Online Supplementary Appendix B

Transcription

1 Online Supplementary Appendix B Uniqueness of the Solution of Lemma and the Properties of λ ( K) We prove the uniqueness y the following steps: () (A8) uniquely determines q as a function of λ () (A) uniquely determines θ () We define the total quantity sold y the monopolist (the LHS of (A)) y Q q f d We can show that Q( λ ) is continuous and ( λ) ( θ, λ) ( θ) θ θ= θ strictly decreasing in λ () By using results in step () and assuming lim λ Q( λ) =, we can conclude that Q( λ ) = K when K is not too large As a result, the inverse function of λ is well-defined: λ ( K) Q ( K) = The intuition ehind lim λ Q( λ) = is that when λ ecomes very large, the monopolist should sell fewer units ecause the virtual variale cost is very high If we interpret λ as the marginal revenue resulted from tight capacity constraint, the argument is that the monopolist has to sell very small numer of units so that the marginal revenue could reach this high level Step : In (A), the uniqueness of q is guaranteed y the concavity assumptions ecause the LHS of (A) is decreasing As a result, the LHS of (A) is a one-to-one, continuous (all functions of RHS are continuous) mapping from the compact metric space of q to the compact metric space of λ Hence, the inverse function is also continuous In other words, q is continuous in λ Step : To show the uniqueness of θ, it can e verified that q( θ ) = is always a solution of (A) Define the LHS of (A) as Lqθ ( ( ), θ ) Fully differentiating Lqθ ( ( ), θ ) with respect to q, it follows that which simplifies into U U H U U H d dq U dh dq, U U H d dq U dh dq ecause U U H( θ) = λ y (A8) Here, we suppress the arguments of U for ease of exposition Note that dl( q, θ )/ dq equals to λ at θ ecause q( θ ) =, U = and U =

2 at θ Hence, dl( q, θ )/ dq is strictly greater than λ θ > θ ecause the second term is positive from the following assumptions: U >, U <, H ( θ ) >, and dθ / dq> The last term is also positive given the fact that H ( θ ) is non-increasing in θ and q is non-decreasing θ Since dl( q, θ ) / dq > λ and dl( q, θ)/ dq = λ, it follows that Lq (, θ) > λq θ > θ, which implies the only solution of (A) is θ = q () Step : To show that (A) uniquely determines λ ( K), we first define the total quantity in terms of λ as Q q,fd We first show that oth Q( λ ) and q( θλ, ) are decreasing in λ Applying implicit function theorem on (A8) yields dq, d U U H, The last inequality results from the gloal concavity assumptions on U and U Since dq( θλ, )/ dλ < θ, it follows that dq( λ) / dλ < Next, we can show that Q( λ ) is also continuous in λ For any ε >, we can find a pair of λ and λ close enough so that in which the RHS equals to Q Q q, q, fd, q, q, fd : qq q, : qq q, fd This term is smaller than max q, q, P : q, q, By the capacity constraint, (max q( θλ, ) q( θλ, ) ) is ounded aove y some constant Also, q( θλ, ) q( θλ, ) can e aritrarily small y the continuity of q( θλ, ) As a result, the second term disappears and we have Q( λ) Q( λ ) < ε, which completes the proof QED

3 Online Supplementary Appendix C By (A8), it follows that the optimal quantity schedule is given y q, max,q,whereq is the solution of q () Zero-marginal cost pricing: when λ=, q max, The marginal customer who feels indifferent etween uying or not is given y θ = / and thus the market size is -/=/ Before deriving the price function, we need to derive the consumer surplus function first: s qtdt / It follows that the optimal price schedule is given y p( θ) = U( q( θ), θ) s( θ) = ( θ )( θ) Next, we are ready to derive the maximal revenue, which is given y / pd / d () Nonlinear pricing with discontinuous cost: The procedure for deriving solutions is similar to that in the zero-marginal-cost pricing case (A8) yields q max, The marginal customer who feels indifferent etween uying or not is given y θ = ( + λ) / and thus the market size is θ = ( λ)/ Next, we need to derive how the demand capacity constraint may affect λ It follows that ( + λ ) / (θ λ) dθ = ( λ) = K, λ = K The consumer surplus function is given y

4 θ s( θ) = q( t) dt = ( θ λ) ( + λ ) / The optimal total price schedule is given y p( θ) = U( q( θ), θ) s( θ) = ( θ λ)( λ+ θ) Therefore, the total revenue is given y p( θ) dθ = ( λ) ( λ+ ) ( + λ ) / () Linear pricing with discontinuous cost: given the linear pricing at per unit, each consumer type θ will self-select a level of usage, q( θ ), that maximizes his/her utility as follows max q q q q, The optimal solution of this prolem is q ( θ) = θ, which implies that consumers with θ will have non-negative usage Back to the monopolist's pricing prolem, it follows that max q fd C qfd, max p d Total Revenue C qfd, Total Cost Applying the same idea of our aseline nonlinear pricing model, it follows that there exists a shadow variale cost λ such that the optimal monopoly price is a function of λ max d Applying the Leieniz's rule, the necessary condition is given y p d Thisterm Hence,

5 d Sustituting ack to the expression of q ( θ) = θ yields q Next, we can derive the equation etween K and λ It follows that, which is equivalent to ( θ ) K = dθ, ( λ ) = 9 Lastly, the total revenue is given y K λ + λ θ θ = λ λ+ ( ) ( ) d ( ) ( ) 7 The closed-form solutions of all three cases are summarized in Tale C Unit p( θ)/ q( θ ) Price, Case Case Case max ( θ + λ ), max ( θ ), max ( λ + ), Usage Plan, q( θ ) max[ θ λ,] max[ θ,] max θ λ Price Schedule, ( ), p q ( λ + ) ( ) ( λ + ) q q Capacity, K ( ) λ q q q λ 9 ( ) Virtual Cost, ( K) λ 8 K 9 K Market (-θ) Coverage, ( ) λ ( ) λ Total Revenue ( λ ) ( λ+ ) ( ) 7 λ ( λ+ ) Tale C Numerical Solutions of the Case with Quadratic Utility Function and Uniformly Distriuted Customer Types Case : Nonlinear Pricing; Case : Zero-Marginal Cost Pricing; Case : Linear Pricing 5

6 Finally, to conduct a numerical analysis on the profit loss due to mispricing, we consider the impacts of two parameters: the numer of locks to achieve Q and the average infrastructure cost ( ci ()/ ki ()) It can e verified that the revenue-maximizing total quantity ( Q ) equals / in the present example Hence, if the numer of locks to achieve Q is, the lock size is / under the assumption that each lock has the same size For this numerical analysis, we consider the numer of locks e,, 5,, and 5 The constant average costs examined are,, or The profit ratios reported in Tale C and Tale C are defined as the profit from the su-optimal pricing schedule divided y the profit from the optimal nonlinear pricing with discontinuous cost Numer of Blocks\Average Cost % 85% 96% 98% 78% 7% 5 96% 786% 99% 96% 786% 96% 5 96% 78% 96% Tale C Numerical Results: Profit from Zero-Marginal-Cost Pricing as a Percentage of Profit from Nonlinear Pricing with Discontinuous Cost Numer of Blocks\Average Cost 857% 9% 879% 97% 879% 959% 5 889% 8895% 98% 8885% 8895% 887% 5 889% 8887% 889% Tale C Numerical Results: Profit from Linear Pricing with Discontinuous Cost as a Percentage of Profit from Nonlinear Pricing with Discontinuous Cost 6