Assuring finite moments for willingness to pay in random coefficient models

Abstract

Random coefficient models such as mixed logit are increasingly being used to allow for random heterogeneity in willingness to pay (WTP) measures. In the most commonly used specifications, the distribution of WTP for an attribute is derived from the distribution of the ratio of individual coefficients. Since the cost coefficient enters the denominator, its distribution plays a major role in the distribution of WTP. Depending on the choice of distribution for the cost coefficient, and its implied range, the distribution of WTP may or may not have finite moments. In this paper, we identify a criterion to determine whether, with a given distribution for the cost coefficient, the distribution of WTP has finite moments. Using this criterion, we show that some popular distributions used for the cost coefficient in random coefficient models, including normal, truncated normal, uniform and triangular, imply infinite moments for the distribution of WTP, even if truncated or bounded at zero. We also point out that relying on simulation approaches to obtain moments of WTP from the estimated distribution of the cost and attribute coefficients can mask the issue by giving finite moments when the true ones are infinite.

Appendices

Appendix 1

Proof of main theorem

• If a random variable $\theta$ has an absolutely continuous probability density $f\left( \theta \right)$ , then for any positive integer k the inverse moment $E\left( {{1 \mathord{\left/ {\vphantom {1 {\theta^{k} }}} \right. \kern-\nulldelimiterspace} {\theta^{k} }}} \right)$ exists if and only if $\lim_{\theta \to 0} {\frac{f\left( \theta \right)}{{\theta^{h} }}}$ exists for some $h > k - 1$.

We first prove the following Lemma.

Let $f\left( \theta \right)$ be the probability density function of a random variable $\theta$ , where $f\left( \theta \right)$:

• 1. is absolutely continuous.

• 2. has support only on the positive half-line

• 3. is monotonic (either non-decreasing or non-increasing) in an interval $\left( {0,r} \right)$ for some r

Then for a positive integer k, $E\left( {{1 \mathord{\left/ {\vphantom {1 {\theta^{k} }}} \right. \kern-\nulldelimiterspace} {\theta^{k} }}} \right)$ exists if and only if $\lim_{\theta \to 0} {\frac{f\left( \theta \right)}{{\theta^{h} }}}$ exists for some $h > k - 1$.

Suppose $\lim_{\theta \to 0} {\frac{f\left( \theta \right)}{{\theta^{h} }}} = H$ for some non-negative value of h. Certainly the limit exists for h = 0 because of the continuity of f.

Define $S_{n}^{k} \left( r \right) = \int\nolimits_{{{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r {n + 1}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${n + 1}$}}}}^{{{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r n}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$n$}}}} {\theta^{ - k} f\left( \theta \right)d\theta }$. Then, because of the monotonicity property, we know that $S_{n}^{k} \left( r \right)$ is contained in a closed interval

$$ S_{n}^{k} \left( r \right) \in \left[ {\left( {\frac{r}{n} - {\frac{r}{n + 1}}} \right)\left( {{\frac{r}{n + 1}}} \right)^{ - k} f\left( {{\frac{r}{n + 1}}} \right)\,\,,\,\,\left( {\frac{r}{n} - {\frac{r}{n + 1}}} \right)\left( {\frac{r}{n}} \right)^{ - k} f\left( {\frac{r}{n}} \right)} \right] $$

i.e.

$$ S_{n}^{k} \left( r \right) \in \left[ {\left( {\frac{1}{n}} \right)\left( {{\frac{n + 1}{r}}} \right)^{k - 1} f\left( {{\frac{r}{n + 1}}} \right),\,\,\left( {{\frac{1}{n + 1}}} \right)\left( {\frac{n}{r}} \right)^{k - 1} f\left( {\frac{r}{n}} \right)} \right] = \left[ {L_{n}^{k} (r),U_{n}^{k} (r)} \right], {\text{say}}$$

and

$$ \lim_{n \to \infty } L_{n}^{k} \left( r \right) = \left( {\frac{1}{n}} \right)\left( {{\frac{n + 1}{r}}} \right)^{k - 1} H\left( {{\frac{r}{n + 1}}} \right)^{h} = H\left( {\frac{1}{n}} \right)\left( {{\frac{n + 1}{r}}} \right)^{k - h - 1} $$

$$ \lim_{n \to \infty } U_{n}^{k} \left( r \right) = \left( {{\frac{1}{n + 1}}} \right)\left( {\frac{n}{r}} \right)^{k - 1} H\left( {\frac{r}{n}} \right)^{h} = H\left( {{\frac{1}{n + 1}}} \right)\left( {\frac{n}{r}} \right)^{k - h - 1} $$

Then, providing $k < h + 2$, this interval shrinks to zero and, because of the continuity and limit properties of f,

$$ \lim_{n \to \infty } S_{n}^{k} \left( r \right) = \left( {{\frac{H}{{r^{k - h - 1} }}}} \right)\,n^{k - h - 2} $$

Then define $T_{N}^{k} = \sum\nolimits_{n \,=\, 1}^{N} {S_{n}^{k} }$. If the limit exists, $\lim_{N \to \infty } T_{N}^{k} = \int\nolimits_{0}^{r} {\theta^{ - k} f\left( \theta \right)d\theta }$, and the kth inverse moment also exists. Conversely, if the limit does not exist, the kth inverse moment does not exist.

It is a classical result that $\sum {n^{\alpha } }$ converges if and only if $\alpha < - 1$, so that the series $T$ converges if and only if $\left( {k - h - 2} \right) < - 1$, i.e. $k - 1 < h$, proving the Lemma.

The Lemma has the following Corollary

• If a random variable $\theta$ has an absolutely continuous probability density $f\left( \theta \right)$ defined on the positive half-line and $\lim_{\theta \to 0} f\left( \theta \right) > 0$ , then none of the inverse moments $E\left( {{1 \mathord{\left/ {\vphantom {1 {\theta^{k} }}} \right. \kern-\nulldelimiterspace} {\theta^{k} }}} \right)$ exists.

The corollary follows immediately by noting that the limit in the Lemma fails to exist for any $h > 0$ and so the moments do not exist for any $k \ge 1$.

We can now conclude that the condition of monotonicity in the Lemma is not required. If $\lim_{\theta \to 0} f\left( \theta \right) > 0$, then the inverse moments and the limit fail to exist, as in the Corollary. However, if lim  _θ→0 f(θ) = 0, then because of continuity the function must be non-decreasing in a neighbourhood of 0 (it must remain non-negative). That is, any function for which inverse moments or the limit might exist must be monotonic close to zero.

Suppose the function f is defined over the negative half line. For negative $\theta$, the limit is defined for integer $h$ only but with that reservation the same existence result then applies to the negative half-line as for the positive half-line.

Finally, if f is defined over the whole line then the kth moment exists for the whole line if and only if the moments over both positive and negative half-lines exist, completing the proof of the theorem.

Note that in the case when $\lim_{\theta \to 0 - } {\frac{f\left( \theta \right)}{{\theta^{h} }}} \ne \lim_{\theta \to 0 + } {\frac{f\left( \theta \right)}{{\theta^{h} }}}$, i.e. there is some sort of ‘kink’, then if both limits exist the kth moment exists for $k < h - 1$; but if one does not exist then the kth moment does not exist.

Appendix 2

On the ratio of correlated variables

Suppose we are interested in the ratio of two correlated random variables, A and B. In the main text it is indicated that we can always define a variable

$$ A^{*} = A - \alpha B $$

which allows us the express investigate the distribution of the ratio by setting

$$ \frac{A}{B} = \alpha + {\frac{A^{*}}{B}} $$

with A* and B uncorrelated, irrespective of the distribution of the variables A and B. In the case of jointly normal variables A and B this implies that A* and B are independent, but in the case of variables with other distributions independence does not follow from lack of correlation, though it might be considered to hold approximately, except in special cases.If the variables A and B can be related by a linear dependence, which we can define by

$$ A = f_{1} \left( B \right) + f_{2} \left( B \right)A^{*} $$

with A* and B truly independent, then, if $f_{1} \ne 0$, we can calculate the ratio $\frac{A}{B} = {\frac{1}{{B_{1}^{*} }}} + {\frac{A*}{{B_{2}^{*} }}},\,\,\,{\text{with}}\,\,\,B_{i}^{*} = {\frac{B}{{f_{i} \left( B \right)}}}$. This ratio exists only if both components exist and these can be tested by the theorem in the usual way, since A*  is independent of $B_{2}^{*}$. If f ₁ = 0, then we just have the second term and that can be tested as usual. For the joint normal distribution, f ₁ is a constant times B and f ₂ is a constant, so that the tests can be made directly on B.

The concept of linear dependence thus defines a fairly wide class of joint distributions for which the existence of ratio moments can be tested using the theorem presented in this paper.