Capacity Management and Price Discrimination under Demand Uncertainty Using Option Contracts

Abstract

This chapter considers the use of option contracts as a price discrimination tool under demand uncertainty to improve supplier profit and supply chain efficiency. Option contracts have long been used to manage demand or supply uncertainty, and the cost of the option is simply considered as the cost to avoid uncertainties. We give an example in a supply chain setting where a supplier has more than one downstream customer with private information. Under such a scenario, our game theoretical model shows that the option price shall be set taking into account the fact that only the customers who are more concerned about the demand uncertainty will purchase. Therefore, the supplier should be able to charge more for a unit of option contract compared to the traditional pricing method where simple expectations are taken. The supplier’s profit improves in three ways. First, the high type customers pay higher marginal prices on average. Second, the high type customers’ demand is satisfied as a first priority, guaranteeing allocation efficiency. Third, the supplier can observe the number of options being purchased and so determine customer types, improving capacity decision efficiency. We compare our results to those of classical second degree price discrimination literature. We show that the use of option contracts guarantee the same level of supplier profit as the level of second degree price discrimination. The overall supply chain efficiency improves to the full information benchmark.

Appendices

Appendix 1: Notation Table

i :

Customer index

D _i :

The realized demand of customer i. D _i  ∈ {D ^H, D ^L}

D :

D = D ₁ + D ₂ the aggregate demand

${\mathbb{D}}$ :

${\mathbb{D}} = ({D_1},{D_2})$ is the demand vector of both customers

$D_i^s$ :

The demand customer i submits to the supplier

${{\mathbb{D}}^s}$ :

${{\mathbb{D}}^s} = (D_1^s,D_2^s)$ is the vector of customers’ submitted demand

$D_i^e$ :

The demand of customer i, being satisfied by the supplier

D ^e :

${D^e} = D_1^e + D_2^e$ is the aggregated demand satisfied by the supplier

${{\mathbb{D}}^e}$ :

${{\mathbb{D}}^e} = (D_1^e,D_2^e)$ is the vector of customers’ satisfied demand

$D_i^o$ :

The amount of options executed by customer i when capacity is tight

$D_{ - i}^o$ :

The amount of options executed by the customer other than i

D ^o :

${D^o} = D_1^o + D_2^o$ the aggregated option demand submitted by both customers

f _o :

The high type customer’s valuation of one unit of option contract

K :

The supplier’s capacity level

m _i :

The amount of monetary transfer made from customer i to the supplier

O _i :

The amount of option contracts customer i will purchase

${\mathbb{O}}$ :

${\mathbb{O}} = ({O_1},{O_2})$ the vector of customers’ options purchase

p :

The unit price for the regular demand

p _o :

The option price

p _e :

The option execution price

p _b :

The option buy back price

t _i :

Type of customer i. t _i  ∈ {l,h}

u _i :

The utility customer i receives

v ^t :

The marginal value of demand satisfaction for each type t

α :

The probability that a customer’s realized demand is high (D _i  = D ^H)

λ:

The probability that a customer is a high type one

Π :

The supplier’s profit

π :

The supplier’s profit gained after period 1, excluding the sale from the option contracts

φ :

The probability that regular demand is satisfied under the option framework

Appendix 2: Proof of Lemmas and Propositions

Proof of Proposition 1

If p ⩽ v ^l, both customers submit all their demand to the supplier,

$$ D_i^s = {D_i} = \left\{ {\begin{array}{llll} {{D^H}} & {{\hbox{with prob}} = \alpha } \hfill\\{{D^L}} & {{\hbox{with prob}} = 1 - \alpha.} \\\end{array} } \right. $$

The supplier’s expected profit:

$$ E\Pi = \hfill p\left[ {{\alpha^2}\min \{ K,2{D^H}\} + 2\alpha (1 - \alpha )\min (K,{D^H} + {D^L}) + {{(1 - \alpha )}^2}\min \{ K,2{D^L}\} } \right] - {c_0}K. $$

Maximizing the profit under the condition p ⩽ v ^l, we have p ^* = v ^l and K ^* = 2D ^L. The supplier’s expected profit is EΠ(p = v ^l) = (v ^l − c ₀)2D ^L. Customer i’s expected utility is u _i  = (v _i  − v ^l)D ^L.

If p ∈ (v ^l, v ^h), only high type customers will submit the demand. Therefore, $D_i^s = {D_i}$ when t _i  = h and $D_i^s = 0$ for t _i  = l.

When v ^l < p ⩽ v ^h, the supplier’s expected profit

$$ \begin{array}{lllll} E\Pi (p) ={\lambda^2}p\left[ {{\alpha^2}\min \{ K,2{D^H}\} + 2\alpha (1 - \alpha \} \min \{ K,{D^H} + {D^L}\} + {{(1 - \alpha )}^2}\min \{ K,2{D^L}\} } \right] \\ \qquad+ 2\lambda (1 - \lambda )p\left[ {\alpha \min \{ K,{D^H}\} + (1 - \alpha )\min \{ K,{D^L}\} } \right] - {c_0}K \hfill \\\end{array} $$

Maximizing the expected profit and applying the assumption $\lambda < 1 - \sqrt {{\frac{{{v^h} - {c_0}}}{{{v^h}}}}}$, we have p ^* = v ^h and K ^* = 0. Thereby, EΠ ^*(p = v ^h) = 0 and it is not worthwhile to build capacity and serve high type customers only, due to the low probability of a high type customer’s existence.

Compare the two cases, we conclude that the supplier’s best strategy is to set p ^ND = v ^l and serve both types of customers. The optimal capacity will be K ^ND = 2D ^L. The expected profit EΠ ^ND = (v ^l − c ₀)2D ^L and the overall efficiency:

$$ {W^{ND}} = E{\Pi^{ND}} + \lambda Eu_i^{ND}({t_i} = h) + (1 - \lambda )Eu_i^{ND}({t_i} = l) = \left[ {\lambda {v^h} + (1 - \lambda ){v^l} - {c_0}} \right]2{D^L}. $$

□

Proof of Lemma 1

The optimal capacity K ^FI is made contingent on the customer types ${\mathbb{T}} = ({t_1},{t_2})$. When ${\mathbb{T}} = (h,h)$, the supplier’s expected profit

$$ \begin{array}{lllll} E{\Pi^{FI}}(K,{\mathbb{T}}) = {v^h}\Big[ {{\alpha^2}\min \{ K,2{D^H}\} + 2\alpha (1 - \alpha )\min \{ K,{D^H} + {D^L}\}} \cr \qquad+ {{(1 - \alpha )}^2}\min \{ K,2{D^L}\} \Big] - {c_0}K \end{array} $$

which is maximized when K(h,h) = D ^H + D ^L due to the assumption αv ^h < c ₀ < (2α − α ²)v ^h.

When ${\mathbb{T}} = (l,l)$, similarly, we can have K ^FI(l,l) = 2D ^L, since (2α − α ²)v ^l < c ₀ < v ^l.

When ${\mathbb{T}} = (h,l)$ or (l,h) and K > D ^H, the supplier’s expected profit

$$ \begin{array}{lll} E{\Pi^{FI}}(K,(h,l)) = {\alpha^2}\big( {{v^h}{D^H} + {v^l}\min \{ K - {D^H},{D^H}\} } \big) + \alpha (1 - \alpha ) {{v^h}{D^H} }\\ \quad +{v^l}\min \{ K - {D^H},{D^L}\} + \alpha (1 - \alpha )\left( {{v^h}{D^L} + {v^l}\min \{ K - {D^L},{D^H}\} } \right) \hfill \\ \quad+ {(1 - \alpha )^2}\left( {{v^h}{D^L} + {v^l}\min \{ K - {D^L},{D^L}\} } \right) - {c_0}K \hfill \\ \end{array} $$

which is maximized when K ^* = 2D ^L. It can also be shown that K ⩽ D ^H cannot be optimal. Therefore, K ^FI(h,l) = K ^FI(l,h) = 2D ^L. □

Proof of Proposition 2

For the supplier, the probability that both customers are high types is λ². The expected profit

$$ E\Pi (h,h) = \left( {{v^l} - {c_0}} \right)2{D^L} + \left( {{v^h} - {v^l}} \right)2{D^L} + \left( {(2\alpha - {\alpha^2}){v^h} - {c_0}} \right)\left( {{D^H} - {D^L}} \right). $$

With probability 2λ(1 − λ), one customer is of high type and the other is of low type. The expected profit EΠ(h, l) = (v ^l − c ₀ )2D ^L + (v ^h − v ^l) (αD ^H + (1 − α)D ^L). With probability (1 − λ)², both customers are of low type. The expected profit EΠ(l,l) = (v ^l − c ₀)2D ^L. Since u _i (t = h) = u _i (t = l) = 0,

$$ \begin{array}{lll} {W^{FI}} = E{\Pi^{FI}} \\ = {\lambda^2}E\Pi (h,h) + 2\lambda (1 - \lambda )E\Pi (h,l) + {(1 - \lambda )^2}E\Pi (l,l) \\ = ({v^l} - {c_0})2{D^L} + \left( {{v^h} - {v^l}} \right)2\alpha \lambda (1 - \lambda )\left( {{D^H} - {D^L}} \right) + \left( {{v^h} - {v^l}} \right)\lambda 2{D^L} \\ \quad+ {\lambda^2}\left( {(2\alpha - {\alpha^2}){v^h} - {c_0}} \right)\left( {{D^H} - {D^L}} \right) \\\end{array} $$

compared to EΠ ^ND = (v ^l − c ₀)2D ^L and W ^ND = (λv ^h + (1 − λ)v ^l − c ₀)2D ^L, we can easily conclude that

$$ \left\{ {\begin{array}{llll} {E{\Pi^{FI}} = E{\Pi^{ND}} + {\Delta_1} + {\Delta_2} + {\Delta_3}} \\{{W^{FI}} = {W^{ND}} + {\Delta_1} + {\Delta_3}} \hfill\\\end{array} } \right. $$

□

Proof of Lemma 2

The proof is straightforward since we can show that the second order condition of the above objective function is non-negative. It means that the objective function is a convex function. The optimal solution of the maximization problem would exist on the boundary. That is, it is either 0 or min{O _i , D _i }. □

Proof of Proposition 3

In this stage, the supplier determines the capacity level to maximize his future revenue less the capacity investment. That is,

$$ \begin{array}{lllll} \pi ({\mathbb{O}},{\mathbb{D}},K,{p_e}) = {m_1} + {m_2} - {p_o}\cdot ({O_1} + {O_2}) - {c_0}K \\ = {p_e}{D^o}({\mathbb{O}},{\mathbb{D}},K,{p_e}) + {v^l}{(\min (K,D) - {D^o}({\mathbb{O}},{\mathbb{D}},K,{p_e}))^{+} } \\{ } \quad- {v^h}{({D^o}({\mathbb{O}},{\mathbb{D}},K,{p_e}) - K)^{+} } - {c_0}K \\\end{array} $$

Applying the outcomes from Tables 1–6, we can derive the profit function with parameters ${\mathbb{O}},{\mathbb{D}},K$, and p _e . Taking expectation over the realized demand ${\mathbb{D}}$, we obtain the expected profit function for each ${\mathbb{O}},{p_e}$, and K. Maximizing those the expected profit function by choosing K, we can obtain the optimal capacity as stated in Proposition 3 as a function of ${\mathbb{O}}$ and option strike price p _e . □

Proof of Lemma 3

From the result of

• 1. When ${p_e} \geqslant {v^h} - ({v^h} - {v^l})\tfrac{{{D^L}}}{{{D^H}}}$, optimal capacity K ^* = 2D ^L for all the possible configurations of ${\mathbb{O}}$. No options will be exercised for all possible realization of ${\mathbb{D}}$. Therefore, the option has no value. In other words, ${f_o}\left( {{O_i},{D^H},{p_e}} \right) = {f_o}\left( {{O_i},{D_L},{p_e}} \right) = {f_o}\left( {{O_i},0,{p_e}} \right) = 0$

• 2. When ${v^h} - ({v^h} - {v^l})\tfrac{{{D^H}}}{{2{D^H} - {D^L}}} \leqslant {p_e} < {v^h} - ({v^h} - {v^l})\tfrac{{{D^L}}}{{{D^H}}}$, if the customer has bought O _i  = D ^L, she will never exercise the options no matter what type the other customer is. Therefore, at the first stage, the value of the options contract would be 0 and the customer should not purchase any options with a positive price

• 3. When ${v^h} - ({v^h} - {v^l})\tfrac{{2{D^L}}}{{{D^H} + {D^L}}} \leqslant {p_e} < {v^h} - ({v^h} - {v^l})\tfrac{{{D^H}}}{{2{D^H} - {D^L}}}$: if the customer has bought O _i  = D ^L units of options, she will only exercise it when ${\mathbb{D}} = ({D^H},{D^H})$ and $D_{ - i}^o \ne {D^H}$. Hence if the other customer is of high type, she is always better off if the other customer has purchased O _−i  = D ^H rather than O _−i  = D ^L units of options. However, given O _−i  = D ^H, customer i will never exercise her options and the value of the options is 0. If the other customer is of low type, the customer’s expected utility from exercising the options are

  • (a) ${O_i} = {D^H}:{u_i}{(}{D^H}{)} = {\alpha^2}({v^h} - {p_e}){D^H} + (1 - {\alpha^2}){D^L} - {p_o}{D^H}$

  • (b) ${O_i} = {D^L}:\,{u_i}({D^L}) = {\alpha^2}\left( {({v^h} - {p_e}){D^L} + ({v^h} - {v^l})\tfrac{{{D^L}{D^H}}}{{2{D^H} - {D^L}}}} \right) + (1 - {\alpha^2}){D^L} - {p_o}{D^L}$

  • (c) ${O_i} = 0:{u_i}{(}0{)} = ({v^h} - {v^l}){D^L}$

From the results above, we can show that $\tfrac{{{u_i}({D^H}) - {u_i}(0) + {p_o}{D^H}}}{{{u_i}({D^H}) - {u_i}(0) + {p_o}{D^L}}} > \tfrac{{{D^H}}}{{{D^L}}}$, meaning that the customer is better off by purchasing O _i  = D ^H units of options.

Following the same calculation, we can show that D ^L is not the optimal choice when p _e  > v ^l. □

Proof of Proposition 4

We need to discuss the profit according to the different p _e segments:

• 1. When ${p_e} \,\geqslant\, {v^h} - ({v^h} - {v^l})\,\tfrac{{{D^L}}}{{{D^H}}}$, no options will be exercised, p _o  = 0 and EΠ = (v ^l − c ₀)2D ^L

• 2. When ${v^h} - ({v^h} - {v^l})\tfrac{{{D^H} + {D^L}}}{{2{D^H}}} \leqslant {p_e} < {v^h} - ({v^h} - {v^l})\tfrac{{{D^L}}}{{{D^H}}}$, we have ${K^*}({D^H},{D^H}) = \tfrac{{{v^h} - {p_e}}}{{{v^h} - {v^l}}}2{D^H}$ from Proposition 3. The supplier’s profit

  $$ \begin{array}{lllll} E\Pi = E\pi + 2\lambda {p_o}{D^H} \\ = {\lambda^2}\left[ {\left( {(2\alpha - {\alpha^2}){v^l} - {c_0}} \right)\frac{{{v^h} - {p_e}}}{{{v^h} - {v^l}}}2{D^H} + (1 - {\alpha^2}){v^l}2{D^L}} \right] \\ \quad+ 2\lambda (1 - \lambda )\left[ {{\alpha^2}({p_e} - {v^l}){D^H} + ({v^l} - {c_0})2{D^L}} \right] + {(1 - \lambda )^2}({v^l} - {c_0})2{D^L} \\ \quad+ 2\lambda \left[ {Eu_i^h({D^H}) - Eu_i^h(0)} \right] \\\end{array} $$

  which we can show $\tfrac{{dE\Pi }}{{d{p_e}}} < 0$.

• 3. When ${v^l} \leqslant {p_e} < {v^h} - ({v^h} - {v^l})\tfrac{{{D^H} + {D^L}}}{{2{D^H}}}$, we have optimal capacity K ^*(D ^H,D ^H) = D ^H + D ^L and K ^*(D ^H,0) = K ^*(0,0) = 2D ^L from Proposition 3. One can get

  $$ E\Pi = ({v^l} - {c_0})2{D^L} + 2\alpha \lambda ({v^h} - {v^l})({D^H} - {D^L}) + {\lambda^2}\left( {(2\alpha - {\alpha^2}){v^h} - {c_0}} \right)({D^H} - {D^L}). $$

  In this case, $\tfrac{{dE\Pi }}{{d{p_e}}} = 0$.

  In summary, we can conclude that the optimal option exercise price p _e should be ${v^h} - ({v^h} - {v^l})\tfrac{{{D^H} + {D^L}}}{{2{D^H}}}$ □

Proof of Proposition 5

The proof is straightforward from the proof of Proposition 4. □