\ge \varepsilon (Q_{HH}R_L-Y_{HH}) + (1-\varepsilon ) (Q_{LH}R_L-Y_{LH})\\&\ge \varepsilon (Q_{HH}R_L-Y_{HH}+\gamma (1-\varepsilon )(R_L-C_H)) \\&\quad + (1-\varepsilon ) (Q_{LH}R_L-Y_{LH})\\&= \varepsilon ({\tilde{Q}}_{HH}R_L-{\tilde{Y}}_{HH}) + (1-\varepsilon ) ({\tilde{Q}}_{LH}R_L-{\tilde{Y}}_{LH}). \end{aligned}$$

Thus, for any mechanism $\mu ^{(Y,Q)}$ with $Q_{LH}<1$ we can construct a mechanism ${\tilde{\mu }}^{({\tilde{Y}},{\tilde{Q}})}$ that Pareto dominates $\mu ^{(Y,Q)}$.

To prove (ii), we use the axiomatization in Weidner (1992) stating that the generalized Nash bargaining solution is Pareto optimal. Precisely, any mechanism for which the Nash product F is maximal cannot be Pareto dominated. Hence, by part (i), $Q_{LH} = 1$. $\square$

Proof

(Proof of Proposition 6) We first show that constraints (IC2) and (IC3) are binding in the generalized Nash bargaining solution. In the next step, we identify conditions under which both constraints (IC1) and (IC2) (and analogously (IC3) and (IC4)) cannot be binding simultaneously. Let the generalized Nash bargaining solution be attained by a mechanism $\mu ^{*(Y,Q)}$.

Step 1 We first establish that if (IC2) is not binding we have

$$\begin{aligned} U_1(\mu ^{*(Y,Q)}|L)>U_1(\mu ^{*(Y,Q)}|H). \end{aligned}$$

(5)

This can be seen as follows:

$$\begin{aligned}&U_1(\mu ^{*(Y,Q)}|L)-U_1(\mu ^{*(Y,Q)}|H) \\&\quad = \delta (Y_{LH}-Q_{LH}C_L) + (1-\delta ) (Y_{LL}-Q_{LL}C_L) - \delta (Y_{HH}-Q_{HH}C_H) \\&\quad > \delta (Y_{HH}-Q_{HH}C_L) - \delta (Y_{HH}-Q_{HH}C_H) \\&\quad = \delta Q_{HH}(C_H-C_L)\ge 0. \end{aligned}$$

Hereby, the strict inequality comes from the strict inequality of (IC2).

Analogously, if (IC3) is not binding then

$$\begin{aligned} U_2(\mu ^{*(Y,Q)}|H)>U_2(\mu ^{*(Y,Q)}|L). \end{aligned}$$

(6)

We establish that if the incentive constraints (IC2) and (IC3) are not both binding simultaneously, then for given transfer quantities $Q_{HH}$, $Q_{LH}$ and $Q_{LL}$ we can modify the transfer payments $Y_{HH}$, $Y_{LH}$ and $Y_{LL}$ in such a way that IR and IC constraints are still met, but the generalized Nash product increases. We distinguish the two cases in which (IC2) or (IC3) are not binding, respectively.

Case 1 Suppose (IC2) is not binding. We increase the transfer payment $Y_{HH}$, decrease $Y_{LH}$ and leave $Y_{LL}$ unchanged. This is done in such a way that the l.h.s. of the incentive constraint (IC3) does not change. Therefore, while increasing $Y_{HH}$ by $\frac{k}{\delta \varepsilon }$ we decrease $Y_{LH}$ by $\frac{k}{\delta (1-\varepsilon )}$ for small $k>0$. It can be easily seen that the remaining incentive constraints (IC1) and (IC4) as well as the IR constraints (IR1) to (IR4) are not violated for k small enough. In order to show that the generalized Nash product F increases, we take the corresponding directional derivative of its logarithm.¹⁵ Formally, this amounts to

$$\begin{aligned}&\frac{1}{\delta \varepsilon } \frac{\partial \log [F(\mu ^{*(Y,Q)})]}{\partial Y_{HH}} - \frac{1}{\delta (1-\varepsilon ) }\frac{\partial \log [ F(\mu ^{*(Y,Q)})]}{\partial Y_{LH}}\\&\quad = \frac{1}{U_1(\mu ^{*(Y,Q)}|H)}- \frac{1}{U_2(\mu ^{*(Y,Q)}|H)}- \frac{1}{U_1(\mu ^{*(Y,Q)}|L)}+ \frac{1}{U_2(\mu ^{*(Y,Q)}|H)}\\&\quad = \frac{U_1(\mu ^{*(Y,Q)}|L)-U_1(\mu ^{*(Y,Q)}|H)}{U_1(\mu ^{*(Y,Q)}|H)U_1(\mu ^{*(Y,Q)}|L)}>0, \end{aligned}$$

which holds by (5).

Case 2 Suppose (IC3) is not binding. Now we increase the transfer payment $Y_{LH}$, decrease $Y_{LL}$ and leave $Y_{HH}$ unchanged so that (IC2) is unaltered and the remaining IC and IR conditions are still valid. Precisely, increase $Y_{LH}$ by $\frac{k}{\delta (1-\varepsilon )}$ and decrease $Y_{LL}$ by $\frac{k}{(1-\delta )(1-\varepsilon )}$ for small enough $k>0$. To see that the generalized Nash product F increases, we again take the directional derivative of its logarithm and use (6), to get

$$\begin{aligned}&\frac{1}{\delta (1-\varepsilon )} \frac{\partial \log [F(\mu ^{*(Y,Q)})]}{\partial Y_{LH}} - \frac{1}{(1-\delta )(1-\varepsilon )}\frac{\partial \log [ F(\mu ^{*(Y,Q)})]}{\partial Y_{LL}}\\&\quad = \frac{1}{U_1(\mu ^{*(Y,Q)}|L)}- \frac{1}{U_2(\mu ^{*(Y,Q)}|H)} - \frac{1}{U_1(\mu ^{*(Y,Q)}|L)}+ \frac{1}{U_2(\mu ^{*(Y,Q)}|L)}\\&\quad = \frac{U_2(\mu ^{*(Y,Q)}|H)-U_2(\mu ^{*(Y,Q)}|L)}{U_2(\mu ^{*(Y,Q)}|L)U_2(\mu ^{*(Y,Q)}|H)}>0. \end{aligned}$$

Taking the two cases together, both incentive constraints (IC2) and (IC3) have to be binding for the maximizer of the generalized Nash product under IR and IC constraints.

Step 2 Consider first (IC1) and (IC2) and suppose both constraints are binding. Adding (IC1) and (IC2) and rearranging implies

$$\begin{aligned} \left[ \delta (Q_{LH}-Q_{HH})+(1-\delta )Q_{LL}\right] \left( C_H-C_L \right) =0. \end{aligned}$$

(7)

As $C_H>C_L$ and $Q_{LH}=1$ hold in the generalized Nash bargaining solution, the above equation is satisfied if and only if $Q_{HH}=1$ and $Q_{LL}=0$. Therefore, if $Q_{HH}<1$ or if $Q_{LL}>0$, (IC1) and (IC2) cannot be binding simultaneously. Hence in this case, (IC1) cannot be binding as we already established in Step 1 that (IC2) binds. Conversely, if $Q_{HH}=1$ and $Q_{LL}=0$ hold, then it is easily verified that (IC1) is satisfied if and only if (IC2) holds. Since (IC2) is binding in the Nash solution, this establishes the first equivalence in part (ii) of the proposition.

Analogous arguments demonstrate the second equivalence on constraints (IC3) and (IC4). Note that if both are binding, their sum amounts to

$$\begin{aligned} \left[ \varepsilon (Q_{LH}-Q_{LL})+(1-\varepsilon )Q_{HH}\right] \left( R_H-R_L \right) =0, \end{aligned}$$

which is the analogue to (7).

We summarize the above observations. At least one of the two constraints (IC1) and (IC4) is not binding in the generalized Nash bargaining solution. Put differently, at least two incentive constraints, namely (IC2) and (IC3), and at most three incentive constraints, namely either (IC2), (IC3), (IC1) or (IC2), (IC3), (IC4), are binding in the generalized Nash bargaining solution.

We close the proof with the remark that the subsequent examples (Examples 1, 2) demonstrate that the equivalence in part (ii) is not trivial in the sense that exactly two or three IC constraints might be binding.$\square$

B Remarks on the Proof of Proposition 3

Remark 3

(Regularity: IR and IC mechanisms) The mechanism of the proof for Proposition 3 (Case 1) is the convex combination (with equal coefficients $\frac{1}{3}$) of the following three mechanisms:

• $\mu ^{(Y^1,Q^1)}$ with the transfer payments and quantities:

  $$\begin{aligned} (Y^1_{HH},Q^1_{HH})&=\left( \frac{\left( 1-\varepsilon \right) Y^1_{LL}+\delta R_H}{\delta },1\right) , \qquad (Y^1_{HL},Q^1_{HL}) =\left( 0,0\right) , \\ (Y^1_{LH},Q^1_{LH})&=\left( \frac{\left( \delta -\varepsilon \right) Y^1_{LL}+\delta R_H}{\delta },1\right) , \qquad (Y^1_{LL},Q^1_{LL}) =\left( -\delta \left( R_H-C_H\right) ,0\right) , \end{aligned}$$

• $\mu ^{(Y^2,Q^2)}$ with the transfer payments and quantities:

  $$\begin{aligned} (Y^2_{HH},Q^2_{HH})&=\left( \frac{\left( 1-\varepsilon \right) Y^2_{LL}-(1-\varepsilon )C_L}{\delta },0\right) ,\qquad (Y^2_{HL},Q^2_{HL})=\left( 0,0\right) ,\\ (Y^2_{LH},Q^2_{LH})&=\left( \frac{\left( \delta -\varepsilon \right) Y^2_{LL}+\varepsilon C_L}{\delta },1\right) , \qquad (Y^2_{LL},Q^2_{LL})=\left( \frac{R_L+C_L}{2},1\right) . \end{aligned}$$

• $\mu ^{(Y^3,Q^3)}$ with the transfer payments and quantities:

  $$\begin{aligned} (Y^3_{HH},Q^3_{HH})&=\left( \frac{\left( 1-\varepsilon \right) Y^3_{LL}+\delta (1-\varepsilon )\left( C_H-C_L\right) }{\delta },0\right) , \qquad (Y^3_{HL},Q^3_{HL})=\left( 0,0\right) ,\\ (Y^3_{LH},Q^3_{LH})&=\left( \frac{\left( \delta -\varepsilon \right) Y^3_{LL}+\left( 1-\varepsilon \right) \delta C_H+\varepsilon \delta C_L}{\delta },1\right) ,\\ (Y^3_{LL},Q^3_{LL})&=\left( -\varepsilon \delta \left( C_H-C_L\right) ,0\right) . \end{aligned}$$

Remark 4

(Regularity: IR, IC and EPE mechanisms) The mechanism of the proof for Proposition 3 (Case 2.1) is a convex combination (with equal factors $\frac{1}{2}$) of the following two mechanisms:

• $\mu ^{(Y^4,Q^4)}$ with the transfer payments and quantities:

  $$\begin{aligned} (Y^4_{HH},Q^4_{HH})&=\left( C_H,1\right) , \qquad (Y^4_{HL},Q^4_{HL})=\left( 0,0\right) ,\\ (Y^4_{LH},Q^4_{LH})&=\left( \frac{\varepsilon (1-\delta )R_H + (1-\delta )(1-\varepsilon )C_L+(\delta -\varepsilon )C_H}{1-\varepsilon },1\right) ,\\ (Y^4_{LL},Q^4_{LL})&=\left( \frac{(1-\delta )(1-\varepsilon )C_L+\delta C_H-\delta \varepsilon R_H}{1-\varepsilon },0\right) . \end{aligned}$$

  We observe using $\mu ^{(Y^4,Q^4)}$ that the constraints (IR2) to (IR4) hold with strict inequality while (IR1) is equal to zero.

• $\mu ^{(Y^5,Q^5)}$ with the transfer payments and quantities:

  $$\begin{aligned} (Y^5_{HH},Q^5_{HH})&=\left( \frac{\delta \varepsilon R_H+(1 - \varepsilon )R_L-(1-\delta )(1-\varepsilon ) C_L}{\delta },1\right) ,\quad (Y^5_{HL},Q^5_{HL})=\left( 0,0\right) ,\\ (Y^5_{LH},Q^5_{LH})&=\left( \frac{\delta \varepsilon R_H+(\delta - \varepsilon )R_L+(1-\delta )\varepsilon C_L}{\delta },1\right) , \qquad (Y^5_{LL},Q^5_{LL})=\left( R_L,0\right) . \end{aligned}$$

  We observe using $\mu ^{(Y^5,Q^5)}$ that the constraints (IR1) to (IR3) hold with strict inequality while (IR4) is equal to zero. Therefore, taking a convex combination (e.g., with $\frac{1}{2}$) of $\mu ^{(Y^4,Q^4)}$ and $\mu ^{(Y^5,Q^5)}$ yields a mechanism that is strictly individually rational.

C Lemma 1 Used in the Proof of Proposition 4

Lemma 1

Let $f:{\mathbb {R}}_+^n \rightarrow {\mathbb {R}}_+$ be defined by

$$\begin{aligned} f(x_1,\ldots ,x_n)=\prod _{i=1}^n x_i^{l_i} \quad \text {with } l_1,\ldots ,l_n>0 \end{aligned}$$

and consider the constrained maximization problem

$$\begin{aligned} \max _{x_1,\ldots , x_n} f(x) \quad \text{ s.t. } \quad \sum _{i=1}^n l_i \cdot x_i\le c \end{aligned}$$

with $c\in {\mathbb {R}}_+$. Then there is a unique maximizer $(x_1^*,\ldots ,x_n^*)$ and $x^*$ satisfies $x_1^*=\ldots =x_n^*$.

Proof

(Proof of Lemma 1) Inserting the constraint

$$\begin{aligned} x_n=\frac{c-\sum _{i=1}^{n-1} l_i \cdot x_i}{l_n} \end{aligned}$$

into the objective function leads to

$$\begin{aligned} f(x_1,\ldots ,x_n)=\prod _{i=1}^{n-1} x_i^{l_i} \cdot \left( \frac{c-\sum _{i=1}^{n-1} l_i \cdot x_i}{l_n} \right) ^{l_n}. \end{aligned}$$

The function does not depend on $x_n$. We obtain for $j \ne n$ the first order condition

$$\begin{aligned}&\frac{\partial f(x_1,\ldots ,x_n)}{\partial x_j}\\&\quad = l_j x_j^{l_j-1} \cdot \prod _{i=1,i\ne j}^{n-1} x_i^{l_i}\cdot \left( \frac{c-\sum _{i=1}^{n-1} l_i \cdot x_i}{l_n} \right) ^{l_n}\\&\qquad + \prod _{i=1}^{n-1} x_i^{l_i}\cdot \left( \frac{-l_j}{l_n} \right) \cdot l_n \cdot \left( \frac{c-\sum _{i=1}^{n-1} l_i \cdot x_i}{l_n} \right) ^{l_n-1}\\&\quad =l_j x_j^{l_j-1} \cdot \prod _{i=1,i\ne j}^{n-1} x_i^{l_i}\cdot \left( \frac{c-\sum _{i=1}^{n-1} l_i \cdot x_i}{l_n} \right) ^{l_n-1} \left( \frac{c-\sum _{i=1}^{n-1} l_i \cdot x_i}{l_n}-x_j \right) \overset{!}{=}0. \end{aligned}$$

Thus, in order to obtain a maximum we need to have for all $j \ne n$

$$\begin{aligned} x_j=\frac{c-\sum _{i=1}^{n-1} l_i \cdot x_i}{l_n}. \end{aligned}$$

Since the right-hand side will be the same for each $x_j$ and is equal to $x_n$, we have for the maximizer $(x_1^*,\ldots ,x_n^*)$ of $f(x_1,\ldots ,x_n)$ under the constraint $\sum _{i=1}^n l_i \cdot x_i=c$ that $x_1^*=\ldots =x_n^*$ holds. $\square$