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Vertical Relations in the Presence of Competitive Recycling

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Abstract

This paper studies the vertical relations between a manufacturer and one or more retailers over two periods in the presence of a competitive recycling sector. In a bilateral monopoly, two-part tariffs are always efficient, i.e. the manufacturer will produce the joint-profit-maximizing output. Under downstream oligopoly, instead, retailers compete to acquire the recycled good which allows the recycling sector to appropriate some of the industry profits. Under two-part tariffs, the manufacturer has an incentive to distort her output choices to reduce this rent loss: She will discriminate among her retailers, and she will either overproduce in the second period or underproduce in the first period. Vertical restraints that restore profit maximization (e.g. loyalty rebates) will harm consumers whenever the manufacturer would overproduce otherwise.

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Correspondence to Liliane Giardino-Karlinger.

Appendix

Appendix

Proof of Lemma 1

If only good i = A, B is supplied at price p i , then the demand for this good is equal to the mass of consumers with taste parameter θ such that θ s i p i ≥ 0. In other words, all consumers with \(\theta \geq \frac {p_{i}}{s_{i}}\) will buy, so that demand for good i is:

$$ D_{i}\left( p_{i}\right) =1-F\left( \frac{p_{i}}{s_{i}}\right) =1-\frac{p_{i} }{s_{i}} $$
(35)

With demand for good i given by Eq. 35, the industry-profit maximizing output and the corresponding profits are given by (where sp denotes the optimal solution under single-product retailing):

$$\begin{array}{@{}rcl@{}} q_{A}^{sp} &=&\frac{1}{2}\left( \frac{s_{A}-c_{A}}{s_{A}}\right) ,{\Pi}_{A}^{sp}=\frac{1}{4s_{A}}\left( s_{A}-c_{A}\right)^{2} \end{array} $$
(36)
$$\begin{array}{@{}rcl@{}} q_{B}^{sp} &=&\min \left\{ \frac{1}{2},\sigma q_{a}\right\} ,{\Pi}_{B}^{sp}=\min \left\{ \frac{1}{4s_{B}}{s_{B}^{2}},s_{B}\left( 1-\sigma q_{a}\right) \sigma q_{a}\right\} \end{array} $$
(37)

Let us now compare these single-product profits with those that arise under multi-product retailing:

  1. 1.

    If the interior solution applies under multi-product retailing, the resulting equilibrium profits are:

    $$\begin{array}{@{}rcl@{}} {\Pi} \left( q_{A}^{\ast },q_{B}^{\ast }\right) &\equiv &p_{A}^{\ast }q_{A}^{\ast }+p_{B}^{\ast }q_{B}^{\ast }-c_{A}q_{A}^{\ast } \\ &=&\frac{1}{4}\left( s_{A}-c_{A}\right) \frac{\Delta s-c_{A}}{\Delta s}+ \frac{1}{4}s_{B}\frac{c_{A}}{\Delta s} \end{array} $$

    where Δs = s A s B .

    1. (i)

      To verify that \({\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) >{\Pi }_{A}^{sp}\), insert for the corresponding expressions from above, and simplify the inequality to \(c_{A}{s_{B}^{2}}>0\), which is of course always true.

    2. (ii)

      To verify that \({\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) >{\Pi }_{B}^{sp}=\frac {1}{4s_{A}}\left (s_{A}-c_{A}\right )^{2}\), insert for the corresponding expressions from above, and simplify the inequality to \(\left ({\Delta } s-c_{A}\right )^{2}>0\), which is of course always true. Since the corner solution for \(q_{B}^{sp}\) always yields lower profits than the interior solution, \({\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) >{\Pi }_{B}^{sp}=s_{B}\left (1-\sigma q_{a}\right ) \sigma q_{a}\) is implied by \({\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) >\frac {1}{4s_{A}}\left (s_{A}-c_{A}\right )^{2}\).

  2. 2.

    If instead the corner solution applies under multi-product retailing, so that \(q_{B}^{\ast }=\sigma q_{a}\), then this implies that \( q_{B}^{sp}>q_{B}^{\ast }=\sigma q_{a}\), so that the corner solution also applies under single-product retailing.

    Then, inserting the corner solutions for the first-best output levels of good A and good B, expressions (13), we obtain equilibrium profits under multi-product retailing as

    $$\begin{array}{@{}rcl@{}} {\Pi} \left( q_{A}^{\ast },q_{B}^{\ast }\right) &\equiv &p_{A}^{\ast }q_{A}^{\ast }+p_{B}^{\ast }q_{B}^{\ast }-c_{A}q_{A}^{\ast } \\ &=&\frac{1}{4s_{A}}\left( s_{A}-c_{A}\right)^{2}+\sigma s_{B}q_{a}\frac{1}{ s_{A}}\left( c_{A}-\left( s_{A}-s_{B}\right) \sigma q_{a}\right) \end{array} $$

    while single-product retailing yields either

    $${\Pi}_{A}^{sp}=\frac{1}{4s_{A}}\left( s_{A}-c_{A}\right)^{2}\mathrm{ or }{\Pi}_{B}^{sp}=s_{B}\left( 1-\sigma q_{a}\right) \sigma q_{a} $$
    1. (i)

      To show that \({\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) >{\Pi }_{A}^{sp}\), simplify this inequality to \(\sigma s_{B}q_{a}\frac {1}{s_{A}} \left (c_{A}-\left (s_{A}-s_{B}\right ) \sigma q_{a}\right ) >0\), which is satisfied whenever \(c_{A}>\left (s_{A}-s_{B}\right ) \sigma q_{a}\). By assumption, we have that the interior solution for good B is not feasible, so that \(q_{B}^{\ast }=\frac {1}{2}\frac {c_{A}}{s_{A}-s_{B}}>\sigma q_{a}\), implying that \(c_{A}>2\sigma q_{a}\left (s_{A}-s_{B}\right ) \), so that \( c_{A}>\left (s_{A}-s_{B}\right ) \sigma q_{a}\) follows immediately.

    2. (ii)

      To show that \({\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) >{\Pi }_{B}^{sp}\), simply note that the corner solution for \({\Pi }_{B}^{sp}\) yields lower profits than the interior solution:\(s_{B}\left (1-\sigma q_{a}\right ) \sigma q_{a}\leq \frac {1}{4s_{B}}{s_{B}^{2}}\), and that \(\frac {1}{4s_{B}} {s_{B}^{2}}<{\Pi }_{A}^{sp}=\frac {1}{4s_{A}}\left (s_{A}-c_{A}\right )^{2}\). We showed before that \({\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) >{\Pi }_{A}^{sp}\), so that \({\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) >{\Pi }_{B}^{sp}\) follows immediately.\(\square \)

Proof of Proposition 1

The proof is constructive and fully characterizes the optimal contracts.

Contracting in Period 2

Let us first consider the retailer’s problem in period 2.

  1. (i)

    Multi-product retailing: Suppose the retailer accepted M’s offer to retail good A under tariff \(\left (w_{A},T_{A}\right ) \). Then, being the monopoly buyer of the recycled good, the retailer can acquire any level of good B (up to σ q a ) at marginal cost zero. Therefore, the retailer’s problem is to maximize his payoff \({\Pi }_{R}\left (q_{A},q_{B}\right ) \) by choosing how much to sell of goods A and B:

    $$\begin{array}{@{}rcl@{}} \max_{q_{A},q_{B}}{\Pi}_{R}\left( q_{A},q_{B}\right) =p_{A}q_{A}+p_{B}q_{B}-w_{A}q_{A}-T_{A} \\ \mathrm{s.t. }q_{B}\leq \sigma q_{a} \end{array} $$
    (38)

    This problem is identical to the vertical chain’s joint profit maximization problem in period 2 as given in Eq. 6, up to constant T A and unit cost w A instead of c A for good A. Clearly, the manufacturer can always induce the retailer to choose the first-best output levels, \(\left (q_{A}^{\ast },q_{B}^{\ast }\right ) \) as in Table 1, by setting \(w_{A}^{\ast }=c_{A}\). How much of the resulting first-best profits \( {\Pi } \left (q_{A}^{\ast },q_{B}^{\ast }\right ) \) M can appropriate through T A depends on R’s outside option, to which we turn next.

  2. (ii)

    Outside option: If R refuses to retail good A, R’s outside option is to sell good B only. Thus, R’s disagreement payoff is the single-product profit from retailing good B only:

    $${\Pi}_{B}^{sp}\left( q_{a}\right) =\min \left\{ \frac{1}{4s_{A}}\left( s_{A}-c_{A}\right)^{2},s_{B}\left( 1-\sigma q_{a}\right) \sigma q_{a}\right\} $$

    as derived in the Proof of Lemma 1; this is the minimum payoff that M must offer for R to accept M’s contract. We therefore obtain the maximum franchise fee M can offer in t = 2 as:

    $$T_{A}^{\ast }={\Pi} \left( q_{A}^{\ast },q_{B}^{\ast }\right) -{\Pi}_{B}^{sp}\left( q_{a}\right) $$

Contracting in Period 1

Without any recycled good available, R’s only outside option in t = 1 is not to sell anything at all. However, rejecting M’s contract in t = 1 also implies reducing the disagreement payoff in t = 2 to zero: If none of the primary good is produced and sold today, there is no supply of the recycled good tomorrow, and so the retailer has no fall-back option tomorrow. M can therefore extract R’s entire first-period profit plus the discounted disagreement payoff in period 2:

$$T_{a}^{\ast }={\Pi}_{R}\left( q_{a},0\right) +\delta {\Pi}_{B}^{sp}\left( q_{a}\right) $$

We now have to identify the unit wholesale price w A that induces R to choose the first-best output level \(q_{a}^{\ast }\) in period 1. R’s problem in period 1 is to maximize his stream of profits:

$$\max_{q_{a}\geq 0}p_{a}q_{a}-w_{a}q_{a}-T_{a}+\delta {\Pi}_{B}^{sp}\left( q_{a}\right) $$

Note that this problem differs in its last term from that of the vertical chain as a whole: In choosing q a , the retailer seeks to maximize his outside option in t = 2, not the joint profits in t = 2. If \(q_{B}^{sp}=\min \left \{ \frac {1}{2},\sigma q_{a}\right \} =\frac {1}{2}\), so that \({\Pi }_{B}^{sp}\) is not a function of q a , M can induce the first-best output level \(q_{a}^{\ast }\) in period 1 by setting w a = c A . (Note that \(q_{B}^{sp}=\frac {1}{2}>q_{B}^{\ast }\), so that \(\sigma q_{a}>\frac {1}{2 }\) implies \(\sigma q_{a}>q_{B}^{\ast }\), i.e. the resource constraint is not binding at the multi-product solution either, and hence \(q_{a}^{\ast }=\arg \max p_{a}q_{a}-c_{A}q_{a}\).)

If instead \(q_{B}^{sp}=\sigma q_{a}\), then R’s demand for good A in period 1 is given by the solution to

$$\max_{q_{a}\geq 0}p_{a}q_{a}-w_{a}q_{a}-T_{a}+\delta s_{B}\left( 1-\sigma q_{a}\right) \sigma q_{a} $$

which we can find as

$${q_{a}^{R}}=\frac{1}{2}\frac{s_{A}-w_{a}+\delta \sigma s_{B}}{s_{A}+\delta \sigma^{2}s_{B}} $$

M then needs to set the unit wholesale price w a that induces R to choose \({q_{a}^{R}}=q_{a}^{\ast }\) as given by either (17) or (16). The corresponding solutions are

$$w_{a}^{\ast }=\left\{ \begin{array}{cc} \delta \sigma s_{B}-\delta \sigma^{2}\frac{s_{B}}{s_{A}}\left( s_{A}-c_{A}\right) +c_{A} & \mathrm{if }\sigma \geq \hat{\sigma} \\ \frac{c_{A}s_{A}+\delta \sigma s_{B}\left( s_{A}-c_{A}\right) -\delta \sigma^{2}s_{B}\left( s_{B}-c_{A}\right) +\delta^{2}\sigma^{3}\frac{{s_{B}^{2}}}{ s_{A}}\left( \left( s_{A}-s_{B}\right) -c_{A}\right) }{s_{A}+\delta \sigma^{2}\frac{s_{B}}{s_{A}}\left( s_{A}-s_{B}\right) } & \mathrm{if }\sigma <\hat{ \sigma} \end{array} \right. $$

M’s discounted stream of profits from the sequence of contracts \(\left (w_{a}^{\ast },T_{a}^{\ast }\right ) ,\left (w_{A}^{\ast },T_{A}^{\ast }\right ) \) is

$$\begin{array}{@{}rcl@{}} {\Pi}^{M} &=&\left( w_{a}^{\ast }-c_{A}\right) q_{a}^{\ast }+T_{a}^{\ast }+\delta \left[ \left( w_{A}^{\ast }-c_{A}\right) q_{A}^{\ast }+T_{A}^{\ast } \right] \\ &=&{\Pi} \left( q_{a}^{\ast },0\right) +\delta {\Pi} \left( q_{A}^{\ast },q_{B}^{\ast }\right) \end{array} $$

so that M’s payoffs do indeed coincide with those of the first-best solution. □

Proof of Lemma 2

Lemma 2 is a direct application of Anton and Yao (1989), inverting the roles of buyers and sellers: In Anton and Yao (1989), a single procurement agency buys a divisible product from two undifferentiated suppliers, in our case, the recycling sector sells a divisible quantity of good B to two undifferentiated retailers.

Let \({\Delta } {{\Pi }_{1}^{R}}\left (\alpha \right ) \) denote Retailer 1’s incremental profit from obtaining a share \(\alpha \in \left [ 0,1\right ] \) of good B, over getting α = 0:

$${\Delta} {{\Pi}_{1}^{R}}\left( \alpha \right) ={{\Pi}_{1}^{R}}\left( \bar{q} _{A1},q_{B1}|_{\alpha },\bar{q}_{A2},q_{B2}|_{\alpha }\right) -{\Pi}_{1}^{R}\left( \bar{q}_{A1},0,\bar{q}_{A2},q_{B2}|_{\alpha =0}\right) $$

where \(q_{Ai}^{CE}=\bar {q}_{Ai}\) for both retailers i = 1,2 by assumption A2, and define \({\Delta } {{\Pi }_{2}^{R}}\left (\alpha \right ) \) analogously. By Proposition 1 of Anton and Yao (1989), if \({\Delta } {{\Pi }_{1}^{R}}\left (1\right ) >{\Delta } {{\Pi }_{2}^{R}}\left (0\right ) \), then α =1 is an equilibrium outcome, and Retailer 1 will acquire the entire stock of good B. Equilibrium bids satisfy \(g^{\ast }={\Delta } {{\Pi }_{2}^{R}}\left (0\right ) =B_{1}^{\ast }\left (1\right ) =B_{2}^{\ast }\left (0\right ) \). Furthermore if \({\Delta } {{\Pi }_{1}^{R}}\left (1\right ) >{\Delta } {{\Pi }_{1}^{R}}\left (\alpha \right ) +{\Delta } {{\Pi }_{2}^{R}}\left (\alpha \right ) \) for all \(\alpha \in \left [ 0,1\right ) \), then α =1 is the unique equilibrium outcome (see Anton and Yao (1989), p. 550 to establish the proof of this proposition).

It is straightforward to calculate \({\Delta } {{\Pi }_{1}^{R}}\left (\alpha \right ) \) and \({\Delta } {{\Pi }_{2}^{R}}\left (0\right ) \) as

$$\begin{array}{@{}rcl@{}} {\Delta} {{\Pi}_{1}^{R}}\left( 1\right) &=&{{\Pi}_{1}^{R}}\left( \bar{q} _{A1},q_{B1}\left( \sigma q_{a}\right) ,\bar{q}_{A2},0\right) -{\Pi}_{1}^{R}\left( \bar{q}_{A1},0,\bar{q}_{A2},q_{B2}\left( \sigma q_{a}\right) \right) \\ &=&\frac{1}{2}s_{B}\left( \frac{1}{2}-\bar{q}_{A1}-\bar{q}_{A2}+\bar{q} _{A1}^{2}+\frac{1}{2}\bar{q}_{A2}^{2}\right) \\ {\Delta} {{\Pi}_{2}^{R}}\left( 0\right) &=&{{\Pi}_{2}^{R}}\left( \bar{q}_{A1},0,\bar{ q}_{A2},q_{B2}\left( \sigma q_{a}\right) \right) -{{\Pi}_{2}^{R}}\left( \bar{q} _{A1},q_{B1}\left( \sigma q_{a}\right) ,\bar{q}_{A2},0\right) \\ &=&\frac{1}{2}s_{B}\left( \frac{1}{2}-\bar{q}_{A1}-\bar{q}_{A2}+\frac{1}{2} \bar{q}_{A1}^{2}+\bar{q}_{A2}^{2}\right) \end{array} $$

by inserting the stocks of good A, \(\bar {q}_{A1},\bar {q}_{A2}\), and the best responses for good B as defined in expression (22), into Retailer i’s final-good stage profit function (19). Note that we use here assumption A3, i.e. \(q_{Bi}\left (\sigma q_{a}\right ) =\frac {1}{2}\left (1-2\bar {q}_{Ai}- \bar {q}_{Aj}\right ) \leq \bar {q}_{Bi}\) whenever \(\bar {q}_{Bi}=\sigma q_{a}\).

The inequality \({\Delta } {{\Pi }_{1}^{R}}\left (1\right ) >{\Delta } {\Pi }_{2}^{R}\left (0\right ) \) can be simplified to \(\bar {q}_{A1}>\bar {q}_{A2}\), which holds by assumption A1. We thus established that α =1 is an equilibrium outcome, and Retailer 1 will acquire the entire stock of good B at equilibrium bids \(B_{1}^{\ast }\left (1\right ) =B_{2}^{\ast }\left (0\right ) ={\Delta } {{\Pi }_{2}^{R}}\left (0\right ) \).

To establish uniqueness of this equilibrium, we need to show that \({\Delta } {{\Pi }_{1}^{R}}\left (1\right ) >{\Delta } {{\Pi }_{1}^{R}}\left (\alpha \right ) +{\Delta } {{\Pi }_{2}^{R}}\left (\alpha \right ) \) for all \(\alpha \in \left [ 0,1\right ) \). To do this, let us insert the best responses for good B as defined in expression (22) into the following inequality:

$$\begin{array}{@{}rcl@{}} &&{\Pi}_{1}^{R}\left( \bar{q}_{A1},q_{B1}|_{\alpha =1},\bar{q}_{A2},0\right) +{{\Pi}_{2}^{R}}\left( \bar{q}_{A1},q_{B1}|_{\alpha =1},\bar{q}_{A2},0\right)\\ &>&{{\Pi}_{1}^{R}}\left( \bar{q}_{A1},q_{B1}|_{\alpha <1},\bar{q} _{A2},q_{B2}|_{\alpha <1}\right) +{{\Pi}_{2}^{R}}\left( \bar{q} _{A1},q_{B1}|_{\alpha <1},\bar{q}_{A2},q_{B2}|_{\alpha <1}\right) \end{array} $$

and simplify this expression to \(\bar {q}_{A2}>-\frac {1}{2}q_{B2}|_{\alpha <1} \) which is of course always satisfied since \(\bar {q}_{A2}\geq 0\), while \( -\frac {1}{2}q_{B2}|_{\alpha <1}<0\). □

Proof of Proposition 3

In the text . □

Proof of Proposition 4:

(Scenario 1:) \(q_{B1}\left (\cdot \right ) =\frac {1}{2}\left (1-2\bar {q}_{A1}- \bar {q}_{A2}\right ) \leq \sigma q_{a}\), but \(q_{B2}\left (\cdot \right ) = \frac {1}{2}\left (1-2\bar {q}_{A2}-\bar {q}_{A1}\right ) >\sigma q_{a}\)

Retailers’ incremental profits from winning the entire stock of good B in the split-award auction are again

$$\begin{array}{@{}rcl@{}} {\Delta} {{\Pi}_{1}^{R}}\left( 1\right) &=&{{\Pi}_{1}^{R}}\left( \bar{q} _{A1},q_{B1}\left( \sigma q_{a}\right) ,\bar{q}_{A2},0\right) -{\Pi}_{1}^{R}\left( \bar{q}_{A1},0,\bar{q}_{A2},q_{B2}\left( \sigma q_{a}\right) \right) \\ {\Delta} {{\Pi}_{2}^{R}}\left( 0\right) &=&{{\Pi}_{2}^{R}}\left( \bar{q}_{A1},0,\bar{ q}_{A2},q_{B2}\left( \sigma q_{a}\right) \right) -{{\Pi}_{2}^{R}}\left( \bar{q} _{A1},q_{B1}\left( \sigma q_{a}\right) ,\bar{q}_{A2},0\right) \end{array} $$

just that now, \(q_{B2}\left (\sigma q_{a}\right ) =\sigma q_{a}\) instead of \( q_{B2}\left (\cdot \right ) =\frac {1}{2}\left (1-2\bar {q}_{A2}-\bar {q}_{A1}\right ) \), while \(q_{B1}\left (\sigma q_{a}\right ) =\frac {1}{2}\left (1-2\bar {q}_{A1}-\bar {q}_{A2}\right ) \leq \sigma q_{a}\) remains unchanged (note that \(\bar {q}_{A1}>\bar {q}_{A2}\) implies \(q_{B2}\left (\cdot \right ) = \frac {1}{2}\left (1-2\bar {q}_{A2}-\bar {q}_{A1}\right ) >q_{B1}\left (\cdot \right ) =\frac {1}{2}\left (1-2\bar {q}_{A1}-\bar {q}_{A2}\right ) \)). It is straightforward that Retailer 2’s incremental profits are lower when the constraint is binding than when it is slack, i.e.

$${{\Pi}_{2}^{R}}\left( \bar{q}_{A1},0,\bar{q}_{A2},\sigma q_{a}\right) -{\Pi}_{2}^{R}\left( \bar{q}_{A1},q_{B1}\left( \cdot \right) ,\bar{q} _{A2},0\right) <{{\Pi}_{2}^{R}}\left( \bar{q}_{A1},0,\bar{q}_{A2},q_{B2}\left( \cdot \right) \right) -{{\Pi}_{2}^{R}}\left( \bar{q}_{A1},q_{B1}\left( \cdot \right) ,\bar{q}_{A2},0\right) $$

by the general principles of maximization. Retailer 1’s incremental profits instead increase when Retailer 2 cannot sell as much of good B as he would want to:

$$\begin{array}{@{}rcl@{}} {{\Pi}_{1}^{R}}\left( \bar{q}_{A1},q_{B1}\left( \cdot \right) ,\bar{q} _{A2},0\right) -{{\Pi}_{1}^{R}}\left( \bar{q}_{A1},0,\bar{q}_{A2},\sigma q_{a}\right) >{{\Pi}_{1}^{R}}\left( \bar{q}_{A1},q_{B1}\left( \cdot \right) , \bar{q}_{A2},0\right) \\ -{{\Pi}_{1}^{R}}\left( \bar{q}_{A1},0,\bar{q} _{A2},q_{B2}\left( \cdot \right) \right) \end{array} $$

because Retailer 2 cannot steal as much of Retailer 1’s business when \( q_{B2}\left (\sigma q_{a}\right ) =\sigma q_{a}\) than if his resource constraint was slack instead.

We therefore have that \({\Delta } {{\Pi }_{1}^{R}}\left (1\right ) >{\Delta } {\Pi }_{2}^{R}\left (0\right ) \); in fact, as we just showed, the difference increases once the resource constraint becomes binding for Retailer 2. This means Retailer 1 can still outbid Retailer 2 in the split-award auction when \(q_{B2}\left (\sigma q_{a}\right ) =\sigma q_{a}\), and so Retailer 1 will win the split award auction paying a price of \(B_{2}^{\ast }\left (0\right ) ={\Delta } {{\Pi }_{2}^{R}}\left (0\right ) \).

Note that \(B_{2}^{\ast }\left (0\right ) ={\Delta } {{\Pi }_{2}^{R}}\left (0\right ) \) is now a function of period-1 output σ q a . Recall that \( B_{2}^{\ast }\left (0\right ) \) represents the rent that is appropriated by the recycling sector in period 2, so it reduces M’s stream of profits from period 1’s perspective as follows:

$$\begin{array}{@{}rcl@{}} &&\max\limits_{q_{a}}\left\{ {\Pi} \left( q_{a},0\right) +\delta {\Pi}^{M}\left( q_{a}\right) \right\} \\ &=&{\Pi} \left( q_{a},0\right) +\delta \left[ \left( p_{A}\left( \mathbf{q} \right) -c_{A}\right) \left( q_{A1}^{c\ast }+q_{A2}^{c\ast }\right) +p_{B}\left( \mathbf{q}\right) q_{B}^{c\ast }-B_{2}^{\ast }|_{\alpha =0}\left( q_{A1}^{c\ast },q_{A2}^{c\ast },\sigma q_{a}\right) \right] \end{array} $$
(39)

Simple calculations show that

$$B_{2}^{\ast }|_{\alpha =0}\left( q_{A1}^{c\ast },q_{A2}^{c\ast },\sigma q_{a}\right) =s_{B}\left[ \sigma q_{a}-2\sigma q_{a}q_{A2}^{c\ast }-\sigma q_{a}q_{A1}^{c\ast }-\left( \sigma q_{a}\right)^{2}+\frac{1}{2}\left( q_{A2}^{c\ast }-2q_{A1}^{c\ast }q_{A2}^{c\ast }-\left( q_{A2}^{c\ast }\right)^{2}\right) \right] $$

and so

$$\frac{\partial B_{2}^{\ast }|_{\alpha =0}\left( q_{A1}^{c\ast },q_{A2}^{c\ast },\sigma q_{a}\right) }{\partial q_{a}}=s_{B}\sigma \left[ 1-2q_{A2}^{c\ast }-q_{A1}^{c\ast }-2\sigma q_{a}\right] >0 $$

where the last inequality is implied by \(q_{B2}\left (\cdot \right ) =\frac {1 }{2}\left (1-2\bar {q}_{A2}-\bar {q}_{A1}\right ) >\sigma q_{a}\).

M’s optimal period-1 output level under retailer competition solves (39); recall that at the first-best level \(q_{a}^{\ast }\), we have \(\partial {\Pi } \left (q_{a},0\right ) /\partial q_{a}=0\), so that the first derivative of Eq. 39, evaluated at \(q_{a}^{\ast }\) , is negative:

$$\left. \frac{\partial {\Pi} \left( q_{a},0\right) }{\partial q_{a}}\right\vert_{q_{a}=q_{a}^{\ast }}-\delta \left. \frac{\partial B_{2}^{\ast }|_{\alpha =0}\left( \cdot \right) }{\partial q_{a}}\right\vert_{q_{a}=q_{a}^{\ast }}=-\delta s_{B}\sigma \left[ 1-2q_{A2}^{c\ast }-q_{A1}^{c\ast }-2\sigma q_{a}^{\ast }\right] <0 $$

Since M’s problem (39) is a well-behaved, single-peaked program, we can infer that its solution must be \(\arg \max _{q_{a}}\left \{ {\Pi } \left (q_{a},0\right ) -\delta B_{2}^{\ast }|_{\alpha =0}\left (\cdot \right ) \right \} <q_{a}^{\ast }\). In other words, M will underproduce good A in period 1, compared to the first-best solution.

(Scenario 2:) \(q_{Bi}\left (\cdot \right ) =\sigma q_{a}\) for both i = 1,2

If both retailers are resource constrained when obtaining the entire stock of good B, so that \(q_{B1}\left (\cdot \right ) =q_{B2}\left (\cdot \right ) =\sigma q_{a}\), then the period-2 prices of goods A and B are independent of the outcome of the split-award auction. This implies that the retailers’ incremental profits are exactly the same:

$$\begin{array}{@{}rcl@{}} {\Delta} {{\Pi}_{1}^{R}}\left( 1\right) &=&p_{A}\left( \mathbf{q}\right) \bar{q} _{A1}+p_{B}\left( \mathbf{q}\right) \sigma q_{a}-p_{A}\left( \mathbf{q} \right) \bar{q}_{A1}=p_{B}\left( \mathbf{q}\right) \sigma q_{a} \\ {\Delta} {{\Pi}_{2}^{R}}\left( 0\right) &=&p_{A}\left( \mathbf{q}\right) \bar{q} _{A2}+p_{B}\left( \mathbf{q}\right) \sigma q_{a}-p_{A}\left( \mathbf{q} \right) \bar{q}_{A2}=p_{B}\left( \mathbf{q}\right) \sigma q_{a} \end{array} $$

Directly applying Proposition 2 of Anton and Yao (1989) to our setup, we can conclude that the set of equilibrium splits α comprises the entire interval \(\left [ 0,1\right ] \), and the revenue generated for the recycling sector will be \(g^{\ast }=B_{1}^{\ast }\left (\alpha =1\right ) =B_{2}^{\ast }\left (\alpha =0\right ) =p_{B}\left (\mathbf {q}\right ) \sigma q_{a}\). In other words, the recycling sector will appropriate the entire revenue from retailing good B.

M’s period-2 profits as function of wholesale prices w now simplify to \({\Pi }^{M}\left (\mathbf {w}\right ) =\left (p_{A}\left (\tilde {\mathbf {q}}\right ) -c_{A}\right ) \left (q_{A1}\left (\mathbf {w}\right ) +q_{A2}\left (\mathbf {w}\right ) \right ) \), which is constant over all possible splits of good A across retailers. We can therefore simplify the problem further by letting M contract with only one of the two retailers, q A1 = q A , so that M’s profits can be written as

$${\Pi}^{M}\left( q_{A},q_{B}\right) =\left[ p_{A}\left( q_{A},q_{B}\right) -c_{A}\right] q_{A} $$

which solves for

$$q_{A}^{c\ast }=\frac{1}{2s_{A}}\left( s_{A}-s_{B}\sigma q_{a}-c_{A}\right) $$

yielding period-2 profits for M as

$${\Pi}^{M}\left( q_{a}\right) =\frac{1}{4s_{A}}\left( s_{A}-s_{B}\sigma q_{a}-c_{A}\right)^{2} $$

M’s period-1 stream of profits can then be written as

$$\begin{array}{@{}rcl@{}} &&\max\limits_{q_{a}}\left\{ {\Pi} \left( q_{a},0\right) +\delta {\Pi}^{M}\left( q_{a}\right) \right\} \\ &=&{\Pi} \left( q_{a},0\right) +\delta \frac{1}{4s_{A}}\left( s_{A}-s_{B}\sigma q_{a}-c_{A}\right)^{2} \end{array} $$
(40)

Now, period-2 profits are decreasing in period-1 output of good A:

$$\frac{\partial {\Pi}^{M}\left( \sigma q_{a}\right) }{\partial q_{a}}=-\frac{1 }{2s_{A}}\left( s_{A}-s_{B}\sigma q_{a}-c_{A}\right) s_{B}\sigma <0 $$

and the first derivative of (39), evaluated at \( q_{a}^{\ast }\), is thus negative as well:

$$\left. \frac{\partial {\Pi} \left( q_{a},0\right) }{\partial q_{a}}\right\vert_{q_{a}=q_{a}^{\ast }}+\delta \left. \frac{\partial {\Pi}^{M}\left( \sigma q_{a}\right) }{\partial q_{a}}\right\vert_{q_{a}=q_{a}^{\ast }}<0 $$

This implies \(\arg \max _{q_{a}}\left \{ {\Pi } \left (q_{a},0\right ) +\delta {\Pi }^{M}\left (q_{a}\right ) \right \} <q_{a}^{\ast }\). Again, M will underproduce good A in period 1, compared to the first-best solution. □

Proof of Corollary 5

in the text □

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Giardino-Karlinger, L. Vertical Relations in the Presence of Competitive Recycling. J Ind Compet Trade 16, 25–49 (2016). https://doi.org/10.1007/s10842-015-0200-1

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