When to sell to the retailer: a game-theoretic analysis T. T. Niranjan Management Development Institute , 122 001 India Email: [email protected] B. Mukhopadhyay Management Development Institute , 122 001 India Abstract Is superior information necessarily better for a firm in a supply chain? To analyse this, we model a one-period, two-echelon supply chain facing stochastic demand. When the retailer buys before (after) demand uncertainty is resolved, the retailer (manufacturer) faces the entire risk. It would seem that the retailer always prefers to buy late. However, a manufacturer who knows the retailer is going to buy after observing demand, may possibly produce so much less (to reduce her risk of overstocking) than before, that the retailer is worse off by buying late. We derive and compare the equilibrium quantities in the two cases, and develop insights into the factors they depend on. Keywords: Supply chain coordination, analytical modelling, two echelon, game theory

INTRODUCTION The setting we consider is a two level supply chain with a manufacturer supplying a highly perishable good to a retailer who faces a stochastic demand, common in several industries such as fashion goods and toys industry (Iyer and Bergen, 1997; Cachon, 2004). The transaction between them is under price only contracts (Lariviere and Porteus 2001). The demand faced by the retailer (manufacturer) would be deterministic (stochastic) if he (she) buys after that event revealing market state, and vice versa. Any units left unsold are salvaged by discounted sales, after the season is over, or by selling in a different market. It might seem that the retailer would prefer to wait until the event before making his stocking decision. However, because of the strategic interactions of the manufacturer and retailer, it is not evident if, by waiting for demand information to be revealed, the retailer is always better off. This is because, in the second case, when the manufacturer is made to face the entire risk of excess inventory, she may manufacture much less than in the first case, thus severely restricting the quantity that the retailer could have sold in the better states. Further, it is far from obvious under what conditions the quantities produced in the two cases exceed each other. In this paper we investigate this problem by studying two specific demand functions and deriving the quantities produced under equillibrium. LITERATURE REVIEW Literature review is limited to the works closest in spirit to ours. In a generalized model, Taylor (2006) addresses similar issue as us, which is the timing of wholesale transactions: when the retailer buys from the manufacturer. The late sales in Taylor (2006) are distinct from the pull contract for the following reason. In late sales, in Taylor (2006), the manufacturer observes actual demand and then sets (offers) her wholesale price. In other words, there is symmetric information. In contrast, in Cachon (2004), even under pull contracts, the manufacturer is unaware of the actual demand at the time when she negotiates her wholesale price for pull contract. In fact, the bargaining process which decides the wholesale prices under both push and pull contracts occurs before production begins. This implies asymmetric information in both push, and pull, contracts. Our late sales is similar to the at-once (pull) contract in Cachon (2004). Taylor (2006) assumes a Stackelberg leader manufacturer and addresses the question: “when would the manufacturer choose to sell early?” analyzes situations when retailer exerts sales efforts or when retailer possesses superior information. But the concept of Stackelberg leader is usually untenable in reality. Therefore, we do not make this assumption. In our model the wholesale prices are fixed outside the model because we do not predict the outcome of the negotiation. We borrow the arguments provided by Cachon (2004) that “The particular contract adopted by the firms is the outcome of some bargaining process. More likely, firms engage in some alternating offer bargaining process. …there is no contract that is the outcome of a wide range of bargaining processes.” Therefore, we assume that factors outside our model, like shift of risk due to timing and relative market power would be handled within the bargaining process. In our model, wholesale prices evolve from the ‘black-box’ of negotiation. The question we address is “what are the inputs (analyses) to the bargaining process, from which the firms agree on the contract (early or late sales, and the corresponding wholesale price).”

1

MODEL A risk-neutral manufacturer sells a product to a risk-neutral retailer who sells that product over a single selling season. Demand during the selling season is stochastic. Let q be the quantity sold. Let ‘ a ’ represents the

[ ]

stochastic component of demand. Assume a ∈ a, a

f (a) and distribution F (a)

with density

such

that F ( a) = 0 and F ( a) = 1 . We assume F to be strictly increasing and twice differentiable in a . The market state a is revealed at t3 and this is symmetrically observed by both retailer and manufacturer. There is one production opportunity which occurs well before the selling season due to the long manufacturing lead time . Notations

cm α 2 ,α 4

=

ct = (1 + α )cm p q2 , q4 qˆ , ~ q vˆ , ~ v ,v r

r

Manufacturer’s cost per unit

=

Manufacturer’s margin when she sells at times t2 ,t 4 Manufacturer’s selling price = Retailer’s cost (at t = 2, 4 )

=

Retail price

=

Quantities produced when retailer orders at t2 ,t 4 Quantities sold at t5 by retailer in the main market, when he buys at t2 ,t 4 Salvage value of retailer who buys at t2 ,t 4 , and of manufacturer

=

= m

~ hˆr , hr

=

Corresponding discount at salvage sales vˆr = c2 (1 − hr )

π 2 ,π 4

=

Retailer’s profits when he buys at t2 ,t 4

Π 2, Π4

=

Manufacturer’s profits

a ∈ a, a

=

Demand shock

=

First derivative of Retailer’s profit w.r.t. q2*

( )

V2r (x ) =

∂π 2 ∂q2*

At time t0 , the distribution of a is commonly known. The manufacturer enters into a negotiation process with the retailer to fix the terms of trade (the time of sales and wholesale prices). The bargaining process that generates these prices is treated as outside of our model. Based on this, the retailer chooses an option to agree to buy early or late, and enters into a contract with the manufacturer to fix it. Production begins at time t1 which is essentially soon after t0 . Let q2* , q*4 denote the optimal quantities produced by the manufacturer, and sold at wholesale prices (retailer’s costs) c2 & c4 . Retail sales are realized soon after the demand state is revealed. Let the variables qˆ and q~ represent the quantities sold by the retailer, in the Production begins

Early sale: retailer stocks

Retailer’s pricing decision

Market state ‘a’ is revealed

t0

t4 t1

t2

Negotiation to fix timing of sale & wholesale prices

t5

t3 Late sale: retailer stocks

Selling season begins

Figure 1: Timeline of events main market, in the cases when he orders before (at t2 ) and after (at t4 ) observing demand state a . Unsold stock is salvaged by the manufacturer/retailer by selling it at prices vm , vr which are strictly lower than their respective costs. The salvage values are a fraction of the respective costs. Let hm , hr be the percentage of discount at which the units are sold in the salvage market. i.e. vm = (1 − hm )cm , vˆr = 1 − hˆ r c2

(

2

)

(

)

~ and v~r = 1 − hr c4 . Different notations have been used for the retailer’s salvage prices in the two cases to maintain generality, and pose no special significance. Let α 2,4 be the manufacturer’s mark-ups, (in other words, they represent the wholesale prices offered) i.e. ci = (1 + α i )cm where i = 2,4 . Let Π and π represent the profits of manufacturer and retailer respectively. We consider the equilibrium quantities and profits under two cases: when the retailer buys before knowing demand state, and when he buys after knowing it. Further, it is evident that the wholesale price in late sales should be greater than that in early sales (c4 ≥ c2 ) because of the shift in risk to the manufacturer. Retail demand is price sensitive. We assume a > v r i.e. the main market is confined to demand that fetches a price higher than the salvage market price. For concreteness, we consider his demand function to follow the additive form p = a − bq where p is the price at which quantity q is sold. a represents the demand shock. It captures the income level and attractiveness of the market, and is stochastic. b relates the price to demand. b is treated as a given exogenous parameter, constant throughout this paper. Parameters a and b are commonly known. We will break down the analysis into two distinct cases : when the retailer has agreed to order at t 2 and at t 4 . At t0 , the wholesale prices fixed for the two cases are known symmetrically. ANALYSIS When the retailer agrees to order early, at t0 , the retailer agrees to buy the goods from the seller at t 2 , while the seller agrees to sell the goods to the retailer at a unit price of c2 . Therefore, during the time of contracting, both the parties try to agree on a price only contract based on what they expect the demand shock to be in the future. The model will be solved using backward induction method. We first solve for the expected profits of the retailer- working backwards from t5 . Let, the quantity ordered by the retailer at t 2 be denoted by q2 . The choice of the retailer at t5 is how much to sell in the main market. Note that, this is constrained by the amount he has ordered at t 2 . Denoting, qˆ * as the optimum amount he chooses to sell in the market at t5 , q *2 ≥ qˆ * . The profits under different demand conditions are when a is low (a − bq)q + vˆ r ( q2 − q ) − c2 q2 π2 =  when a is high (a − bq)q − c 2 q2 It is a standard result that maximization of the above gives two possible solutions of

 a − vˆ r    q =  2b   q2 ˆ*

q *2 ; they are

if vˆ r < a ≤ 2bq2 + vˆ r

(1)

if a > 2bq2 + vˆ r

Denoting a& ≡ 2bq2 + vˆ r ,

 a 2 − vˆ 2   r  + q − a − vˆ r vˆ - c q   2  r 2 2 *   π 2 =  4b   2b   q2 .(a − bq2 ) − c2 q2

if vˆr < a < a&

(2)

if a ≥ a&

Note that, for any (realized) a , the retailer can sell the entire amount q2 (ordered by him at t 2 ) provided the market demand is sufficiently high. Ho wever, with each additional unit sold in the main market, the price depresses (the inverse relationship between price and quantity determines this), thereby possibly lowering his profits. Therefore, if the demand intercept is sufficiently high, he sells the entire amount in the market at a per unit price higher than the salvage value, vˆr . However, for low realized values of the demand intercept the retailer will sell only a certain proportion of his ordered quantity in the main market and the remainder in the salvage market. a& represents the point of inflexion. At t 2 , the retailer knows that given his optimum choice,

q*2 , the possible profits at t5 is either of the scenarios given in equation (1). Therefore, his expected profits at t 2 , denoted by E (π 2 ) is given by,

3

E (π 2 ) =

a&

 a 2 − vˆ 2 r   4b a

∫

a     +  q − a − vˆ r vˆ  f (a)da + q (a − bq ) f (a )da - c q r 2 2 2 2   2 2b     a&

∫

(3)

In the right hand side of the above equation, the first two terms under integral sign correspond to the expected revenue to the retailer earned from two possible future states at t5 : the low demand and high demand, respectively. The last term is the cost of ordering q 2 units and is non-stochastic. We now solve for the optimum stocking output by the retailer Interior solution is guaranteed as long as the probability distribution follows a monotone hazard rate. Assumption regarding monotone hazard rate is standard in information economics. Proposition 1: The equilibrium quantity is increasing in vr and decreasing in c2 when the retailer orders before observing demand. The implications of this proposition are that from among the various parameters modeled here, the equilibrium quantity depends only on the retailer’s salvage price (directly) and the manufacturer’s selling price (inversely). Further, solution of (1.a) in Appendix yields the equilibrium quantity for this case. We now focus on the manufacturer’s output decision. The manufacturer chooses the production quantity at t1 . She will try to infer the amount the retailer is going to order at t 2 - given by q2* . Note that, from (3), the output that the retailer will order at t 2 depends only upon the observable parameters in the model- b, a, vˆ r and c2 . Therefore, she will correctly infer the order quantity and produce the equilibrium quantity q*2 . She does not have to worry about overstocking in this case. Denoting Π 2 as the expected profits to the manufacturer corresponding to the case where the retailer orders at t 2 ,

Π 2 = (c2 − cm )q*2 = α 2 cm q*2

When the retailer agrees to order late, in this case, the retailer postpones his purchase decision till demand state is revealed at t 4 . At t1 , the manufacturer faces newsvendor situation, as the retailer will order only what he can optimally sell. Let q*4 be the optimal quantity produced by the manufacturer when she plans to sell at t 4 . After demand uncertainty is resolved, the retailer buys his stock as a cost ofc4 . His profit function is:

π 4 = {(a − bq) − c 4 }q

for any quantity q sold. Denote q~* , ~p* as the optimal quantity sold by the retailer, given a demand state. The first order condition of the above equation results in

 a − c4   a + c4  , ≡ ql , pl * ~ *  2b   2  ~ q , p =      * * ≡ qh , ph  q4 , a − bq4 Denote a&& ≡ 2bq*4 + c 4 . The retailer’s profit

E (π 4 ) is:

(

a&&

)

if ~ v r < a ≤ 2bq*4 + c4 if a

> 2bq*4

+ c4

a

∫

∫

E (π 4 ) = ql ( pl − c4 ) f ( a)da + qh .{ph − c4 } f (a) da a

The manufacturer’s profit function E (Π 4 ) is given by

E (Π 4 ) =

a&&

∫ [q . p + { l

l

q4*

} ]

a&&

− ql vm f (a) da +

a

∫ [q .p ]f (a)da − q c h

h

* 4 m

(5)

&a&

a

Proposition 2: Ceteris-paribus, the equilibrium quantity produced is increasing in manufacturer’s mark-up and salvage value. It is intuitive that with increasing margins, the manufacturer would tend to stock more under uncertainty. Further, as holding cost increases (salvage value decreases) the manufacturer would tend to hold less, rather than more, under uncertainty.

4

(4)

We now compare the equilibrium quantities in the two cases (early and late sales). We now explicitly recognize that a (the manufacturer’s margins) are necessarily different in the two cases i.e. c2 < c4 , strictly. We make the assumption that the retailer and manufacturer both have equal salvage sales discounts factors i.e. hm = hr = h . This is a necessary assumption for mathematical analysis, and is mild in comparison to those made in literature. For example, Cachon (2004) assumes that salvage value is equal for both players . In reality, overstocking is more expensive for downstream players; a fact that we have incorporated. Proposition 3: Ceteris paribus, for sufficiently low values of manufacturer’s margins, in the late sales case, the optimal quantity produced in early sales would exceed that in the late sales case. From (3.c) in Appendix, we observe that for comparison of the equilibrium quantities, the only relevant parameters are those in G and H. Clearly, (3.c) holds forα 4 → 0 . For a most stringent test, assume equal manufacturer margin in the early sale case as well (although, in reality, it would be lower in early sale). Intuitively, in the late sale case in which the manufacturer faces newsvendor situation, when manufacturer margins are lowered, with margins falling further down, her speculative quantity produced can only reduce. This is evident because in such a situation, the profits in good states would be inadequate to compensate the overstock costs in bad states. In the early sale case, the lowering of manufacturer margin has no effect on the manufacturer’s quantity produced because she faces a non-negative profit with every additional unit. In fact, she would be better off producing any higher quantity that the retailer could demand, under certainty (which is the case here). With falling manufacturer margins, the cost of procurement, and hence, cost of possible overstocks diminishes, and hence, the retailer would order a higher quantity. Thus, q*2 > q*4 for very low values of α 4 . Having derived conditions where quantities produced are greater in one case than in another, we now evaluate the quantities actually sold in the retail market. In the early sales case, the manufacturer faces a deterministic *

demand and sells the quantity q 2 . The retailer faces a stochastic demand and sells in the retail market, a quantity whose expected value is: E (qˆ2 ) =

a&

∫ a

a

 a − vr   . f (a )da + qˆ *2 f (a )da 2 b  

∫ a&

*

In the late sales case, the manufacturer faces a stochastic demand and sells an expected quantity q 4 as given by (3). The retailer meets a known demand (because he orders after observing retail demand), and sells a quantity whose expected value is q *4

a − c4 E (q~4 ) = ∫ . f (a )da + q*4 ∫ f (a )da 2b a q* a

4

Contracting at t0 : At this stage, both the manufacturer and the retailer have symmetric knowledge of the demand density distribution f (a) , the market parameter b , and the salvage prices of each player. They enter a bargaining process at this time, to agree upon the timing and pricing of wholesale transaction. The bargaining process would assume knowledge by both players of the results of the foregoing analyses. In addition, the relative power of each enters the process. One instance would be the manufacturer enjoying full power and dictating the wholesale prices as take it or leave it offers. We do not analyze this process for reasons explained in section 2. Here, we assume the outcome of this process to be a set of values c2 and c4 . Based on this, the retailer signs the contract to promise to buy early or late. What would be the profit if there was the best possible co-operation between the retailer and manufacturer? This profit would correspond to the integrated case i.e. the two players merge as one. In this case, the wholesale price would be irrelevant and double marginalization would be avoided. There would also be only one case: the early and late sales would be impossible. Letting q I be the quantity produced in the integrated case, the profits are: a&

( ) ∫

 a 2 − vˆ 2   r  + q I − a − vˆ r E Π I =      4 b 2b   a 

∫a& (

)

  I I  vˆ r  f (a )da + q 2 a − bq f (a )da - cm q   a

This profit would necessarily be higher than the early and late sales cases.

5

DISCUSSIONS AND CONCLUSIONS: We modeled a risk neutral monopolist retailer facing stochastic demand, who is supplied by a manufacturer at fixed wholesale price. We derived the equilibrium quantities produced by the manufacturer if the retailer buys before and after resolution of demand uncertainty. The wholesale prices in the two cases were treated as given. If the retailer buys before observing the demand state, he would face all the risk of overstocking. If on the other hand the retailer buys after observing the demand, the manufacturer would face all the risk of unsold units, and might therefore produce a lower quantity to reduce her cost of overstocking. In the latter case, the wholesale price would also be higher and the retailer would still face certain sales, limited to the lesser of the available demand and the quantity stocked. Thus, when the retailer buys late, both the manufacturer and retailer face a stochastic demand. Because of these conflicting forces, it is not evident if the quantity produced by the manufacturer is higher in one case than in the other. We derived conditions for the equilibrium quantity and profits for the players under the two cases. Comparative statics revealed how equilibrium quantities vary with parameter values under different conditions. The analyses provided would be the input to the negotiation process that takes place before production begins. The players would make offers and counter-offers regarding both early and late sales. The outcome of the negotiation will be a particular contract (either early, or late sales, and the wholesale price). We modeled the outcome to arise from a black-box (the negotiation). This is similar to the modeling assumption of Ferguson et al. (2003). In the late sales, the players can possibly engage in renegotiation after demand resolution, if it is not too costly. However, Iyer and Bergen (1997) report that there is strong resistance to changing the wholesale price after demand state is revealed. For example, if the demand turns out to be very low, in case the retailer can set his retail price, the manufacturer would sell at a lower price to clear her stock. This would distribute demand shock, and make both of them better off. This, we argued, is the middle ground between the two types of contracts we study. We did not analyze this because our focus was at the time before production begins, when the negotiation takes place. However, it would be useful to analyze this, and compare how supply chain performance under this case, as against pure early or late sales. It would also be useful to assume some outcomes of the negotiations (wholesale prices) and study individual/supply chain profits. We leave this for future research. We showed conditions when the quantity produced is higher when the retailer buys after knowing the demand state, than when the retailer buys before that. A numerical example provided in Appendix shows the quantity produced and retailer’s profit in early sales exceeding that in late sales. This illustrates that better information of one player (e.g. retailer) is not necessarily better for that player, and can hurt an uncoordinated supply chain’s performance. We showed the importance of analyzing the strategic actions of other players in the supply chain, when individual players have to make their purchase timing decisions under uncertainty. REFERENCES: Cachon, G. 2004. The Allocation of Inventory Risk in a Supply Chain: Push, Pull, and Advance Purchase Discount Contracts. Management Science. 50 (2), 222-238. Cachon, G. 1998. Competitive Supply Chain Inventory Management. In S. Tayur, M. Magazine, R. Ganeshan (Eds.). Quantitative Models for Supply Chain Management. Kluwer Academic Publishers. Norwell, MA. Ferguson, M. 2003. When to commit in a serial supply chain with forecast updating. Naval Research Logistics. 50 (8), 917-936 Frazier, Robert. 1986 (Sep). Quick Response. The KSA Perspective. Iyer A.V., Bergen, M.E. 1997. Quick Response in Manufacturer-Retailer Channels. Management Science. 43 (4), 559-570. Lariviere, M., Porteus, E. 2001. Selling to the newsvendor: An analysis of price-only contracts. Manufacturing & Service Operations Management. 3 (5), 293-305. Taylor, T. 2006. Sale timing in a supply chain: When to sell to the retailer. Manufacturing & Service Operations Management (8) 1, 23-42, 20p

6

APPENDIX Proofs for proposition 1: From (3), the retailer chooses, q*2 at this stage. The relevant first order condition is, a&

∫a

V2r = vr f (a)da + Putting vr = p2 (1 − hr ) the first order condition reduces to a&

V2r

( ) = −h ∫ q*2

r

r From the assumption of monotone hazard rate, V 22 ≡

r

∫a& (a − 2bq2 ) f (a)da − c2 = 0

1 f (a)da + c2

a

We have V2rv =

a

∂ 2 E(π 2 ) ∂q22

a

∫a& (a − 2bq2 − c2 ) f (a )da = 0

(1.a)

< 0. This satisfies the second order condition.

( )

∂V2r dq*2 − V2rv * > 0 . Total differentiation of V2rv gives = > 0 . Thus, q2 is strictly decreasing r ∂ vr dvr V22

in v r . Similarly, we have V2rc2

 ∂V r − 2  ∂ c2 ∂V r dq2* = 2 < 0 .The total differentiation of V2rp gives =  r ∂c2 dc2 V22

    < 0.

Thus, q 2* is strictly decreasing in c 2 .

Proofs for proposition (2) From (5), putting vm = cm (1 − hm ) , the relevant first order condition is a&&

V 4m = − h m

∫

a

f ( a ) da +α 4

∫ f ( a ) da = 0

(2.a)

a&&

a

Second order condition is satisfied. The solution to the above yields the optimal value, q*4 produced by the manufacturer. We have,

∂V4m = ∂α 4

a

∫&a&

a

∂V4m f (a )da > 0 and = − f ( a) da < 0 ∂hm

 ∂V m − 4  ∂α dq* 4 ∴ 4 =  m dα 4 V44

∫&a&

    >0.

 ∂V m − 4  ∂ hm dq4* Similarly, =  m dhm V44

   <0

Proofs for proposition (3) With the assumption in Subsection 4.3 that h = hr = hm , (1.a) is written as

V 2r

( ) q2*

a&

∫a

1 = − h f (a )da + c2

∫a& (a − 2bq*4 − c 2 )f (a)da = 0 a

(3.a) can be rearranged as:

7

(3.a)

2 bq*2 + c2

V2r

a&

( ) = −h ∫ f (a).da − h ∫ f (a ).da + c1 ∫ (a − 2bq a

q2*

2bq*2 + c2

a

Replacing q *2 with q *4 , it follows that

( ) = −h ∫ f (a).da − h ∫ q4*

&a&

a

2

− c2 ) f (a )da = 0

a&

(&a&− hc 2 )

a&&

V2r

2

1 f (a).da + c2

a

(a − &a&) f (a )da ≠ 0 ∫ ( &a&− hc )

(3.b )

2

We have from (2.a), &a&

h

a

∫ f (a ) da =α ∫ f (a )da 4

&a&

a

Replacing the first part in (3.b) using the above equality, and re-arranging it, we have

V2r

a

a&&

&a&

a&& −hc 2

( ) = −α ∫ f (a)da + h ∫ q*4

4

1 f (a).da + c2

a

∫ (a − &a&) f (a )da ≠ 0

&a& − hc2

| - - - - -G - - - - | - - - - - - - - - - - - -H - - - - - - - - - - - - - - | G is strictly negative. The first part of H is strictly positive, while the second is usually positive. V2r q*4 > 0 iff |G| <|H|. Also, V2r (q ) is decreasing in q (by concavity). Therefore

( )

( )

( )

V2r q*4 > V 2r q*2 ⇒ q*2 > q*4 i.e. G < H ⇒ q *2 > q *4 From the above, for α 4 → 0 , (3.c) holds. The proof follows.

(3.c)

Numerical example Consider a uniform distribution with a = 5; a = 15 . In the demand function, let b = 0. We evaluate the most stringent case, when manufacturer’s price is same in both cases. Even in this case, the retailer is worse off by buying after demand state is revealed i.e. his profit reduces from 11.25 to 8.57 by waiting for demand uncertainty resolution. In reality, when the retailer buys late, he would likely pay a higher price, and therefore, his profit would be still lower, so the effect would be much more pronounced than indicated by this example. Parameters

cm 1

α2 =α4 0.5

c 2 = c4 1.5

vm 0.1

γ

vr 0.6

2

pr 3

Numerical results Retailer orders at

Qty produced

Expected Profit of Retailer

Manufacturer

Supply Chain

t2

11.25

11.91

5.63

17.54

t4

8.57

11.73

3.23

14.96

8