SUPPLY CHAIN CONTRACTING AND COORDINATION WITH SHELF-SPACE-
3. Two Competitive Retailers with Demand a Function of Aggregate Inventory
3. Two Competitive Retailers with Demand a
R1 has a lower holding cost (i.e., ), the best centralized decision will be to stock only at R1, and the optimal quantity will thus be the same as when R1 alone owns the whole market. Furthermore, we have that if the two retailers are identical (i.e., ) and are centrally controlled, their total profit will depend only on their total inventory level (i-e., it does not matter how any given total inventory is allocated between the retailers). The optimal total inventory will be the same as when one of the retailers is the only one operating on the given marketplace.
Let us now return to the competitive retailers setting. For this sup- ply chain, we are interested in the following questions: For any given manufacturer contract what are the retailers’ equilibrium inven- tory decisions and their properties? Is the equilibrium unique? Can this two-parameter contractual arrangement coordinate the supply chain?
The general problem in (7.16) and (7.17) is too complex to analyze.
To gain some concrete insights, we will consider the specific demand function of example 7.1:
Such form of inventory-level-dependence was previously used (in a sin- gle retailer case) by Wang (1992) and Parlar et al. (1994). Note that the demand’s inventory-elasticity, equals Sec- ond, we will use the proportional demand allocation model, which was motivated in the Introduction. That is,
and so
Finally, we consider the case of two identical retailers; so we let With these specifications, problem (7.16) - (7.17) now reduces to
The following theorem characterizes the competitive equilibrium for the game defined by (7.18)-(7.19):
Theorem 7.4 The unique Nash equilibrium for each of the two retailers is to stock
and so
At the equilibrium, the system-wide inventory of both retailers will be
Now, had one of the retailers occupied the entire market (and faced the same manufacturer contract), her optimal inventory would have been (specializing (7.9) to our particular demand function):
Comparing this with (7.21), since for we
have That is,
Corollary 7.5 For any given manufacturer contract the total inventory displayed by two competitive retailers is always higher than that of a single retailer (or two centrally controlled retailers). Thus, competition generates inefficiency at the retail stage of the supply chain.
Corollary 7.5 assumes that the prices in the monopolistic and duopolis- tic markets, which are not modelled, will be equal. In practice, the latter is likely to be lower due to competition. It is easy to check that in order for the Corollary’s conclusion to be reversed it will have to be lower than
where is the monopoly price.
The inventory displayed by one of the competitive retailers can be higher or lower than that of a single retailer depending on the value of the inventory elasticity parameter Note that the conclusions here also hold for a price-only contract, since that is merely a contract specialized to
Can a contractual arrangement coordinate this manufacturer- competitive retailers supply chain? The answer is yes! To see this we only need to show that such a contract can bring the two retailers to choose the centralized or system-optimal inventory levels. But, from Proposition 7.6, we know that the total inventory of two retailers (inde- pendent of inventory allocation between them) maximizing the system- wide performance will be the same as that maximizing the performance of a system with a single retailer. Thus, to achieve channel coordina- tion, one only needs to design such that the decentralized total inventory given in (7.21) be equal to the centralized optimal inventory, which can be derived from (7.2) for a single retailer system (specializing to our specific demand function) as
Thus, one can easily show,
then
In light of Corollary 7.5, this coordinating property of a con- tract is particularly valuable: it can actually eliminate the inefficiency generated by the presence of competing retailers within the supply chain!
When coordination is achieved, each retailer’s inventory is
Substituting (7.24) into either (7.18) or (7.19), we can show that the retailers’ profits will be
Thus, again, not only can a contract coordinate the supply chain, but it can also achieve, by varying any desired allocation of the total channel profit between the manufacturer and the retailers.
In concluding this section, we compare the coordination mech- anisms for a single retailer to that for two competitive retailers. First, we see from (7.10) that in the single retailer case the manufacturer does not need to know anything about the demand function in order to design a coordinating contract, assuming profit allocation is not a con- cern. In a sharp contrast, to coordinate a supply chain with competing retailers, he does need to know the demand pattern, captured via the parameter as shown in (7.24). Second, for a given wholesale price the two-retailer supply chain needs a smaller inventory subsidy to be coordinated than the single retailer chain does. (With
this can be seen by a direct comparison of (7.10) with (7.24).) This second point is not surprising since we know from Corollary 7.5 that two competing retailers will always hold more inventory with the same contract. This observation seems to suggest that as the number of retailers grows the optimal subsidy declines. Future research needs to explore that behavior, and in particular whether the subsidy declines to zero as the number of retailers grows to infinity.