An allocatively efficient auction for pollution permits

Abstract

This article proposes a new auction design for the efficient allocation of pollution permits. We show that if the auctioneer restricts the bidding rule of the uniform-price auction—coupled with a simple ex-post supply adjustment rule—then truthful bidding is obtained. Consequently, the uniform-price auction is more allocatively efficient than conventional formats that are currently observed in pollution markets.

Appendix

Proof of Proposition 1

Suppose bidder i with value $v_i$ submits a bid $b_i<v_i$ and the price at which all permits are sold is p. We claim that bidding $b_i=v_i$ weakly dominates $b_i<v_i$. If $p>v_i$ in both situations the payoffs are zero. If $p<v_i$, then bidding $b_i<v_i$ either results in a zero payoff or results in a payoff equal to $v_i-p$. However, bidding equal to $v_i$ would always result in a positive payoff equal to $v_i-p$. Therefore, bidding $b_i=v_i$ weakly dominates bidding $b_i<v_i$.

Now suppose bidder i submits a bid equal to $b'_i>v_i$. We show this is also weakly dominated by a bid equal to her value. If $v_i<b'_i<p$, then both cases would result in zero payoff. If $v_i<p<b'_i$, then $b'_i$ wins but results in a negative payoff, since $v_i<p$. In this case bidding equal to $v_i$ results in zero payoff. Finally, if $p<v_i<b'_i$ both bids would result in similar payoffs. $\square$

Proof of Corollary 1

Efficiency is a straightforward result of truthful bidding. Since the mechanism allocates the objects to those with the highest values, then it is efficient. However, since there are instances that the seller may sell less than $\bar{Q}$, then it is weakly efficient. $\square$

Proof of Proposition 2

To show that bidding truthfully is a weakly dominant strategy we start by a representative buyer and show that this buyer cannot benefit from demand reduction if all other buyers act truthfully. Suppose buyer i bids its true demand schedule, that is, $p=v_i -\frac{v_i}{\lambda _i} q_i$. First, calculate the aggregate demand without the consideration of buyer i. Note that each buyer by submitting a demand schedule, creates a kink on the aggregate demand at their reported value. Define $c_{-i}$ as the value of the buyer who sets the last kink before $\bar{Q}$ when excluding buyer i in the aggregate demand. In other words $c_{-i}$ is the lowest reported value that can win some permits, excluding i. There are two possible scenarios with respect to the relation of $c_{-i}$ and buyer i’s value $v_i$.

S1: If $v_i > c_{-i}$, then buyer i wins some permits, say $q_i$, and pays a price $p<v_i$. Thus, the payoff for buyer i is,

$$\begin{aligned} \pi _t = \frac{(v_i-p)q_i}{2}. \end{aligned}$$

(5)

S2: If $v_i < c_{-i}$, then two outcomes are possible. The kink at $v_i$ on the aggregate demand could be either to the left of $\bar{Q}$ or to the right and the last one before $\bar{Q}$.

S2.1: When the kink is at the left of $\bar{Q}$, but lower than $c_{-i}$, they win $q'_i$ permits and pay a price equal to $p'$ with a positive payoff equal to,

$$\begin{aligned} \pi _{t'} = \frac{(v_i-p')q'_i}{2}. \end{aligned}$$

(6)

S2.2: When the kink is at the right of $\bar{Q}$, buyer i receives zero payoff.

Now, suppose buyer i submits a value $z_i <v_i$ in its demand schedule then in scenario S2.2 they obviously receive zero payoff again. But now in scenario S1 and S2.1 they would also receive a zero payoff due to the supply adjustment. To see this, suppose the kink at $z_i$ is the kth highest kink. The slope of the aggregate demand after this kink is,

$$\begin{aligned} slope = \frac{\frac{\rho }{(k-1)}\rho '}{\frac{\rho }{(k-1)}+\rho '}, \end{aligned}$$

(7)

where $\rho '<\rho$ is the slope of the buyer i’s demand function with $z_i$. The above slope is less than $\frac{\rho }{k}$ because,

$$\begin{aligned} slope = \frac{\frac{\rho }{(k-1)}\rho '}{\frac{\rho }{(k-1)}+\rho '}< \frac{\rho }{k} \Leftrightarrow \frac{\rho '}{(k-1)}< \frac{\rho + (k-1)\rho '}{k(k-1)} \Leftrightarrow k\rho ' <\rho + (k-1)\rho '. \end{aligned}$$

(8)

The last inequality is true because $\rho '<\rho$. According to the supply adjustment, when the slope of aggregate demand is less than $\frac{\rho }{k}$ after kink k, the supply would reduce to kink k and buyer i receives zero permits.

Thus it is a weakly dominant strategy for buyers to bid their true demand. $\square$

Proof of Proposition 3

Given the result of Proposition 2, if all buyers except buyer i play their weakly dominant strategy, then there is no incentive for buyer i to deviate from this strategy. So submitting the true value is a dominant strategy equilibrium of this game. The efficiency part is straightforward because the mechanism allocates the objects to those with the highest values, since in equilibrium the aggregate demand represents the true demand of each buyer. Also, since the supply adjustment does not take place in equilibrium, all quantities available would be allocated to buyers. Thus the mechanism is fully efficient. $\square$

Proof of Proposition 4

Suppose all buyers except buyer i submit their true demands. We want to show it is not optimal for buyer i to follow the same strategy. First, rewrite Equation (4) as follows,

$$\begin{aligned} \pi _i = \frac{1}{2} (v_i-p) \left( \bar{Q} - \sum _{j\ne i} \left( \lambda _j - \frac{\lambda _j}{v_j}p\right) \right) . \end{aligned}$$

(9)

The first-order condition is

$$\begin{aligned} - \, \bar{Q} + \sum _{j\ne i} \left( \lambda _j - \frac{\lambda _j}{v_j}p\right) + \sum _{j\ne i} \frac{\lambda _j}{v_j} (v_i-p) =0. \end{aligned}$$

(10)

Given the market-clearing price condition, $q^*_i =\bar{Q} - \sum _{j\ne i} (\lambda _j - \frac{\lambda _j}{v_j}p)$, one can rewrite the first-order condition as follows

$$\begin{aligned} q_i^* = \sum _{j\ne i} \frac{\lambda _j}{v_j} (v_i-p). \end{aligned}$$

(11)

Define $\psi _j = \sum _{j\ne i} \frac{\lambda _j}{v_j}$, then the equilibrium demand schedule of bidder i becomes,

$$\begin{aligned} p^* = v_i - \frac{q_i}{\psi _j }. \end{aligned}$$

(12)

Since $\frac{1}{\psi _j} < \rho$ for $n>2$, the best response of player i is to submit a demand function with a strictly lower slope than their actual demand. Thus there exist no equilibrium with truthful demands.

What remains to show is that bidding above the actual demand is not an optimal decision for buyers. To see this, suppose bidder i submits a demand schedule that is above their actual demand function. The only situation that matters is the one where buyer i is pivotal and wins some permits. When the submitted demand is higher than the actual demand, buyer i wins more permits but pays a higher price. Let us focus on the margin and a case where buyer i is only marginally above their true demand. For any extra unit buyer i wins, there will be an extra loss to the surplus at the same price. Now given that price will also go up, the loss in the surplus will be even higher. Therefore, submitting the true demand dominates submitting any demand above the true demand. $\square$

Proof of Proposition 5

Given the result in Proposition 4, in any undominated equilibrium of the standard uniform-price at least some buyers reduce their demands. Therefore, any equilibrium aggregate demand of the standard uniform-price auction is strictly lower than the actual demand. Since we show that in the proposed mechanism the aggregate demand is the same as the sum of the true demands, then the auction clearing price for the proposed mechanism is strictly larger than the clearing price of a standard uniform-price auction. Thus, the revenue of the proposed auction, which is $p\bar{Q}$, is also strictly higher than the revenue of a standard uniform-price auction. $\square$