A Note on Probability Distributions with Increasing Generalized Failure Rates

X has an increasing generalized failure rate (IGFR) and is an IGFR distribution if g is weakly increasing for all such that < 1. Decreasing failure rate (DFR) or decreasing generalized failure rate (DGFR) distributions can be defined analogously. Clearly, if X is IFR it is also IGFR, but the reverse need not hold; many DFR distributions are IGFR. Pricing a service illustrates the use of IGFR distributions. One customer arrives per period, and service takes one period. The cost of service is zero. Customers privately observe their valuations, which are independent and identically distributed according to . A firm posting price p then faces demand D p = p and sets p to maximize revenue, p = pD p . An optimal price p∗ must solve ′ p∗ =D p∗ +p∗D′ p∗ = p∗ 1− g p∗ = 0 The uniqueness of p∗ depends on the generalized failure rate g . In particular, if g is increasing, it can equal one at only a single point, and the unique p∗ must solve g p∗ = 1. This analysis assumes lim ↓ g 1 and lim ↑ g > 1. Fortunately, g has an economic interpretation that offers guidance when these assumptions fail: g p = −pD′ p /D p , so the generalized failure rate is also the elasticity of demand. If lim ↓ g > 1, demand is always elastic, and one is best off charging p∗ = and serving all customers. If lim ↑ g < 1, demand is always inelastic, and a price increase always increases profits. Hence, p∗ = . Realistic problems must have g > 1 for all > y for some finite y. Section 3 gives a mild condition guaranteeing the existence of such a y. The above is a simplification of Ziya et al. (2004a) and has a direct analogue in the supply chain contracting literature (Lariviere and Porteus 2001). Ziya et al. (2004b) compares the IGFR assumption with other commonly imposed conditions for unimodality. The IGFR property is useful, but often difficult to verify. Here, we examine alternative characterizations of IGFR distributions that simplify checking that X is IGFR. We then relate the limit of the generalized failure rate to the moments of a distribution.