SUPPLY CHAIN CONTRACTING AND COORDINATION WITH SHELF-SPACE-
2. A Manufacturer and Single Retailer Supply Chain
2.1 The Model and Centralized Control
A single product is produced by a manufacturer and then sold to con- sumers through (for now) a single retailer. The marginal production cost and retail price are constant at $/unit and $/unit respectively, where The demand rate for the item will depend on the amount
of inventory displayed at the retailer’s shelf. Specifically, a constant inventory level of I units generates a demand of D(I) units/year. In general, D(I) can be assumed to be an increasing and concave function (i.e., D'(I) > 0 and D(I) < 0) to reflect the motivational effect of inven- tory on demand and the “diminishing returns”. For technical purposes, we also assume that D(I) is continuous and twice differentiable with and D(I) > 0 for any I > 0. Displaying in- ventory at the retailer is costly. Assume that a constant inventory cost of $/unit/year is charged. So the key decision here is to choose the displayed inventory level I to trade-off increased sales against inventory costs. Note that, once I is chosen, the system is assumed to keep the inventory at level I all the time by continuously replenishing it.
Now, if this supply chain is centrally owned and controlled, the objec- tive is to maximize the long-run average channel profit (i.e., the profit rate):
where the first term is the sales revenue net of production cost and the second term is the inventory holding cost. One can easily verify that is concave and thus the unique solution is given by the first
We next consider a decentralized system where the manufacturer, through contractual arrangements, wholesales the product to the retailer who then chooses its displayed inventory level and, hence, the demand rate. We consider two specific contractual arrangements: a price-only contract, in which the manufacturer selects only the wholesale price, and a price-plus-inventory-subsidy contract where a wholesale price plus an inventory cost subsidy are offered by the manufacturer.
2.2 Price-Only Contract
Here, the manufacturer offers the retailer a take-it-or-leave-it contract which specifies only a wholesale price, say, $/unit. If the retailer takes the contract, she then selects a (permanent) displayed inventory level I which determines the sales rate D(I). For simplicity, assume that both the manufacturer and the retailer have an opportunity cost of zero. So, as long as is chosen such that it will be a viable contract for both parties. In determining the inventory level, the retailer wishes to maximize her own profit (rate):
The unique optimal inventory level for the retailer is thus given by
Since D'(I) is a decreasing function, by simply comparing (7.4) with (7.2), we see that, as long as the manufacturer charges a wholesale price above his marginal cost the inventory level in a price-only con- tractual arrangement will always be lower than the inventory level in a centralized system. This phenomenon is essentially the “double marginalization” problem studied in the economics and industrial orga- nization literature (Spengler 1950, Cachon 1999 and Lariviere 1999). We assume that the manufacturer is not able to charge the retailer a fixed fee. If that was feasible, then by setting the manufacturer could coordinate the system and extract all the channel profit.
Knowing that the retailer will choose the inventory level according to (7.4), the manufacturer chooses the wholesale price so as to maximize his own profit:
order condition:
Example 7.1 Let Then, so Thus,
We observe that the optimal value of does not depend on Thus, whether or not the manufacturer has information about retailer’s holding cost, the optimal wholesale price is not affected. Maximizing in this example amounts to maximizing
Now,
For solving for we get the unique solution
Furthermore, as and as Thus, is the unique maximizer of and, hence, of
Example 7.2 Let Here,
so Thus,
We then have and
The optimality condition is then
So, here does depend on
For the general optimization problem of (7.5), the concavity of
is not guaranteed for all demand function forms To generate some
further insights into the uniqueness of the solution to (7.5), we note that there is a one-to-one correspondence between and – they are related to each other through (7.4). So, when the manufacturer chooses a value for he is equivalently choosing a value for
Thus, substituting the optimization problem over in (7.5) can be written as the following optimization over
(A similar approach was employed by Lariviere and Porteus 1999).
Taking the derivative of we have,
Now, the second term in (7.7) is always positive and decreasing and the first always negative. Thus, for a given demand function if the first term in (7.7), i.e., is non-increasing, then the manufacturer’s profit function is unimodal and, hence, the solution to (7.6) is unique. For the demand functions of Examples 7.1 and 7.2, one can check that this condition is always satisfied.
2.3 Price-Plus-Inventory-Subsidy Contract
Suppose now that the manufacturer offers the retailer a wholesale price of $/unit and an inventory holding subsidy of $/unit/year towards any inventory the retailer chooses to hold on shelf. The retailer’s problem then becomes
and her optimal inventory level is given by
Comparing (7.9) with (7.2), we have the following important observation:
Proposition 7.3 For any contract such that
we have
In other words, for any wholesale price offered by the manufacturer, if he accordingly chooses an inventory subsidy
then the retailer will always be induced to choose the centralized-system-optimal inventory level and, hence, such a contract coordinates the decentralized supply chain. Thus, there exists a continuum of contracts that coordinate the supply chain.
In choosing the manufacturer will want to maximize his own profit. So, presumably, he solves the following problem:
where is determined through (7.9). The first term in (7.11) is his sales revenue net of production cost, and the second term is her inventory subsidy to the retailer.
Instead of solving (7.11) directly, the following argument (Pasternack 1985) illustrates how the manufacturer can find his best strategy: Focus on the set of contracts which satisfy (7.10) with
We know from Proposition 7.3 that any contract within this set will coordinate the channel and, hence, achieve the maximum channel profit.
But, as we will show next, different contracts within this set, represented by different values of provide the retailer with a different amount of profit - the rest of the maximal channel profit will go to the manufacturer.
As a consequence, the manufacturer needs simply to choose a value of so as to allocate any amount of profit required by the retailer (so that she will accept the contract) and thus to extract as much profit out of the supply chain as he can! Now, with and after some algebra, the retailer’s profit in (7.8) can be written as
which is linearly decreasing in
The total channel profit can be obtained simply by substituting into (7.1). We can show that
So, the proportion of the channel profit allocated to the retailer is also linearly decreasing in
To summarize, a properly designed price-plus-inventory-subsidy con- tractual arrangement can achieve: 1. coordination of the supply chain channel; 2. any desired allocation of channel profit between the manufacturer and the retailer.
Finally, we discuss the information the manufacturer will need in or- der to coordinate this supply chain. First, we see from (7.10) that for any given wholesale price, the manufacturer only needs the cost param- eters (i.e., and ) to determine a corresponding inventory subsidy
Here he does not need information about the demand function The value of will determine the portion/amount of channel profit allo- cated to the retailer. If he is only interested in the proportion allocated, then equation (7.13) shows that the manufacturer still does not need the demand information. Only if his aim is to achieve a specific allocation of absolute profits does he need this information - equation (7.12).
2.4 Non-Linear Holding Costs
We now relax the linear holding/shelf space cost assumption by con- sidering a general convex cost function, which is arguably more realistic for most retailing situations where shelf space is a limited resource. We derive the retailer’s optimal inventory decision and discuss if and how the manufacturer can coordinate the supply chain.
Assume that, when displayingI units on shelf, the retailer incurs total inventory cost of H(I) $ per year, where H(I) is a general increasing and convex function. For a given contract offered by the manufacturer, the retailer chooses her inventory level so as to maximize her own profit as follows
We can easily check that is concave and, hence, the retailer’s op- timal inventory level can be found by solving the first order condition which yields,
When the channel is centrally controlled, the optimal inventory level is determined by solving problem (7.14) with and That is,
In a decentralized setting, a contract offered by the manufac- turer will coordinate the channel if and only if it induces the retailer to choose the system-optimal inventory Such channel coordinating con- tracts can be characterized by (7.15). That is, for any given wholesale price the corresponding inventory-subsidy is determined by
Note that the manufacturer will have to possess information about the retailer’s demand function (as well as her holding cost function) in order to offer a coordinating contract. In a sharp contrast, he does not need to know the demand function when the retailer’s holding cost is linear, as shown in (7.10).