Question 1,2,3,4

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 differentiable. Use the second order Taylor approximation of r to estimate (E[r(Z)]−r(E[Z]))/r(E[Z]) if d(p) = λP(W ≥ p), W is exponential with mean θ and Z is Poisson with mean µ.

c) Compare your approximation to the true improvement for the following pair of (µ,θ) 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 profits relative to pricing at p(E[Z]).

2. Let h(p) be the hazard rate of the random variable W , defined by h(p) = g(p)/H(p) where H(p) = P(W ≥ p) is the survival probability and g(p) = −H′(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) = Γ(a+b)

Γ(a)Γ(b) pa−1(1 −p)b−1 for p ∈ [0, 1], a,b ≥ 0, and b > 1.

b) H(p) = e−p/θ for θ > 0.

c) H(p) = ( a p

)b 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(e

p) is concave or if pg(p) is increasing in p.

3. Consider the demand function d(p) = λH(p), where H(p) = P(W ≥ p) = 1 for p < 1 and H(p) = p−b for p ≥ 1. In what follows assume that z ≥ 1 is a constant. We will assume that the customers know the distribution of W at the time of booking and learn the realization at the time of consumption.

a) Find the price p(z) that maximizes r(p,z) = (p − z)d(p) and then find r(z) and s(p(z)) = λE[(W −p(z))+] =

∫∞ p(z) d(y)dy.

b) Consider now an option (x,z) where by paying x at the time of booking, gives the customer the right to purchase a unit of capacity at z. The expected surplus per customer from this option is −x + E[(W − z)+]. Find the value of x that results in aggregate surplus s(p(z)).

c) Show that the expected profit from offering the option in part b) is equal to s(p(z))− s(z)).

d) Compute the relative improvement in profits [s(p(z)) − s(z)) − r(z)]/r(z) as a function of b. What happens as b increases?

1

4. Finite Price Menu for Log Linear Demands. Suppose that the demand function is of the form dm(p) = am exp(−p/bmp) for some constants am > 0 and bm > 0,m ∈M. Suppose that am = 100m and bm = 40 + 10m,m ∈M = {1, . . . , 10}.

a) Compute q1, γ1 and the actual performance of R1(q1,z)/R1(z) for values of z ∈ {100, 200, 300}. Notice that you can calculate γ1 just by knowing u = max bj/ min bj and that you can calculate q1 just by knowing z, b1 and u.

b) What is the best common price, the one maximizing R1(p,z) if you have full in- formation about the demands? Compare the performance of this price to q1 for z ∈{100, 200, 300}.

2