Analysis of Objective Function

Note x₁, δ, and r are the key parameters in the firm’s optimization problem. Later we will investigate how these parameters affect the firm’s optimal revenue.

3.3.1 Analysis of Objective Function

As preparation, we first study the property of the firm’s objective function in (3.3).

Close scrutiny of the demand functions di(p1, p2) (i = 1, 2) reveals that the region R in (3.3) can be partitioned into sub-regions based on consumers’ purchase decisions.

In particular, R can be divided into 3 regions, Ri (i = 1, 2, 3), where R1 is the region in which there is demand for product 1 from both types of consumers, R₂ is the region in which the demand for product 1 is only from the naive consumers, and R3 is the region in which no consumer purchases product 1. Consumers’ purchase decisions regarding product 2 further split the Ri’s into sub-regions. For example, R11 is the sub-region in which there is demand for product 2 from both types of consumers.

Table 3.1 summarizes the consumers’ purchase decisions in these different regions, which are also illustrated by Figure 3.1. A full characterization of the sub-regions is lengthy and therefore given in the appendix.

From (3.3) and Table 3.1, it is clear that π(p₁, p₂) has different expressions in different regions. The following lemma summarizes the properties of π(p1, p2).

Consumer Type Opportunistic Naive

Product 1 2 1 2

R₁

R11 X X X X

R₁₂ X X X

R13 X X

R₂ R₂₁ X X X

R₂₂ X X

R₃ R₃ X X

Table 3.1.: Consumer Purchase Decisions (X means positive demand).

Lemma 3.3.1 The objective function π(p₁, p₂) is continuous in R. Moreover, π(p₁, p₂) is continuously differentiable and jointly concave in (p₁, p₂) in R_i (i = 1, 2, 3), respec-tively.

Now we consider the firm’s optimal pricing decisions. From Lemma 3.3.1, the optimal solution of (3.3) is straightforward if the capacity constraint d₁(p₁, p₂) ≤ x₁ does not exist. Let d^∗_i = d_i(p^∗₁, p^∗₂) (i = 1, 2) be the demand of product i when the optimal prices are used, and π^∗ = π(p^∗₁, p^∗₂) the firm’s optimal profit. Actually, it can be readily shown that in the absence of the capacity constraint, the firm’s optimal solution is given by

(p^∗₁, p^∗₂) = (q₁

2, q₂+ δ

2 ), d^∗₁ = 1

2, d^∗₂ = 0, and π = q₁ 4.

That is, the firm only sells the premium product, i.e., there is d^∗₂ = 0 in the optimal solution. Furthermore, there must be d^∗₁ = ¹₂, i.e., only half of the consumers with θ ≥ ¹₂ will be served. This is consistent with the existing results in the product line design literature (see Mussa and Rosen 1978). In the literature, it has been assumed that all consumers have the same valuation for the regular product. So we can show such a result still holds even when the consumers have different valuations for product 2. The intuition is that the firm wants to eliminate product competition between its

own products when there is no usage cost. Thus, the firm’s prices make the premium product more attractive than the regular product (or essentially removing the regular product from the market). From this observation, we will focus on the case x₁ ≤ ¹₂ in the rest of chapter.⁵

Figure 3.1.: Feasible Regions.

The analysis of the optimization problem in (3.3) becomes more involved when the capacity constraint is present, which can be highlighted in Figure 3.1. Note that the dashed lines in Figure 3.1 are the curves of prices (p₁, p₂) under which the capacity x₁ is fully utilized, i.e. d₁(p₁, p₂) = x₁. Intuitively, the premium product capacity is precious to the firm, which implies that the firm should always utilize such capacity to the fullest extent. Recall that only the premium product will be purchased when

5We have assumed the capacities are exogenously given. In reality, the capacity constraint would be determined by the cost of the capacity. For example, a capacity would be more constrained as its cost increases.

the capacity is sufficiently large (i.e., x₁ ≥ ¹₂). As x₁ decreases from ¹₂, the firm needs to divert part of the demand of the premium product to the regular product due to the capacity limit. Since the opportunistic consumers have higher valuation of the regular product, the firm would charge a price to first divert some opportunistic consumers would change their decision and purchase the regular product. As x₁ further decreases, at certain point, even though all the opportunistic consumers have been diverted to the regular product, the limited premium capacity can not satisfy all demand from the naive consumers. Thus, the firm has to change the prices so that some of the naive consumers will also be diverted to the regular product. Define parameters k^j (j = 1, ..., 6) that are independent of x1 as follows:

k¹ = (1 − r)δ q₁ − q₂ , k² = 1

2 1 − δ

q₁− q₂− δ s

rq₁(q₁− (q₂+ δ)(1 − r)) q₂(q₂+ (1 − r)δ)

! ,

k³ = (1 − r)(q₂+ (1 − 2r)δ) 2(q₂+ (1 − r)δ) , k⁴ = (q₂+ δ)(1 − r)

2q₁ , k⁵ = (q₂− δ)(1 − r)

2q₂ , k⁶ = 1

4(q₂+ δ) q₂(1 − r) q₁

 q₂+ δ q₁ − 2



+ q₂

q₂+ (1 − r)δ − r

 .

(3.4)

The firm’s optimal solution can be achieved in one of the four possible regions:

Proposition 3.3.1 characterizes the firm’s optimal solutions.

Proposition 3.3.1 Consider x₁ ≤ ¹₂. The firm’s optimal solution is determined by x₁ and the thresholds k^j (j = 1, · · · , 6) defined in (3.4). Specifically,

Case 2. If k¹ ≥ k², k³ > k⁴ and k⁶ ≥ 0: there exists a threshold ¯k ∈ [k⁴, k³], then

We may further explain the intuition behind Proposition 3.3.1. Recall that the opportunistic consumers have higher valuation of the regular product compared to the naive ones, which implies that there will be more opportunistic consumers buying the regular product than naive consumers under any price scheme. On the other hand, the premium product is less attractive to the opportunistic consumers, who are more sensitive to the price change of the premium product. When capacity is restrictive, the firm tends to increase the price of the premium product, which drives the opportunistic consumers to the regular product. Thus, the number of the opportunistic consumers buying the premium product decreases as the capacity becomes scarce. Table 3.2 illustrates the optimal solutions in each case described by Proposition 3.3.1. For instance, if k¹ < k², the optimal solutions are described in Case 1. As x₁ decreases from ¹₂ to 0, the optimal solution moves from region R₁₂ to R₁₁ to R₂₁, which is shown in Table 3.2a.

Consumer Type Opportunistic Naive

Product 1 2 1 2

R12 X X X

R₁₁ X X X X

R₂₁ X X X

(a) Case 1.

Consumer Type Opportunistic Naive

Product 1 2 1 2

R12 X X X

R21 X X X

(b) Case 2.

Consumer Type Opportunistic Naive

Product 1 2 1 2

R₁₂ X X X

R₂₂ X X

R21 X X X

(c) Case 3 and Case 4.

Table 3.2.: Evolution of the optimal solution (x1 decreases from ¹₂ to 0).