Regulation of non-marketed outputs and substitutable inputs

Abstract

We study the regulation of a monopolistic firm that provides a non-marketed output based on multiple substitutable inputs. The regulator is able to observe the effectiveness of the provision, but faces information asymmetries with respect to the efficiency of the firm’s activities. Specifically, we consider a setting where one input and the output are observable, while another input and related costs are not. Multi-dimensional information asymmetries are introduced by discrete distributions for the functional form of the marginal rate of substitution between the inputs as well as for the input costs. For this novel setting, we investigate the theoretically optimal Bayesian regulation mechanism. We find that the first-best solution cannot be obtained in case of shadow costs of public funding. The second-best solution implies separation of the most efficient type with first-best input levels, and upwards distorted (potentially bunched) observable input levels for all other types. Moreover, we compare these results to a simpler non-Bayesian approach, i.e., a single pooling contract, and hence, bridge the gap between the academic discussion and regulatory practice. In a numerical simulation, we identify certain conditions in which a single contract non-Bayesian regulation can indeed get close to the second-best solution of the Bayesian menu of contracts regulation.

Appendix

1.1 Proof of Proposition 1

Proof

• (i) From Eq. (20), we see that $\frac{\partial C}{\partial x_{lh}}$ is strictly smaller than 0 for $x_{lh}=x^{fb}_{lh}$ and monotonically increasing in $x_{lh}$, which implies that $x^*_{lh}>x^{fb}_{lh}$ must always hold. The same logic applies for $x^*_{hl}$ and $x^*_{hh}$.

• (ii) From the fact that $x^{fb}_{ll}<x^{fb}_{lh}$ and the strict upwards distortion of all other types, it follows that the ll-type can always be separated. In order to investigate whether the types lh, hl and hh can be separated or need to be bunched, we proceed as follows: For each of the possible pairs $lh-hl$, $hl-hh$ and $lh-hh$, we check the derivative of C with respect to the former type at the optimal level of $x^*$ of the latter type (derived from the first order condition). If the change in C is greater than 0 we can conclude that we have already surpassed the optimal level of the former type, which then must be smaller than the optimal level of the latter type. In other words, we check the level of upwards distortion for the lh, hl and hh types while considering the necessary ordering of the types according to Lemma 2. For the pair lh-hl, we find that $x^*_{lh}$ may surpass $x^*_{hl}$ in case of $\nu \rightarrow 1$, while they are otherwise clearly separated from each. For the pair hl-hh, bunching may occur for $g_l'(q,x) \rightarrow 0$ together with $c^y_l$ being large. Furthermore, we find that lh-hh can always be separated, implying that at most two types (i.e., either lh-hl or hl-hh) may be bunched under certain parameter constellations.

Fig. 9

Constraints considered binding for Case (B)

Lastly, it is straightforward to check that the remaining constraints are satisfied under the obtained solution of the relaxed problem. Hence, we have indeed obtained to optimal solution for the full regulatory problem we are facing in Case (A). $\square$

1.2 Proof of Proposition 2

Proof

Written explicitly, Eq. (27) becomes

$$\begin{aligned} \bar{C}&= \mu \nu \left[ \lambda \left( c_h^y g_{h}(\bar{x}) - c_l^y g_{l}(\bar{x}) \right) + (1+\lambda ) \left( c^x \bar{x} + c^y_l g_l(\bar{x})\right) \right] \nonumber \\&\quad + \mu (1-\nu ) \left[ \lambda \left( c_h^y g_{h}(\bar{x}) - c_h^y g_{l}(\bar{x})\right) + (1+\lambda ) \left( c^x \bar{x} + c^y_h g_l(\bar{x})\right) \right] \nonumber \\&\quad + (1-\mu ) \nu \left[ \lambda \left( c_h^y g_{h}(\bar{x}) - c_l^y g_{h}(\bar{x})\right) + (1+\lambda ) \left( c^x \bar{x} + c^y_l g_h(\bar{x})\right) \right] \nonumber \\&\quad + (1-\mu ) (1-\nu ) \left[ (1+\lambda ) \left( c^x \bar{x} + c^y_h g_h(\bar{x})\right) \right] . \end{aligned}$$

(31)

Deriving the above with respect to $\bar{x}$ yields, after a few calculations, $\mathbb {E}(g_{i}'(\bar{x}^*))\mathbb {E}(c_j^y)+c^x+\lambda (c_h^y g_{h}'(\bar{x}^*) + c^x) = 0$. Hence, for $\lambda =0$, $\mathbb {E}(g_{h}'(\bar{x}^*)) = - \frac{c^x}{\mathbb {E}(c_j^y)}$. $\square$

1.3 Two-dimensional asymmetric information, Case (B): cost variation large compared to isoquant variation

To solve the second case following from Lemma 2, we need to apply a different educated guess with respect to the binding constraints. However, we apply a similar reasoning as in Case (A), but take account of the fact that now, cost variation is more relevant than isoquant variation. Hence, we choose a symmetric setting and imply incentive constraints $ll \rightarrow hl$, $hl \rightarrow lh$ and $lh \rightarrow hh$ to be binding. Again, we assume the participation constraint of the hh-type to be binding. Figure 9 illustrates this setting.

After having determined the results and checked all remaining constraints, we find the setting of binding constraints as in Fig. 9 indeed to be optimal for Case (B). Results are summarized in the following Proposition 3.

Proposition 3

For case (B),

• (i) Optimal regulation is achieved under the following set of observable input levels:

  $$\begin{aligned} x^*_{ll}&= x^{fb}_{ll} \end{aligned}$$

  (32)
  $$\begin{aligned} x^*_{lh}&\ge x^{fb}_{lh} \end{aligned}$$

  (33)
  $$\begin{aligned} x^*_{hl}&\ge x^{fb}_{hl} \end{aligned}$$

  (34)
  $$\begin{aligned} x^*_{hh}&\ge x^{fb}_{hh}, \end{aligned}$$

  (35)

  while respecting $x^*_{ll} < x^*_{hl} \le x^*_{lh} \le x^*_{hh}$.

• (ii) The most efficient (ll) type can always be separated. Moreover, separation of at least three types is always possible, while bunching of the hl and lh types is unavoidable in case of $\mu \rightarrow 1$.The lh and hh types may be bunched in case of $c^y_l$ being small and $g_h'(q,x)$ large.

Proof

• (i) Under the constraints considered binding for Case (B)—as discussed and shown in Fig. 9—the social cost function (4) becomes

  $$\begin{aligned} C =&\, \mu \nu \left[ \lambda \left( c^y_h (g_h(x_{hh})-g_l(x_{hh})) + c^y_h g_l(x_{lh}) - c^y_l g_h(x_{lh})\right. \right. \nonumber \\&\left. \left. + \,c^y_l (g_h(x_{hl})-g_l(x_{hl}))\right) + (1+\lambda ) \left( c^x x_{ll} + c^y_l g_l(x_{ll})\right) \right] \nonumber \\&+\, \mu (1-\nu ) \left[ \lambda \left( c^y_h (g_h(x_{hh})-g_l(x_{hh}))\right) + (1+\lambda ) \left( c^x x_{lh} + c^y_h g_l(x_{lh})\right) \right] \nonumber \\&+\, (1-\mu ) \nu \left[ \lambda \left( c^y_h (g_h(x_{hh})-g_l(x_{hh}))\right) + c^y_h g_l(x_{lh}) \right. \nonumber \\&\left. -\, c^y_l g_h(x_{lh}) + (1+\lambda ) \left( c^x x_{hl} + c^y_l g_h(x_{hl})\right) \right] \nonumber \\&+\, (1-\mu ) (1-\nu ) \left[ (1+\lambda ) \left( c^x x_{hh} + c^y_h g_h(x_{hh})\right) \right] . \end{aligned}$$

  (36)

  Minimizing C with respect to $x_{ll}$ yields

  $$\begin{aligned} g'_l(x_{ll}^*) = -\frac{c^x}{c^y_l}, \end{aligned}$$

  (37)

  which implies that $x_{ll}^* = x_{ll}^{fb}$. Derivation of C with respect to $x_{lh}$, $x_{hl}$ and $x_{hh}$ yields:

  $$\begin{aligned} \frac{\partial C}{\partial x_{lh}}&= \underbrace{\mu \lambda (c^y_h g'_l(x_{lh}) -c^y_l g'_h(x_{lh})}_{<0} + \underbrace{\mu (1-\nu ) (1+\lambda ) (c^x + c^y_h g_l'(x_{lh}))}_{\begin{array}{c} =0 \text { for } x_{lh}=x^{fb}_{lh} \\<0 \text { for } x_{lh}<x^{fb}_{lh} \\>0 \text { for } x_{lh}>x^{fb}_{lh} \end{array}} \end{aligned}$$

  (38)
  $$\begin{aligned} \frac{\partial C}{\partial x_{hl}}&= \underbrace{\mu \nu \lambda (c^y_l g'_h(x_{hl}) - c^y_l g'_l(x_{hl}))}_{<0} + \underbrace{(1-\mu ) \nu (1+\lambda ) (c^x + c^y_l g_h'(x_{hl}))}_{\begin{array}{c} =0 \text { for } x_{hl}=x^{fb}_{hl} \\<0 \text { for } x_{hl}<x^{fb}_{hl} \\>0 \text { for } x_{hl}>x^{fb}_{hl} \end{array}} \end{aligned}$$

  (39)
  $$\begin{aligned} \frac{\partial C}{\partial x_{hh}}&= \underbrace{(\mu + (1-\mu ) \nu ) \lambda c^y_h (g'_h(x_{hh})-g'_l(x_{hh}))}_{<0}\nonumber \\&\quad + \underbrace{(1-\mu ) (1-\nu ) (1+\lambda ) (c^x + c^y_h g_h'(x_{hh}))}_{\begin{array}{c} =0 \text { for } x_{hh}=x^{fb}_{hh} \\<0 \text { for } x_{hh}<x^{fb}_{hh} \\ {>0 \text { for } x_{hh}>x^{fb}_{hh}} \end{array}}. \end{aligned}$$

  (40)

  From Eq. (38), we see that $\frac{\partial C}{\partial x_{lh}}$ is strictly smaller than 0 for $x_{lh}=x^{fb}_{lh}$ and monotonically increasing in $x_{lh}$, which implies that $x^*_{lh}>x^{fb}_{lh}$ must always hold. The same logic applies for $x^*_{hl}$ and $x^*_{hh}$.

• (ii) From $x^{fb}_{ll}<x^{fb}_{lh}$ and the strict upwards distortion of all other types, it follows that the ll-type can always be separated. $x^*_{hl}$ may surpass $x^*_{lh}$ in case of $\mu \rightarrow 1$. If the low costs $c^y_l$ are small and $g_h'(q,x)$ becomes large, lh and hh types may need to be bunched, without impacting the separation of the other types.

The remaining constraints are satisfied under the obtained solution. $\square$

As in Case (A), the first-best solution can be obtained for $\lambda =0$, while the solution is second-best and incurring an increasing level of inefficiency for increasing levels of $\lambda$. Also again, the most efficient type can always be separated, while bunching of the hl and lh types (lh and hh types) may occur for very high occurrence probability of low isoquants, or if $g_h(q,x)$ is very steep and $c^y_l$ small.

Fig. 10

Numerical results of x, for different $\lambda$, a and $\rho$

1.4 Impact of the production function specification: numerical results for different levels of $\lambda$

See Figs. 10 and 11.

Fig. 11

Numerical results of the expected costs C, for different $\lambda$, a and $\rho$

Fig. 12

Numerical results of x, for different $\lambda$, $\mu$ and $\nu$

1.5 Impact of the random draw probabilities: numerical results for different levels of $\lambda$

See Figs. 12 and 13.

Fig. 13

Numerical results of C, for different $\lambda$, $\mu$ and $\nu$