Optimal regulation of lumpy investments

Abstract

We study optimal timing of regulated investment in a real options setting, in which the regulated monopolist has private information on investment costs. In solving the ensuing agency problem, the regulator trades off investment timing inefficiency against the dead-weight loss arising from high price caps. We show that optimal regulation is implemented by a price cap that decreases as a function of the monopolist’s chosen investment time.

Appendices

Appendix 1: proofs

Proof of lemma 2

For future reference, we solve the slightly more general problem where the monopolist’s discounted expected profits (i.e. revenues minus investment costs) at time 0 are constrained to be at some non-negative fixed level $\Pi \ge 0$, rather than zero. We have to solve the programme

$$\begin{aligned} \max _{\bar{A},p}\left[\left(\frac{A_0}{\bar{A}}\right)^\lambda \left(\frac{\bar{A}\left(\bar{p}^{1-\eta }-p^{1-\eta }\right)}{(1-\eta )(r-\mu )}\right) +\Pi \right] \end{aligned}$$

s.t.

$$\begin{aligned} \left(\frac{A_0}{\bar{A}}\right)^\lambda \left(\frac{\bar{A}p^{1-\eta }}{r-\mu } -c\right)=\Pi . \end{aligned}$$

With $\nu$ the Lagrange multiplier on the constraint, the first-order condition for $p$ yields

$$\begin{aligned} \nu =\frac{1}{1-\eta }. \end{aligned}$$

(20)

For the first-order condition for the threshold demand level $\bar{A}$ we find

$$\begin{aligned} (\lambda -1)\frac{\bar{A}\left(\bar{p}^{1-\eta }-p^{1-\eta }\right)}{(1-\eta )(r-\mu )} +\nu (\lambda -1)\left(\frac{\bar{A}p^{1-\eta }}{r-\mu }\right)=\lambda \nu c. \end{aligned}$$

(21)

Substituting $\nu$ we obtain the desired second-best threshold $\bar{A}^{sb}$ (9) from the lemma, which turns out to be independent of the value of $\Pi$. The optimal price $p$ then follows from substituting this threshold $\bar{A}^{sb}$ in the constraint, and setting $\Pi =0$.$\square$

Proof of Proposition 1

Consider a price cap $p_c\le \bar{p}$. The monopolist will choose investment threshold $\bar{A}$ to optimize his expected profits

$$\begin{aligned} \Pi (\bar{A})=\left(\frac{A_0}{\bar{A}}\right)^\lambda \left(\frac{\bar{A}p_c^{1-\eta }}{r-\mu }-c\right), \end{aligned}$$

and solving the first-order condition gives

$$\begin{aligned} \frac{\bar{A}p_c^{1-\eta }}{r-\mu }=\frac{\lambda }{\lambda -1}c. \end{aligned}$$

(22)

Comparing with second-best, (9), we find $\bar{A}\ge \bar{A}^{sb}$, with equality only when $p_c=\bar{p}$. The optimal price cap involves trading off timing inefficiency against lower ex-post pricing, maximizing social surplus:

$$\begin{aligned} \left(\frac{A_0}{\bar{A}(p_c)}\right)^\lambda \left(\frac{\bar{A}(p_c) (\bar{p}^{1-\eta }-p_c^{1-\eta })}{(1-\eta )(r-\mu )}+\frac{\bar{A}(p_c)p_c^{1-\eta }}{r-\mu }-c\right), \end{aligned}$$

with $\bar{A}(p_c)$ the solution to (22). Solving the first-order condition for $p_c$ gives

$$\begin{aligned} \left(\frac{p_c}{\bar{p}}\right)^{1-\eta }=\frac{\lambda -1}{\lambda -1+\eta }. \end{aligned}$$

$\square$

Proof of lemma 3

The proof of the lemma follows standard arguments (see e.g. Laffont and Tirole 1993, Sect. 1.4). Suppose the monopolist has costs $c$. By choosing the contract designed for a monopolist with costs $\hat{c}$ (which may be different from his real costs $c$), the monopolist’s profits are

$$\begin{aligned} \Pi (\hat{c},c)=\left(\frac{A_0}{\bar{A}(\hat{c})}\right)^\lambda \left(\frac{\bar{A}(\hat{c})p(\hat{c})^{1-\eta }}{r-\mu }-c\right). \end{aligned}$$

Note that, in this expression, the contract parameters are governed by the monopolist’s choice $\hat{c}$, while his actual cost upon investing equals his true cost $c$. Incentive compatibility of the scheme means that the monopolist maximizes his profits by announcing his true costs, $\hat{c}=c$. But this implies, by the envelope theorem, that

$$\begin{aligned} \left.\frac{d\Pi (\hat{c},c)}{dc}\right|_{\hat{c}=c}=\left.\frac{\partial \Pi (\hat{c},c)}{\partial c}\right|_{\hat{c}=c}=- \left(\frac{A_0}{\bar{A}(c)}\right)^\lambda . \end{aligned}$$

Using the short-hand notation

$$\begin{aligned} \Pi (c)\equiv \Pi (\hat{c}=c,c) \end{aligned}$$

this proves that incentive compatibility requires condition (12). To show the need for condition (13), we observe that

$$\begin{aligned} \Pi (c,c)\ge \Pi (\hat{c},c)=\Pi (\hat{c},\hat{c})+(\hat{c}-c) \left(\frac{A_0}{\bar{A}(\hat{c})}\right)^\lambda . \end{aligned}$$

Combining this with a similar inequality where $\hat{c}$ and $c$ are reversed,

$$\begin{aligned} \Pi (\hat{c},\hat{c})\ge \Pi (c,\hat{c})=\Pi (c,c)-(\hat{c}-c)\left(\frac{A_0}{\bar{A}(c)}\right)^\lambda , \end{aligned}$$

we obtain

$$\begin{aligned} (\hat{c}-c)\left(\frac{A_0}{\bar{A}(c)}\right)^\lambda \ge (\hat{c}-c)\left(\frac{A_0}{\bar{A}(\hat{c})}\right)^\lambda . \end{aligned}$$

Since $\lambda >1$, if $\hat{c}>c$ then $\bar{A}(\hat{c})\ge \bar{A}(c)$, or

$$\begin{aligned} \frac{d\bar{A}(c)}{dc}\ge 0. \end{aligned}$$

$\square$

Proof of Proposition 2

For convenience we recall the programme to be solved:

$$\begin{aligned} \max _{\bar{A}(c),p(c)} \int _{c_L}^{c_H} \left[\left(\frac{A_0}{\bar{A}}\right)^\lambda \left(\frac{\bar{A} (\bar{p}^{1-\eta }-p^{1-\eta })}{(1-\eta )(r-\mu )}\right)+\Pi (c)\right]\mathrm{d}F(c) \end{aligned}$$

(23)

s.t.

$$\begin{aligned}&\Pi (c)=\left(\frac{A_0}{\bar{A}}\right)^\lambda \left(\frac{\bar{A}p^{1-\eta }}{r-\mu }-c\right)\end{aligned}$$

(24)

$$\begin{aligned}&\frac{\mathrm{d}\Pi (c)}{\mathrm{d}c}=-\left(\frac{A_0}{\bar{A}}\right)^\lambda \end{aligned}$$

(25)

$$\begin{aligned}&\Pi (c_H)= 0 \end{aligned}$$

(26)

This is an optimal control problem with constraints, with $\Pi (c)$ the state variable. To solve it, we consider the related Hamiltonian $H$, with a co-state variable $\omega (c)$ that is adjoint to the state variable $\Pi (c)$. Furthermore, we use a Lagrange multiplier $\nu (c)$ for the definition of $\Pi (c)$ in terms of the control variables $\bar{A}(c),p(c)$ (as we did before, in the proof of lemma 2). Optimization in this Hamiltonian framework then requires

$$\begin{aligned} \frac{\mathrm{d}\omega (c)}{\mathrm{d}c}&=-\frac{\partial H}{\partial \Pi }=\nu (c)-f(c),\\ \omega (c_L)&=0 \quad \text{(the} \text{ transversality} \text{ condition)},\\ \Pi (c_H)&=0, \end{aligned}$$

as well as the first-order conditions for $p(c)$ and $\bar{A}(c)$. Considering first the first-order condition for $p(c)$:

$$\begin{aligned} -f(c)+(1-\eta )\nu (c)=0 \end{aligned}$$

or

$$\begin{aligned} \nu (c)=\frac{f(c)}{1-\eta }. \end{aligned}$$

Hence

$$\begin{aligned} \omega (c)=\frac{\eta F(c)}{1-\eta }. \end{aligned}$$

Secondly, the first-order condition for $\bar{A}(c)$ is

$$\begin{aligned} (\lambda -1)\left(\frac{\bar{A}\left(\bar{p}^{1-\eta }-p^{1-\eta }\right)}{(1-\eta )(r-\mu )}+\frac{\nu }{f}\left(\frac{\bar{A}p^{1-\eta }}{r-\mu }\right)\right) =\lambda \frac{\omega }{f}+\lambda \frac{\nu }{f}c, \end{aligned}$$

(27)

which reduces to the desired optimal threshold

$$\begin{aligned} \frac{\bar{A}(c)\bar{p}^{1-\eta }}{r-\mu }=\frac{\lambda }{\lambda -1} \left(c+\eta \frac{F}{f}(c)\right). \end{aligned}$$

(28)

Note that as a result of the assumed monotonicity of $F/f(c), \,\bar{A}(c)$ is duly increasing in $c$. Given $\bar{A}(c)$, we can next explicitly compute $p(c)$ from the constraint,

$$\begin{aligned} \frac{\bar{A}(c)p(c)^{1-\eta }}{r-\mu }=c+\left(c+\eta \frac{F}{f}(c)\right)^\lambda \int \limits _c^{c_H} \left(c^{\prime }+\eta \frac{F}{f}(c^{\prime })\right)^{-\lambda }dc^{\prime }. \end{aligned}$$

(29)

$\square$

Proof of Corollary 2

Note first that the menu of contracts $\bar{A}(c),p(c)$ from Proposition 2 defines a parametrized curve that is described by a function $p(\bar{A})$, since $\bar{A}(c)$ is strictly increasing in $c$.

To prove that $p(\bar{A})$ is downward sloping, since $\bar{A}$ is increasing in $c$, it is sufficient to show that $p(c)$ is downward sloping. Combining (28) and (29) we write

$$\begin{aligned} \frac{\lambda }{\lambda -1}\left(\frac{p(c)}{\bar{p}}\right)^{1-\eta } =c\tilde{c}(c)^{-1}+\tilde{c}(c)^{\lambda -1}\int \limits _c^{c_H}\tilde{c}(c^{\prime })^{-\lambda }dc^{\prime }. \end{aligned}$$

with $\tilde{c}(c)=c+\eta \frac{F}{f}(c)$ the virtual cost function. Taking derivatives on both sides, we find that decreasing $p(c)$ is equivalent to

$$\begin{aligned} (\lambda -1)\tilde{c}(c)^{\lambda }\int \limits _c^{c_H}\tilde{c}(c^{\prime })^{-\lambda }dc^{\prime }<c. \end{aligned}$$

(30)

To verify that this inequality holds, note that since $\bar{p}>p(c)$, we have

$$\begin{aligned} \frac{\lambda }{\lambda -1}\tilde{c}(c)=\frac{\bar{A}(c) \bar{p}^{1-\eta }}{r-\mu }>\frac{\bar{A}(c){p}^{1-\eta }}{r-\mu } =c+\tilde{c}(c)^{\lambda }\int \limits _c^{c_H}\tilde{c}(c^{\prime })^{-\lambda }dc^{\prime }, \end{aligned}$$

so that

$$\begin{aligned} \lambda (\tilde{c}(c)-c)+c>(\lambda -1)\tilde{c}(c)^{\lambda } \int \limits _c^{c_H}\tilde{c}(c^{\prime })^{-\lambda }dc^{\prime }. \end{aligned}$$

Since $\tilde{c}(c)>c$ this implies inequality (30) and we conclude that $p(\bar{A})$ is downward sloping. $\square$

Appendix 2: extensions

In this appendix we briefly sketch how our main result, the optimal regulatory scheme outlined in Proposition 2, is affected if we relax some of the assumptions on the model.

The assumption of Geometric Brownian motion for demand process $A_t$, Eq. (2), allowed us to express the results in simple closed form. If we allow for more general stochastic processes (but excluding explicit time-dependence of the coefficients of the process), it will in general not be possible to obtain such explicit expressions. In this case, though, we may reach similar conclusions. Let us introduce the shorthand

$$\begin{aligned} \mathcal{{A}}(\bar{A})\equiv E\left(\int \limits _0^\infty e^{-rt}A_t dt\mid A_0=\bar{A}\right), \end{aligned}$$

so that aggregate net consumer welfare at the moment of investment equals $v(p)\mathcal{A}$. Note that in our main model, $\mathcal{{A}}(\bar{A})=\bar{A}/ (r-\mu )$. In the general case, replication of the proof of Proposition 2 leads us to the first-order equation for the investment threshold

$$\begin{aligned} \frac{d}{d\bar{A}}\left[ E\left(e^{-rT(\bar{A})}\mid A_0\right)(\bar{p}^{1-\eta }\mathcal{A}\left(\bar{A})-\tilde{c}\right)\right]=0, \end{aligned}$$

where we defined virtual costs $\tilde{c}=c+\eta \frac{F}{f}(c)$. For the case of Geometrical Brownian motion considered in the paper, substituting the expression for the expected discount factor, Eq. (5), together with the appropriate $\mathcal{A}$ as above, leads to the familiar expression from Proposition 2. In the general case, again the optimal investment under asymmetric information occurs at a threshold that would be (second-best) efficient for a firm for which costs would equal $\tilde{c}$, rather than $c$ itself. Hence at the lower cost bound, where $\tilde{c}_L=c_L$, the threshold replicates the (second-best) real option result for general stochastics. For all higher cost types, since virtual costs $\tilde{c}$ rise monotonically in $c$, investment is delayed until demand hits a threshold that would be efficient for a firm with costs strictly above those actual costs.

Next consider what happens if we instead relax the assumption of constant elasticity demand: we consider more general forms of net consumer surplus $v(p)$, with demand $q(p)$ satisfying $v^{\prime }(p)=-q(p)$, and elasticity $\eta (p)=-\frac{p}{q}\frac{dq}{dp}$ in general price-dependent. Again, we can redo the analysis of Proposition 2 to find the condition for optimal $\bar{A}$:

$$\begin{aligned} \bar{A}\left((1-\eta (p))\frac{v(p)}{r-\mu }+\frac{pq}{r-\mu }\right) =\frac{\lambda }{\lambda -1}\left( c + (1-\eta (p))\int \limits _{c_L}^{c} \frac{\eta (p(c^{\prime }))}{1-\eta (p(c^{\prime }))}f(c^{\prime }) dc^{\prime }\right). \end{aligned}$$

while $p(c)$ is determined jointly with this equation to make sure that the firm’s surplus satisfies the incentive compatibility constraint. Comparing with the constant elasticity result, we see two effects of allowing general elasticities. On the right-hand side of the equation, the expression for the virtual costs is more complex, reflecting also the changes in elasticities for the lower types. On the left-hand side, with general consumer surplus, the combination of consumer surplus does not reduce to the simple, $p$-independent expression that we found for the constant elasticity case.

Finally, we assumed that variable costs after investment are zero. In fact, expressions for the situation with positive marginal costs of production $m$ are related to those without such costs by a transformation of the surplus, $v(p)\rightarrow v(p+m)$, and hence, $q(p)\rightarrow q(p+m)$. Starting from the base case with constant elasticity demand, after the marginal cost shift the analysis becomes equivalent to one with non-constant elasticity, as considered above.