Regulating a firm with bilateral unknown demands

Abstract

We consider the impact of bilateral unknown information about the market capacity on the optimal regulatory policies in a regulation problem. We first analyze how to solve such problem when the market capacity is full information, a case which allows us to obtain most of key insights from regulatory models. We then extend the analysis to the bilateral unknown information case. We do this by assuming that neither the regulator nor the firm knows exactly the true market capacity, but they can make respective estimates about it, the same or different. The results show that when the regulator and the firm make the same estimate about the true market capacity, the optimal price is distorted downwards from that under full information, and the transfer payment is distorted upwards from that under full information, but bilateral unknown information does not necessarily result in the distortion of the firm’s output. When the regulator and the firm make the different estimates about the true market capacity, and if the firm’s estimate is more optimistic than the regulator’s, the optimal price and output are more than those under the case that they have the same estimate. Contrary to this case, if the firm’s estimate is more pessimistic than the regulator’s, the optimal price and output are less than those under the case that they have the same estimate.

Appendix

Proof Proof of Theorem 1.

The constraint of Model (4) is binding at the optimum; otherwise, the regulator can decrease t until it binds, i.e., π(p,t,y) = 0. Therefore, the regulator allocates all surplus to the consumer. By π(p,t,y) = 0, we can obtain

$$ t=C(y-bp)-p(y-bp). $$

(23)

Substituting t into the objective function of Model (4) yields

$$ W(p,t,y)= V(y-bp)-C(y-bp). $$

(24)

By $V^{\prime \prime }(\cdot )\leq 0$ and $C^{\prime \prime }(\cdot )>0$, we obtain W(p,t,y) is concave with respect to p. Therefore, the optimal price p^∗ satisfies the first-order condition

$$ V^{\prime}(y-bp^{*})= C^{\prime}(y-bp^{*}). $$

(25)

The optimal output q^∗ follows q^∗ = y − bp^∗ from Eq. 25. It also follows from Eq. 23 that t^∗ = C(y − bp) − p(y − bp). The proof is complete. □

Proof Proof of Theorem 2.

We firstly verify that the objective function (13) is jointly concave with respect to (p,q). The first-order and second-order partial derivatives of E[W(p,q,Y )] with respect to (p,q) can be shown as

$$ \begin{array}{@{}rcl@{}} \frac{\partial{E[W(p,q,Y)]}}{\partial{p}}&=& -bV^{\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F(y)\mathrm{d}y\right)F(q+bp),\\ \frac{\partial{E[W(p,q,Y)]}}{\partial{q}}&=& V^{\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F(y)\text{d}y\right)\left( 1-F(q+bp)\right)\\ &&-C^{\prime}(q),\\ \frac{\partial^{2}{E[W(p,q,Y)]}}{\partial{p^{2}}}&=&b^{2} V^{\prime\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F(y)\mathrm{ d}y\right)F^{2}(q+bp)\\&&-b^{2}V^{\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F(y)\text{d}y\right)F^{\prime}(q+bp),\\ \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} \frac{\partial^{2}{E[W(p,q,Y)]}}{\partial{q^{2}}}&= &V^{\prime\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F(y)\mathrm{ d}y\right)\left( 1 - F(q+bp)\right)^{2}\\ &&-V^{\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F(y)\text{d}y\right)F^{\prime}(q+bp)\\ && -C^{\prime\prime}(q), \end{array} $$

(27)

$$ \begin{array}{@{}rcl@{}} \frac{\partial^{2}{E[W(p,q,Y)]}}{\partial{p}\partial{q}}&= &-bV^{\prime\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F(y)\mathrm{ d}y\right)\\ &&~~~\left( 1-F(q+bp)\right)F(q+bp)\\&&-bV^{\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F(y)\text{d}y\right)F^{\prime}(q+bp).\\ \end{array} $$

(28)

By Eqs. 26, 27 and 28, the Hessian matrix

$$H=\left( \begin{array}{cc} \frac{\partial^{2}{E[W(p,q,Y)]}}{\partial{p^{2}}} & \frac{\partial^{2}{E[W(p,q,Y)]}}{\partial{p}\partial{q}}\\ \frac{\partial^{2}{E[W(p,q,Y)]}}{\partial{q}\partial{p}} & \frac{\partial^{2}{E[W(p,q,Y)]}}{\partial{q^{2}}} \end{array} \right).$$

It follows from $V^{\prime }(\cdot )>0$ and $V^{\prime \prime }(\cdot )\leq 0$ that $\frac {\partial ^{2}{E[W(p,q,Y)]}}{\partial {p^{2}}}<0$, $\frac {\partial ^{2}{E[W(p,q,Y)]}}{\partial {q^{2}}}<0$, $\left |H\right |=b^{2}V^{\prime }F^{\prime }C^{\prime \prime }>0$, i.e., H is negative definite, thus E[W(p,q,Y )] is jointly concave with respect to (p,q).

Therefore, the optimal price $p_{1}^{*}$ and the optimal output $q_{1}^{*}$ satisfy

$$ \left\{\begin{array}{ll} \displaystyle \frac{\partial{E[W(p_{1}^{*},q_{1}^{*},Y)]}} {\partial{p_{1}}}= V^{\prime}\left( q_{1}^{*}-{\int}_{\underline{y}}^{q_{1}^{*}+bp_{1}^{*}}F(y)\text{d}y\right)F(q_{1}^{*}+bp_{1}^{*})=0,\\[0.5cm] \displaystyle \frac{\partial{E[W(p_{1}^{*},q_{1}^{*},Y)]}}{\partial{q_{1}}}= V^{\prime}\left( q_{1}^{*}-{\int}_{\underline{y}}^{q_{1}^{*}+bp_{1}^{*}}F(y)\text{d}y\right)\left( 1-F(q_{1}^{*}+bp_{1}^{*})\right)=C^{\prime}(q_{1}^{*}). \end{array} \right. $$

(29)

It follows from Eq. 29 that $p_{1}^{*}=\frac {1}{b}(\underline {y}-q_{1}^{*})$ and $V^{\prime }(q_{1}^{*})=C^{\prime }(q_{1}^{*})$. Thus, the proof of Theorem 4 is complete. □

Proof Proof of Proposition 1.

It follows from Eq. 5 that $V^{\prime }(q^{*})=C^{\prime }(q^{*})$, together with Eq. 15 and $C^{\prime \prime }>0$, we can obtain $q_{1}^{*}=q^{*}$. Note that $p_{1}^{*}=\underline {y}-bq_{1}^{*}$ and p^∗ = y − bq^∗, due to ${y}\geq {\underline {y}}$, it is clear that $p_{1}^{*}\leq p^{*}$. Comparing Eq. 7 with Eq. 16, applying $p_{1}^{*}\leq p^{*}$ and $q_{1}^{*}=q^{*}$, we find immediately $t_{1}^{*}\geq t^{*}$. The proof is complete. □

Proof Proof of Corollary 1.

It follows from Eq. 24 that

$$ E[W^{*}(p^{*},q^{*},t^{*})]= V(y-bp^{*})-C(y-bp^{*}), $$

i.e.,

$$ E[W^{*}(p^{*},q^{*},t^{*})]= V(q^{*})-C(q^{*}), $$

(30)

and by Eq. 13, we know

$$ E[W_{1}^{*}(p_{1}^{*},q_{1}^{*},t_{1}^{*})]= V\left( q_{1}^{*}-{\int}_{\underline{y}}^{q_{1}^{*}+bp_{1}^{*}}F(y)\text{d}y\right)-C(q_{1}^{*}). $$

(31)

By Eq. 17$q^{*}=q_{1}^{*}$, together with $V^{\prime }>0$, comparing Eq. 30 with Eq. 31, it is clear that $E[W^{*}(p^{*},q^{*},t^{*})]\geq E[W_{1}^{*}(p_{1}^{*},q_{1}^{*},t_{1}^{*})]$. The proof of Corollary 1 is complete. □

Proof Proof of Proposition 2.

For the regulator’s objective function (22), if the optimal price $p_{2}^{*}$ exists, it should satisfy the first-order condition

$$ \begin{array}{@{}rcl@{}} &&V^{\prime}\left( q_{2}^{*}-{\int}_{\underline{y}}^{q_{2}^{*}+bp_{2}^{*}}F_{1}(y)\text{d}y\right)F_{1}(q_{2}^{*}+bp_{2}^{*})\\ &&\quad= (1-\alpha)\left( {\int}_{\underline{y}}^{q_{2}^{*}+bp_{2}^{*}}[F_{1}(y)-F_{2}(y)]\text{d}y\right)\\&&\quad+(1-\alpha)p_{2}^{*}\left[F_{1}(q_{2}^{*}+bp_{2}^{*})-F_{2}(q_{2}^{*}+bp_{2}^{*})\right]. \end{array} $$

(32)

It follows from Eq. 29 that $p_{1}^{*}$ satisfies

$$ V^{\prime}\left( q_{1}^{*}-{\int}_{\underline{y}}^{q_{1}^{*}+bp_{1}^{*}}F_{1}(y)\text{d}y\right)F_{1}(q_{1}^{*}+bp_{1}^{*})=0. $$

(33)

To compare $p_{2}^{*}$ and $p_{1}^{*}$, let

$$K(p)=V^{\prime}\left( q-{\int}_{\underline{y}}^{q+bp}F_{1}(y)\text{d}y\right)F_{1}(q+bp).$$

By $V^{\prime }>0$, $V^{\prime \prime }\leq 0$ and $F_{1}^{\prime }>0$, differentiating K(p) with respect to p yields

$$-V^{\prime\prime}{F_{1}^{2}}+V^{\prime}F_{1}^{\prime}>0.$$

Therefore, K(p) is increasing with respect to p. Comparing Eq. 32 with Eq. 33, we can obtain $p_{2}^{*}>p_{1}^{*}$ when F₁(y) > F₂(y). Correspondingly, $p_{2}^{*}<p_{1}^{*}$ holds when F₁(y) < F₂(y).

Similarly, if the optimal output $q_{2}^{*}$ exists, it should satisfy the first-order condition

$$ \begin{array}{@{}rcl@{}} &&V^{\prime}\left( q_{2}^{*} - {\int}_{\underline{y}}^{q_{2}^{*}+bp_{2}^{*}}F_{1}(y)\mathrm{ d}y\right)\left[1 - F_{1}(q_{2}^{*} + bp_{2}^{*})\right] - C^{\prime}(q_{2}^{*})\\&&=(1-\alpha)p_{2}^{*}\left[F_{2}(q_{2}^{*}+bp_{2}^{*}) -F_{1}(q_{2}^{*}+bp_{2}^{*})\right]. \end{array} $$

(34)

It follows from Eq. 29 that $q_{1}^{*}$ satisfies

$$ V^{\prime}\left( q_{1}^{*} - {\int}_{\underline{y}}^{q_{1}^{*}+bp_{1}^{*}}F_{1}(y)\text{d}y\right)\left[1 - F_{1}(q_{1}^{*} + bp_{1}^{*})\right]-C^{\prime}(q_{1}^{*}) = 0. $$

(35)

To compare $q_{2}^{*}$ with $q_{1}^{*}$, let

$$H(q) = V^{\prime}\left( q - {\int}_{\underline{y}}^{q+bp}F_{1}(y)\text{d}y\right)\left[1 - F_{1}(q + bp)\right]-C^{\prime}(q).$$

By $V^{\prime }>0$, $V^{\prime \prime }\leq 0$ and $C^{\prime \prime }>0$, and differentiating with respect to q yields

$$V^{\prime\prime}(1-F_{1})^{2}-V^{\prime}F_{1}^{\prime}-C^{\prime\prime}<0.$$

Hence, H(q) is decreasing with respect to q. Comparing Eq. 34 with Eq. 35, we can obtain $q_{2}^{*}>q_{1}^{*}$ when F₁(y) > F₂(y). Correspondingly, when F₁(y) < F₂(y), $q_{2}^{*}<q_{1}^{*}$ holds. The proof of Proposition 2 is complete. □

Lemma 1

Both the optimal price $p_{2}^{*}$ and the optimal output $q_{2}^{*}$ are increasing with respect to $\left [F_{1}(\cdot )-F_{2}(\cdot )\right ]$. In particular, $p_{2}^{*}$ and $q_{2}^{*}$ are down to minimum or up to maximum at F₁(⋅) = F₂(⋅), i.e., $p_{2}^{*}=p_{1}^{*}$, $q_{2}^{*}=q_{1}^{*}$.

Proof Proof of Proposition 3.

Similar to the proof of Proposition 2, it is easy to verify $p_{2}^{*}= p_{1}^{*}$ and $q_{2}^{*}= q_{1}^{*}$. By Eqs. 12 and 21, together with $p_{2}^{*}= p_{1}^{*}$ and $q_{2}^{*}= q_{1}^{*}$, we can obtain $t_{1}^{*}>t_{2}^{*}$ when F₁(y) > F₂(y), and $t_{1}^{*}<t_{2}^{*}$ when F₁(y) < F₂(y), respectively. The proof of Proposition 3 is complete. □