Pricing strategy in the presence of reference prices with thresholds

Abstract

Reference price models allow for the explicit incorporation of inter-temporal effects of pricing decisions. We introduce a reference price model with thresholds within which there are no reference effects. For frequently purchased items such as groceries, a pricing action in this period may affect demand in this period and in the next period, and reference price models build that effect into an expected price. Previous reference models have been structurally limited to either single-price or cyclical price strategies but not both. The incorporation of thresholds allows for both types of pricing strategies, depending on the specific parameters for the model. We investigate the conditions under which a price cycles and those under which a single price is optimal. We also present propositions for the reduction of the search space to improve computational performance and present results of computational experiments to highlight key results. Understanding the nature of the product demand can help managers develop pricing strategies that maximize their profits.

APPENDIX

Proofs of propositions

Proof of Proposition 1 It is clear that:

This can be expanded to:

Proof of Proposition 2 This proof is based on the definition of a reference price (equation (1)). We use p_c(i)−1 and r_c(i)−1, i=1, …, M to denote the price and reference price in the period that immediately precedes the period where p_c(i) and r_c(i) occurred, respectively. By definition of the reference price we have αr_c(1)−1=r_c(1)−(1−α)p_c(1)−1, and by the ordering we have αr_c(1)<αr_c(1)−1. It follows that αr_c(1)<r_c(1)−(1−α)p_c(1)−1, or p_c(1)−1<r_c(1). This establishes part (1). Similar arguments can be used to show part (2). Part (3) follows immediately from parts (1) and (2). For part (4), r_c(1)⩽p_c(1) has to hold, otherwise the resulting reference price in the subsequent period would be smaller than r_c(1), contradicting the fact that r_c(1)=min_1⩽m⩽M r _cm . A similar argument can be used to show that r_c(M)⩾p_c(M).

Thus the period corresponding to r_c(1) and r_c(M) corresponds to a loss and gain, respectively.

Proof of Proposition 3 This follows easily from the fact that the reference price in period t is a convex combination of the reference price and price in period t−1, and that r _t ⩽p _t and r _t ⩾p _t for a loss and gain period, respectively.

Knowing that it suffices to establish that to show that

Proof of Lemma 1 We first assume that we have a one-period cycle and show, by contradiction, that only p^* will maximize the profit. Assume that price charged ≠p. We consider two cases.

Case 1: < p^*.

As price is constant, we have . We know by definition that < g(p^*). There is a finite loss in profit, owing to increasing price, from to p^* that is equal to (p^*−c)β _L ( −p^*). On the other hand, there is a long-term loss in profit as p^*=arg max _p g(p). Clearly, that is the short-term loss in profit by moving to p^* is less than the long-term loss in profit by staying at Thus a single-price strategy will never stabilize below p^*.

Case 2: When > p^*.

In this case, decreasing the price to p^* generates a gain equal to (p^*−c)β _G ( −p^*)>0, and as g(p^*)>

we conclude that a single-price strategy will never stabilize above p^*.

Proof of Theorem 1 We must show that for any M-period cycle, M>1, the profit (equation (5)) is not superior to that obtained for a one-period cycle charging p^*. We do not consider the initial adjustment that might be required to achieve the steady-state cycle. We consider the long-term steady-state pricing policy and the associated cycle of repeating prices. Let the vectors (p_c1, p_c2, …, p _cM ) and (r_c1, r_c2, …, r _cM ) denote the prices and reference prices in an M-period cycle. Note that p _cM , the last price in the cycle, will be followed by p_c1, the first price in the next cycle. We also define x⁺=max(x, 0) and x⁻=min(x, 0). Our goal is to show that for M>1.

We now consider an arbitrary cycle of length M, with K consecutive gains followed by J consecutive losses. We define all gains, (r _cm −p _cm )⁺, G _k , and losses (r _cm −p _cm )⁻, L _j .

We know from Proposition 3 that the losses will occur in sequence with ascending reference price, where , and similarly that the gains will occur in order of descending reference price and specify them such that We know from Proposition 1 that

We now consider L₁ (the loss with the lowest reference price) and G _K (the gain with the lowest reference price). As L₁ is a loss, we know that and using a similar logic that We assume, without loss of generality, that c=0. We recall that β _L ⩾β _G .

For an M=2 cycle, we know that ∣L₁∣=G₁ and can easily see that as and β _L >β _G . There is, therefore, no case where M=2 is optimal. It is easy to apply that same logic to M>2.

Step 1: If ∣L₁∣=G _K , we know from above that the lost profit from L₁ exceeds the profit gained from G _K . And now consider L₂ and G_K−1. It is worth noting that as ∣L₁∣=G _K , and , we get . Then

This means that . We can now start this process again by comparing L₂ and G_K−1. We also see that and therefore we have the same starting condition as we did where the first loss under consideration has a lower reference price than the first gain under consideration.

Step 2: If ∣L₁∣>G _K , we consider S additional gains such that but . If we partition G_K−S into G_K−S′ and G_K−S″ such that . We know that because they are gains, ∀i=K−S … K, and it follows that . Now

and so

which means that ∀i=K−S…K, which gives us so the profit lost due to the loss is greater than the profit gained owing to the matching gains. If we do not partition G_K−S, the component disappears but the profit impacts are the same. The next step is to compare L₂ and G′_K−S or G_K−(s+1). We note before proceeding that as

but

it is easily shown that and therefore we have the same beginning condition where the first loss under consideration has a lower reference price than the first gain under consideration.

Step 3: If ∣L₁∣<G _K , we consider S additional losses such that but . If we partition L _S into L _S ′ and L _S ″ such that . We know that as they are losses, ∀i=1 … S. We also know that because of the definition of reference price. It follows then that . If we do not partition L _S , the L _S ″/L _S component disappears but the profit impacts are the same. Once again, the next step is to compare G_K−1 and L _S ′ or L_s+1. We can consider the finishing condition with respect to reference prices.

but

and show that has the same beginning condition where the first loss under consideration has a lower reference price than the first gain under consideration.

This iterative process where step 1, 2 or 3 is applied sequentially as appropriate to the entire M-period cycle results in definitive proof that there is no cycle M>1 that provides more profit than the single price.

In the case where the K gains and/or J losses are not consecutive, we approach the problem similarly by considering the sub-cycles of the large price cycle within which one or more gains precede one or more losses. The steps (1 through 3) outlined above are applied to the sub-cycles, beginning with the highest reference price that is formed by a gain but involves a loss (an inflection point). We can show that each sub-cycle, and subsequently the entire cycle, results in a net loss of profit.

There is, therefore, no cycle M>1, which is superior to M=1, when β _L ⩾β _G .

Proof of Lemma 2 We know from Lemma 1 that if a single price is optimal, the price is p^* in the absence of thresholds. The same is true with thresholds. Recall that we considered a price policy with a single price, , that is not equal to p^*.

Case 1: When <p^*, we have , as price is constant. There is a finite loss in profit (⩾ 0) owing to moving to p^* that is (p^*−c)β _L ( +ρ−p^*)⁻. There could be no loss in profit owing to moving to p^* because the price increases can happen in increments such that the loss threshold is never exceeded, but it may also be worthwhile to incur the loss about ρ in the short term to increase the profits in the long term. We know g( ) < g(p^*). Profit will always increase by moving the price up to p^*. Thus a single-price strategy will never stabilize below p^* even within the presence of thresholds.

Case 2: When >p^*, the proof is straightforward. We know that g()<g(p^*) and (p^*−c)β _G ( −τ−p^*)⁺⩾0. Thus we gain by moving to p^* as we will either have a positive reference gain effect or no reference effect and the price effect is positive.

Proof of Theorem 2 Consider an M=2 cycle in which we subtract a small increment, δ, from p^* for one price and add the increment to p^* for the second price. We choose a value such that δ⩽0.5ρ. We specify: p_c1=p^*+δ and p_c2=p^*−δ. Reference prices are: r_c2=αr_c1+(1−α)p_c1 and r_c1=αr_c2+(1−α)p_c2. Then:

Similarly

We now define Y, the loss effect, such that

for 0⩽α⩽1, ((1+α²−2α)/(1−α²))⩽1 then

It is clear, therefore, that Y=0. For the single-price strategy to be optimal:

But Y=0, thus

Now we divide both sides by δ and take the limits as δ → 0 and as g is continuous and differentiable

which is impossible as the right-hand side is always positive. Thus, with a loss threshold only, the price always cycles.

Proof of Theorem 3 We begin with the same arbitrary M-period cycle we specified in the proof for Theorem 1. We have cycle profit:

where

We showed in the proof of Theorem 1 that:

for all cycle lengths. We know τ>0, and therefore

and

Thus in no case where there is a gain threshold only would a cyclical price be optimal.

Proof of Lemma 3 Consider an optimal solution in which the range [p_min, p_max] does not include p^*. If p_max<p^*, each price in the M-period price cycle can be increased by p^*−p_max. In the steady-state case, this would increase each of the reference prices in the M-period cycle by the same amount. The gaps between price and reference price in each period will be unchanged, and therefore there would be no change in the reference component of the profit function, as the reference gaps for price decreases and price increases would not change. Owing to the concavity of g(p) we know that the non-reference component of each period's profit will increase as prices move closer to p^*.

In the case where p_min>p^*, we similarly subtract p_min−p^* from each price in the M-period price cycle. In all cases, the solution is improved if the range includes p^*.

Proof of Lemma 4 We note that for M>1:

We will use the absolute threshold example for illustration, but the proportional threshold example is analogous. If this were not true, then M=1 and a single-price strategy is optimal.

First, for simplification let us specify the individual elements of the equation above.

Now, consider a portion of an optimal price cycle of length M with a sequence p _ci >p _cj >p _ck , where 0<i<j<k⩽M and i, j, k are integer. This sequence has two consecutive price decreases where we assume that r _cj −p _cj < τ, and the threshold is not exceeded in that interval. If we eliminate p _cj we have a cycle of length M−1. Component B will decrease by g(p^*)−g(p _cj ). If p _cj =p^*, component B will not change. Component C may or may not change. We know that, at worst, this will stay the same if there is no threshold effect. Note the reference price after charging p _ck in the original sequence is r _ck =α²r _cj +(α−α²)p _cj +(1−α)p _ck . If we skip the interim price p _cj , the new reference price is αr _cj +(1−α)p _ck , which then would lead to the potential for decrease in component C. Now we consider component A. The step of eliminating the intermediate price would not change if r _cj −τ−p _ck ⩽0, but would increase if r _cj −τ−p _ck >0. By eliminating the interim price, in which the reference threshold was not exceeded, we have increased profit.

Proof of Lemma 5 The proof builds on that for Lemma 3. We relax the assumption that r _cj −p _cj < τ, so that the threshold may be exceeded in the interim step. Otherwise, our specification of p _ci > p _cj > p _ck remains the same. The impacts on components B and C are identical. We now consider the impact on Component A. With the interim step, the reference gains are (r _cj −τ−p _cj )+(r _ck −τ−p _ck ) if (r _cj −τ−p _cj ) > 0, and (r _ck −τ−p _ck ) > 0. Both are, by definition, no lower than zero.

Lemma 3 establishes that we will not consider price paths in which either of the components are less than zero. Without the interim step, the reference gains are r _cj −τ−p _ck . The difference between the two is the gain effect lost by eliminating the interim step. We find the difference in the reference gaps between the two scenarios (with and without interim step p _cj ): (r _cj −τ−p _cj )+(r _ck −τ−p _ck )−(r _cj −τ−p _ck ) that then can be simplified:

It is worth noting that α(r _cj −p _cj )−τ may be negative, depending on the size of τ. The difference in reference gap may be smaller than having to overcome one more gain threshold. Removing the interim step, p _ck , reduces component B by (g(p^*)−g(p _cj )). The amount lost would be from component A and is at least (p _cj −p _ck )β _G (r _cj −τ−p _cj )+(p _ck −c)β _G (α(r _cj −p _cj )−τ), where the second term may, in fact, be negative. If

for any p _ck < p _cj , then there will be no price decrease from p _cj to p _ck .

Proof of Lemma 6 Consider an optimal price path in which a price increase to price p _cj falls below the minimum of r _cj +ρ and p^* in the price path p _i <p _j <p _k . We assume in the first case that p _ck is greater than the minimum of r _cj +ρ and p^*. Now, if we increase p _cj to the minimum of r _cj +ρ and p^*, we need to evaluate the impact on each of the components A, B and C. M will not change, as we are not inserting or removing a price but rather just increasing p _cj .

• Component A will increase. A higher reference price, r _cj , will result because p _cj is higher. Subsequent reference prices will also be higher. We know that r _ct −τ−p _ct >0 at least once or M=1 and a constant price would be optimal.

• Component B will decrease. An increase in p _cj will decrease g(p^*)−g(p _cj ) owing to convexity and the definition of p^*.

• Component C will stay the same or decrease. We remain within the loss threshold for the time period where we charge the higher p _cj . This also results in a higher reference price for the subsequent period. If r _ck ′+ρ−p _ck > 0, then Component C will decrease. The same is true for subsequent price increases.

In all cases, the increase to at least the minimum of r _cm +ρ and p^* increases average profit.