Proof of Lemma 15 with Our New Method

Since the standard IFT and MCS approaches are not applicable to conducting com- parative statics analysis in (6.1), we develop a new method to prove Lemma 15. Before

presenting the proof of Lemma 15 and our new method in detail, we introduce a lemma that plays a key role therein:

Lemma 16 Let Fi(z, Zi) be a first-order differentiable function in (z, Zi) for i = 1, 2,

where z ∈ [z, ¯z] (z and ¯z might be infinite) and Zi ∈ Zi, where Zi is the feasible set of

Zi. For i = 1, 2, let

(z_i∗, Z_i∗) := argmax_{(z,Zi)∈[z,¯z]×Zi}Fi(z, Zi),

be the optimizer of Fi(·, ·). If z1∗ < z∗2, we have: ∂zF1(z1∗, Z1∗)≤ ∂zF2(z2∗, Z2∗).

Proof. z₁∗ < z₂∗, so z ≤ z₁∗ < z₂∗ ≤ ¯z. Hence, ∂zF1(z∗1, Z1∗)      = 0 if z₁∗ > z, ≤ 0 if z₁∗ = z; and ∂zF2(z2∗, Z2∗)      = 0 if z₂∗ < ¯z, ≥ 0 if z₂∗ = ¯z, i.e., ∂zF1(z1∗, Z1∗)≤ 0 ≤ ∂zF2(z2∗, Z2∗). Q.E.D.

Lemma 16 is straightforward, but it is a powerful tool in our new comparative statics method, as illustrated by the proof of Lemma 15:

Proof of Lemma 15. We show by contradiction, i.e., we derive a contradiction

under the assumption that y_i∗(γ) > y_i∗(ˆγ) for some 1≤ i ≤ p and γ < ˆγ. Without loss of

generality, we choose i = 1, i.e.,

y∗₁(γ) > y₁∗(ˆγ). (6.3)

Denote y₀∗(γ) := ∑p+q_j=1λjyj∗(γ) for all γ ∈ Γ. Lemma 16 implies that ∂y1F (y∗(γ)|γ) ≥ ∂y1F (y∗(ˆγ)|ˆγ), i.e., ∂y1f1(y ∗ 1(γ)) + λ1∂y0h(y ∗ 0(γ)|γ) = ∂y1F (y ∗_{(γ)|γ) ≥∂} y1F (y ∗_(ˆγ)|ˆγ) =∂y1f1(y ∗ 1(ˆγ)) + λ1∂y0h(y ∗ 0(ˆγ)|ˆγ). (6.4)

The strict concavity of f1(·) yields that ∂y1f1(y1∗(γ)) < ∂y1f1(y1∗(ˆγ)). Hence,

∂y0h(y

∗

0(γ)|γ) > ∂y0h(y

∗

0(ˆγ)|ˆγ). (6.5)

Since h(·|·) is supermodular in (y0, γ) and concave in y0, y0∗(γ) < y0∗(ˆγ). Therefore,

there exists a j, 2≤ j ≤ p + q, such that y_j∗(γ) < y_j∗(ˆγ). (6.6) If 2≤ j ≤ p, we invoke Lemma 16 again, so y_j∗(γ) < y_j∗(ˆγ) implies that ∂yjF (y∗(γ)|γ) ≤

∂yjF (y ∗_{(ˆγ)|ˆγ), i.e.,} ∂yjfj(y ∗ j(γ)) + λj∂y0h(y ∗ 0(γ)|γ) = ∂yjF (y ∗_{(γ)|γ) ≤∂} yjF (y ∗_(ˆγ)|ˆγ) =∂yjfj(y ∗ j(ˆγ)) + λj∂y0h(y ∗ 0(ˆγ)|ˆγ).

Since ∂y0h(y∗0(γ)|γ) > ∂y0h(y0∗(ˆγ)|ˆγ) by (6.5), ∂yjfj(y∗j(γ)) < ∂yjfj(yj∗(ˆγ)). Since fj(·) is strictly concave, y_j∗(γ) < y_j∗(ˆγ) implies that

∂yjfj(y ∗

j(γ)) > ∂yjfj(y ∗

j(ˆγ)), which contradicts ∂yjfj(yj∗(γ)) < ∂yjfj(y∗j(ˆγ)). (6.7)

Analogously, if p + 1 ≤ j ≤ p + q in (6.6), Lemma 16 implies that ∂yjF (y

∗_{(γ)|γ) ≤} ∂yjF (y∗(ˆγ)|ˆγ), i.e., ∂yjgj(y ∗ j(γ)|γ) + λj∂y0h(y ∗ 0(γ)|γ) = ∂yjF (y ∗_{(γ)|γ) ≤∂} yjF (y ∗_(ˆγ)|ˆγ) =∂yjgj(y ∗ j(ˆγ)|ˆγ) + λj∂y0h(y ∗ 0(ˆγ)|ˆγ).

Since ∂y0h(y∗0(γ)|γ) > ∂y0h(y0∗(ˆγ)|ˆγ) by (6.5), ∂yjgj(yj∗(γ)|γ) < ∂yjgj(y∗j(ˆγ)|ˆγ). Since

gj(·|·) is submodular in (yj, γ) and strictly concave in yj, yj∗(γ) < yj∗(ˆγ) implies that

∂yjgj(y ∗ j(γ)|γ) > ∂yjgj(y ∗ j(ˆγ)|ˆγ), which contradicts ∂yjgj(y∗j(γ)|γ) < ∂yjgj(yj∗(ˆγ)|ˆγ). (6.8) Combining the contradictions of (6.7) and (6.8), we have y₁∗(γ) ≤ y∗₁(ˆγ). Repeat the

above argument for 1 < i≤ p, it follows that y∗_i(γ)≤ y∗_i(ˆγ) for all 1≤ i ≤ p. Q.E.D.

As we can see from the proof of Lemma 15, our new method employs Lemma 16 to make componentwise comparisons between the optimizers under different parameter values. More specifically, the method consists of five steps: Step (a). For each of the focal decision variable with some potential comparative statics result, we first assume, to the contrary, that the comparative statics prediction of this decision variable is reversed for some parameter values (e.g., inequality (6.3) in the proof of Lemma 15). Step (b). We invoke Lemma 16 to characterize some monotone relationships of the partial deriva-

tives of the objective function with respect to this decision variable at these parameter

values (e.g., inequality (6.4) in the proof of Lemma 15). Step (c). Using some model specific properties of the objective function (e.g., the supermodularity in one decision variable and the parameter, componentwise concavity, and first-order differentiability), such monotone relationships of the partial derivatives can be translated back into the monotone relationship of another optimal decision variable at the given parameter values (e.g., inequality (6.6) in the proof of Lemma 15). Step (d). Repeating steps (b) - (c), we employ Lemma 16 to iteratively establish the monotone relationship of partial deriva- tives and that of some other optimal decision variables at the given parameter values. This iterative procedure is stopped when either (i) the desired comparative statics result

for the focal decision variable is proved by contradiction (e.g., inequalities (6.7) and (6.8) in the proof of Lemma 15), or (ii) no further monotone relationship can be established (e.g., the case in which we assume that y_i∗(γ) > y_i∗(ˆγ) or y∗_i(γ) < y_i∗(ˆγ) for γ < ˆγ and p + 1 ≤ i ≤ p + q, see Appendix E.2). Step (e). We repeat the same iterative pro-

cedure, i.e., steps (a) - (d), for each focal decision variable to obtain its corresponding comparative statics result.

Note that there are two stopping conditions for the iterative procedure in Step (d). When the stopping condition (ii) applies, by our experience, it is very likely that there exist some model specifications such that the desired comparative statics result for the focal decision variable does not hold. For example, in the optimization problem (6.1), no contradiction can be reached under any monotone comparative statics prediction of

y∗_i(·) (p + 1 ≤ i ≤ p + q) with respect to γ for general {fi(·), gi(·|·), h(·|·)} functions. In Appendix E.2, we discuss in detail on how the iterative procedure in Step (d) is stopped without reaching a contradiction in this case, and give an example in which

y∗_i(γ) (p + 1≤ i ≤ p + q) is not monotone in γ. Hence, our new method not only helps

prove the comparative statics results when they exist, but also helps identify cases in which comparative statics results do not hold for some decision variables.

Our method proves the desired comparative statics result by contradiction. The

essence is to construct a contradiction by iteratively linking the monotone relationship between the optimizers and that between the partial derivatives. Though simple, Lemma 16 plays a crucial role in this process, because, in Step (b), it translates the monotonic- ity of the focal decision variable (in the parameter) into that of the partial derivative of the objective function with respect to this decision variable at the optimizing point. Hence, in Step (d), Lemma 16 enables us to iteratively link the monotone relationship of optimizers and that of partial derivatives, which is the key to establish a contradiction in our method. The main benefit of Lemma 16 is that the monotonicity of the partial derivatives with respect to the focal decision variable is irrelevant to the values of other decision variables at the optimizing point. This benefit allows us to perform comparative statics analysis componentwisely in Step (e). Hence, our method enables us to perform comparative statics analysis in a model where only part of the optimal decision variables are monotone in the parameter, and it is scalable. The componentwise comparison be- tween the optimizers is also the key difference between our method and the IFT and

MCS approaches, both of which involve the analysis of some properties of the objective function in terms of the whole decision vector (e.g., the Hessian and/or the joint super- modularity of the objective function). Moreover, since the objective functions in Lemma 16, Fi(·, ·) (i = 1, 2), can be completely different, our method can be used to compare the optimal decisions in different models. See, e.g., the proofs of Theorems 6.4.7, 6.4.8, and 6.5.5 in Appendix E.1.

Although our method is fundamentally different from the IFT and MCS approaches, it shares some similarity with these two standard approaches. As the IFT approach, the proposed method studies the first-order (KKT) condition at the optimizer of interest. Hence, the objective function needs to satisfy the first-order continuous differentiabil- ity condition, but not necessarily the second-order continuous differentiability condition. Analogous to the MCS approach, our new method analyzes the impact of the parameter upon the marginal value of each decision variable in detail, so that we can translate the monotonicity of partial derivatives with respect to one decision variable back into the monotonicity of another optimal decision variable. Thus, in order to obtain a contradic- tion (and a comparative statics result), our method requires the objective function to be

supermodular in the parameter and each of the focal decision variables (e.g., F (y|γ) is

supermodular in (yi, γ) for each 1 ≤ i ≤ p in our example), but not necessarily jointly

supermodular or satisfying the single crossing property. The above two condition relax- ations enhance the applicability of our method in the general joint pricing and inven- tory management model with demand segmentation, supply diversification, and market environment fluctuation, where the second-order continuous differentiability and/or, in particular, the joint supermodularity of the objective function in each decision epoch are hard, if not impossible, to establish. In the next section, we discuss in detail how the new method facilitates the comparative statics analysis in this model. We also demon- strate the applicability of our method in game theoretic models of joint price and effort competition in Section 6.5.

6.4 Application of the New Comparative Statics Method in a General Joint

Pricing and Inventory Management Model

In this section, we employ our new comparative statics method to study a general joint pricing and inventory management model with demand segmentation, supply di-

versification, and market environment fluctuation. The comparative statics analysis is essential to studying this model, because it enables us to characterize the optimal pricing and replenishment policy, and the impact of demand segmentation, supply diversification, and market environment fluctuation therein. The analysis in this section demonstrates the applicability of our new method in joint pricing and inventory management models.