Intertemporal pricing with boundedly rational consumers

Abstract

Flexible pricing plans are commonly observed in service industries. In this article, we argue that the presence of flexible pricing plans can be attributed to consumers being boundedly rational – these consumers do not always select the best available option; rather, they select better options more often. In our model, the seller faces consumers who are heterogeneous in their degrees of intertemporal inconsistency – their ultimate actions can be different from their intended actions. We show that, in response to these boundedly rational consumers the seller may be able to extract more profit by setting different prices in different periods and allowing the consumers to self-select which period to pay. Moreover, a single pricing plan may emerge as an optimal pricing scheme even when the consumers are heterogeneous in their degrees of rationality and the seller is not fully aware of the consumers’ types. We further show that the pricing patterns depend primarily on the relative discounting factor between the seller and the consumers.

APPENDIX

Proofs of Propositions 1, 2 and 3

Recall that the participation constraint can be written as

which, after replacing u_i2 by v _i +u_i1, can be simplified as follows: . The seller's problem is therefore to find a pricing plan that solves the following problem:

Since u_i2=v _i +u_i1 and δ _s p_i2=δ(p_i1−v _i ), for a fixed value of v _i , u_i2 decreases as u_i1 decreases and p_i2 increases in p_i1. Suppose that the participation constraint does not bind, the seller can decrease u_i1 while keeping v _i unchanged. u_i2 decreases as a result. Since p_i1=μ−u_i1 and p_i2=(μ−u_i2)/δ _c , both p_i1 and p_i2 increase and so does the seller's expected profit. Thus (PC-i) binds:

We can therefore represent utilities and prices in terms of v _i :

Substituting these terms in the seller's objective, the seller's optimization problem becomes an unconstrained one:

where the third equation follows from . The first-order condition on v _i leads to

Let and observe that D(0)=0 and D(v _i ) is negative for all other values of v _i . Therefore, and thus:

• When δ=1, (A.2) is satisfied for any v _i , thus the seller's expected profit is μ−c regardless of the first and second period prices.

• When δ<1 and γ _i >1/μ, first order derivative with respect to v _i is positive for all values of v _i . Thus the optimal value of v _i is −∞, the optimal pricing plan is {p_i1^F, p_i2^F}={μ,∞}, and the associated seller's expected profit is μ−c.

• When δ>1 and γ _i >1/μ, first order derivative with respect to v _i is negative for all values of v _i . Thus the optimal value of v _i is ∞, the optimal pricing plan is {p_i1^F, p_i2^F}={∞, δμ}, and the associated seller's expected profit is δμ−c.

Notice that D(v _i ) is symmetric with respect to the origin:

Its derivative,

is negative for positive values of v _i and positive for negative values of v _i . Thus, for all γ _i <1/μ, there are exactly two solutions to (A.2). Let v _i ^P>0 be the positive solution and v _i ^N=−v _i ^P be the negative solution to (A.2). Now we examine the second-order conditions on v _i :

We see that (A.3) is negative when δ<1 and positive when δ>1 while (A.4) is negative when δ>1 and positive when δ<1. Thus v _i ^P is a local maximizer when δ<1 while v _i ^N is a local maximizer when δ>1. We now show that Π _i ^F monotonically decreases in γ _i :

Thus, there exists a cutoff type, γ _lp , such that the corresponding v _lp (>0) satisfy Π _i ^F(v _lp )=μ−c. When δ<1, v _i ^P is the global maximizer for all γ _i <γ _lp . we now look for the condition for γ _lp and v _lp :

Thus, γ _lp and v _lp satisfy the following equations simultaneously:

Similarly, there exists a cutoff type, γ _mp , such that the corresponding v _mp (<0) satisfy Π _i ^F(v _mp )=δμ−c. When δ>1, γ _mp is the global maximizer for all γ _i <γ _mp ,

Thus, γ _mp and v _mp satisfy the following equations simultaneously:

Proof of Proposition 5

First, we re-write the optimization problem as

Notice that when δ<1, the {p_i1, p_i2}={μ,∞}, ∀i=1,2, is a feasible solution to (A.7). The seller's expected profit associated with this candidate solution is μ−c. We thus focus on searching for a solution that yields an expected profit that is higher than μ−c. Similarly, {p_i1, p_i2}={∞, δμ}, ∀i=1,2, is a feasible solution to (A.7) when δ>1. The seller's expected profit associated with this candidate solution is δμ−c, and we search for a solution that yields a higher than δμ−c expected profit. We now show that both (PC-2) and (IC-1,2) bind at optimality by contradiction. Suppose (PC-2): does not bind. In this case, the seller can increase his expected profit by decreasing u₂₁ and u₂₂ while keeping v₂ unchanged (v₂=u₂₂−u₂₁). This will increase the objective value without violating any constraint. Thus (PC-2) binds, that is

Given this, we can write type-2 consumers’ utilities and payments in terms of a single variable v₂:

We then consider (IC-1,2):

This constraint must bind at optimality as well; otherwise, the seller can increase his expected profit by decreasing u₁₁ and u₁₂ while keeping v₂ unchanged. When v₂>0, and thus When v₂<0, and thus (IC-1,2) can then be written as the following equality constraint:

In the remaining proof, we divide our analysis into two cases: (1) v₂ is positive and (2) v₂ is negative. There are two subcases in each case: max{u₁₁, u₁₂}=u₁₁ and max{u₁₁, u₁₂}=u₁₂ in each case. We now look at each of them separately:

Case 1: v₂ is positive.

Subcase 1.1: . In this case, A fully rational consumer purchases in the period when the price is lower, and thus the first-period price has to be non-negative in the optimal solution; otherwise, the seller can set the second-period price to be negative instead to obtain a smaller loss. Thus, p₁₁⩾0, that is Notice that if we set v₁=v₂, we obtain u₁₁=u₁₂−v₁=u₁₂−v₂=u₂₂−v₂=u₂₁. In this case,

that is, (IC-2,1) is satisfied. The seller's expected profit function can be written as an unconstrained optimization function with a single variable (v₂):

First, let us compare the seller's expected profit with μ−c – obtained from a candidate solution {p_i1, p_i2}={μ,∞}, ∀i=1,2, when δ<1:

where the inequality follows from the fact that p₁₁⩾0 and δ<1. Since , setting the first-period price to is never optimal when δ<1. Next, we compare the seller's expected profit with δμ−c – obtained from a candidate solution {pi₁, pi₂}={∞, δμ} when δ>1:⁴

Thus, it is optimal to set v₂ to infinity when the first-period price to

Subcase 1.2: In this case, Moreover, (PC-1):

is satisfied as well. Substituting all the utilities and payments by the aforementioned functions of v₂, the seller's expected profit function can be written as:

Observe that when δ>1,

Since Π^S−(δμ−c)<0, it is optimal to set v₂ to ∞. When δ<1, (A.8) is greater than equation (A.9) and in which case setting the second-period price to is never optimal. When δ<1 and the first-order condition on v₂ yields

We can rewrite (A.10) as follows:

Let v₂^P be the positive solution of (A.11), the second-order condition is as follows⁵:

The equation above is only negative when ρ₁ is sufficiently small, that is if

we obtain that

The equation above shows that Π^S monotonically decreases in γ _i .

Case 2: v₂ is negative.

Subcase 2.1: With the same reasoning as Subcase 1.1, The seller's expected profit function can be written as

Let us first compare the seller's expected profit with μ−c – obtained from a candidate solution {p_i1, p_i2}={μ,∞}∀ i=1,2 when δ<1:

Therefore, setting the second-period price v₂ to −∞ is optimal. We now compare the seller's expected profit with δμ−c, which is obtained from the candidate solution {p_i1, p_i2}={∞, δμ}, ∀ i=1, 2 when δ>1:

Thus, setting first-period price to is never optimal.

Subcase 2.2: In this case, The seller's optimization problem can be written as

Observe that when δ<1,

Thus, setting the second-period price v₂ to −∞ is optimal. When δ>1, The expression of Π^S in (A.12) is greater than that in (A.13) and Therefore, setting the second-period price to is never optimal. When δ>1 and the first-order condition on v₂ yields:

We can re-write (A.14) as

Let v₂^N be the negative solution of equation (A.15), the second-order condition is as follows:

Equation (A.16) is negative when ρ₁ is sufficiently small. Specifically,

We now show that Π^S monotonically decreases in γ _i :

Finally, we can pin down the cutoff thresholds ρ₁^c and v^c. They are the solutions that solve (A.17) and (A.18) for δ<1:

and for δ>1:

Proof of Proposition 6

Since u_i2=u_i1+v _i , (PC-i) and (IC-i, j) can be simplified as follows:

Now we show that (PC-2) and (IC-1,2) imply (PC-1):

Recall that both (PC-2) and (IC-1,2) bind, otherwise the seller can raise prices to decrease consumers’ utilities until they bind while increasing his expected profit. Define a positive function for all values of v. From the binding (PC-2) and (IC-1,2), we obtain that

Recall that the objective function is

After substituting the prices and utilities by functions of {v _i }'s, we can express the reduced program of the seller's optimization problem as follows:

Π^S can then be written as follows:

Let

represent the maximum expected profit extracted from type-1 consumer when the seller is able to identify the consumer's type. Recall that γ _c is the cutoff value of γ above which the seller's expected profit is bounded above by μ−c. Thus, R₁^*⩽μ−c for all γ₁⩾γ _c . Since F(v₂)⩾0 for all values of v₂,

Similarly, let

represent the maximum revenue extracted from type-2 consumer when the consumer's type is known. Since γ₂<γ _c , R₂^*>μ. From the above discussions, we obtain an upper bound of the excess expected profit the seller can obtain from offering a menu of contracts:

Notice that when ρ₁=1, only type-1 consumers are present in the market. Π^S becomes Π^F and thus Π^S=μ−c. When ρ₁=0, Π^S=R₂^*>μ−c.

Let and be the cutoff value such that For any value of Thus, it is optimal for the seller to simply offer a single pricing plan {p_i1, p_i2}={μ, ∞}. □

Proof of Proposition 7

The participation constraint can be simplified as

This constraint must bind, thus we can re-write utilities and prices in terms of v _i :

Substituting these terms in the seller's objective, the seller's optimization problem becomes an unconstrained one:

The first-order condition on v _i leads to

Notice that the above equation is the same as equation (A.2) in the basic case if we replace μ−μ_i0 by μ. Thus, we can directly borrow results from the proofs of Propositions 1, 2 and 3:

• When δ=1, equation (A.19) is satisfied for any v _i , thus the seller's expected profit is μ−μ_i0−c regardless of the first and second period prices.

• When δ<1 and γ _i >1/μ−μ_i0, first-order derivative with respect to v _i is positive for all values of v _i . Thus the optimal value of v _i is −∞, the optimal pricing plan is (p_i1^C, p_i2^C)=(μ−μ_i0,∞), and the associated seller's expected profit is μ−μ_i0−c.

• When δ>1 and γ _i >1/μ, first-order derivative with respect to v _i is negative for all values of v _i . Thus the optimal value of v _i is ∞, the optimal pricing plan is (p_i1^F, p_i2^F)=(∞,δ(μ−μ_i0)), and the associated seller's expected profit is δ(μ−μ_i0)−c.

Let v _i ^CP>0 be the positive solution to equation (A.19) and v _i ^CN=−v _i ^CP be the negative solution. Use the same mathematical deduction as that in the proofs of Propositions 1, 2 and 3, v _i ^CP is a local maximizer when δ<1 while v _i ^CN is a local maximizer when δ>1. Thus, there exists a cutoff type γ _lp ^C such that its corresponding ν _lp ^C satisfy Π _i ^C(v _lp ^C)=μ−μ_i0−c and that v _i ^CP is the global maximizer for all γ _i <γ _lp ^C. The cutoff values, γ _lp ^C and v _lp ^C, satisfy the following equations simultaneously:

It is worth noting that if we replace the term (μ−μ_i0) with μ, equation (A.20) is the same as equation (A.5). This means that when the best outside option gives type-i consumer a utility of μ_i0, the pricing plan is the same as the case of no competition and the consumer's valuation is μ−μ_i0. Combining the two equations above, we obtain a relationship between γ _lp ^C and v _lp ^C:

The equation above shows that γ _ip ^Cv _ip ^C is constant regardless the value of μ−μ_i0, thus v _lp ^C decreases in μ_i0 while γ _lp ^C increases in μ_i0.

Similarly, there exists a cutoff type γ _mp ^C such that the corresponding v _mp ^C satisfy Π _i ^C(v _mp ^C)=δ(μ−μ_i0)−c and that γ _mp ^C is the global maximizer for all γ _i <γ _mp ^C. The cutoff values, γ _mp ^C and v _mp ^C, satisfy the following equations simultaneously:

With the same reasoning as above, we can conclude that the absolute value of v _mp ^C decreases in μ_i0 while γ _mp ^C increases in μ_i0. Combine both cases we can conclude that the price dispersion decreases in μ_i0 while the cutoff valuations increase in μ_i0 increases. □

Proof of Proposition 8

Before we solve the optimization problem, let us first compare the consumers’ expected utility for a given pricing plan {p₁, p₂}. The expected utility for a low-valuation consumer is

where v=p₁−δ _c p₂. The expected utility for a high-valuation consumer is

Let μ _d =μ₁−μ₂, the above equation can be written as

Owing to the common degree of rationality, the probability of a consumer chooses to purchase in period 1 depends only on the adjusted price difference. A high-valuation consumer behaves the same as the low-valuation consumer does provided that the pricing plan induce both types of consumer to participate. Moreover, the expected utility of a high valuation consumer is higher than that of a low-valuation consumer by a fixed amount (μ _d ) for a given pricing plan. With that insight in mind, we can move on to solve the problem. Let v _i =u_i2−u_i1=p_i1−δ _c p_i2. Since μ₁−p₂₁=(μ₁−μ₂)+(μ₂−p₂₁)=μ _d +u₂₁ and μ₁−δ _c p₂₂=(μ₁−μ₂)+(μ₂−δ _c p₂₂)=μ _d +u₂₂, (IC-1,2) can thus be simplified as

and similarly (IC-2,1) can be written as

It is clear that (PC-2) binds, and thus We can write p₂₁ in terms of μ₂ and v₂: Similarly, δ _c p₂₂ can be written as Suppose (IC-1,2) binds, then , that is, Consequently, p₁₁ equals and δ _c p₁₂ equals

It is easy to verify that both (PC-1) and (IC-2,1) are satisfied:

The expected profit function can then be written as

The derivatives with respect to v₁ and v₂ are as follows:

Let v₁^p>0 and v₁ⁿ<0 be the two solutions for (A.25) and v₂^p>0 and v₂ⁿ<0 for (A.26). Since (A.25) and (A.26) must be satisfied simultaneously, v₁^p=v₂^p=v^p and v₁ⁿ=v₂ⁿ=vⁿ. For the expected profit function to achieve its maximum, we also need

Thus, the local maximizers are (v₁, v₂)=(v^p, v^p) when δ<1 and (v₁, v₂)=(vⁿ, vⁿ) when δ>1.

Case 1: δ<1

A candidate solution is the corner solution (v₁, v₂)=(−∞,−∞), with a profit of μ₂−c. Π^H(v^p, v^p) is a global maximum if it is greater than μ₂−c. We need to compare to the case where the seller only serve type-1 consumers. Here only (PC-1) needs to be satisfied; thus, Let Π₁ be the seller's expected profit, which can be written as

where v₁ satisfies Since Π₁(v₁) decreases in ρ₁, there exists a cutoff value for ρ₁ such that Π₁(v₁)=Π^H(v^p, v^p),

To summarize, the seller is interested in serving high-valuation consumers when the proportion is large enough (ρ₁>ρ₁^c), and when the proportion of high-valuation consumers is low, the seller sets the price as if it only serves type 2 consumers.

Case 2: δ>1

In this case, we obtain

The corner solution (v₁, v₂)=(∞,∞) gives rise to the expected payoff Π^H(∞,∞)=δμ₂−c. Thus, Π^H(vⁿ, vⁿ) is a global maximum if it is greater than δμ₂−c. □