Political pressures and the credibility of regulation: can profit sharing mitigate regulatory risk?

Abstract

When price-cap rules determine the structure of prices for a long period, they suffer a credibility problem and introduce an element of risk especially if a firm’s profits are “too large”. Profit sharing may be seen as a device to pre-determine price adjustments and thus to decrease regulatory risk. We analyse the effects of profit sharing on the incentives to invest, using a real option approach. Absent credibility issues, a well designed profit sharing system may be neutral relative to a pure price cap. With regulatory risk, profit sharing is preferable to a pure price-cap one, if it intervenes for high enough profit levels.

Appendix

1.1 The profit dynamics

In order to determine the dynamics of profit, we first derive the dynamics of the demand function. Using (1), (2), and (6), and applying Itô’s Lemma one obtains

$$ dq_{t}=-\eta \left(\hbox{RPI}-x_{j}\right) q_{t}dt+\sigma_{q}q_{t}dz_{t}. $$

(15)

Using (3), (4), and (15) one can obtain the profits’ dynamics under price cap regulation

$$ d\Uppi_{t}=\alpha \left(x_{j}\right) \Uppi_{t}dt+\sigma \Uppi_{t}dz_{t}, $$

(16)

where α(x _j ) ≡ (1−η) (RPI−x _j ) + α _m is the expected growth rate of per-period profits.

1.2 The derivation of (8) and (9)

Using dynamic programming, the firm’s value O _PC (Π(t)) can be written as

$$ O_{PC}(\Uppi (t))=\Uppi (t)dt+e^{-rdt}E\left[O_{PC}(\Uppi (t)+d\Uppi (t))\right]. $$

Expanding the right-hand side and using Itô’s lemma one obtains

$$ rO_{PC}(\Uppi (t))=\Uppi (t)+(r-\delta (x_{l}))\Uppi O_{PC_{\Uppi}}(\Uppi (t))+\frac{ \sigma^{2}}{2}\Uppi^{2}O_{PC_{\Uppi \Uppi}}(\Uppi (t)), $$

(17)

where $O_{PC_{\Uppi}}=\partial O_{PC}/\partial \Uppi (t)$ and $O_{PC_{\Uppi \Uppi}}=\partial^{2}O_{PC}/\partial \Uppi^{2}(t),$ respectively.

To compute the value function, it is assumed that O _PC (0, x _l ) = 0, namely when Π is very small, the project is almost worthless. Thus, Eq. (17) yields the function (8), where β₁(x _l ) is the positive root of the following characteristic equation $\frac{\sigma^{2}}{2}\beta (\beta -1)+(r-\delta (x_{l}))\beta -r=0.$

Let us next turn to the firm’s value after expansion. Using dynamic programming, the firm’s value V _PC (Π(t)) can be written as

$$ V_{PC}(\Uppi(t))=\Uppsi\Uppi(t)dt+e^{-rdt}E\left[V_{PC}(\Uppi(t)+d\Uppi(t))\right]. $$

Following the same procedure, assuming that V _PC (0, x) = 0 and that no speculative bubbles exist yields function (9).²⁷

1.3 The derivation of (11) and (12)

To derive function (11) we follow the same procedure used to compute (8) and (9). We can thus write the general solution as

$$ O_{PS}(\Uppi ,x_{l},x_{h})=\left\{ \begin{array}{*{20}l} \frac{\Uppi}{\delta(x_{l})}+\sum\limits_{i=1}^{2}C_{i}\Uppi^{\beta_{i}(x_{l})} & \hbox{if}\quad\pi \leq \Uppsi{\widetilde{\Uppi}},\\ \frac{\Uppi}{\delta(x_{h})}+\sum\limits_{i=1}^{2}B_{i}\Uppi^{\beta_{i}(x_{h})} & \hbox{if}\quad\pi > \Uppsi {\widetilde{\Uppi}},\\ \end{array} \right. $$

(18)

where parameters β₁(x _h ) and β₂(x _h ) are the roots of the characteristic equation $\frac{\sigma^{2}}{2}\beta (\beta -1)+(r-\delta (x_{h}))\beta -r=0,$ with β₁(x _h ) > 1 > 0 > β₂(x _h ).²⁸ In the $(0,{\widetilde{\Uppi}})$ region, condition O _PS (0, x _l , x _h ) = 0 holds: this implies that C ₂ = 0. In the $({\widetilde{\Uppi}},\infty)$ region, instead, there are no boundary conditions. To compute B ₁ and B ₂, we thus apply the VMC and SPC to the two components of (10) at the switch point $\Uppi ={\widetilde{\Uppi}}.$ Both parameters depend on the regulatory coefficients x _l and x _h . It is easy to ascertain that $B_{1}\;{\varpropto}\;C_{1}$ and $B_{2}\;{\varpropto}\;C_{1}.$ Function (10) is thus obtained.

To compute (12) we follow the same procedure. Notice that, after undertaking I, the firm loses any flexibility. Therefore, V _PS (Π, x _l , x _h ) does not embody any real option. In this case, the general solution is given by the sum of a perpetual rent, with discount rate δ(x _h ), and a homogeneous (exponential) part. Again, it is assumed that V _PS (0, x _l , x _h ) = 0 and that no speculative bubbles exist. Given these conditions, it is straightforward to obtain (12). To compute V ₁ and V ₂, we let the two branches of (12) meet tangentially at the switch point $\Uppi ={\widetilde{\Uppi}}.$

1.4 Proof of Proposition 1

Substituting Eqs. (11) and (12) into the VMC and SPC one obtains the following two-equation system

$$ \begin{aligned} &\frac{\left(\Uppsi -1\right) \Uppi}{\delta (x_{l})}+\left(V_{1}-C_{1}\right) \Uppi^{^{\beta_{1}(x_{l})}}-I=0,\\ &\frac{\Uppsi -1}{\delta (x_{l})}+\left(V_{1}-C_{1}\right) \beta_{1}(x_{l})\Uppi^{^{\beta_{1}(x_{l})-1}}=0,\\ \end{aligned} $$

(19)

in the $(0,{\widetilde{\Uppi}})$ region. Solving (17) yields the trigger point

$$ \pi_{PS}^{*}(x_{l})\equiv \frac{\beta_{1}(x_{l})}{\beta _{1}(x_{l})-1} \frac{\Uppsi}{\Uppsi -1}\delta (x_{l})I, $$

(20)

which is equal to the pure price cap one (10). The equality π _PS *(x _l ) = π _PC *(x _l ) proves Proposition 1.□

1.5 Proof of Corollary 1

Recall Eqs. (11) and (12). When the x factor is characterised by a permanent upward shift, the switch level represents an absorbing barrier. Since no downward jump in the x factor is allowed, we have B ₂ = V ₂ = 0. It is worth noting that the two-equation system obtained with the VMC and SPC is equal to (19). This implies that the trigger point is equal to (20). The Corollary is thus proven.□

1.6 Proof of Proposition 2

To prove Proposition 2 let us first show that investment is never undertaken at any point π ≤ π _PC *(x _l,0). To do so, suppose ab absurdo that in the case at hand (without profit sharing and with regulatory risk) the optimal investment timing, π**, is less than π _PC *(x _l,0). Given Proposition 1, we thus would have π** ≤ π _PC *(x _l,0) = π _PS *(x _l,0).

Using the per-period benefit arising from an incremental investment, (Ψ −1)Π, and applying standard techniques one obtains the present value in the (0, π _PS *) region of the net gain from investing as

$$ NV_{i}=\frac{\Uppsi -1}{\delta (x_{l,0})}\Uppi +H_{i}\Uppi^{\beta_{1}(x_{l,0})} $$

where i = 0,1. When i = 0, the profit is such that the starting factor x _l,0 does not change, while i = 1 means that the x factor jumps upward when $\pi =\Uppsi {\widetilde{\Uppi}}.$ The term $H_{i}\Uppi^{\beta_{1}(x_{l,0})}$ measures the present value of the expected loss due to the increase in x.

The x factor may or may not jump upwards. If this does not happen, the firm’s loss is nil. Therefore for NV ₀ this second term is nil (i.e., H ₀ = 0):

$$ NV=\left\{\begin{array}{*{20}l} NV_{0}=\frac{\Uppsi -1}{\delta (x_{l,0})}\Uppi & \hbox{if}\;x={x_{l,0}}, \\ NV_{1}=\frac{\Uppsi -1}{\delta (x_{l,0})}\Uppi +H_{1}\Uppi^{\beta_{1}(x_{l,0})} & \hbox{if}\;x={x_{l,1}}.\\ \end{array}\right. $$

(21)

Let us next solve the following optimal investment timing problem

$$ \max_{t}E\left\{\left[\lambda NV_{1}+\left(1-\lambda \right) NV_{0}-I \right] e^{-rt}\right\} $$

(22)

It is worth noting that a Brownian motion satisfies the Markov property. Namely, the probability distribution for all future values of Π depends only on its current value. Applying this property, and following Harrison (1985), one can rewrite (22) as

$$ \max_{\Uppi^{**}}\left(\frac{\Uppi}{\Uppi^{* *}}\right)^{\beta_{1}(x_{l,0})}\left[\lambda NV_{1}\left(\Uppi^{* *}\right)+\left(1-\lambda \right) NV_{0}\left(\Uppi^{**}\right) -I\right]. $$

(23)

Solving problem (23), it is straightforward to obtain

$$ \pi^{**}\equiv \frac{\beta_{1}(x_{l,0})}{\beta _{1}(x_{l,0})-1} \frac{\Uppsi}{\Uppsi -1}\phi I, $$

where β₁(x _l,0) is the positive root of the characteristic equation $\frac{\sigma^{2}}{2}\beta (\beta -1)+(r-\delta (x_{l,0}))\beta -r=0,$ and where $\phi^{-1}\equiv \frac{\lambda} {\delta (x_{l,1})}+\frac{\lambda}{\delta (x_{l,0})}.$

Remember that we assumed ab absurdo that π** ≤ π _PC *(x _l,0) = π _PS *(x _l,0). However, given x _l,0 < x _l,1 we have δ(x _l,0) < δ(x _l,1). This implies that δ(x _l,1) > ϕ > δ(x _l,0), and thus π** > π _PC *(x _l,0) = π _PS *(x _l,0). This inequality contradicts the starting assumption, and thus proves point (a).

To prove points (b) and (c) let us next apply Proposition 1: when π reaches π _PC *(x _l,0), and the x factor does not rise, then investment is undertaken. Otherwise, the firm will wait until π = π _PC *(x _l,1) > π _PC *(x _l,0). This concludes the proof. □

1.7 Proof of Proposition 3

Let us define O _PS,i (Π) as the firm’s value before expansion, under the old (i = 0) and the new (i = 1) regime. Similarly, define V _PS,i (Π) and W _PS,i (Π) as the value functions after, respectively, undertaking investment I and F, under state i. We first compute the closed-form solutions in state 1. Then we turn to the pre-change case (state 0). Finally, we compare the two trigger points obtained and show that they are equal.

1.7.1 State 1 (the new regime)

Let us first compute the firm’s value under state 1 i.e under the new regime. Before expansion, the firm’s value can be written as

$$ O_{PS,1}(\Uppi)=\Uppi dt+e^{-rdt}E\left[O_{PS,1}(\Uppi +d\Uppi)\right]. $$

Expanding the right-hand side and using Itô’s lemma one obtains

$$ rO_{PS,1}(\Uppi)=\Uppi +(r-\delta (x_{l}))\Uppi \frac{\partial O_{PS,1}(\Uppi)}{\partial \Uppi}+\frac{\sigma^{2}}{2}\Uppi^{2}\frac{\partial ^{2}O_{PS,1}(\Uppi) }{\partial \Uppi^{2}}, $$

(24)

where x = x _l , x _h,1. The Eq. (21) has the following solution

$$ O_{PS,1}(\Uppi ,x)=\left\{ \begin{array}{*{20}l} \frac{\Uppi}{\delta (x_{l})}+C_{1}\Uppi^{\beta_{1}(x_{l})}\quad \hbox{if}\quad\Uppsi \Uppi < \Uppsi {\widetilde{\Uppi}}_{1},\\ \frac{\Uppsi \Uppi}{\delta (x_{h,1})}+B_{1}\Uppi^{\beta_{1}(x_{h,1})}+B_{2}\Uppi ^{\beta_{2}(x_{h,1})}\quad \hbox{if}\quad\Uppsi \Uppi > \Uppsi {\widetilde{\Uppi}}_{1}.\end{array} \right. $$

(25)

Let us turn to the post-investment value function V _PS,1(Π), which can be written as

$$ V_{PS,1}(\Uppi)=\Uppsi \Uppi dt+e^{-rdt}E\left[V_{PS,1}(\Uppi +d\Uppi)\right]. $$

Expanding the right-hand side and using Itô’s lemma one obtains

$$ rV_{PS,1}(\Uppi)=\Uppsi \Uppi +(r-\delta (x))\Uppi \frac{\partial V_{PS,1}(\Uppi )}{\partial \Uppi}+\frac{\sigma^{2}}{2}\Uppi^{2}\frac{\partial ^{2}V_{PS,1}(\Uppi)}{\partial \Uppi^{2}}, $$

(26)

where x = x _l , x _h,1. Solving (26) yields

$$ V_{PS,1}(\Uppi)=\left\{ \begin{array}{*{20}l} \frac{\Uppsi \Uppi}{\delta (x_{l})}+V_{1}\Uppi^{\beta_{1}(x_{l})}&\quad \hbox{if}\quad\Uppsi \Uppi < \Uppsi {\widetilde{\Uppi}}_{1},\\ \frac{\Uppsi \Uppi}{\delta (x_{h,1})}+V_{2}\Uppi^{\beta_{2}(x_{h,1})}&\quad \hbox{if}\quad\Uppsi \Uppi > \Uppsi {\widetilde{\Uppi}}_{1}. \end{array} \right. $$

(27)

Let us focus on the ${\Uppi}{\in}\left(0,{\widetilde{\Uppi}}_{1}\right)$ region. Using the VMC and SPC, we obtain the trigger point²⁹

$$ \Uppi^{*}\equiv \frac{\beta_{1}(x_{l})}{\beta _{1}(x_{l})-1}\frac{\delta (x_{l})I}{\Uppsi -1}, $$

(28)

and

$$ V_{1}-C_{1}=-\frac{\Uppsi -1}{\delta (x_{l})}\frac{1}{\beta _{1}(x_{l})}\Uppi^{*^{1-\beta_{1}(x_{l})}}. $$

(29)

Let us next focus on the second-stage investment F. In this case the value function can be written as

$$ W_{PS,1}(\Uppi)=\left(\Uppsi +\Uptheta \right) \Uppi dt+e^{-rdt}E\left[W_{PS,1}(\Uppi +d\Uppi)\right] $$

(30)

Using the boundary condition W _PS,1(0) = 0, and following the usual procedure we obtain the solution to (30),

$$ W_{PS,1}(\Uppi)=\frac{\left(\Uppsi +\Theta \right) \Uppi}{\delta (x_{l})} +U_{1}\Uppi^{\beta_{1}(x_{l})} $$

(31)

where x = x _l , x _h,1. Substituting (27) and (31) into the VMC and SPC yields the trigger point

$$ \Uppi^{**}\equiv \frac{\beta_{1}(x_{l})}{\beta _{1}(x_{l})-1}\frac{ \delta (x_{l})}{\Theta}F. $$

(32)

Manipulating (32) one easily obtains (14).

Finally, remember that the switch levels ${\widetilde{\Uppi}}_{0}$ and ${\widetilde{\Uppi}}_{1}$ must be sufficiently high to ensure that a tighter regulation is applied only under good states. This implies that $\Uppi^{**}\in \left(0,\min ({\widetilde{\Uppi}}_{0},{\widetilde{\Uppi}}_{1})\right).$ If, otherwise, at least one of the two switch points were low, the profit-sharing regulation would be implemented in the bad-news region. This would cause a distortion.

1.7.2 State 0 (the old regime)

Let us start with the pre-investment value. The Bellman equation is

$$ O_{PS,0}(\Uppi)=\Uppi dt+(1-\lambda dt)e^{-rdt}E\left[O_{PS,0}(\Uppi +d\Uppi) \right] +\lambda dte^{-rdt}E\left[O_{PS,1}(\Uppi +d\Uppi)\right]. $$

Expanding its right-hand side and using Itô’s lemma yields

$$ (r+\lambda)O_{PS,0}(\Uppi)=\Uppi +(r-\delta (x_{l}))\Uppi \frac{\partial O_{PS,0}(\Uppi)}{\partial \Uppi}+\frac{\sigma^{2}}{2}\Uppi^{2}\frac{\partial ^{2}O_{PS,0}(\Uppi)}{\partial \Uppi^{2}}+\lambda O_{PS,1}(\Uppi). $$

(33)

Defining Y(Π) ≡ O _PS,0(Π)−O _PS,1(Π), and subtracting (24) from (33) yields

$$ (r+\lambda)Y(\Uppi)=(r-\delta (x_{l}))\Uppi \frac{\partial Y(\Uppi )}{\partial \Uppi}+\frac{\sigma^{2}}{2}\Uppi^{2}\frac{\partial^{2}Y(\Uppi )}{\partial \Uppi ^{2}}. $$

Given the condition Y(0) = 0, Y ₂ is nil. Using the solution of Y(Π) and Eq. (28) we obtain the option function under regulatory risk

$$ O_{PS,0}(\Uppi)=\frac{\Uppi}{\delta (x_{l})}+C_{1}\Uppi^{\beta_{1}(x_{l})}+Y_{1}\Uppi^{\beta_{1}(x_{l},\lambda )}, $$

(34)

where the parameter Y ₁ is an unknown.

After investment I, the firm’s value is

$$ V_{PS,0}(\Uppi)=\Uppsi \Uppi dt+(1-\lambda dt)e^{-rdt}E\left[V_{PS,0}(\Uppi +d\Uppi) \right] +\lambda dte^{-rdt}E\left[V_{PS,1}(\Uppi +d\Uppi)\right]. $$

Expanding the right-hand side and using Itô’s lemma one obtains

$$ (r+\lambda)V_{PS,0}(\Uppi)=\Uppsi \Uppi +(r-\delta (x_{l}))\Uppi \frac{\partial V_{PS,0}(\Uppi)}{\partial \Uppi}+\frac{\sigma^{2}}{2}\Uppi^{2}\frac{\partial ^{2}V_{PS,0}(\Uppi)}{\partial \Uppi^{2}}+\lambda V_{PS,1}(\Uppi). $$

(35)

Define next X(Π) ≡ V _PS,0(Π)−V _PS,1(Π) and subtract (26) from (35), so as to obtain

$$ (r+\lambda)X(\Uppi)=(r-\delta (x_{l}))\Uppi \frac{\partial X(\Uppi )}{\partial \Uppi}+\frac{\sigma^{2}}{2}\Uppi^{2}\frac{\partial^{2}X(\Uppi )}{\partial \Uppi ^{2}}. $$

The function X(Π) has a standard solution

$$ X(\Uppi)=\sum_{i=1}^{2}X_{i}\Uppi^{\beta_{i}(x_{l},\lambda)}, $$

where β _i (x _l , λ) are the roots of the characteristic equation $\frac{\sigma^{2}}{2}\beta (\beta -1)+(r-\delta (x_{l}))\beta -(r+\lambda)=0.$ Given the condition X(0) = 0, X ₂ is nil.

Using the solution of X(Π) and Eq. (22) we obtain

$$ V_{PS,0}(\Uppi)=\frac{\Uppsi \Uppi}{\delta (x_{l})}+V_{1}\Uppi^{\beta_{1}(x_{l})}+X_{1}\Uppi^{\beta_{1}(x_{l},\lambda )}, $$

(36)

where the parameter X ₁ is an unknown.

Finally, given the boundary condition W _PS,0(0) = 0, the post-investment function will have the following form

$$ W_{PS,0}(\Uppi)=\frac{\left(\Uppsi +\Uptheta \right) \Uppi}{\delta (x_{l})} +U_{1}\Uppi^{\beta_{1}(x_{l})}+Z_{1}\Uppi^{\beta_{1}(x_{l},\lambda)}, $$

(37)

where Z ₁ is an unknown parameter.

1.7.3 The trigger points

Let us now compute the trigger point above which investment I is profitable under policy risk. Substituting (34) and (36) into the (VMC) and (SPC) we obtain a two-equation system

$$ \frac{\left(\Uppsi -1\right) \Uppi}{\delta (x_{l})}+\left( V_{1}-C_{1}\right) \Uppi^{\beta_{1}(x_{l})}+\left(X_{1}-Y_{1}\right) \Uppi^{\beta_{1}(x_{l},\lambda)}-I=0, $$

(38)

$$ \frac{\Uppsi -1}{\delta (x_{l})}+\beta_{1}(x_{l})\left( V_{1}-C_{1}\right) \Uppi^{\beta_{1}(x_{l})-1}+\beta_{1}(x_{l},\lambda )\left(X_{1}-Y_{1}\right) \Uppi^{\beta_{1}(x_{l},\lambda)-1}=0. $$

(39)

Divide (39) by β₁(x _l ,λ) and substitute it into (38) so as to obtain

$$ \left[\frac{\beta_{1}(x_{l},\lambda)-1}{\beta_{1}(x_{l},\lambda )}\right] \frac{\left(\Uppsi -1\right) \Uppi}{\delta (x_{l})}+\left[\frac{\beta _{1}(x_{l},\lambda)-\beta_{1}(x_{l})}{\beta_{1}(x_{l},\lambda )}\right] \left(V_{1}-C_{1}\right) \Uppi^{\beta_{1}(x_{l})}-I=0. $$

(40)

Substitute (29) into (40) and multiply it by $\frac{\beta_{1}(x_{l})-1}{\beta_{1}(x_{l})}\frac{1}{I}.$ We thus obtain

$$ \left[\frac{\beta_{1}(x_{l},\lambda)-1}{\beta_{1}(x_{l},\lambda )}\right] \left(\frac{\Uppi}{\Uppi^{*}}\right) -\left[\frac{\beta _{1}(x_{l},\lambda)-\beta_{1}(x_{l})}{\beta_{1}(x_{l},\lambda )}\right] \left(\frac{\Uppi}{\Uppi^{*}}\right)^{\beta_{1}(x_{l})}-\frac{\beta _{1}(x_{l})-1}{\beta_{1}(x_{l})}=0. $$

(41)

Multiply (41) by $\frac{\beta_{1}(x_{l},\lambda)}{\beta _{1}(x_{l},\lambda)-1}$ so as to obtain

$$ \left(\frac{\Uppi}{\Uppi^{*}}\right) -\frac{\beta_{1}(x_{l},\lambda )-\beta_{1}(x_{l})}{\left[\beta_{1}(x_{l},\lambda)-1\right] \beta _{1}(x_{l})}\left(\frac{\Uppi}{\Uppi^{*}}\right)^{\beta_{1}(x_{l})}- \frac{\beta_{1}(x_{l},\lambda)\left[\beta_{1}(x_{l})-1\right] }{\left[\beta_{1}(x_{l},\lambda)-1\right] \beta_{1}(x_{l})}=0. $$

Adding and subtracting 1 from the LHS yields

$$ \left[\left(\frac{\Uppi}{\Uppi^{*}}\right) -1\right] -\frac{\beta _{1}(x_{l},\lambda)-\beta_{1}(x_{l})}{\left[\beta_{1}(x_{l},\lambda )-1 \right] \beta_{1}(x_{l})}\left[\left(\frac{\Uppi}{\Uppi^{*}}\right)^{\beta_{1}(x_{l})}-1\right] =0. $$

(42)

Define $y\equiv \left(\frac{\Uppi}{\Uppi^{*}}\right)$ and $\phi \equiv \frac{\beta_{1}(x_{l},\lambda)-\beta_{1}(x_{l})}{\left[\beta _{1}(x_{l},\lambda)-1\right] \beta_{1}(x_{l})} < 1.$ Thus, Eq. (42) can be rewritten as

$$ y-1=\phi \left(y^{\beta_{1}(x_{l})}-1\right). $$

(43)

Equation (43) has more than one solution. We thus compute these solutions and identify the optimal one. As can be noted, the equality y = 1 holds in Eq. (43). This entails that Π** = Π*. Substituting Π* into system (38) and (39) one thus obtains (X ₁−Y ₁) = 0. This is the first couple of solutions of system (38) and (39).

Define y′ as any other solution. Given inequalities β₁(x _l ) > 1, ϕ < 1 and β₁(x _l )ϕ < 1, it is easy to show that any other solution is y′ > 1. This implies that the trigger point obtained would be Π** > Π*. Substituting this new solution into system (38) and (39) yields (X ₁−Y ₁) > 0. Thus (Π** > Π*,(X ₁−Y ₁) > 0) is the second couple of solutions. However, this couple is sub-optimal. To show this, assume ab absurdo that the optimal solution is (Π** > Π*,(X ₁−Y ₁) > 0). Then, using the definitions of X(Π) and Y(Π), we define the pre-reform added value arising from investment (net of both the opportunity and the effective costs), as

$$ F(\Uppi)\equiv \left[V_{PS,0}(\Uppi)-O_{PS,0}(\Uppi)-I\right]. $$

(44)

Using (VMC) and (44) we obtain F(Π**) = 0. Rewrite (44) as

$$ F(\Uppi)=\left[V_{PS,1}(\Uppi)-O_{PS,1}(\Uppi)-I\right] +\left( X_{1}-Y_{1}\right) \Uppi^{\beta_{1}(x_{l},\lambda)}. $$

Since in Π = Π* the post-reform added value [V _PS,1(Π)−O _PS,1(Π)−I] is nil, we have $F(\Uppi^{*})=\left(X_{1}-Y_{1}\right) \Uppi^{*\beta_{1}(x_{l},\lambda )} > 0.$ This means that, in the interval Π ∈ (0,Π**), there exists at least one point (Π = Π*) such that the project’s payoff is strictly positive. Thus, a rational firm facing a positive payoff in Π = Π*, would immediately invest instead of waiting until the trigger point Π** is reached. This contradicts the assessment that (Π** > Π*, (X ₁−Y ₁) > 0) is the optimal investment strategy. The remaining solution (Π** = Π*, (X ₁−Y ₁) = 0) is thus the optimal one.

Following the same procedure, we can show that function (37) has the same form as functions (34) and (36). Therefore, substituting (36) and (37) into the VMC and SPC and manipulating the results yields the following equation

$$ \left[\left(\frac{\Uppi}{\Uppi^{**}}\right) -1\right] -\frac{\beta _{1}(x_{l},\lambda)-\beta_{1}(x_{l})}{\left[\beta_{1}(x_{l},\lambda )-1\right] \beta_{1}(x_{l})}\left[\left(\frac{\Uppi}{\Uppi^{**}} \right)^ {\beta_{1}(x_{l})}-1\right] =0, $$

(45)

which has the same form as Eq. (42). Following the same procedure, we can thus show that the post-reform trigger point coincides with Π*. This proves Proposition 3.□