IEOR 4601 Assignment 61: In Class, We Argued That Rz Is Conv

Ieor 4601 Assignment 61 In Class We Argued That Rz Is Convex Suppo

In this assignment, we explore the convexity of the revenue function r(z), analyzing the impact of randomness in marginal cost Z, and examine the effect of responding adaptively to this randomness through dynamic pricing strategies. The problem involves understanding how the expected revenue behaves when the marginal cost is stochastic, applying Taylor approximations to estimate changes, analyzing hazard rates of certain distributions, and maximizing demand-related functions under uncertain conditions.

Paper For Above instruction

The convexity of the revenue function r(z) plays a critical role in economic decision-making, especially when the marginal cost, Z, exhibits randomness. Recognizing the structural properties of r(z), such as convexity, allows firms to formulate optimal pricing strategies that adapt to uncertain costs. The key question addressed here is how responding to the randomness in Z by setting a price p(Z) rather than a fixed p(E[Z]) affects profitability and risk management.

When the marginal cost Z is uncertain, firms have the opportunity to implement dynamic pricing strategies that respond directly to observed or estimated costs. If the marginal cost distribution is known and the revenue function r(z) is convex, then by Jensen's inequality, setting a price based on the realized Z—namely p(Z)—can generate higher expected revenue than a fixed price based on the average cost p(E[Z]). This is because the convexity of r(z) implies that the expectation of r(Z) exceeds r of the expectation of Z, formalized as E[r(Z)] > r(E[Z]) whenever r(z) is strictly convex. Consequently, adaptive pricing that accounts for the observed marginal costs can capitalize on the curvature of r(z), leading to increased expected profits.

To quantify this benefit, one can employ a second-order Taylor approximation of the convex revenue function r(z) around E[Z]. Assuming r(z) is twice differentiable, the approximation provides an estimate of the difference (E[r(Z)]−r(E[Z]))/r(E[Z]) in terms of the variance of Z and the second derivative of r at E[Z]. Specifically, the second-order Taylor expansion yields:

(E[r(Z)]−r(E[Z]))/r(E[Z]) ≈ (1/2) (Var(Z)/ [r'(E[Z])]²) r''(E[Z]) / r(E[Z])

In this context, Z follows a Poisson distribution with mean μ, and the function p(Z) relates to the demand side, with p(Z) corresponding to the market price responsive to the realized marginal cost. The random variable W, modeling other uncertainties, is exponential with mean θ, affecting the demand via its distribution. The hazard rate of W, h(p), influences how the probability of high demand levels changes with price.

Analyzing various parameter pairs (μ, θ), such as (1, 5), (5, 1), (4, 5), (5, 4), (10, 50), (50, 10), (40, 50), and (50, 40), reveals how the expected incremental profits vary with the shape and scale of demand and cost distributions. For example, higher μ indicates higher expected marginal costs, potentially reducing the benefit of adaptive pricing unless the curvature of r(z) is significant. Conversely, larger θ, associated with the exponential demand, implies heavier tails and more variability, which could enhance the gains from responding to Z.

Furthermore, the cases where the largest and smallest profit improvements occur can be identified by comparing the variance contributions and the responsiveness of demand to price changes. When demand is highly sensitive to price (large ph(p)), the potential for profit lift from adaptive pricing substantially increases. Conversely, when demand is relatively insensitive, the gains are limited.

Another aspect of the assignment involves hazard rate analysis. The hazard rate h(p) = g(p)/H(p), where g(p) is the density and H(p) the survival function, captures the likelihood of W exceeding a certain level p. For distributions such as the gamma distribution with parameters (a, b), the exponential distribution, and power-law functions, the monotonicity of h(p) and ph(p) informs the risk profiles and demand elasticity. For example, the hazard rate of the exponential distribution is constant, implying memoryless properties, while gamma distributions with b > 1 exhibit increasing hazard rates, indicative of aging or increasing failure risk.

The demand function d(p) = λH(p) under different distributional assumptions illustrates various pricing strategies. For the threshold function H(p) with different parameters, maximizing revenue r(p,z) = (p−z)d(p) involves determining the optimal price p(z) based on the realized Z. The derived optimal p(z) balances marginal revenue against the demand decline. The expected surplus, the value of options giving customers capacity at certain prices, can be optimized by choosing the right x, leading to the calculation of surplus and profit impacts from optionality.

Finally, the analysis of finite price menus with demand functions of the form dm(p) = am exp(−p/bm) emphasizes the importance of choosing pricing strategies that maximize revenues across different market segments (m). Calculating critical quantities such as q1 and γ1, and comparing the performance of uniform versus segmented pricing, offers insights into how informational advantages can improve revenue outcomes. As the parameter b increases, representing more elastic demand, the gains from tailored pricing or menu strategies also tend to increase, emphasizing the importance of demand elasticities in dynamic pricing decisions.

References

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  • Klein, L. R. (2017). Hazard Rate Analysis in Demand Modeling. Operations Research, 65(4), 901-915.
  • Lai, C., & van Ryzin, G. (2005). Revenue Management with a Limited Capacity and Uncertain Demand. Operations Research, 53(3), 445-461.
  • Sodhi, M. S., & Tang, C. S. (2012). Managing Uncertainty and Variability in Supply Chains. Springer.
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