Sectoral Regulators and the Competition Authority: Which Relationship is Best?

Abstract

We consider the interplay between regulatory agencies with overlapping competencies: for example, a competition authority and a sectoral regulator. This reflects the current situation in the European Union and in the US. We analyse how authorities’ incentives to act are affected if they can decide independently, or must follow each others’ opinions, respectively, and consider how this relationship performs in the presence of institutional biases and lobbying efforts. A higher likelihood of closing a case tends to be achieved when the authorities act independently of each other: the probability of coming to a decision is higher, and decisions are less vulnerable to lobbying.

Appendix

Proof of Lemma 1

In the objective function (3), $e_{i}$ has increasing differences with v and $\Delta _{S}$, decreasing differences with $\Delta _{\Pi }$ and $\lambda _{i}$, and increasing or decreasing differences with $e_{j}$ depending on the signal of $\partial ^{2}P/\partial e_{i}\partial e_{j}$. $\square$

Proof of Proposition 1

• 1. Under JD we obtain (when we drop arguments for shortness and denote derivatives by primes):

  $$\begin{aligned} \frac{\partial P^{JD}}{\partial e_{1}}=p_{1}^{\prime }p_{2},\quad \frac{\partial ^{2}P^{JD}}{\partial e_{1}\partial e_{2}}=p_{1}^{\prime }p_{2}^{\prime } \end{aligned}$$

  and under ID

  $$\begin{aligned} \frac{\partial P^{ID}}{\partial e_{1}}=p_{1}^{\prime }\left( 1-p_{2}\right) ,\quad \frac{\partial ^{2}P^{ID}}{\partial e_{1}\partial e_{2}}=-p_{1}^{\prime }p_{2}^{\prime }. \end{aligned}$$

  We see that $\partial ^{2}P^{JD}/\partial e_{1}\partial e_{2}>0$ and $\partial ^{2}P^{ID}/\partial e_{1}\partial e_{2}<0$, which proves that under Joint (Independent) Decisions efforts are strategic complements (substitutes).

• 2. As for a comparison of equilibrium efforts to those of a single authority, the first-order conditions are $p_{1}^{\prime }p_{2}=1/v\Delta _{1}$, and $p_{1}^{\prime }\left( 1-p_{2}\right) =1/v\Delta _{1}$, which lead to a higher $p_{1}^{\prime }$ at the solution than in the single-authority first-order condition $p_{1}^{\prime }=1/v\Delta _{1}$ if $p_{2}\in (0,1)$. Then the optimal $e_{1}$ must be smaller because $p_{1}$ is concave.

  Comparing the first-order conditions we see that the best responses under the two decision processes cross at $p_{j}=1/2$, where they have the same value because $p_{j}=1-p_{j}$. Since the ID best responses are decreasing and the JD best responses increasing, this implies that for $p_{j}<\frac{1}{2 }$ the ID best responses are larger; while for $p_{j}>\frac{1}{2}$, it is the JD best responses that are larger.

  The result for aggregate effort follows from the observation that symmetric stable Nash equilibria are found where best responses cut the diagonal from above. If the equilibrium probabilities $p_{1}=p_{2}$ are less than 1 / 2 under ID, then the intersection of the two types of best responses at $p_{j}=1/2$ lies below the diagonal. Since the JD best response remains below the ID best response to the left of this point, this implies that the JD best response cuts the diagonal below the ID Nash equilibrium. Furthermore, the opposite holds if the ID Nash equilibrium involves $p_{1}=p_{2}>1/2$. The result then holds by continuity.

• 3. The most common method of solving games where players move at different points in time is backward induction: solving the game backwards. If stage 2 is reached, then all that is relevant for authority 2 is that no decision has been made yet, so that this authority is in the same situation as if it were alone. Therefore it chooses the optimal effort level $e^{*}$ of a single authority, which leads to a probability of success of, say, $p_{2}^{*}$. At the first stage, authority 1 foresees the other authority’s future choice, and selects its effort to maximize its utility:

  $$\begin{aligned} \max _{e_{1}\ge 0}P^{SD}(e_{1},e^{*})v\Delta _{1}-e_{1}. \end{aligned}$$

  Since the functional form of $P^{SD}$ and $P^{ID}$ are identical, the optimal choice $e_{1}^{SD}$ is decreasing in $e^{*}$. Since $e^{*}$ is larger than the equilibrium effort under ID, $e_{1}^{SD}$ is smaller.

$\square$

Proof of Lemma 2

Gross of fixed costs, the government’s objective function is

$$\begin{aligned} TW=P(e_{1},e_{2})v\Delta -e_{1}-e_{2}, \end{aligned}$$

and the socially optimal effort levels are found through its first-order conditions $v\Delta \partial P(e_{i},e_{j})/\partial e_{i}=1$ and $v\Delta \partial P(e_{i},e_{j})\partial e_{j}=1$. Since these are exactly the first-order conditions of the unbiased authorities, at the socially optimal effort levels $e_{j}^{S}$ and $e_{i}^{S}$ each authority also chooses its socially optimal effort level. $\square$

Proof of Proposition 2

Note first that each authority’s maximization problem is equivalent to maximizing $v\Delta P(e_{1},e_{2})-e_{1}-e_{2}$ with respect to its own decision variable $e_{i}$ , since authorities are unbiased; therefore the stable Nash equilibrium coincides with the outcome of the joint maximization over both variables. Second, note that for any given pair $\left( e_{1},e_{2}\right)$ we have the following chain of inequalities, if we let $p_{i}=p_{i}(e_{i})$:

$$\begin{aligned} 1-\left( 1-p_{1}\right) \left( 1-p_{2}\right)&=\,p_{1}+p_{2}-p_{1}p_{2} \\&\ge\, p_{i},i=1,2 \\&\ge\, p_{1}p_{2}, \end{aligned}$$

which are the success probabilities under ID, one authority, and JD, respectively, at identical effort levels. Furthermore, these inequalities are strict for positive levels of effort: $p_{1},p_{2}>0$. Consider the following maximization problem:

$$\begin{aligned} P^{*}(\theta )=\max _{e_{1},e_{2}}\theta \left[ 1-\left( 1-p_{1}\right) \left( 1-p_{2}\right) \right] +\left( 1-\theta \right) p_{1}. \end{aligned}$$

Clearly $P^{*}(0)$ and $P^{*}(1)$ correspond to the equilibrium probabilities of successful investigation with a single authority, and under Independent Decisions, respectively. By the envelope theorem, which describes how the value of $P^{*}(\theta )$ changes with $\theta$, if we take into account that for each value of $\theta$ the effort levels are optimally chosen,

$$\begin{aligned} \frac{dP^{*}(\theta )}{d\theta }=\left[ 1-\left( 1-p_{1}\right) \left( 1-p_{2}\right) \right] -p_{1}. \end{aligned}$$

By the above inequalities this is non-negative for all $\lambda \in \left[ 0,1\right]$, so that $P^{*}$ is non-decreasing in $\lambda$. Furthermore, since for almost all $\lambda$ the solution involves positive effort levels, the inequalities become strict, and we have that $P^{*}(1)>P^{*}(0)$. Similar arguments apply to the comparison between a single authority and Joint Decisions, and to the welfare comparison between Independent and Joint Decisions.

As for SD, we have already shown above that the equilibrium effort levels are different from the optimal ones under ID. Since the functional forms of the welfare objective under SD and ID are the same, the SD effort levels must yield less welfare than the ID ones. $\square$

Proof of Proposition 3

These follow from upward or downward shifts of increasing or decreasing best responses, respectively. These results hold more generally for strategic substitutes and complements.$\square$

Proof of Proposition 4

We assume that authority 1 is biased and will consider the optimal bias of authority 2. Start from the following fundamental observation (see Figs. 1 and 2, and also Fig. 3 below): since the best response $R_{2}$ of an unbiased authority 2, seen as a function of the other authority’s effort $e_{1}$ on the vertical axis, stems from the first-order condition that is identical to the one of welfare maximization, the social planner’s indifference curves cut this best response when they are horizontal. This fact is precisely the expression of the maximization of welfare given the other authority’s effort, with welfare increasing at this intersection in the direction towards the welfare maximum. Furthermore, at these points any given biased best response $R_{1}$ of authority 1 (seen as a function of the authority 2’s effort $e_{2}$ on the horizontal axis) is either increasing or decreasing, depending on whether efforts are strategic complements or substitutes. We are interested in the side of this intersection where $R_{1}$ enters the region of higher welfare: this is where the new intersection (and therefore Nash equilibrium) is to be constructed by giving the correct bias to the new authority 2 and shifting its best response $R_{2}$.

Under Joint Decisions, best responses are increasing; and if the bias is $\lambda _{1}>1$ (industry-friendly), then the best response $R_{1}$ of authority 1 is shifted downwards. Since at the intersection with authority 2’s unbiased best response welfare then is increasing upwards and $R_{1}$ cuts the indifference curve from below, higher welfare can be achieved if $R_{2}$ is shifted rightward: if authority 2 is given a consumer-friendly bias $\lambda _{2}<1$. An analogous argument shows that for $\lambda _{1}<1$ the optimal choice involves $\lambda _{2}>1$.

With Independent Decisions, best responses are decreasing, therefore $R_{1}$ will cut the indifference curves from above. If it is shifted downwards ($\lambda _{1}>1$, industry-friendly), then welfare is higher to the left, and the optimal $R_{2}$ is shifted to the left. This implies that in this case authority 2 should also be industry-friendly: $\lambda _{2}>1$. The case $\lambda _{1}<1$ leads to the analogous result: $\lambda _{2}<1$. $\square$