Uncertainty and speculators in an auction for emissions permits

Abstract

Auctions have become popular as means of allocating emissions permits in the emissions trading schemes developed around the world. Mostly, only a subset of the regulated polluters participate in these auctions along with speculators, creating a market with relatively few participants and, thus, incentive for strategic bidding. I characterize the bidding behavior of the polluters and the speculators, examining the effect of the latter on the profits of the former and on the auction outcome. It turns out that in addition to bidding for compliance, polluters also bid for speculation in the aftermarket. While the presence of the speculators forces the polluters to bid closer to their true valuations, it also creates a trade-off between increasing the revenue accrued to the regulator and reducing the profits of the auction-participating polluters. Nevertheless, the profits of the latter increase in the speculators’ risk aversion.

Appendices

Appendix 1: Derivation of the bidding schedules

Each bidder $j\in \{f,s\}$ chooses her bidding strategy maximizing the utility in (12) if she is a polluter or in (14) if she is a speculator, acting as a monopsonist on the residual supply of permits $\bar{{E}}-D_{-j} (\nu )$, where $D_{-j} (\nu )=\sum \nolimits _i {D_i } (\nu ),i=1,\ldots ,j-1,j+1,\ldots ,N_a +M$. The equilibrium concept is the supply function equilibria (Klemperer and Meyer 1989). A strategy for bidder j is a non-increasing schedule $D_j (\nu )$ which specifies the quantity demanded for every price $\nu$.

Thus, each bidder solves the following problem:

$$\begin{aligned} \hbox {max}_{\nu } \hat{{U}}_j (D_j (\nu ),\nu )\hbox { such that }D_j (\nu )=\bar{{E}}-D_{-j} (\nu ), \quad j=1,\ldots ,N_a ,1\ldots ,M.\nonumber \\ \end{aligned}$$

(21)

The first order conditions read:

$$\begin{aligned} \frac{\partial \hat{{U}}_j (D_j (\nu ),\nu )}{\partial D_j (\nu )} \left( {-\frac{\partial D_{-j} (\nu )}{\partial \nu }} \right) +\frac{\partial \hat{{U}}_j (D_j ,\nu )}{\partial \nu }=0, \quad j=1,\ldots ,N_a ,1\ldots ,M\nonumber \\ \end{aligned}$$

(22)

Focusing on linear strategies, let the demand schedules have the form:

(23)

Substituting (23) in (22), grouping around $\nu$ and identifying the coeffcients yields:

$$\begin{aligned} \begin{array}{l} y_f =(1-\Omega ^{2}\kappa _f y_f )y_{-f} ,y_{-f} =\sum \limits _{i=1,i\ne f}^{Na+M} {y_i }, \quad \forall f=1,\ldots ,N_a , \\ y_s =(1-\Omega ^{2}\rho _s y_s )y_{-s} ,y_{-s} =\sum \limits _{i=1,i\ne s}^{Na+M} {y_i }, \quad \forall s=1,\ldots ,M. \\ \end{array} \end{aligned}$$

(24)

and

$$\begin{aligned} \begin{array}{l} x_f =\left( {\bar{{\lambda }}^{*}-\kappa _f \Omega B_f } \right) y_f,\quad \forall f=1,\ldots ,N_a \\ x_s =\bar{{\lambda }}^{*}y_s, \quad \forall s=1,\ldots ,M \\ \end{array} \end{aligned}$$

(25)

Appendix 2: The case of risk-neutral speculators

Instead of maximizing the CARA utility function of profits, a risk-neutral speculator maximizes the expected profit: $\max _\nu E[\Pi _s ]=E[(\lambda ^{*}-\nu )D_s (\nu )]=(\bar{{\lambda }}^{*}-\nu )D_s (\nu )$ on the residual supply $D_s (\nu )=\bar{{E}}-D_{-s} (\nu ),$ where $-s$ represents bidders $1,\ldots ,N_a$ (i.e. the polluters) plus the $M-1$ speculators except for s. The FOC reads:

$$\begin{aligned} (\bar{{\lambda }}^{*}-\nu )(-D_{-s }^{\prime } (\nu ))-D_s (\nu )=0. \end{aligned}$$

(26)

Assuming linear strategies of the form $D_s (\nu )=x_s -y_s \nu$ and identifying the coefficients in (B.1) it obtains: $D_s (\nu )=y_s (\bar{{\lambda }}^{*}-\nu ),$ where $y_s =\sum \nolimits _{j=1,j\ne s}^{N_a +M} {y_j } ,$ which is exactly the counterpart of the second type of equations in (24), for $\rho _s =0$. The slopes of the polluters will continue to be given by the first type of equations in (24). Note, however, that in this case the system of equations (24) does not have a positive solution for$M\ge 2$. In fact, the only non-negative solution is $y_f =y_s =0$ and this amounts to all firms submitting empty bids. The only non-degenerated equilibrium in linear strategies is obtained for $M=1$. In this case we have $y_s =\sum \nolimits _{f=1}^{N_a } {y_f } ,$ with $y_f$ defined by $y_f =(1-\Omega ^{2}\kappa _f y_f )(y_f +2y_{-f} ),$ where $y_{-f} =\sum \nolimits _{i=1,i\ne f}^{Na} {y_i }$.