Rent Seeking over Tradable Emission Permits

Abstract

The allocation of emission permits at no cost during the establishment of a cap-and-trade program creates opportunities for rent-seeking. I examine the consequences of such rent-seeking by exploiting an unusual feature of the UK’s permit allocation procedure in Phase 1 of the EU’s CO$_{2}$ Emissions Trading Scheme, whereby it is possible to observe both a firm’s actual permit allocation as well as an earlier, technocratically-based provisional allocation that was never implemented. Firms had the opportunity to appeal their provisional allocation. I find that a firm’s financial connections to members of the House of Commons strongly predict its post-appeal allocation. Even after controlling for the provisional allocation, along with industry and financial characteristics, a connection to an additional member is associated with a significant increase in a firm’s actual permit allocation. Using results from a contest-theoretic framework, I estimate the welfare loss from rent-seeking to be over 100 million euros—a significant amount relative to the abatement costs firms incurred to reduce emissions.

Appendices

Appendix 1

Proposition 1

(Nash equilibrium of the n-firm contest):

• (a) Only the k firms with the lowest ranking lobbying costs participate in lobbying, where $k\in \{2,\ldots ,n\}$ and is uniquely determined (See Proposition 2for proof of uniqueness).

• (b) If firm i participates in lobbying, its lobbying cost must be strictly less than the sum of the lobbying costs of the k lobbying firms, divided by $k-1$.

• (c) If firm i does not participate in lobbying, its lobbying cost must be greater than or equal to the sum of the lobbying costs of the k lobbying firms, divided by $k-1$.

• (d) The lobbying efforts of the participating firms are defined by (12).

Proposition 1 formalizes the idea that the number of firms that participate in lobbying depends critically on the dispersion of the marginal costs of lobbying across firms.³⁴ Those firms whose costs are too high relative to those of other firms will choose to refrain from lobbying. In the extreme case where all n firms face equal marginal costs, the equilibrium will have participation by all firms ($k=n$). At the other extreme, a sufficiently high degree of dispersion in costs can produce an equilibrium where only the two firms with the lowest costs will participate ($k=2$).³⁵

Proof

Differentiating (3) with respect to $x_{i}$ yields the following first order conditions for an interior solution:

$$\begin{aligned} \begin{array}{ll} \tau \phi \frac{x_{-i}}{(x_{i}+x_{-i})^{2}}=\omega _{i}\end{array}, \end{aligned}$$

(9)

where $x_{-i}$ refers to the sum of lobbying efforts of firms other than firm i. These conditions state that each firm chooses a level of lobbying effort that equalizes its marginal benefits and marginal cost.³⁶

The first order conditions together with the non-negativity constraints on lobbying efforts lead to the following best response functions:

$$\begin{aligned} x_{i}={\left\{ \begin{array}{ll} \begin{array}{ll} \sqrt{\frac{\tau \phi x_{-i}}{\omega _{i}}}-x_{-i} &{} if\,x_{-i}\in (0,\frac{\tau \phi }{\omega _{i}})\\ 0 &{} if\,x_{-i}\ge \frac{\tau \phi }{\omega _{i}} \end{array}&for\,i=1,2\end{array}\right. },...n. \end{aligned}$$

(10)

A strategy profile in which all firms exert zero lobbying effort cannot constitute a Nash equilibrium. According to the contest function specified in (1), the best response of firm i to $x_{-i}=0$ is to exert an arbitrarily small amount of lobbying effort, $x_{i}=\epsilon >0$, and thereby capture the entire quantity of contested permits. However, this cannot constitute a Nash equilibrium either. Suppose for example that $x_{-i}=0$ and $x_{i}=\epsilon >0$. Although firm i’s choice of an arbitrarily small amount of lobbying effort, $\epsilon$, is a best response to $x_{-i}=0$, the best response function (10) indicates that the other (non-i) firms’ choices of zero lobbying effort are not best responses to a sufficiently small $\epsilon$. This reasoning implies that a Nash equilibrium cannot involve only one firm with strictly positive effort while all other firms refrain from lobbying. Thus at least two firms must exert strictly positive lobbying effort in a Nash equilibrium.

Generally, there exists a unique Nash equilibrium to this contest in which k firms exert strictly positive lobbying effort, with $k\in [2,n]$. To see this, let $j=1,2,...,k$ index the lobbying firms. For these k firms, the first order condition (9) holds with equality. Summing both sides of (9) over $i=1,2,...,k$ and rearranging, the total equilibrium lobbying effort is

$$\begin{aligned} x=\frac{(k-1)\tau \phi }{\sum _{j=1}^{k}\omega _{j}}, \end{aligned}$$

(11)

and combining (11) with the best response function (10),

$$\begin{aligned} x_{i}={\left\{ \begin{array}{ll} \begin{array}{ll} \frac{\tau \phi (k-1)}{(\sum _{j=1}^{k}\omega _{j})^{2}}\cdot \left( \left( \sum _{j=1}^{k}\omega _{j}\right) -\omega _{i}(k-1)\right) &{} for\,i=1,2,...,k\\ 0 &{} for\,i=k+1,k+2,...,n. \end{array}\end{array}\right. } \end{aligned}$$

(12)

A Nash equilibrium can involve $k\le n$ lobbying firms if the lobbying costs of these k firms are sufficiently close to each other, and the lobbying costs of the non-lobbying firms are sufficiently higher than those of the lobbying firms. In particular, for (12) to be a Nash equilibrium, it is required that $\omega _{i}<\frac{1}{k-1}\sum _{j=1}^{k}\omega _{j}\,\,for\,i=1,2,...,k$. Furthermore, the best response function (10) requires that $x_{-i}\ge \frac{\tau \phi }{\omega _{i}}\,for\,i=k+1,k+2,...,n$. Noting that for the non-lobbying firms $x_{-i}=x$, this requirement can be expressed as $\omega _{i}\ge \frac{1}{k-1}\sum _{j=1}^{k}\omega _{j}\,\,for\,i=k+1,k+2,...,n$.

These requirements for a Nash equilibrium imply that the k lobbying firms must be the k firms with the lowest ranking lobbying costs. Also, because $\omega _{i}<\sum _{j=1}^{2}\omega _{j}\,for\,i=1,2$ is trivially true, an equilibrium cannot involve fewer than two lobbying firms. $\square$

Proposition 2

(Uniqueness of equilibrium) The number of lobbying firms in equilibrium, k, is uniquely determined.

Proof

Suppose there exists a Nash equilibrium with k lobbying firms and another Nash equilibrium with $k'$ lobbying firms. Without loss of generality, suppose $k'>k$. The equilibrium conditions stipulate

$$\begin{aligned} \begin{array}{ll} \omega _{i}<\frac{1}{k-1}\sum _{j=1}^{k}\omega _{j} &{} for\,i=1,2,...,k\\ \omega _{i}\ge \frac{1}{k-1}\sum _{j=1}^{k}\omega _{j} &{} for\,i=k+1,k+2,...,n \end{array} \end{aligned}$$

(13)

and

$$\begin{aligned} \begin{array}{ll} \omega _{i}<\frac{1}{k'-1}\sum _{j=1}^{k}\omega _{j} &{} for\,i=1,2,...,k'\\ \omega _{i}\ge \frac{1}{k'-1}\sum _{j=1}^{k}\omega _{j} &{} for\,i=k'+1,k'+2,...,n \end{array} \end{aligned}$$

(14)

where i indexes firms in order of lobbying cost (i.e. firm 1 is the firm with lowest lobbying cost; firm n is the firm with the highest lobbying cost).

Condition (10) implies $(k'-1)\omega _{k'}<\sum _{j=1}^{k'}\omega _{j}$, which can be equivalently expressed as $(k-1)\omega _{k'}+(k'-k)\omega _{k'}<\sum _{j=1}^{k}\omega _{j}+\sum _{j=k+1}^{k'}\omega _{j}$. Condition (9) implies $(k-1)\omega _{k'}>\sum _{j=1}^{k}\omega _{j}$, therefore it must be that $(k'-k)\omega _{k'}<\sum _{j=k+1}^{k'}\omega _{j}$. However, because $\forall \,j<k'$, $\omega _{j}<\omega _{k'}$, $\omega _{k'}>\frac{1}{k'-k}\sum _{j=k+1}^{k'}\omega _{j}$, which implies $(k'-k)\omega _{k'}>\sum _{j=k+1}^{k'}\omega _{j}$ leading to a contradiction. Therefore there cannot exist two Nash equilibria with different numbers of lobbying firms. $\square$

Proposition 3

(Equilibrium lobbying effort and unit lobbying cost) The equilibrium lobbying effort of a lobbying firm is decreasing in the firm’s own unit cost of lobbying. The effect of an increase in another firm’s unit lobbying costs is ambiguous.

Proof

For a lobbying firm i, $x_{i}=\frac{\tau \phi (k-1)}{(\sum _{j=1}^{k}\omega _{j})^{2}}\cdot \left( \left( \sum _{j=1}^{k}\omega _{j}\right) -\omega _{i}(k-1)\right)$.

Differentiating $x_{i}$ with respect to $\omega _{i}$ yields

$$\begin{aligned} \frac{\partial x_{i}}{\partial \omega _{i}}=\tau \phi (k-1)\left[ \frac{2(k-1)\omega _{i}\sum _{j=1}^{k}\omega _{j}-k\left( \sum _{j=1}^{k}\omega _{j}\right) ^{2}}{\left( \sum _{j=1}^{k}\omega _{j}\right) ^{4}}\right] , \end{aligned}$$

(15)

which is strictly negative if and only if $\frac{2\omega _{i}}{k}<\frac{1}{k-1}\sum _{j=1}^{k}\omega _{j}$. Because $k\ge 2$, $\frac{2\omega _{i}}{k}\le \omega _{i}$. The condition for a lobbying firm i is that $\omega _{i}<\frac{1}{k-1}\sum _{j=1}^{k}\omega _{j}$. Together these imply $\frac{2\omega _{i}}{k}<\frac{1}{k-1}\sum _{j=1}^{k}\omega _{j}$.

Differentiating $x_{i}$ with respect to $\omega _{i'}$ (with $i'\ne i$) yields

$$\begin{aligned} \frac{\partial x_{i}}{\partial \omega _{i'}}=\tau \phi (k-1)\left[ \frac{2(k-1)\omega _{i}-\sum _{j=1}^{k}\omega _{j}}{\left( \sum _{j=1}^{k}\omega _{j}\right) ^{3}}\right] . \end{aligned}$$

(16)

The denominator is obviously positive. Because $\omega _{i}$ may be greater or less than $\frac{1}{2(k-1)}\sum _{j=1}^{k}\omega _{j}$, the sign of $\frac{\partial x_{i}}{\partial \omega _{i'}}$ is ambiguous. $\square$

Proposition 4

(Rent dissipation) When the number of lobbying firms, k, is arbitrarily large, total expenditures on lobbying effort equal the value of the rents, $\tau \phi$.

Proof

For a given k, an upper bound on total lobbying expenditure is $\left( \frac{k-1}{k}\right) \tau \phi$. The upper bound is reached when the lobbying costs of the k lobbying firms are equal; higher variance in the lobbying costs of the k firms leads to lower total lobbying expenditure. This is because the quantity $\frac{\sum _{j=1}^{k}\omega _{j}^{2}}{\left( \sum _{j=1}^{k}\omega _{j}\right) ^{2}}$ is monotonically increasing in the variance of the $\omega _{j}$’s and attains a minimum value of $\frac{1}{k}$ when the variance of the $\omega _{j}$’s is zero (i.e. when $\frac{\sum _{j=1}^{k}\omega _{j}^{2}}{k}-\frac{\left( \sum _{j=1}^{k}\omega _{j}\right) ^{2}}{k^{2}}=0$).

When the number of lobbying firms, k, is arbitrarily large, total lobbying expenditures are equal to $\tau \phi$. The reason for this is twofold. First, the larger the value of k, the more stringent are the limitations on the variance of the lobbying costs. Specifically, when k is arbitrarily large, the conditions for a Nash equilibrium imply that the variance of the lobbying costs of the k firms must be zero. (Recall that, for any Nash equilibrium in which k firms lobby, it must be that $\omega _{i}<\frac{1}{k-1}\sum _{j=1}^{k}\omega _{j}\,for\,i=1,2,...,k$.) Second, as the variance of the lobbying costs of the k firms reaches zero, total lobbying expenditure must itself reach its upper bound, $\left( \frac{k-1}{k}\right) \tau \phi$. It is evident that this upper bound reaches $\tau \phi$ for an arbitrarily large k. $\square$

Appendix 2

See Tables 10, 11, 12, 13, 14, 15 and 16.

Table 10 Regressions with number of connected MPs (unweighted)

Table 11 Regressions with number of connected MPs (unweighted, firms in top quintile of provisional allocations)

Table 12 Regressions with number of connected MPs (unweighted, firms in bottom 4 quintiles of provisional allocations)

Table 13 Regressions with number of connected MPs (weighted, levels)

Table 14 Regressions with number of connected MPs (unweighted, levels)

Table 15 Regressions with number of connected MPs (unweighted, levels, firms in top quintile of provisional allocations)

Table 16 Regressions with number of connected MPs (unweighted, levels, firms in bottom 4 quintiles of provisional allocations)