Expected value of a Gumbel variable conditional on Gumbel being the maximum of N iid Gumbel

Question

I found the following results in Hanemann (1984) which I cannot find a proof for. I checked through simulation that it is right, but I would like to see an analytical proof...

Hanemann, W. M. (1984). Discrete/continuous models of consumer demand. Econometrica: Journal of the Econometric Society, 541-561.

He considers a discrete choice with $i=1,...,N$ options. The value of a choice is the sum of an intercept that is specific to each option $v_i$ and a random draw: $V_{i}^{F} = v_i + \zeta_{i} $ where $\zeta_{i} $ is an idiosyncratic taste shock.

He defines with $M_i$ for the set of values of the vector $\zeta$ such that the option $i$ yields to highest to the agent, i.e. $A_i \equiv \left\{\zeta \mid V_{i}^{F}\geq V_{j}^{F}, \; \forall j \right\} $.

He further considers the case where $\zeta$ is a vector of i.i.d. random variables distributed Type 1 Extreme Value with scale/dispersion parameter $\sigma$. Note that ${P}(\zeta \in A_{i})$ is the probability that option $i$ is actually chosen.

Hanemann (1984) states the following results in equation (3.15) and I was wondering if one knew where to find the proof. \begin{equation} {E}\left\{e^{t \zeta_{i}} \mid \zeta \in A_{i}\right\} = \Gamma(1-\sigma t) \times \beta_{i}^{\sigma t}; \quad \text{ where } \beta^{-1}_{i} = {P}(\zeta \in A_{i}). \end{equation}

Answer 1

Let $\boldsymbol{\zeta} = [\zeta_i]_{1 \leq i \leq N}$ be a random vector with i.i.d. components $\zeta_i$ being Gumbel with location $\mu = 0$ and scale $\sigma$. Let $B_i = \{\ \boldsymbol{\zeta} \in A_i\}$ be the event "$\zeta_i$ is the largest of the $N$ r.vs. $\zeta_j$". Then the claim is that $$ \mathbb{E}\{ e^{t\zeta_i} \vert B_i \} = \Gamma(1 - \sigma t) \times \beta_i^{\sigma t} $$ where $\beta_i = 1 / \text{Pr}(B_i)$.

Note that since the r.vs $\zeta_i$ are i.i.d we have $\beta_i = N$ because each of the $N$ r.vs $\zeta_i$ has the same probability of being the largest. With $M := \max_i \{\zeta_i\}$ we have for any $t$ $$ \mathbb{E}\{ e^{t\zeta_i} \vert B_i \} = \mathbb{E}\{ e^{t M} \vert B_i \} = \mathbb{E}\{ e^{t M} \} $$ because the distribution of $M$ conditional on $B_i$ does not depend on the specific index $i$ where the maximum is reached.

Moreover it easy to see that $M$ is Gumbel with location $\mu = \sigma \log N$ and scale $\sigma$. Using the moment generating function of the Gumbel distribution with location $\mu$ and scale $\sigma$ as given in Wikipedia, that is $$ \mathbb{E}\{ e^{tM} \} = \Gamma(1 - \sigma t) \times e^{\mu t}, $$ we get the wanted result because $e^{\mu t} = e^{\sigma t \log N} = N^{\sigma t}$.