A two-stage multi-period negotiation model with reference price effect

Abstract

In this article, we consider the revenue management problem of firms with customers who use historical pricing data to make their purchasing decision. This behavior is modeled using the concept of reference price, which quantifies customers’ perceived ‘fair value’ of a product. We consider a segmented market consisting of two types of customers. One segment uses all price history to form a reference point and the other only uses the current price to do so. We study the revenue maximization problem of a firm who offers a two-price scheme in this market and derive closed form analytical solutions. Finally, we extend the monopoly model to a duopoly setting, and provide a numerical analysis on the added value of offering a two-price scheme compared to a single-price scheme and suggest a novel interpretation of such segmentation regime.

Appendix

Proof of Proposition 1

The Hessian matrix of the one-stage profit function given in equation (2) is as follows:

The first principal minor of this matrix is negative (−2(b₁+c₁)⩽0), and the second principal minor (4(b₁+c₁)(b₂+c₂)−c₂²⩾0) is positive by the regularity assumption. Hence by Sylvester criterion, Gilbert (1991), the Hessian is semi-negative definite and the objective function is jointly concave. The result follows from the first-order conditions. □

Proof of Corollary 1

Each firm will be using a linear function in the form of p₁^*(r)=mr+h to maximize her one-stage profit. The result follows from the observations that both m and h coefficients are increasing (decreasing) in reference price (absolute price) sensitivity, that is, c₂(b₂). □

Proof of Theorem 1

Results used in proof of Theorem 1 are as follows:

Proposition 10:

• The optimal value function J^*(r) is unique.

This follows from Stokey et al (1989) and Bertsekas (2001, p. 11, proposition 1.2.2) from the observation that (a) the state-space set is a convex subset of ℝ¹, (b) the state update function f(r, p₁) is non-empty, compact valued and continuous, and (c) one stage revenue function, π, is bounded and continuous and 0<β<1.

Proposition 11:

• The optimal value function J^*(r) is increasing in r.

This follows from the observation that the one stage profit π _t (r _t , p_1, t, p_2, t) and the next stage reference price r _t =αr _t +(1−α)p_1, t are increasing in r _t . Assume that a firm is implementing the (p ₁ , p ₂ ) policy with the initial reference price r (this policy is not necessarily stationary). Assume r′⩾r to be the new initial reference price. In the first period, implementing the same (p ₁ , p ₂ ) policy will yield a higher profit π₀(r′, p₀¹, p₀²)⩾π₀(r, p₀¹, p₀²) also we will have r₁′⩾r₁. As the optimal solution is bounded (follows from the linear form of the demand function), an inductive argument proves the proposition.

Given the above propositions we now proceed to prove Theorem 1. We postulate that the value function is of the quadratic form presented in equation (4). Hence the Bellman equation (3) becomes

The above maximization problem is concave and the optimal prices (p₁^*(r), p₂^*(r)) can be derived by first-order conditions. Solving the system of equations ∂J/∂p₁=∂J/∂p₂=0 for (p₁, p₂) yields the optimal policies (4) and (5). Substituting these optimal policies in (A1) and collecting the coefficients of the quadratic polynomial, we derive the values of coefficients γ, δ and ω. Now we just have to validate the sign of coefficients.

Choice of γ: γ is one of the roots of the following quadratic equation,

As the value function J^*(r) is unique (Proposition 10) and increasing in r for all values of r>0 (Proposition 11) the coefficients γ, δ and ω should be non-negative real numbers. Assume that equation (A2) has real roots, as the coefficient of γ² and γ⁰ are both positive, then the roots have the same sign. Hence for γ to be positive we require the coefficient of γ to be negative, which means:

The above inequality holds for the ‘regular’ problems. In such setting we have:

where the first inequality follows from minimizing the left side over all feasible value of β (for 0<α<1) and the second inequality follows from 0<α<1 and c₂⩽b₁ and the third inequity follows from b₂ and c₂⩾0. Hence, assuming the real roots exist they are both positive. Now we need to show that real roots exist. In order to show that, we need to show that the discriminant of equation (A2) is always positive. In order to do so, we first establish some preliminary results.

Recall

Lemma 1:

• Monotonicity properties of Γ ₂ :

  1. Γ ₂ is increasing in b₂

  

  We show that

  

  and obviously for 0<α, β<1, max Ψ₂=3. Hence,

  

  where the last inequality follows from b₂⩾0 and the regularity condition.

  2. Γ ₂ is decreasing in c₂, when b₂=0

  

Now we show that the discriminant of equation (A2) is always positive, that is,

The above inequality holds for the discriminant of equation (A2) of regular problems. In such settings we have:

where the first inequality follows from the increasing property of Γ₂ in b₂ and the second inequality follows from the decreasing property of Γ₂ in c₂. Now,

The above inequalities follow from 0<α, β<1. Hence, the real roots always exist for ‘regular’ problems. This derivation gives us two choices for γ. We choose γ as the smaller root of the equation (A2). We will show that this choice will lead to positive values for δ and ω.

Positivity of δ: In order to show that δ, equation (6), is well defined and positive we require that

In order to prove the above inequality we maximize the left-hand side:

The first inequality follows from maximizing the LHS over b₂ and the second follows from maximizing over c₂. The last step holds as the left-hand side of the last inequality is negative.

Positivity of ω: First we show that if the denominator of ω is positive then the enumerator is also positive. This follows from comparing the first two terms in Ω₁ with the denominator. a₂(b₁+c₁)−βγa₂(1−α)²⩾a₂[(b₁+c₁)−βγ(1−α)²−Ω₂]⩾0. It only remains to show that the denominator is always positive. Hence, ω will be defined if and only if

We maximize the LHS of the above inequality.

The first inequality follows from maximizing the LHS as used in the previous steps and the last inequality follows as the left-hand side of the last inequality is negative.

Given the uniqueness result proved in Proposition 10 we have derived the optimal solution. □

Proof of Proposition 2

• a The proof follows from the condition for f^*(r)⩽r.

  

  Hence, if r⩽r^** then f^*(r)⩾r(†). This shows that if the initial reference price is lower than the steady-state reference price, the series of reference prices associated with the optimal path are increasing until they hit the reference price level or go higher. Next, we show that the series never goes above the steady-state reference price. This follows from the condition:

  

  Hence, if r⩽r^** then f^*(r)⩽r^**(‡). The monotone convergence result follows from (†) and (‡).

• b From the reference price update equation (1), we know that the steady-state price p₁^** is equal to r^**. From equation (4), we know that the optimal price p₁^*(t) is increasing in r(t). Hence p₁^*(t) is an increasing series and the result follows for monotone convergence of p₁^*(t) series from part a. From equation (5) we know that p₂^* is an increasing function of p₁^*, hence the result for p₂^*(t) series follows directly. □

Proof of Corollary 2

• a1. The increasing (decreasing) property of the optimal reference prices follows directly from the monotone convergence result in Proposition 2.

• a2. Convexity (Concavity): from equation (6) we know that the progression of the optimal reference prices is characterized by the following series:

  

where A and B are coefficients characterized in equation (6). Hence

We show that 0⩽A⩽1.

the second line follows from positivity of r^** and equation (6). It is easy to see that the enumerator of A is positive by maximizing Δ₂. Hence, we only need to show the denominator is positive to establish positivity of A.

From the increasing (decreasing) property (part a) and observation that 0⩽A⩽1 we know:

The above inequalities coupled with 0⩽A⩽1 show that r_t+1−r _t , equation (A4), is decreasing in t and establish the convexity(concavity) property.

• b1. The increasing (decreasing) property of the optimal first- and second-stage prices follows directly from the monotone convergence result in Proposition 2.

• b2. p_1, t^* is a linear function of the reference price in the form p_1, t^*=Cr+D, where C is positive. This positivity is established in the same manner as part a of this corollary. In addition, p_2, t^* is also a linear function of the first-stage price with positive coefficients. The desired result directly follows as concavity (convexity) is preserved under such transformations. □

Proof of Proposition 3

The proof is same as Proposition 2 and follows from p₁^*(r)⩾r⇔r⩾r^** □

Proof of Theorem 2

In this subsection, we derive the closed-form solution to the optimal pricing policy of a firm operating in the generalized setting discussed in the section ‘Generalized reference price update equation’. Recall J^*(r₀) is the optimal value of the optimization problem of the retailer that is derived from:

We retain all the assumptions of the subsection ‘Analysis: multi-period model’, hence as before the optimal value function J^*(r) is the solution to the following Bellman equation:

Below we state Theorem 2 again fully without any reference to Theorem 1.

Theorem 2:

• The optimal pricing policy for a retailer in the above general setting is

  
  

  where

  

Proof: Propositions 10 and 11 hold same as the case of Theorem 1. We postulate that the value function is of the quadratic form presented in equation (4). Hence the Bellman equation (A1) becomes

The optimal prices (p₁^*(r), p₂^*(r)) can be derived by first-order conditions. Solving the system of equations ∂J/∂p₁=∂J/∂p₂=0 for (p₁, p₂) yields the optimal policies (A5) and (A6). Substituting these optimal policies in (A7) and collecting the coefficients of the quadratic polynomial, we derive the values of coefficients γ, δ and ω. Now we just have to validate the sign of coefficients.

Choice of γ: γ is one of the roots of the following quadratic equation,

The proof for positivity of γ is same as Theorem 1 and is omitted here.

Positivity of δ: The same proof as Theorem 1 goes through. We only need to show that the extra term added in enumerator is always positive.

Positivity of ω: The same proof as Theorem 1 goes through. We only need to show that the extra term added in enumerator is always positive.

In both the above arguments, the first inequalities are trivial and the second inequalities follow from the regularity condition.

Proof of Proposition 4

• a The proof is similar to the proof of Proposition 2 and its details are omitted here. The proof technique is the same and follows from the following observations

  

  and

  

• b From the reference price update equation (7), we know that the steady-state price p₁^** is equal to ((1−k₂)r^**−k₃)/k₁. From equation (A5), we know that the optimal price p₁^*(t) is increasing in r(t). Hence, p₁^*(t) is an increasing series and the result follows for monotone convergence of p₁^*(t) series from part a. From equation (A6) we know that p₂^* is an increasing function of p₁^*, and hence the result for p₂^*(t) series follows directly. □

Proof of Proposition 5

This proposition is a special case of Theorem 2. This follows from the following observation:

Hence, set

and the results follow. □

Proof of Proposition 6

This proposition is a special case of Theorem 2. This follows from the following observation:

Hence, set

and the results follow. □

Proof of Proposition 7

This proposition is a special case of Theorem 2. This follows from the following observation. The myopic optimal decision of firm 2 is as follows (subscript t has been dropped):

Hence

Set

and the results follow. □

Proof of Proposition 9

If firm 1 uses a linear function of the reference price to choose her pricing decision, firm 2 observes a linear update function in terms of her own price and the reference price. Hence by Theorem 2 the optimal decision of firm 2 will be a linear function of the reference price. As a result of this, firm 1 will also observe a linear update function of the reference price and her own price and will keep her pricing function a linear function of the reference price. □