I posted 4 questions please take a look first then decide whether you want to do them or not also I posted some material related to this home hope this will be helpful upload solution to me before deadline I need detailed explaination. The name of this course is dynamic pricing and revenue management thank you.
IEOR 4601 Assignment 6 1. In class we argued that r(z) is convex. Suppose that the marginal cost Z is random. a) Explain why you can make more money by responding to the randomness in Z by charging p(Z) instead of p(E[Z]). b) Suppose that r is twice dierentiable. Use the second order Taylor approximation of r to estimate (E[r(Z)]??r(E[Z]))=r(E[Z]) if d(p) = P(W p) W is exponential with mean and Z is Poisson with mean . c) Compare your approximation to the true improvement for the following pair of (; ) values: (1; 5); (5; 1); (4; 5); (5; 4); (10; 50); (50; 10); (40; 50) and (50; 40). d) Identify the cases where responding to changes in Z provide the largest and the smallest lifts in prots relative to pricing at p(E[Z]). 2. Let h(p) be the hazard rate of the random variable W dened by h(p) = g(p)=H(p) where H(p) = P(W p) is the survival probability and g(p) = ??H0(p) is the density of W. For each of the following distributions check whether or not h(p) or ph(p) are increasing in p. a) g(p) = ??(a+b) ??(a)??(b)pa??1(1 ?? p)b??1 for p 2 [0; 1] a; b 0 and b > 1. b) H(p) = e??p=for > 0. c) H(p) = apb p a for positive numbers a and b. Hint: A non-negative random variable W has increasing ph(p) if and only if WL = ln(W) has increasing hazard rate hL(p). Moreover ph(p) is increasing in p if either ln g(ep) is concave or if pg(p) is increasing in p. 3. Consider the demand function d(p) = H(p) where H(p) = P(W p) = 1 for p