SUPPLY CHAIN CONTRACTING AND COORDINATION WITH SHELF-SPACE-
5. Concluding Remarks
As argued by Moorthy (1993, p.182), “The ... interesting issues in channel competition arise from the effect of downstream (retail) compe- tition on relations between the manufacturer and the retailers...”. Our model indeed attempted to capture such interactions within a concrete
setting. We did so (in the analysis of two competitive retailers) by view- ing the system as that of a Stackleberg leader (the manufacturer) who considers the effect of its actions on the resulting Nash equilibrium of the competing retailers (for a similar modelling philosophy, see Gerchak and Wang 2000).
While our work was motivated and presented through demand’s shelf space (or inventory displayed) dependence, the model is, in fact, rather general; I could correspond to any marketing effort. As such, our model can be viewed as a marketing problem as well as an operations problem.
As pointed out, some components of our model – notably the demand split ratios – are often used by marketing researchers. The coordination issues and mechanism we addressed, however, seem new or different than models explored in the marketing literature.
Future research could deal with more general market-share models.
For example, with competing retailers, when each retailer allocates shelf space retailer has a market share of where the coefficient represent retailers relativeeffectiveness of shelf space utilization in attracting demand (e.g., Cooper 1993). Another would be Our models took retail price as given. A more general setting will have a demand which depends on price as well as inventory, and where price is a decision variable. In the duopolistic setting, that may call for a Bertrand-type approach. Since the prices will then depend on the type of market, the relations among the optimal inventory levels will be affected.
Our current models assumed a setting with complete information (though in the single-retailer setting the manufacturer did not always need to know the demand function and holding costs experienced by the retailer). A natural extension is to consider various scenarios where either the retailers or the manufacturer have some private information regarding costs or demand parameters. For recent work, see Ha (1998), Corbett and Tang (1999), Cachon and Lariviere (1999) and references therein.
Another extension to the current models is to consider stochastic de- mand which is influenced by inventory/shelf space. In a periodic review setting, Gerchak and Wang (1994) studied such models for centralized systems. Interesting issues of channel coordination might arise if one considers decentralized decisions with competition in such settings.
Acknowledgments
The authors wish to thank two anonymous referees for their valuable comments. Part of this work was done when the second author was at the University of Waterloo, and was supported by the Natural Sci- ences and Engineering Research Council of Canada. Some of the results appeared in a condensed form in the short paper by the same authors:
“Supply Chain Coordination When Demand is Shelf-Space-Dependent,”
Manufacturing & Service Operations Management, 3(1), 82-87, 2001.
Appendix
Proof of Theorem 7.4: We note first that Anupindi et al. (1999, Theorem 4.1) provide two sufficient conditions for the existence and uniqueness of symmetric equi- librium. Unfortunately, we found that our model can not satisfy their second condi- tion when Now, to find the Nash Equilibrium, we will first characterize R1’s reaction function denoted by R2 will have an identical reaction function.
Prom (7.18), it follows after some algebra that
and
So, is concave in and, thus, (7.A.1) indeed defines the reaction function.
The following lemma partially characterizes the shape of see Figure 7.A.1.
Lemma A.1: In the plane:
1) passes through the following four points:
2) The horizontal coordinate of P4 is longer than the vertical coordinate of P1.
3) is increasing from P1 to P2, and is decreasing from P2 to P4.
Proof of Lemma A.1:
Part 1) For P1, substituting into (7.A.1), we find P2 is the intersection of with the line so, together with (7.A.1),we can find and
For P3 and P4, using (7.A.1) together with and respectively, we can verify their coordinates as well.
Part 2) Since the result can be verified immediately.
Part 3) By implicit differentiation, one can show from (7.A.1) that
At P2, we have Now, from P1 to P2, we have and,
hence, which indicates that is increasing. But, from P2 to
P4, we know that and so and thus must
be decreasing.
End of Proof for Lemma 1.
With our characterization of the reaction function, we are ready to identify the Nash equilibrium point. If we place the reaction function of R2 on the same plane with they will be symmetric across the line since the two retailers are identical; see Figure 7.A.2. Thus, we immediately identify point P3, where and intersect, as one Nash equilibrium. To complete the
proof of Theorem 7.4, we next show that P3 is the unique equilibrium point.
For any given value of let and be the corresponding points on and respectively; see Figure 7.A.2. We need to show that except at point P3, that is, and do not intersect other than at P3.
Now, satisfies (7.A.1), so we have
Similarly, by deriving from (7.19) and then substituting we have
We next show that, except for P3, the segment P4’-P3 of does not intersect with segment P1-P3 of i.e., (That P3-P4 of
does not intersect with P3-P1‘ of will then follow immediately from the symmetry of and Note that on P1-P3 of we have
Now, suppose that Then, from (7.A.2) and (7.A.3), we must have
which contradicts (7.A.4)!
Proof of Theorem 7.7: We need to show that if R2 chooses the best
choice for R1 is as well, that is, for all To
that end, w e will show that starting at has an “S” shape, and it reaches its maximum at see Figure 7.A.3.
The “S” shape of can be shown by studying its derivative function From (7.30), we have,
Now, since its numerator, namely, is strictly decreasing, will start positive and then become and remain nega- tive, which implies that itself will initially be increasing and then become decreasing (i.e., it is unimodal). Furthermore, we can check from (7.A.5) that
which implies that is at the decreasing portion of This, combined
with indicates that starting at
increases to a positive value at some point before and then decreases to zero at and stay negative after Thus, we have showed that
has the “S” shape.
Now, to have for all and we only need for which can be demonstrated simply by substituting into (7.28). This completes the proof.
Proof of Proposition 7.8: From (7.28) and (7.29), we can show that the total profit of the two retailers is
For any given total inventory, say units, substituting and into the above equation, we have
Thus, it is easy to show that in order to maximize the total profit, the optimal solution is to set to be either zero (i.e., to stock only at R2) or (i.e., to stock only at R1).
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