Interest group incentives for post-lottery trade restrictions

Abstract

The rights to use publicly-managed natural resources are sometimes distributed by lottery, and typically these rights are nontransferable. Prohibition of post-lottery permit transfers discourages applicants from entering the lottery solely for profitable permit sale, so only those who personally value the use of the resource apply. However, because permits are distributed randomly and trade is restricted, permits may not be used by those who value them most. We argue that restrictions on permit transfers is a policy response designed to limit entry when interest group membership is not distinguishable ex ante, and characterize the economic/informational conditions under which post-lottery prohibitions on trade are likely to arise. We develop our model using the specific case of the Four Rivers Lottery used to allocate rafting permits on four river sections in Idaho.

1 Appendix

This appendix provides derivations of several results referred to in the text.

1.1 1.1. Comparisons of number of applicants

1.1.1 1.1.1. The number users for nontradable permits is greater than the number of applicants.

To prove $q_0^n > q^n_a$, note that the marginal applicant for a nontradable permit is $\frac{\bar{q}}{q_a^{n}} \cdot v(\bar{q}) = f_a+(\bar{q}/q_a^{n}) f_p$. Substituting $v(\bar{q})=\alpha -\beta \bar{q}-\gamma \bar{q}+(\eta /\bar{q})M$ and rearranging provides

$$\begin{aligned} \left( \alpha - \gamma \bar{q} + (\eta /\bar{q}) M\right) = \beta q_a^n + (q_a^{n}/\bar{q})f_a + f_p. \end{aligned}$$

By definition, WTP=0 at $q_0^n$: $\alpha -\beta q^0 -\gamma \bar{q} + (\eta /\bar{q})M=0$, which we can rewrite as

$$\begin{aligned} \left( \alpha - \gamma \bar{q} + (\eta /\bar{q}) M\right) = \beta q^0_n. \end{aligned}$$

The left-hand side of the two displayed equations above are identical, so substituting $\beta q_0^n$ from the latter into the former displayed equation and rearranging provides $q_a^n = q_0^n - (1/\beta )((q_a^n/\bar{q})f_a+f_p)$. The term after the minus sign must be positive, implying $q_0^n > q_a^n$.

1.1.2 1.1.2. The number of applicants for nonexclusive tradable permits is greater than the number of applicants for nontradable permits

To show that $q_a^t>q_a^n$, as long as there are any applicants at all, rearrange Eqs. 4 and 9, respectively, to get

$$\begin{aligned} q^n_a(\beta \bar{q}+f_a\psi )&= \bar{q}(\alpha - \gamma \bar{q}-f_p\psi ) \\ q^t_a f_a \psi + \beta \bar{q}&= \bar{q}(\alpha - \gamma \bar{q}-f_p\psi ). \end{aligned}$$

Setting the left sides equal to each other and rearranging provides

$$\begin{aligned} q^t_a = \frac{(q^n_a-1)\beta \bar{q}}{f_a\psi } + q^n_a >0,\qquad \text {for } q^n_a\ge 1. \end{aligned}$$

Thus, for practical purposes, there are always more applicants with tradable permits than nontradable.

1.1.3 1.1.3. The number of users compared to the number of applicants for tradable permits

For $q_0^t \gtrless q^t_a$, consider the marginal applicant’s decision condition $\frac{\bar{q}}{q_a^{t}} \cdot v(\bar{q}) = f_a+\frac{\bar{q}}{q_a^{t}} f_p$. Manipulation and provides

$$\begin{aligned} \frac{\bar{q}}{f_a}\left( \beta (q_0^t-\bar{q})-f_p\right) =q_a^t. \end{aligned}$$

So the relationship between $q_0^t$ and is $q_a^t$ ambiguous. However, for our simulations, $q_a^t$ is larger than $q_0^t$ for reasonable ranges of the policy parameters, as illustrated in Fig. 4.

Fig. 4

$q_0^t$ versus $q^t_a$ as a function of policy parameters. a Applicants $q_a$ and users $\bar{q}_0$ as a function of available permits. b Applicants $q_a$ and users $\bar{q}_0$ as a function of application fees. c Applicants $q_a$ and users $\bar{q}_0$ as a function of permit fees

In fact, a comparison with Fig. 4a–c show that $q^t_a$ is larger than $q_0^t$ for approximately the entire positive surplus space for exclusive tradable lotteries.

1.1.4 1.1.4. The number of applicants for exclusive tradable permits is greater than the number of applicants for nontradable permits

Equation 12 representing $q_a^e$ can be rewritten as

$$\begin{aligned} q^e_a = \frac{[\bar{q}(\alpha -\bar{q}\gamma - f_p\psi )] + \bar{q}f_p}{[\beta \bar{q}+f_a\psi ]-f_a}. \end{aligned}$$

A comparison to $q^n_a$ in Eq. 4 shows that the contents of the brackets in the numerator and denominator are the numerator and denominator of $q^n_a$, respectively. Thus, the numerator of $q^e_a$ is bigger than that of $q^n_a$ and the denominator is smaller, implying $q_a^e > q_a^n$.

1.2 1.2. User surplus

1.2.1 1.2.1. Surplus for nonexclusive tradable permits

For nonexclusive, tradable permits, total surplus can be broken down into six categories

$$\begin{aligned} E[S^t]&= [\text {winning buyers}] + [\text {losing buyers}] \\&\quad + [\text {winning user/sellers}] + [\text {losing user/sellers}] \\&\quad + [\text {winning nonusers}] + [\text {losing nonusers}]. \end{aligned}$$

Letting $\pi =\bar{q}/q^t_a$ equal the probability of winning, and maintaining brackets above,

$$\begin{aligned} E[S^t]&= \left[ \pi \int _0^{\bar{q}} v(q) -f_p-f_a dq\right] + \left[ (1-\pi )\int _0^{\bar{q}} v(q) - \bar{v} - f_a dq\right] \\&\quad + \left[ \pi (q^t_0-\bar{q})(\bar{v}-f_p-f_a)\right] + \left[ (1-\pi )(q^t_0-\bar{q})(-f_a) \right] \\&\quad + \left[ \pi (q^t_a-q^t_0)(\bar{v}-f_p-f_a)\right] + \left[ (1-\pi )(q^t_a-q^t_0)(-f_a)\right] . \end{aligned}$$

This can also be simplified to

$$\begin{aligned} E[S^t]&= [\text {user buyers}] + [\text {user/sellers}] + [\text {nonusers}] \\&= \left[ \int _0^{\bar{q}} v(q) dq - \bar{q}(\pi f_p + (1-\pi )\bar{v} - f_a)\right] \\&\quad + \left[ (q^t_0-\bar{q})(\pi (\bar{v} - f_p) - f_a) \right] \\&\quad + \left[ (q^t_a-q^t_0)(\pi (\bar{v}-f_p) - f_a) \right] . \end{aligned}$$

1.2.2 1.2.2. Surplus for exclusive tradable permits

Without substituting the parameterized version of $q^e_a$, which, after some minor manipulation provides

$$\begin{aligned} E[S^e]&= -(1/2)(\beta +2\gamma )\bar{q}^2 + \alpha \bar{q} + f_p(s\eta -1)\bar{q}+ f_aq^e_a(s\eta -1)\\&= (\beta /2)\bar{q}^2 + \bar{q}(\alpha - \gamma \bar{q}-f_p\psi ) - f_a q^e_a\psi \\&= (\beta /2)\bar{q}^2 + q^n_a\beta \bar{q}+ q^n_af_a\psi - f_aq^e_a\psi \\&= E[S^t] + 2E[S^n] + f_a\psi (q^n_a-q^e_a). \end{aligned}$$