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Regulation of non-marketed outputs and substitutable inputs


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.

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  1. We thank an anonymous reviewer for drawing our attention towards this paper and the results presented therein.

  2. Aguirre and Beitia (2004) show the difference between shadow costs of public funding and distributional welfare preferences based on a model with continuous probability distribution, while we assume a discrete distribution.

  3. Noticeably, with the (discrete) two-dimensional adverse selection problem, our problem setting is technically closest to the model discussed by Armstrong (1999).

  4. Upwards distorted observable input levels coincide with upwards distorted prices for the inefficient type as shown in Laffont and Tirole (1993). They also agree with the results in a setting with unknown cost and demand functions as long as shadow costs of public funding are considered (Aguirre and Beitia 2004). Noticeably, the case of prices below marginal costs, as found in Lewis and Sappington (1988b) and Armstrong (1999), is mainly triggered by using a distributive social welfare function instead of shadow costs of public funding.

  5. We will argue that in our context, this type of regulation is more suitable than other “simple” mechanisms like rate of return or revenue cap regimes.

  6. Although this assumption might seem restrictive at first sight, it may indeed fit many relevant cases very well. For instance, due to the very high societal value of uninterrupted electricity transmission, changes in costs will hardly affect the desired level of the transmission service quality q. In the same vein, access to infrastructure in rural areas, e.g., high-speed internet or telecommunication services, may not be cost-efficient but considered a public service.

  7. As it is well known from production theory, the optimal rate of substitution is determined by equating the marginal rate of technical substitution between the factors (i.e., the slope of the isoquant) with the relative factor costs (i.e., the slope of the isocost line).

  8. For an analysis involving continuous variables, this would require the expansion path to behave like a function with a unique function value y for each x, or, in other words, an expansion path that is not bending backwards.

  9. Stochastic deviations due to force majeure are supposed to be detectable and excludable from the contract framework.

  10. It goes without saying here that the firm is characterized such that she tries to maximize her rent.

  11. This is the reason why q appears as a subscript here. In case of a more complex model with an endogenous demand, q would be a variable for the regulator, and \(S_q\) be replaced by S(q) (with \(\frac{\partial S}{\partial q} >0\)), indicating that social utility is increasing as the regulator chooses higher output q. The additional first order condition would write as \(\frac{\partial W}{\partial q} = 0 \rightarrow \frac{\partial S(q)}{\partial q} = c^y_j \frac{\partial g_i(q,x)_ij}{\partial q}\), expressing the optimal balance of the social value of the provided output and the related social costs. In our model, this would involve the complexity of one additional choice variable and the associated emergence of a continuum of isoquants. Qualitatively, the efficiency of all results obtained in our model would need to be checked against their social value, and adjusted in case of a mismatch. Starting from the requested output in the first-best case, a second-best solution would thus entail the need to reduce the requested output.

  12. Here and in the following, a prime denotes derivation with respect to x.

  13. Note that if Eq. (8) does not hold, the effect is reversed: incentives for the lj-types to claim higher isoquants could then be reduced by a downwards distortion of x. For the sake of conciseness, we omit further discussions of this particular case.

  14. Note that Condition (12) (or its inverse) is a precondition for the subsequent analysis. We assume it holds unambiguously within the entire relevant range covered by the contract variable x. In more practical applications, this will depend on the (parametric or non-parametric) specification of the functions \(g_l(q,x)\) and \(g_h(q,x)\).

  15. We thank an anonymous reviewer for pointing out this interpretation.

  16. Due to the symmetry of the problem, we omit the detailed calculation here.

  17. The ordering and solution of Case (B) is reversed, but similar. The corresponding discussion can be found in the “Appendix”.

  18. See the “Appendix” for a detailed discussion and the corresponding proposition and proof.

  19. Note that the solution for a pure cost-based regulation without quantity restriction would simply reimburse the costs of the observable input. This would incentivize the firm to choose infinitely high values of x (known as the gold-plating effect). Assuming that the regulator restricts her set of choices by an upper level of \(\bar{x} = x^{fb}_{hh}\) in order to limit excessive (socially costly) rents, all types would then choose this level. In contrast to this very simple approach, the regulatory regime considered in this section makes use of being able to use the observable input x as a contracting variable.

  20. I.e., the point where the ratio of the expected costs equals the expected slope of the isoquants.

  21. Otherwise, the isoquants are no longer strictly convex.

  22. For the sake of brevity, we leave out a computational analysis of Case B.

  23. Recall that input factor x was assumed to be more expensive than y.

  24. Technically, both specifications result in low values of \(\frac{\partial g}{\partial x}\).

  25. Noticeably, a commitment problem of the regulator might impede the implementation of an incentive-based approach, which would be welfare-superior compared to a cost-based regulation. If the firm gets an unconditional payment representing the pay-off of the hh-type, i.e., \(\tilde{T} = c^x x^{fb}_{hh} + c_h^y g_{h}(q,x^{fb}_{hh})\), she will realize first-best input quantities \(\{x^{fb}_{ij},y^{fb}_{ij} \}\). In this case, the realized rent of the firm becomes \(R_{ij} = c^x x^{fb}_{hh} + c_h^y g_{h}(q,x^{fb}_{hh}) - c^x x^{fb}_{ij} - c_j^y g_{i}(q,x^{fb}_{ij})\). However, due to the (observable) separation of types via the realized input x, the regulator might be tempted to adjust the regulatory contract ex-post, and hence, jeopardize the regulatory success if the firm anticipates this behavior.

  26. E.g. the cost-based regulation of German TSOs, where the necessary is suspected to be high, but the German public heavily discusses the related costs for consumers.


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Correspondence to Joachim Bertsch.

Additional information

The authors want to thank Felix Höffler and Christian Tode for their helpful comments, Clara Dewes for her support, the participants of the IAEE 2014 in Rome and the Spring Meeting of Young Economists 2015 in Ghent for valuable discussions, and an anonymous referee for his detailed comments. Funding of the German research society DFG through Grant HO 5108/2-1 is gratefully acknowledged.



1.1 Proof of Proposition 1


  1. (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 \(x^*_{hh}\).

  2. (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
figure 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


Written explicitly, Eq. (27) becomes

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

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),

  1. (i)

    Optimal regulation is achieved under the following set of observable input levels:

    $$\begin{aligned} x^*_{ll}&= x^{fb}_{ll} \end{aligned}$$
    $$\begin{aligned} x^*_{lh}&\ge x^{fb}_{lh} \end{aligned}$$
    $$\begin{aligned} x^*_{hl}&\ge x^{fb}_{hl} \end{aligned}$$
    $$\begin{aligned} x^*_{hh}&\ge x^{fb}_{hh}, \end{aligned}$$

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

  2. (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.


  1. (i)

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

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

    Minimizing C with respect to \(x_{ll}\) yields

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

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

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

    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 \(x^*_{hh}\).

  2. (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
figure 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
figure 11

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

Fig. 12
figure 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
figure 13

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

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Bertsch, J., Hagspiel, S. Regulation of non-marketed outputs and substitutable inputs. J Regul Econ 53, 174–205 (2018).

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