Competition and Contracting in Supply Chains

Engelnrecht-Wiggans, Katok and Pavlov: Competition and Contracting Article submitted to Management Science; MS-00923-2007

Competition and Contracting in Supply Chains Richard Engelbrecht-Wiggans College of Business, University of Illinois, Urbana-Champaign, Champaign, IL 61820, [email protected].

Elena Katok Smeal College of Business, Penn State University, University Park, PA 16802 [email protected]

Valery Pavlov Smeal College of Business, Penn State University, University Park, PA 16802 [email protected]

It is well known that in theory wholesale price contracts lead to double marginalization. At the same time, in spite of its inefficiency, many manufacturers use wholesale price contracts. In this study we examine competition as a potential explanation and show that it can eliminate double marginalization by altering retailers’ incentives and revealing useful information. The existing literature suggests that under full information a channel can be coordinated by means of coordinating contracts, such as minimum order quantity. We extend the literature by developing a new model for the minimum order quantity contract under incomplete information, test this model in the laboratory, and find that this coordinating contract does not perform as well as the theory predicts either under full or under incomplete information. In contrast, we find that competition performs well in the laboratory, and under incomplete information a wholesale price contract with competition outperforms the minimum order quantity contract without competition. We also find that supply chain profits are divided between the supplier and the retailer(s) in a way that is consistently more equitable than is predicted by standard theory.

Key words: supply chain coordination, contracts, auctions History: This paper was first submitted September 26, 2007.

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1. Introduction “Even though the wholesale price contract does not coordinate the supply chain, it is worth studying because it is commonly observed in practice. That fact alone suggests it has redeeming qualities” – Cachon (2003, p. 238). We suggest that an important, practical redeeming quality is that wholesale pricing can actually work much better in realistic, practical settings than the theory for simple settings would suggest. Specifically, many practical situations also include some amount of horizontal competition. 1

2 Engelbrecht-Wiggans, Katok and Pavlov: Competition and Contracting Article submitted to Management Science; MS-00923-2007

This competition may be explicit; for example, a retailer may solicit bids from multiple potential suppliers and use this to decide who wins the supply contract. Or, the competition may be implicit; for example, a manufacturer may have the option to replace an “authorized dealer” by another retailer if the manufacturer would benefit from doing so. While wholesale pricing does not, by itself, coordinate the supply chain, we will see that wholesale pricing does quite well, in general, in the presence of competition. Indeed, wholesale pricing with competition can approach a first best solution as the number of competitors increases. Moreover, just as an auction elicits a market clearing price, competition in supply chains may, in effect, elicit the information needed in order to improve coordination. Indeed, a wholesale price contract based on such information may be more robust to asymmetric information than some of the other, more sophisticated mechanisms which might otherwise fully coordinate the supply chain. In short, we suggest that the apparent theoretical limitations of wholesale pricing may simply be an artifact of ignoring the affect of the competition that is present in many situations where wholesale pricing is used. The problem of double marginalization, first described by Spengler (1950), has gained a significant attention in the economics, marketing and supply chain literatures.1 Typically, double marginalization results in an inefficient outcome (from the system wide perspective), and therefore is of great practical concern. The best known example, and the one that concerns us in this paper, is that of using wholesale pricing in a bilateral monopoly, namely the case of a single buyer and a single seller.2 The result, citing Cachon (2003, p. 238), is that “ … in this serial supply chain there is coordination failure because there are two margins and neither firm considers the entire supply chain’s margin when making a decision”. Since the possibility of double marginalization exists in any decentralized system, the issue has been widely studied. The scope of analyzed settings ranges from the simplest bilateral monopoly with deterministic or stochastic demand to rather involved ones that involve price dependent stochastic de-

1 2

For representative publications and reviews see Tirole (1988), Moorthy (1993) and Cachon (2003). See Machlup and Taber (1960) and Williamson (1979) for more information on bilateral monopolies.


mand, competition on one or both sides of the supply chain, multiple periods, different replenishment options, asymmetric information, etc. A variety of different schemes have been suggested to address the problem of double marginalization. Indeed, in the case of full information, parties can always attain the first-best outcome by using an appropriate pricing scheme, regardless of the number of firms involved, their roles and decision spaces; such schemes are commonly known as “coordinating contracts.” However, such schemes tend to be rather complicated, as in Chen et al (2001), and potentially, difficult or costly to implement. Still, the issue is far from settled. Furthermore, coordinating contracts designed for a particular environment tend to fail when transplanted into a different one. For example, Bernstein and Federgruen (2005) demonstrate that a buy-back contract that works perfectly when retailer’s only decision is the quantity to order fails if pricing becomes an extra option. Therefore, there has been a tendency for the literature to address the issue of double marginalization on a case-by-case basis. In this paper, we would like to show how competition, quite generally, addresses the problem of double marginalization in the case of wholesale prices. Competition changes the player's incentives; in particular, it can temper the incentives that push prices above the first best solution. As a result, competition may reduce the double marginalization problem. Competition can, however, also serve an additional role. In particular, it can elicit information. Just as simple auctions, in effect, elicit market-clearing prices, horizontal competition in a supply chain may elicit information needed to write a fully coordinating contract. In other words, wholesale pricing with competition can result in a relatively simple yet robust mechanism...a mechanism that does not require the contract writer to have as much situation specific information as many of the other, more complex, competition free, alternative schemes would. We are not the first to consider competition in supply chains. Indeed, Moorthy (1993) notes (p. 182): "The more interesting issues in channel competition arise from the effect of downstream (retail) competition on relations between the manufacturer and the retailers". Mahajan and van Ryzin (2000) analyze a model of the two-echelon supply chain in which a monopolist retailer stocks products from horizontally competing suppliers. When the retailer has rights to manage inventory (RMI) then two suppliers


suffice to fully coordinate the channel. They engage in Bertrand (1883) competition and in the equilibrium sell their products to the retailer at cost. Double marginalization disappears. Competition on the supply side is not that severe when it is suppliers who manage the retailer’s inventory (called vendormanaged inventory, or VMI) but, nevertheless, the channel profit approaches first-best as the number of suppliers increases. Mahajan and van Ryzin (2001), considering a model parallel to that of Lippman and McCardle (1997), show that in the equilibrium retailers overstock, and as the number of retailers increases their profits approach zero. The model does not have a supplier in it, though. Cachon (2003), studies a model with one supplier and many identical retailers. In his (and similar) models, competition takes place because the product(s) goes to one market. He finds that the supplier can coordinate the channel with just a wholesale price contract and make a positive profit. Although in the equilibrium the supplier does not offer a channel-efficient contract, as the number of retailers increases the channel profit approaches first-best. This shows that horizontal competition may help overcome inefficiencies introduced by double marginalization. Notice that even if products are not fully identical, as in Mahajan and van Ryzin (2000), or if they are even complements, as in Netessine and Rudi (2003), parties on one side of the supply chain affect each other because, broadly speaking, they share the same marketplace. Trivedi (1998) makes a remarkable observation (p. 907): "The introduction of competition in the marketplace removes this bilateral monopoly. ... Retail competition thus becomes a substitute for other methods of channel coordination". Our goal is to understand the role of competition in wholesale pricing in general. Therefore, we avoid making any assumptions about how, for example, retailers compete for customers. Instead, our model goes to the extreme of having no competition for customers.3 Each retailer is a monopolist—they have their own customers—and when competition excludes a retailer from the market, it also eliminates that retailer’s customers from the market.

3

In this respect, a model of Ingene and Parry (1995) is similar to ours. The difference is that they study channel coordination under full information only and retailers are heterogeneous. Although they allow the supplier to ex-


Most real world situations are less extreme than our model. Typically, not all of the excluded retailers' customers are lost. But, our extreme case strips away many irrelevant details, and thereby clarifies the role of competition. In the end, however well competition resolves the double marginalization problem for wholesale pricing in our extreme case, it will do even better in any less extreme—more realistic— case. In short, our extreme clearly identifies the role of competition in resolving the double marginalization problem for wholesale pricing, while it, at the same time, provides a lower bound on the beneficial effects of competition in practical situations. Our simple model allows for a relatively simple intuition of our theoretical results. In our model, potential retailers compete against each other to determine who becomes an actual (or, “active”) retailer and who gets excluded from participating in the market. Excluding a retailer also eliminates all of that retailer’s customers; this is a downside to competition. However, competition changes the retailers’ incentives and moves the retailers’ decisions toward the first best solution. If the total marketplace consists of N markets, and the supplier can induce competition by excluding K of those markets, then competition excludes at most K/N of the total market, while moving the remaining (N-K)/N of the market closer to full coordination. How much the remaining retailers move toward the first best solution may, in general, depend on the level of competition, and therefore on K. However, even a little competition may go a long way. In particular, in the case of homogenous retailers, excluding even just one retailer fully coordinates the remaining retailers. In this case, wholesale pricing with competition can fully coordinate (N-1)/N of the market; for large numbers of homogeneous retailers, wholesale pricing with competition approaches first best. For heterogeneous retailers, the optimal K may be larger—and, therefore, a larger fraction of the market may be excluded—and the effect of competition my not be extreme—the remaining retailers need not be fully coordinated. However, the same forces will still be at work. Therefore, for any large enough number of retailers who are close enough to being homogeneous, wholesale pricing with competition will clude retailers from trade they find that a two-part tariff generically cannot cope with double marginalization: the


still approximate full coordination, and will do so even in the extreme case where competition completely eliminates all of the excluded retailers’ customers from the market. Now let us compare wholesale pricing with competition to wholesale pricing without. Without competition, the N retailers look like N independent, parallel, single retailer supply chains. In this case, double marginalization prevents such systems from being fully coordinated. For example, as we show below, in the case of linear demand functions, double marginalization typically leaves the supplier exactly half as much revenue as from a fully coordinated mechanism. One might expect that the total effect of double marginalization increases proportional to N as N increases; our experimental results support this prediction. However, competition more fully coordinates the remaining retailers, and this may more than make up for losing some of the market. For example, in the case of homogeneous retailers, competition fully coordinates the remaining retailers, and the supplier will prefer N-K fully coordinated retailers to N less coordinated retailers—whatever the loss of coordination from double marginalization in the absence of competition—so long as N is large enough relative to K. In short, for large enough N, wholesale pricing with competition can outperform wholesale pricing without competition even in the extreme case where competition completely eliminates all of some retailers’ customers from the market. Even very small numbers of retailers can be “large enough.” Exactly how large N needs to be depends on the functional form of the retail demand function. For the commonly used example of linear demand, wholesale pricing with competition should, in theory, outperform wholesale pricing without competition whenever there are at least three homogeneous retailers. We experimentally test the case of three homogeneous retailers and find strong support for our theoretical prediction. In short, even in our extreme model, modest amounts of competition have dramatic effects on supply chain coordination; in the face of moderate competition, wholesale pricing may perform much better than suggested by the simple one supplier-one retailer model.

supplier can coordinate the channel by selling at cost but in the equilibrium he chooses not to.


Now let us turn to the case of asymmetric information; for example, imagine that retailers have better information about customer demand than does the supplier. Although schemes such as minimum order quantity and two-tier pricing fully coordinate in the case of full information, they typically fail to do so in the face of asymmetric information. However, wholesale pricing with competition can still coordinate the active retailers in the case of asymmetric information. Similar to the effect of competition of settings with full information, the supplier may end up preferring N-K more coordinated retailers to N less coordinated retailers. Therefore, if N is large enough relative to K, wholesale pricing with competition can do better for the seller than, for example, minimum order quantity contracts. Again, N does not need to be very large for this to happen in theory, and our experimental results suggest that wholesale pricing with competition does at least as well relative to minimum order quantity contracts as the theory predicts for the case of asymmetric information. In short, in the case of asymmetric information, wholesale pricing with competition can outperform schemes that are first best in the case of full information; wholesale pricing may indeed be an appropriate mechanism in real world situations where there is competition and asymmetric information. The rest of the paper is organized as follows. Section 2 formally defined our model and derives our theoretical results. In Section 3 we develop a numeric example to illustrate the theory in a simple manner. In Section 4, we develop our research hypotheses and the design of the laboratory experiment. We present results of the laboratory study in Section 5 and summarize the paper, discuss the implications of our results, and offer managerial implications in Section 6.

2. The Model 2.1. Environment, notation and terminology In our model, a supplier produces a single good and has no direct access to the marketplace. Instead, he sells the product to one or more homogeneous retailers, each a monopolist on an isolated market. As discussed above, this extreme model is very useful in that it 1) avoids making specific assumptions about how, for example, retailers compete for customers, 2) more clearly reveals the role of information than


more complex models might obscure, and 3) provides a lower bound on the beneficial effects of competition in more realistic situations. The manufacturer incurs a cost c > 0 and we normalize retailers’ costs to 0 (all costs are public information). Each market is characterized by a downward-sloping demand function, q = d (r ) , where

r > 0 is the retail price per unit. If the trade takes place, the supplier (S) and the retailer (R) earn S and

R so that the channel (C) total surplus is sum of the two profits: C = S + R . We will also use superscripts to denote values resulting from various optimization problems, so for example, C (r FB ) is the firstbest channel surplus, whereas E[ SMOQ I ] is the supplier’s expected profit resulting from the Minimum Order Quantity (MOQ) contract under incomplete information (I). When we call an outcome efficient or inefficient, we mean that the channel surplus in that case is equal to or, respectively, smaller than the first-best channel surplus. We model the process as a two-stage game in which the supplier moves first and makes a “take-it-or-leave-it” offer to the retailer, and the retailer moves second by specifying an order quantity q. We use the subgame-perfect equilibrium as the solution concept. In what follows, we assume that parameters of this environment are such that contract variables can be chosen in a way that makes trade a (weakly) better outcome for both parties than no-trade.

2.2. The First-Best Benchmark A useful benchmark, the first-best solution, obtains in the situation when the manufacturer can access markets directly. The game reduces to the classic monopoly problem of choosing the optimal price p*:

p * (c) = arg max S , where S = C (r c)d (r )

(1)

r

We assume that the demand function is strictly monotone, which insures that the objective function of (1) is unimodal.


For any contract in which the retailer is paid only a lump-sum amount R0 > 0 (which can be thought of as the expected value of the retailer’s outside option), the supplier’s problem remains equivalent to (1) and the supplier’s profit is merely the first-best channel profit less the lump-sum payment to the retailer: SFB = CFB R0 . But when the retailer acts autonomously and earns profit proportional to sales, it is well known that the decentralized decision-making causes double marginalization and special contractual arrangements are required to avoid an inefficient outcome. 2.3. The Wholesale Price Contract One of the possibilities for the supplier to arrange the trade is to offer a constant wholesale price per unit, w c . We refer to this as a wholesale price contract, or more simply, wholesale pricing (WP). When w is small enough so that the retailer finds it beneficial to accept the offer, the retailer's bestresponse is to order q = d (r ) where the retail price is

r ( w) = arg max( x w)d ( x)

(2)

x

Given the retailer's best-response function r ( w) , the supplier chooses the optimal wholesale price w* by solving:

w* = arg max (w c)d(r(w)).

(3)

w

Note that for the retailer to make a positive profit, the retail price must be strictly higher than the wholesale price: r ( w) > w . Also, the retailer's problem (2) is structurally the same as (1), with w instead of c, which leads to the following conclusion (all proofs are in the Appendix): Proposition 1. The retailer places a first best order if and only if the wholesale price faced by the retailer is equal to the supplier’s cost. More precisely, qWP = d (r FB ) , if and only if w = c. As a result of this proposition, we can argue that WP will, in general, yield less profit to the supplier than the first best solution. Specifically, in order to make positive profit from WP, the supplier must set w>c, but Proposition 1 assures that this results in something other than a first best outcome. Formally, we state the following:


Proposition 2. If the first best profit is strictly positive, then any wholesale price contract gives the supplier a strictly lower profit. In addition, if the supplier makes a positive profit from a wholesale price contract with price w, then that wholesale price contract is inefficient; in particular, d (r ( w)) < d (r FB ).

2.4. The Minimum Order Quantity Contract One way to cope with the inefficiencies of the decentralized channel is to employ a minimum order quantity contract (MOQ). The supplier's offer now consists of two values, the wholesale price and the minimum order quantity: {w, qmin }. Since for a given w > c the retailer typically orders less than the first-best, the supplier can induce the retailer to order q = d (r FB ) only by setting qmin = d (r FB ) , giving us the following straightforward proposition: Proposition 3. In a minimum order quantity contract, the supplier should set the minimum order quantity qmin equal to the demand d (r FB ) that would be realized from using the first best retail price r FB . More precisely, for any qmin and w > c , such that the retailer’s participation is secured, the retailer orders first-best if and only if qmin = d (r FB ) . The retailer's participation can be insured by a proper choice of w r FB (for example, if the retailer's reservation level is zero, then the supplier should set w = r FB ). More specifically, the supplier chooses w based on (r FB w)d (r FB ) = R0 . This way the supplier induces the retailer to serve the market efficiently and manages to extract all (but for the retailer’s reservation level) profit from the retailer. 2.5. Incomplete Information In this section we introduce incomplete information (I). Suppose that the supplier knows the structure of the retail market but not its size, so that the market demand is D(r ) = Zd (r ) , where Z > 0 is a non-degenerate random variable with a finite mean E[ Z ] . We assume that Z is not ex post verifiable and cannot be contracted upon. The retailer knows the realization of Z and her reservation level is also scaled by Z to Z R0 . The supplier does not know the realization of Z but does know its distribution.


This uncertainty in market size does not significantly change our analysis of the wholesale price contract. Problems (1), (2) and (3) are simply multiplied by Z, and the solutions (prices) remain the same. Importantly, for the wholesale price contract, it does not matter for the supplier whether he knows Z or not. Thus, the wholesale price contract is robust to this type of uncertainty. However, when Z is the retailer's private information, the supplier is no longer able to achieve efficiency by means of an MOQ contract. This is not surprising. In essence, the supplier is facing an adverse selection problem. Formally, we state the following: Proposition 4. The supplier’s maximum expected profit that can be achieved under information asymmetry by means of a minimum order quantity contract is strictly smaller than the expected profit that can be achieved under full information:

E[ SMOQ I ] = E[ Z ] SFB , where 0 < 1

(4)

where “MOQ-I” denotes the minimum order quantity contract under incomplete information. The term

E[ Z ] SFB appears on the right-hand-side of (4) because under full information the supplier can always achieve the first-best solution, scaled by a factor of Z. Intuitively, by Proposition 2, the only way to induce the retailer to order the first-best quantity,

Zd (r FB ) , is to set qmin = Zd (r FB ) . However, this is not possible because Z is the retailer’s private information. For the same reason, it is no longer possible for the supplier to extract the entire retailer’s surplus. 2.6. Wholesale Price Contract with Competition A supplier who faces N 2 potential retailers, each serving a separate market as a monopolist, could make the retailers compete for a contract. We call this arrangement the wholesale price contract with competition, or, more simply, wholesale pricing with competition (WPC). Specifically, the supplier offers a wholesale price, letting retailers freely choose order quantities, but announces that the K retailers (where

1 K < N ) with the smallest orders, will be excluded from trade. Ties get broken according to any rule that gives each of the tied retailers a strictly positive chance of winning.


In general, retailers could differ on a variety of dimensions, including, for example, their costs or their market size. However, if the retailers are similar enough, then—by continuity—the underlying forces will be similar to those in the case of homogeneous retailers. Assuming homogeneity considerably simplifies the analysis, and, at the same time, help illuminate the underlying forces. Therefore, we focus on the case of homogeneous retailers; throughout our theory, multiple bidders are always presumed to be homogeneous. At equilibrium, homogeneous retailers compete their profits away (down to the reservation level) and the supplier randomly selects ( N K ) of them to deal with.4 Now the supplier can obtain the firstbest from each of the chosen (or active) retailers. Formally,

Proposition 5. The supplier’s expected profit E[ SWPC ( N K )] from wholesale pricing when N potential retailers compete to become one of ( N K ) active retailers is

E[ SWPC ( N K )] = ( N K ) E[ Z ] SFB .

(5)

Note that this expected profit increases as K decreases. In order for there to be competition, K must be at least one. Therefore, the optimal K = 1. However, our claims for the potential superiority of WPC still hold even if the supplier were to set K > 1.5 Corollary (to Proposition 5): In the absence of additional constraints on K, the optimal K = 1. We now compare the supplier’s expected profits using MOQ versus WPC. The seller’s expected profit from using the optimal MOQ contract with N identical retailers is simply N times the amount specified in Proposition 4. How this compares with the expected profit, specified in Proposition 5, from using 4

Heterogeneous retailers who were nearly homogeneous would nearly compete away all their profits and would

make nearly the same decisions as homogeneous retailers, and the supplier would realize nearly the same surplus. 5

For example, there may be considerations outside our model—such as the supplier’s long term relationship with

the retailers and never wanting to single out just one retailer as a loser—that further constrain the supplier’s choice of K.


wholesale pricing with competition depends on specific details such as the number of available retailers N, the number of active retailers, the demand function, and distribution of Z . However, when N is large enough, the supplier’s expected profit under WPC is higher than under the MOQ. Formally,

Proposition 6. For any non-degenerate distribution of the market scaling factor Z (i.e. the information is actually incomplete) and any fixed number of losing retailers K (and fixed values of the other parameters except N), if the number of potential retailers N is big enough, the supplier can do strictly better using wholesale pricing with competition (WPC) than using any minimum order quantity contract. For the example considered below, = 1050/1400. In this case, WPC with K = 1 and MOQ yield the same expected profit of 4200 to the supplier when N = 4, and MOQ yields a strictly higher expected profit whenever N > 4. Note that for both WP and WPC, the supplier’s decisions are independent of the state of the markets. Moreover, he need not even be aware of the information asymmetry at all (let alone values of the scaling factor and their distribution). While this simplifies the supplier’s decision, it also limits the supplier’s ability to tailor the contract to the state of the market. In general, this can limit the effectiveness of wholesale pricing. However, competition results in the retailers’ offer reflecting the market state; in effect, competition allows the contract to depend on the market state even though the supplier’s decision does not. As a result, with WPC the supplier will make the first-best profit on every market he did not exclude from trade. Summarizing, theory argues that: (i) Under full information double marginalization can be avoided, for example, by means of an MOQ contract, (ii) Under incomplete information no contract achieves firstbest and, (iii) By making retailers compete for the right to trade, first-best can be achieved asymptotically with just a wholesale price contract regardless of the information structure.


3. Example for the Laboratory Setting To illustrate the ideas, consider a specific example. The retailer is facing market demand

d (r ) = Z (100 r ) and her reservation level is R0 Z = 200 Z , where Z is a scaling factor. The supplier's per unit cost is c = 20. To begin with, let us look at the full-information case: Z 1 . It is easy to show that in order to maximize the channel profit, (r 20)(100 r ) , parties need to set the retail price to

r=

1 (100 + 20 ) = 60 sell q = d (r ) = 100 60 = 40 units . The resulting (channel maximal) profit is 1600. 2

Since the supplier cannot access the market directly, that is the retailer's reservation level must be taken into account, he can make at most 1600 200 = 1400. For the supplier it is the (constraint) first-best. One of the ways the supplier can secure this profit is to offer a MOQ contract. To find the optimal parameters for the MOQ contract supplier solves:

max( w c)qmin w,qmin

s.t .

qmin (r (qmin ) w) Z R0 ((100 qmin ) w )qmin 200 0 4 200 MOQ F = 12 (100 20) = 40. The retailer acWhich gives us wMOQ F = 12 100 + 20 = 55 and qmin 100 20 cepts this offer because by selling q = 40 units at a price of 60 she earns (60 55) 40 = 200 = R0 , i.e. the participation constraint is not violated. The supplier then earns (55 20) 40 = 1400. If the supplier offers a WP with the wholesale price w, the retailer's best-response (derived by maximizing the retailer's profit R ( w) = (r w)(100 r ) ) is to order q = 12 (100 w). The resulting supplier's profit is then S = 12 ( w 20)(100 w) , maximized at w =

1 2

(100 + 20 ) = 60.

The retailer then orders

q = 20 units and sells them on the market at price r = 100 20 = 80. Her profit is (80 60) 20 = 400 , that is the retailer accepts this offer. The supplier's profit is only (60 20) 20 = 800 . If the supplier can induce competition among retailers by announcing that there will be no trade with one of the retailers whose order is the smallest, then we have a wholesale price contract with competition.


The supplier maximizes his profit by setting w so that by ordering q = 40 (the first best order quantity) and selling it at the retail price of 60, the retailer earns his reservation profit of 0 = 200 . The reader can verify that this happens for w = 55. In the unique equilibrium of the resulting game all retailers order

q = 40 and one of them is excluded from trade. The supplier loses one market completely but earns 1400 (the first-best profit) in each remaining market. Therefore, with just three potential retailers it is strictly better for the supplier to use WPC (in this case he earns 1400 2 = 2800 ) than to deal with all three retailers and offer an optimal WP (in this case he makes only 800 3 = 2400 ). Still, under full information, an MOQ is a better option because in this case the supplier can make first-best profit on every market. However, this is no longer true if we look at the problem in the incomplete information environment. Consider a two-type case (generalization to the continuum of types is immediate) where Z {z L = 12 , z H = 32 } and Pr( Z = z L ) = Pr( Z = z H ) =

1 2

(Note that the private information

parameters were chosen so that E [Z ] = 1 , just as in the full information case). As we argued above, if the supplier offers a WP both the optimal wholesale price and the resulting market price are invariant with respect to the realization of Z (the same is true for WPC). The resulting order quantity and the supplier’s profit (from every active retailer) are simply scaled by Z and, therefore, in expectation the supplier earns the corresponding full-information profit multiplied by E[ Z ] :

E[ SWP I ] = E[ Z ] SWP F = 800 and E[ SWPC I ] = E[ Z ] SWPC F = 1400 . Considering MOQ, one can show that an optimal contract either 1. extracts all the retailers’ profit when markets are high but retailers do not accept it when markets are low (shutdown of inefficient types) or 2. makes retailers’ participation possible in both market states but extracts all profit from the retailers when markets are low. We omit details because they are quite similar to the standard analyses of the adverse selection problem and refer interested readers to Salanie (1997) or Laffont and Tirole (1993). In fact, the whole argument applies to the optimal screening contract as well. The main insights are that in the presence of private information designing an optimal contract requires knowledge of the distribution


of Z but, nevertheless, full-information first-best is no longer achievable. In our environment, it is optimal to offer {w = 55, qmin = 3 40 = 60} . Retailers accept this offer only when markets are high. The 2 supplier, when the offer is accepted, earns the full-information first-best,

3 2

1400 = 2100 , but this hap-

pens only with probability 12 , and when markets are low, this contract gives zero, and therefore the supplier’s expected profit is 12 2100 = 1050 . Alternatively, if the supplier offers {w = 57.6, qmin = 18.5}, retailers should accept this contract in both high and low markets, and each retailer orders q = 18.5 when markets are low and

q = 31.8

when markets are high. The supplier's expected profit is

(12 18.5 + 12 31.8)(57.6 20 ) = 945.6

, which is smaller than the profit from the contract designed to

deal with high markets only. For the reader’s convenience, we summarize the optimal contract parameters and the resulting expected profits in the Table 1.

4. Research Hypotheses and Experimental Design In the laboratory experiment that we use to test the model we set the parameters as in the example in section 3: c = 20 , R0 = 200 , N = 3 and d (r ) = Z (100 r ) . We manipulate two factors in our design: (1) Information can be either Full (F) or Incomplete (I); under full information Z = 1 and under incomplete information Z {z L = 12 , z H = 32 } , and (2) The contracting arrangements we consider are the wholesale-price contract without competition (WP) and with competition (WPC), as well as the minimum-order-quantity contract (MOQ). The reader will recall that in this setting, under full information, the first-best outcome is the order quantity of q = 40 and the total supply chain profit of 1600. The maximum attainable expected profit for the supplier is N (1600 – 200) = 3 1400 = 4200. In all treatments, the supplier is the first mover. In WP and WPC treatments the supplier offers the per unit wholesale price w. In the MOQ treatments, the supplier offers w and qmin. The software transmits the supplier’s offer to the retailer and the retailer responds with the order quantity. The


retailer also has an option to reject the contract, which causes the supplier to earn the profit of 0 from this retailer, and the retailer to earn 200Z. Both, the supplier and the retailer have access to a simple calculator that we programmed to help them with their decisions. Suppliers can enter different w’s (and qmin’s in MOQ treatments) and the system calculates the retailer’s best reply to that offer. Retailers can enter different q’s and the system calculates the retailer and the supplier profit corresponding to this decision. See Appendix 3 for sample instructions and screen shots. In Table 1, we summarize our design and theoretical predictions based on the model in the previous section for contract parameters and the resulting profits in each treatment. For the MOQ contract with incomplete information we provide theoretical benchmarks for arrangements designed to deal with the high markets only, as well as with both market types. Note that although the supplier maximizes his expected profit by only transacting in the high market, the differences in expected profit are small. In addition to the four treatments in Table 1 we conducted a control treatment with the wholesale-price contract in which each supplier was matched with a single retailer only (labeled WP-1). The retailer profit, w and q in WP-1 is the same as in WP, and the supplier expected profit is 800.

Contract Wholesale Price (WP)

Information Full (F) Incomplete (I) 1. Treatment WP w = 60, q = 20, E [ S ] = 3 800 = 2400 , E [ R ] = 400

Wholesale Price with Competition (WPC)

2. Treatment WPC w = 55, q = 40, E [ S ] = 2 1400 = 2800 , E [ R ] = 200

Minimum Order Quantity (MOQ)

3. Treatment MOQ-F w = 55, qmin = 40, E [ S ] = 3 1400

= 4200, E [ R ] = 200 .

4. Treatment MOQ-I High Market: w =55, qmin = E[q] = 40zH = 401.5 = 60, E [ S ] = 3 1400 z H (1 p ) =

3 1050 = 3150, E [ R ] = .5 300 = 150

Both Markets: w = 57.6, qmin = 18.5, E[q] = 0.5 18.5 + 0.5 30 = 24.25 E [ S ] = 3 945.59 = 2836.74,

E [ R ] = 0.5 100 + 0.5 600 = 350 .

Table 1. Experimental design and theoretical predictions.


Each treatment included three cohorts of 8 participants. Each cohort included two participants in the supplier role and six in the retailer role. Each round one supplier was randomly matched with three retailers (in WP-1 each cohort included four supplier and four retailers, randomly re-matched each period). Each participant kept the same role for duration of the session, which included 40 periods. Thus in total, 120 subjects participated in our study. Participants were randomly assigned to one of the five treatments and to one of the two roles within a treatment. The experiment was programmed and conducted with the software z-Tree (Fischbacher 2007). Participants were Penn State University students, mostly undergraduates, from a variety of majors. We recruited them using on-line recruitment system, and offered earning cash as the only incentive. Each subject participated only once. We conducted all sessions at the Laboratory for Economic Management and Auctions (LEMA) at the Penn State Smeal College of business, during November 2006 through June 2007. We formulate the following research hypotheses dealing with the performance of these contracts. Hypothesis 1: Contract parameters and resulting (expected) profits will be as summarized in Table 1. Under the MOQ contract with incomplete information suppliers will transact in high market types only. Hypothesis 2: Supplier profits will be higher under the wholesale-price contract with competition than without competition. Retailer profits will be lower with competition than without competition. These comparisons are independent of information. Hypothesis 3: The MOQ contract will deliver higher supplier profits and lower retailer profits than the wholesale price contract with or without competition, under both, full and private information.

5. Results 5.1 Statistical Analysis In Table 2 we summarize the average wholesale prices, order quantities, acceptance rates, and profits for each treatment over 40 rounds. In each case, we also indicate whether the average quantity is


significantly different from its corresponding theoretical benchmark from Table 1 according to a t-test. The unit of observation is a cohort, and each treatment consists of three independent cohorts. Profit Treatment WP-1 WP WPC MOQ-F MOQ-I

w

q

53.55** (1.59) 54.13** (3.34) 53.71+ (1.58) 49.31** (3.34) 52.63+.+ (4.47)

19.92 (2.07) 22.19** (1.22) 33.96** (2.74) 29.78** (3.25) 23.28**,+ (2.74)

Acceptance Rate 0.93 (0.02) 0.96 (0.03) 0.99 (0.02) 0.85 (0.04) 0.46 (0.03)

Retailer

Supplier

518.64** (43.15) 498.69* (78.01) 310.99** (3.88) 498.17** (82.94) 543.48**,+ (120.44)

636.46** (42.27) 2,157.61* (162.76) 2,359.71** (45.91) 2,462.17** (92.53) 2,181.72**,** (41.34)

Comparisons to theoretical benchmarks in Table 1 (two-sided t-test): * p < 0.05; **p < 0.1; + p > 0.1. For MOQ-I the first set of asterisks refers to comparisons to high market benchmarks and the second set to comparisons with both markets benchmarks. Table 2. Averages and standard deviations (in parenthesis) of wholesale prices, order quantities, acceptance rates, and profits, by treatment.

In Figure 1, we plot the wholesale prices (w) on the x-axis and the resulting order quantities (q) on the yaxis for each transaction in our study (all 40 periods). We plot transactions in the WP-1 and WP treatments on the same graph, and the transactions in other treatments on separate graphs. The solid line represents the retailers’ best reply for a given w. 120

100

80 Q

60

40

20

0 25

30

35

40

45

50 W

(a) WP-1 and WP treatments

(b) WPC treatment

55

60

65

70


(c) MOQ-F treatment

(d) MOQ-I treatment

WP-1 WP,WPC, MOQ-F, MOQ-I --- retailer best reply ––best reply of the high demand retailer to low demand contract in the MOQ-I treatment. Figure 1. Wholesale prices and order quantities in the five treatments.

The data is largely inconsistent with Hypothesis 1. Average wholesale prices under the WP-1 and WP are significantly below 60 and under the MOQ-F they are significantly below 55. Under the WPC and MOQI, average wholesale prices are not significantly different from 55, however, the average order quantities are significantly below 40 under WPC and 60 under MOQ-I. The average order quantity under MOQ-F is also significantly below 40, but under WP it is significantly above 20 (and also slightly below the best reply to the average w). To better understand how the contracts that are offered affect profits, we plot, in Figures 2 and 3, wholesale prices and the retailer and supplier profits that result from these transactions. First let us examine Figure 2, which displays the full information treatments. The solid concave lines represent the theoretical supplier’s profit for each wholesale price, and the dashed line represents the theoretical retailer profit for each wholesale price. The solid vertical lines represent the optimal wholesale prices.


(a) WP-1

(b) WP

(c) WPC

(d) MOW-F

Retailer Profit Supplier Profit –– Supplier Theoretical Profit --- Retailer Theoretical Profit

Figure 2. Supplier and Retailer profits in full information treatments. In the WP-1 and WP treatments suppliers tend to offer wholesale prices that are on average below 60. Retailers, in turn, behave very close to the best reply. We can tell that this is so because the majority of the supplier and retailer profit observations are located around their theoretical prediction lines. As a result, suppliers earn less than the theory predicts and the retailers earn more. The situation is different in the WPC and MOQ-F treatments. In the WPC treatment the average wholesale price is right around 55 (although there is a substantial amount of variability). However, retailers do not fully compete away their excess profits, and consequently supplier profits generally lie below the optimal theoretical line and retailer profits are above 200. Also note that in the WPC treatment there are almost no rejections (over 99% of the contracts are not rejected). The pattern of behavior in the MOQ-F treatment differs from the WPC because average wholesale prices are significantly below 55. This may well be due to much higher rejec-


tion rates (almost 15% of offers are rejected). On average, retailers earn more than 200, which means that suppliers do not set qmin as high as they should to extract all excess supplier profit. Again, this is likely due to rejections of contracts that attempt to extract too much of the surplus. We now turn to Figure 3 to examine behavior in the MOQ-I treatment.

Retailer Profit Supplier Profit –– Supplier Theoretical Profit --- Retailer Theoretical Profit Figure 3. Supplier and Retailer profits in the MOQ-I treatment. The three solid concave lines represent theoretical supplier profits in the high market (top), in the low market (bottom) and in the high market when the contract is set to deal with the low market (middle). The bottom dashed lines represent retailer profit of 200, which is the theoretical profit when the supplier deals with the high market only or with the low market and the realized market type is low. The top dashed line represents retailer profit when the supplier deals with the low market but the realized market type is high.

The main feature to note about the behavior in the MOQ-I treatment is that there is virtually no attempt by suppliers to extract most of the profit from high type markets. Instead, suppliers appear to set w and qmin in a way that does not exclude low markets. High market retailers appear to behave close to the best reply to those contracts (the average order is not significantly different from the best reply to the contract designed for both markets), as evidenced by clusters of data around the middle concave line and the top dashed line. Of course when high market retailers’ best reply to contracts designed to extract most of the low market surplus, these retailers make a substantial amount of profit above their reservation levels. Turning to hypothesis 2, we compare average supplier profits under WP and WPC. We find that the data supports the hypothesis. Although not as high as the theory predicts, supplier profits under WPC


are nevertheless higher than under WP (p= 0.0783). When we make the comparison over the last 10 rounds of the session, the differences become strongly significant (p= 0.0303). Also consistent with hypothesis 2, retailer profits, although above theoretical predictions in both treatments, are lower under WPC than under WP (p=0.0253). The data is mostly inconsistent with hypothesis 3. Under full information we find comparisons of supplier profits to be in line with the hypothesis. Although lower than theoretical benchmarks, supplier profits under MOQ-F are significantly higher than under WP (p=0.0314). Interestingly, average supplier profits under MOQ-F are only marginally higher than under WPC (p=0.0934) and the differences do not become much more significant when we look at the last 10 rounds. Average retailer profits under MOQF are not different, from retailer profits under WP (contrary to hypothesis 3) or under WPC (consistent with hypothesis 3, but note that average profits in both treatments are significantly above their theoretical benchmarks of 200). Under incomplete information we find no support for hypothesis 3. We find no difference in average supplier profits (p=0.4122) or average retailer profits (p=0.3101) between the MOQ-I and WP treatments. Average supplier profits are actually significantly higher under the WPC treatment than under the MOQ-I treatment (p=0.0039) and the average retailer profits are significantly lower (p=0.0394). The conclusion is that the MOQ contract performs much worse than it should under both, full and incomplete information. What is surprising is that although the WPC contract does not perform as well for the supplier as the theory suggests, it is nevertheless extremely robust to the type of incomplete information we investigate. It performs so well for suppliers that under incomplete information (contrary to standard theory) it even outperforms the MOQ-I contract. In the next section, we discuss a potential explanation. 5.2 Discussion The key to understanding the results of our study, and why standard theory largely fails to predict several key features of contract performance has to do with how suppliers and retailers split profits. Figure 4 plots supplier profit share on the x-axis and the retailer profit share on the y-axis. For clarity, we combine


the data in the WP-1 and WP treatments in Figure 4a, and data in MOQ-F and MOQ-I treatments in Figure 4b. For comparison, we plot data from WPC treatments in both parts of the figure.

WP and WP-1 prediction

WPC/MOQ-F predictions MOQ-I prediction

WP-1 - WP WPC --- 50/50 split

MOQ-F - MOQ-I WPC --- 50/50 split

— First Best

— First Best for zL (bottom) zH (top) WPC (middle)

(a) WP-1, WP and WPC treatments

(b) MOQ-F, MOQ-I and WPC treatments Figure 4. Profit distribution

Participants in every treatment of our study divide profits closer to the 50/50 split than the theoretical prediction. In WP-1 and WP treatments, the differences between the data and theoretical prediction are smallest (although still significant) largely because theoretical split is less extreme, with the supplier commanding 2/3 of the profits and the retailer commanding 1/3. Under the WPC and MOQ contracts, theory implies that the supplier receives 87.5% of the total profit, with retailers merely receiving their outside option, and the actual divisions do not resemble this theory. In the WPC treatment, supplier average share is just under 76%--significantly below 87.5%. In all other treatments, average supplier share is just around 60%, significantly lower than the average supplier share under WPC. The main idea that organizes our results and explains why contracting mechanisms systematically yield lower supplier profits than the theory predicts has to do with social utility. There is a large and growing literature in economics that deals with social preferences (see for example Bolton and Ockenfels


2000) and these ideas are also gaining traction in the operations management literature (Loch and Wu 2007). There is a great deal of laboratory evidence that people are motivated not only by absolute gains, but also by relative gains. For example, in ultimatum games 6 small offers are generally rejected, while 50/50 splits are never rejected. Loch and Wu (2007) report on a study in which subjects in the laboratory play the wholesale contract game. These experiments are very close to our WP-1 treatment, with the exception that our subjects were randomly re-matched each round, while the subjects in the Loch and Wu (2007) study played repeatedly with the same partner for the duration of the session. Interestingly, even without the emphasis on the repeated interaction, behavior in our WP-1 treatment is remarkably similar to the behavior Loch and Wu (2007) report in their baseline treatment: suppliers offer lower wholesale prices than the theoretical prediction, and retailers place larger orders than the best reply. In the Loch and Wu (2007) study, this behavior is more pronounced in treatments in which the saliency of the relationship is increased and is less pronounced in treatments in which the saliency of status is increased, providing additional evince of social preferences.7 The MOQ mechanisms in our study resemble the ultimatum game more than it does the wholesale-price contracting game, because suppliers, in effect, propose a profit division, and retailers can (and do) reject unfair offers.8 Rejection rates in the MOQ-F treatments are around 15%, significantly higher than in all other treatments, save for MOQ-I (rejections in MOQ-I are often due to the low market type). In the WPC treatment retailers compete, and even though they do not compete their profits all the way down to the outside option, as the theory suggests, supplier obtain significantly larger shares. An interesting effect of competition is that it virtually eliminates rejections (rejection rates in the WPC treatment are below 1%). Suppliers in that treatment offer wholesale prices that are not, on average, different from op-

6

In the ultimatum game (Gueth et al. 1982) the first moved proposes a division of a fixed sum of money and the second mover accepts or rejects this division. If the second mover rejects, both players receive 0. 7 Keser and Paleologo (2004) report similar results in the wholesale-price contract game in which the retailer is modeled as a newsvendor, and Wu (2007) finds similar regularities with the buyback and revenue-sharing contracts, and also explains them using social preferences. 8 Ho and Zhang (2005) report laboratory results similar to ours with different coordinating contracts and use loss aversion as the explanation for deviations from standard theoretical predictions.


timal. Competition is used to induce retailers to “voluntarily” increase orders above the WP best reply (the WP best reply to the wholesale price of 55 is the order quantity of 22.5), and competition does so. The average order quantity is increased to 33.96. The order quantity that would drive retailer excess profits all the way down to their outside option is 40, so retailers do not compete away all their excess profits as the standard theory predicts—social preferences continue to play a role even in the face of competition.9

6. Summary and Conclusions The wholesale-price contract is widely used in spite of the fact that in theory it results in double marginalization. We suggest that this may be because practical situations often include horizontal competition and asymmetric information. Competition goes a long way to coordinating a supply chain with wholesale pricing even in the case of full information. Furthermore, in the case of asymmetric information, wholesale pricing (with competition) can actually coordinate the supply chain better than more complex mechanisms that fully coordinate in the case of full information can. In short, while, for example, minimum quantity contracts look very good compare to wholesale pricing in simple models, wholesale pricing can actually outperform minimum quantity contracts in more realistic settings. Competition eliminates double marginalization at a cost of giving up a part of the market. We demonstrate, both in theory and in the laboratory, that under full information and with as few as three potential retailers, a wholesale-price contract with competition yields higher supplier profit than the wholesale-price contract without competition. Of course with full information other mechanisms coordinate supply chains without foregoing a part of the market. Minimum order quantity contract is one such coordinating mechanism, and in our experiments supplier profits under the MOQ contract are indeed higher than under the wholesale-price contract. However, supplier profits are substantially below theoretical predictions because the actual con9

This result may well be in line with the Bolton and Ockenfels (2000) model because in a game with 1 supplier and three retailers, it is reasonable to expect the social reference point to imply that the retailer receives of the total


tracts we observe in our study are substantially less favorable to suppliers than the theory predicts they should be. In fact, even though in theory, under full information the MOQ contract should yield substantially higher supplier profits than the wholesale-price contracts with competition, in our experiment the differences are only marginally larger. In order to illustrate the effects of asymmetric information, we consider a specific simple model. Our model involves only a single stochastic parameter, and this parameter simply scales each retailer’s market. When the supplier needs to set only this single parameter, the wholesale price, the optimal wholesale price is not affected by the incomplete information. Competition under the wholesale-price contract induces the retailers to reveal their private information, so the performance of the competitive wholesale-price contract is not affected by incomplete information. However, the performance of the MOQ contract is degraded by the incomplete information. Therefore, if there are enough retailers, wholesale pricing with competition should outperform the MOQ contract. In our experiments, the number N of potential retailers is three. In theory, this is not enough competition in order for wholesale pricing with competition to outperform the MOQ contract. However, in our experiments, the wholesale-price contract with competition actually performs significantly better than the MOQ contract; wholesale pricing wins out over MOQ contracts even though we had fewer competitors than our theory suggested were necessary. This suggests that competition has yet another affect on decision makers. Specifically, without competition, retailers demand a substantially higher portion of the supply chain profit than the theory predicts they should. Consequently, retailers reject contracts that provide insufficient profits in favor of a smaller outside option. As a result, suppliers tend to offer more favorable contracts and earn less than the theory predicts. In contrast, under competition, retailers act more nearly as simple utility theory would suggest that they do.

profit, which is exactly the average profit share retailers earn in our WPC treatment.


The main lesson from our study is that mechanisms that involve competition are more robust than mechanisms that do not. In bilateral negotiations, parties tend to split the profits more equitably than the theory suggests, even when the rules of the game clearly give one side an advantage. Competition somewhat mitigates social preferences and retailers compete more of their profits away, benefiting suppliers. Competition also makes contracting more robust to information asymmetry by inducing retailers to reveal more of their private information. Acknowledgments The authors gratefully acknowledge support from the National Science Foundation and thank Axel Ockenfels and the Deutsche Forschungsgemeinschaft for financial support through the Leibniz-Program. Katok gratefully acknowledges the support from the Smeal College of Business and the Center for Supply Chain Research (CSCR) at Penn State’s Smeal College of Business through the Smeal Summer Grants Program.

References Bernstein, F. and Federgruen, A. 2005. Decentralized Supply Chains with Competing Retailers Under Demand Uncertainty. Management Science, January 1, 2005; 51(1): 18 - 29. Bertrand, J. 1883. Theory mathematique de la Richesse Sociale. Journal des Savants, 67, pp. 499-508. Bolton, G., Ockenfels, A., 2000. ERC: A theory of equity, reciprocity, and competition. American Economic Review. 90, 166–193. Cachon G. 2003. Supply Chain Coordination with Contracts. In the Handbooks in Operations Research and Management Science: Supply Chain Management edited by Steve Graves and Ton de Kok and published by North-Holland. Chen, F., A. Federgruen and Y. S. Zheng. 2001. Coordination Mechanisms for a Distribution System with One Supplier and Multiple Retailers. Management Science, 47, 5, 693-708. Fischbacher, U. 2007. z-Tree: Zurich Toolbox for Ready-made Economic Experiments, Experimental Economics 10(2), 171-178. Gueth, W., Schmittberger, R., Schwarz, B., 1982. An experimental analysis of ultimatum bargaining. Journal of Economic Behavior and Organization 3, 367 – 388. Ho, T-H, J. Zhang. 2005. Does format of pricing contract matter? Haas School Working Paper. Ingene, C. and Parry, M. 1995. Channel Coordination When Retailers Compete. Marketing Science. Vol. 14, No. 4, pp. 360-377 Iyer, G. 1998. Coordinating Channels under Price and Nonprice Competition. Marketing Science, 17 (Winter), 338–355.


Lippman, S. A. and McCardle, K. F. 1997. The competitive newsboy. Operations Research. Vol. 45, pp. 54-65. Laffont, J. and Tirole, J. 1993. A Theory of Incentives in Procurement and Regulation. Cambridge. The MIT Press Loch, C. H., Y. Wu. 2007. Social Preferences and Supply Chain Performance: An Experimental Study. INSEAD Working Paper. Machlup F, Taber M. 1960. Bilateral monopoly, successive monopoly, and vertical integration. Economica. 27(106):101-19. Mahajan, S. and van Ryzin, G. 2000. Supply Chain Coordination under horizontal competition. Working Paper, DRO-2000-01. Columbia University. Mahajan, S. and van Ryzin, G. 2001. Inventory competition under dynamic consumer choice. Operations Research. Vol. 49, No. 5, pp 646-657 Moorthy, K. S. 1993. Competitive marketing strategies: Game-theoretic models. J. Eliashberg and G. Lilien, eds. Marketing. Handbooks in OR/MS, Amsterdam, North Holland. 5 143--190. Netessine, S. and N. Rudi. 2003. Centralized and competitive substitution. Operations Research. Vol. 51, No. 2, March 2003, pp. 329-335 Salanie, B. 1997. The Economics of Contracts. Cambridge. The MIT Press. Tirole, J., 1988. Theory of Industrial Organization. MIT Press. Trivedi, M. 1998. Distribution channels: An Extension of Exclusive Retailership. Management Science, Vol. 44, No. 7. (Jul., 1998), pp. 896-909 Williamson, O.E. 1979. Transaction-cost Economics: The Governance of Contractual Relations. Journal of Law and Economics, 22: 233-261. Wu, Y. D. (2007), The impact of long-term relationship on supply chain contracts: a laboratory study, University of Kansas Working Paper.


Appendix 1. Lemma 1.

(i)

r (c) = arg max( x c)d ( x) is a strictly increasing function of c and (ii) x

(c) = max( x c)d ( x) is a strictly decreasing function of c . x

Proof of the Lemma 1. Consider c1 < c2 . Let r1 and r2 be maximizers of the corresponding problems. Part (i) is proven as follows: Uniqueness of r (c) follows from the definition of unimodality. Then, from the uniqueness and unimodality, the following two inequalities hold:

(r1 c1 )d (r1 ) > (r2 c1 )d (r2 ) (r2 c2 )d (r2 ) > (r1 c2 )d (r1 )

(

)

Adding them gives (c1 c2 ) d (r1 ) d (r2 ) < 0 . Hence, d (r1 ) > d (r2 ) . Since d ( x)

is monotone,

the latter implies r1 < r2 .

For the part (ii), uniqueness implies:

(r1 c1 )d (r1 ) > ( x c1 )d ( x) = ( x c2 )d ( x) + (c2 c1 )d ( x), x r1 (r1 c1 )d (r1 ) > (r2 c2 )d (r2 ) + (c2 c1 )d (r2 ) (r1 c1 )d (r1 ) > (r2 c2 )d (r2 ) Proof of Proposition 1. “If” part:

If w = c , then problem (2) is fully equivalent to problem (1). Hence both give rise to the

same retail price, and therefore also to the same demand.


“Only if” part:

The demand d (r ) is strictly decreasing in r . Therefore, qWP = d (r FB ) implies

that r ( w) = r FB = r (c). By Lemma 1, the retailer’s best response r ( w) is strictly increasing in w . Therefore, r ( w) = r FB = r (c) in turn implies that w = c . Proof of the Proposition 2. To start, we argue that the solution to (3) can not deliver a first-best profit if the first best profit is positive. To see this, by adding and subtracting the same term, re-write the objective function as

SWP ( w, c) = ( w c)d ( w) + [( w c)(d (r ( w)) d ( w)) ] Now, notice that in this expression (i) the first term can only be as high as the first-best, whereas (ii) the term in the square brackets is negative as long as w c . If w = c , then the whole expression turns to zero. Therefore, the supplier makes either zero profit or strictly less than the first best profit. Finally, we argue the inefficiency of wholesale pricing. Since the supplier is presumed to make a positive profit, first best profit must also be positive and it must be the case that d (r FB ) > 0. If the wholesale price is so high that the retailer foregoes the market, then d (r ( w)) = 0 and it follows immediately that d (r ( w)) = 0 < d (r FB ). So, imagine that the retailer actually sets a meaningful retail price r ( w) . Since the supplier is presumed to make a positive profit, it must be the case that w>c. By Lemma 1,

r ( w) arg max( x w)d ( x) is a strictly increasing function of w, and therefore r ( w) > r (c) = r FB . x

Then, since d(x) is a strictly decreasing function of x, it follows that d (r ( w)) < d (r FB ) for any w such that the supplier makes a positive profit.

Proof of the Proposition 3. To begin with, notice that the retailer’s problem is now different from (2) in that it has an extra constraint:


max( x w)d ( x) x

s.t. d ( x) qmin “If” part:

When qmin = d (r FB ) the retailer cannot order less than d (r FB ) . To prove that the re-

tailer orders d (r FB ) let us suppose the opposite. That is suppose he finds it optimal to order more than d (r FB ) , what means the constraint is not binding. However, if the constraint is irrelevant, the problem reduces to (2), which solution, r ( w) , implies d (r ( w)) < d (r FB ), w > c (by Lemma 1). This contradicts our previous assumption. “Only if” part: Suppose in the optimum the constraint is not binding. Then, by Proposition 1 and Lemma 1, the resulting order quantity is less the first-best. Therefore, the retailer’s order of d (r FB ) implies that the constraint is binding. Again, we have a contradiction. Hence, qmin = d (r FB ) .

Proof of the Proposition 4. Under full information, the supplier knows Z

qmin = Zd (r FB ) .

Therefore,

for

the

and achieves the first-best profit of Z FB by setting

expected

profit

under

full

information

one

ob-

tains E[ SMOQ F ] = E[ z ] SFB . Under information asymmetry, it is no longer possible for the supplier to set

qmin = Zd (r FB ) because he knows only distribution of Z

but not

Z

itself. Therefore, for any

{w > c, qmin > 0} with a positive probability either the retailer does not participate or she prefers not to serve the market efficiently. Hence, E[ SMOQ I ] = E[ Z ] SFB , where 0 < < 1 (superscripts MOQ-I and MOQ-F indicate whether the supplier's profit belongs to the information asymmetry or the full information case).


Proof of the Proposition 5. The supplier has to solve the following problem:

(

)

max N K Z(w c)d(r) w

s.t.

(

)

Z (r w)d(r) 0R = 0 where R0 denotes the retailer's reservation level. Note that Z can be dropped from the objective and from the constraint without changing the solution to the problem; In particular, it does not matter whether the supplier knows Z or not. In addition, (N-K) can also be dropped from the objective. Since the constraint right hand side is zero, the constraint may be added to the objective. Then, rearranging the constraint yields the following transformation of the problem:

max(r c)d (r ) r

s.t. 0

w = r d (Rr ) The objective to this transformed problem is actually the first-best benchmark problem. The constraint simply defines w in terms of the optimal r. Therefore the solution is the first best solution and the supplier' expected profit is:

E[ SWPC ( N K )] = ( N K ) E[ Z ] SFB

Proof of Proposition 6: It follows from equations (4) and (5) that

E[ SWPC ( N K )] (N K ) 1 = lim = >1 MOQ I N N NE[ S ] N lim


Experimental Instructions and Screen Shots (on-line appendix) Appendix 3. All participants were given sufficient time to read instructions and compute answers to the quiz questions. After that, the experimenter read instruction aloud and went over numeric examples and quiz questions using PowerPoint slides. All details of the supplier’s and the retailer’s decision screens and the way to use them were also explained to the participants.

Instructions used in experiments You are about to participate in decision-making experiment. If you follow these instructions carefully you can earn a considerable amount of money. Your earnings depend on your decisions as well as well as on the decisions of other participants. The experiment lasts 40 periods. You will be randomly matched with three other people in the room in each period. You are NOT allowed to communicate with the other participants during the session. If you have any questions, raise your hand and the experimenter will come to help you. The Game Flow In this experiment you will have one of two roles, either a supplier or a retailer. Each round one supplier and three retailers are matched together. The matching will change randomly every round. You will have the same role for the duration of the session. You will learn your role after you log into the game software. The supplier produces a product that costs 20 tokens per unit. Retailers sell this product on the market. During each period the markets of all three retailers can be either of Type HIGH or of Type LOW. There is a 50/50 chance that the markets are all HIGH or all LOW in any given period. A HIGH or LOW market in any period has no effect on the type of the market in any other period (in other words, market type is random and independent). Each retailer will know the type of his own market before he has to make a decision. The supplier does not know the type of the market for the retailers, but only that each type, HIGH or LOW, is equally likely. Each round starts with the supplier announcing a wholesale price W and a minimum order quantity MOQ to all retailers. After the supplier submits his wholesale price W and minimum order quantity MOQ, all three retailers simultaneously make their decisions. Each retailer has two options, either order some quantity Q that is at least as high as MOQ (but can be higher) or reject the supplier’s proposal. When a retailer rejects a proposal the retailer earns 100 tokens in a LOW market and 300 tokens in a HIGH market, in that round. The supplier earns 0 from the transaction with this retailer (but may well have positive earnings from transactions with the other two retailers)


When a retailer accepts the supplier’s contract by placing an order Q (which is at least as high as MOQ) earnings are computed as follows: For each retailer, the retail price P at which the units are sold depends on the order quantity Q and the type of market as follows: In the HIGH type market: P = (150-Q) / 1.5 In the LOW type market: P = (50- Q) / 0.5 The retailer’s profit is then: Retailer’s Profit = (P-W) x Q The supplier’s total profit from the interaction with the three retailers is: Supplier’s Profit = (W-20) x (Q1 + Q2 + Q3) Where Q1 is the first retailer’s order quantity, Q2 is the second and Q3 is the third. Note that if any of the retailers rejected the supplier’s contract, this simply causes the corresponding Q’s to be 0. Example Suppose the supplier offers W = 50 and MOQ = 20. Suppose retailers have a HIGH type market. Suppose the three retailers order Q1 = 20, Q2 = 30 and Q3 = 40. Supplier’s total profit is (50 – 20) x (20 + 30 + 40) = 30 x 90 = 2700. Retailer 1’s retail price from Q1 = 20 is (150 – 20)/1.5 = 86.67 and his profit is (86.67 – 50) x 20 = 1333.33 Retailer 2’s retail price from Q2 = 30 is (150 – 30)/1.5 = 80 and his profit is (80 – 50) x 30 = 900 Retailer 3’s retail price from Q3 = 40 is (150 – 40)/1.5 = 73.33 and his profit is (73.33 – 50) x 40 = 933.20 Suppose instead retailer 1 chooses to reject the offer and retailers 2 and 3 place orders as above. Then: Supplier’s total profit is (50 – 20) x (0 + 30 + 40) = 30 x 90 = 2100. Retailer 1’s profit from rejecting the contract is 300 Retailer 2’s retail price from Q2 = 30 is (150 – 30)/1.5 = 80 and his profit is (80 – 50) x 30 = 900 Retailer 3’s retail price from Q3 = 40 is (150 – 40)/1.5 = 73.33 and his profit is (73.33 – 50) x 40 = 933.20

Information to help suppliers make their decisions

On the supplier’s screen we provide a calculator that allows supplier to enter in possible W and MOQ and click the CALCULATE button. The calculator will then show the Retailer’s order that will


maximize the retailer’s earnings given those W and MOQ for each market Type (or if the retailer is likely to reject because his or her earnings will be below 100 in LOW market and below 300 on HIGH market). The calculator will also show supplier’s average profit which is simply the average of the supplier’s profit from a LOW market type and a HIGH market type. Suppliers can use this calculator as many times as they wish in order to come up with the contract they would like to submit. The contract is transmitted to retailers by using the SUBMIT button.

Information to help retailers make their decisions After the supplier makes his decision, the computer will display the wholesale price W and the minimum order quantity MOQ. The retailers’ computer screens have a calculator that will allow them to enter in different order quantities. Pressing the CALCULATE button and automatically calculate their own profit, and the supplier’s profit from their order. If you are a retailer, you can use your calculator as many times as you want. When you are finished with the calculator you can make your decision by pressing the “Place Order” button to place the order or the “Reject Contract” button to reject the contract. How you will be paid The session will involve 40 periods. In each period you will be randomly matched with other people in the room, but you will have the same role, retailer or supplier, for all 40 periods. Your total earnings from the 40 periods will be converted to US dollars at the rate of 1600 experimental tokens per dollar, added to your participation fee of $5 and paid to you in private and in cash at the end of the session. All earnings are confidential.


Quiz Question 1 Suppose a supplier offers the wholesale price W = 70 and MOQ = 10. If one retailer orders Q1 = 20, what is his retail price IN LOW market? _________In HIGH market? __________ What is his profit in LOW market? ________________ In HIGH Market? Suppose another retail orders Q2 = 10, what is his retail price IN LOW market? _________In HIGH market? __________ What is his profit in LOW market? ________________ In HIGH Market? Suppose another retail orders Q3 = 15, what is his retail price IN LOW market? _________In HIGH market? __________ What is his profit in LOW market? ________________ In HIGH Market? What is the supplier’s profit if retailers 1 and 3 are in HIGH market and retailer 2 is in LOW market? ___________________

Question 2 Suppose the supplier offers W = 40 and MOQ = 20. In LOW market would a retailer make higher profit from ordering Q = 20 or Q = 50? Explain. In HIGH market would a retailer make higher profit from ordering Q = 20 or Q = 50? Explain.


Sample screen shots