Two-part tariffs set by a risk-averse monopolist

Abstract

This paper revisits the classical issues of two-part tariffs by considering risk aversion of a monopolistic seller. Under demand uncertainty, equilibrium unit price declines and approaches towards marginal cost as the seller becomes more risk averse. Marginal-cost pricing prevails, irrespective of the seller’s risk attitude, if clients are homogenous. Under cost uncertainty, unit price is higher than marginal cost and monotonically increases in risk aversion. The model is then extended to accommodate buyers’ risk aversion and it is found that demand uncertainty makes unit price decline in the seller’s risk aversion again but increase in buyers’ risk aversion.

Appendix

Proof of Proposition 1

Because $u_{i,q\varepsilon } >0$, we have $q_{i,\varepsilon } >0$ and in turn $\pi _\varepsilon (p,\varepsilon )>0$ for $p>c$. Let $P$ be the optimal unit price set by a less-risk-averse seller and define $\varepsilon _0$ such that $\pi _p (P,\varepsilon _0 )=0$. We evaluate the derivative of $E[\varphi (V(\pi ))]$ with respect to $p$ at $p=P$. Noting $P>c, \pi _{p\varepsilon }>0$ and $\varphi ^{\prime }(V(\pi ))$ decreases in $\pi$, we have

$$\begin{aligned}&E[ {\varphi ^{\prime }(V(\pi (P,\varepsilon )))V^{\prime }(\pi (P,\varepsilon ))\pi _p (P,\varepsilon )}] \\&\quad =E_{\varepsilon >\varepsilon _0 } [ {\varphi ^{\prime }(V(\pi (P,\varepsilon )))V^{\prime }(\pi (P,\varepsilon ))\pi _p (P,\varepsilon )}]\\&\qquad +E_{\varepsilon <\varepsilon _0 } [ {\varphi ^{\prime }(V(\pi (P,\varepsilon )))V^{\prime }(\pi (P,\varepsilon ))\pi _p (P,\varepsilon )} ] \\&\quad <E_{\varepsilon >\varepsilon _0 }[ {\varphi ^{\prime }(V(\pi (P,\varepsilon _0 )))V^{\prime }(\pi (P,\varepsilon ))\pi _p (P,\varepsilon )} ]\\&\qquad +E_{\varepsilon <\varepsilon _0 }[ {\varphi ^{\prime }(V(\pi (P,\varepsilon _0 )))V^{\prime }(\pi (P,\varepsilon ))\pi _p (P,\varepsilon )}] \\&\quad =\varphi ^{\prime }(V(\pi (P,\varepsilon _0 )))E[ {V^{\prime }(\pi (P,\varepsilon ))\pi _p (P,\varepsilon )} ]=0 \end{aligned}$$

Since the derivative is negative at $P$, it means that $p=P$ is too high for a more-risk-averse seller so that it sets the optimal unit price lower than $P$. That the entry fee decreases in risk aversion can be obtained by noting that ${de}/{dp}=-q_1 <0$.

Proof of Proposition 2

Applying the identity $E[xy]=\text{ cov}(x,y)+E[x]E[y]$ to (3) yields

$$\begin{aligned}&\text{ cov}( {V^{\prime }(\pi ),Q})+E[ {V^{\prime }(\pi )}]\bar{Q}+(p-c)\text{ cov}( {V^{\prime }(\pi ),Q_p })\\&\quad +(p-c)E[ {V^{\prime }(\pi )}]\bar{Q}^{\prime } =E[ {V^{\prime }(\pi )}]\bar{q}_1. \end{aligned}$$

By assumption, both $\pi$ and $Q$ are normally distributed. Applying Stein’s lemma to the equation leads to²¹

$$\begin{aligned}&E[ {V^{\prime \prime }}]\text{ cov}( {\pi ,Q})+E[ {V^{\prime }}]\bar{Q}+(p-c)E[ {V^{\prime \prime }} ]\text{ cov}( {\pi ,Q_p })\\&\quad +(p-c)E [ {V^{\prime }}]\bar{Q}^{\prime }=E[ {V^{\prime }}]\bar{q}_1. \end{aligned}$$

Therefore,

$$\begin{aligned} (p-c)R[ {\text{ var}( Q)+\text{ cov}(\pi ,Q_p )} ]=\bar{Q}-\bar{q}_1 +(p-c)\bar{Q}^{\prime }. \end{aligned}$$

Since the right-hand side of the equation is finite when $R$ tends infinity, it requires $p=c$ for the left-had side to be finite.

Proof of Proposition 3

We prove the proposition by three steps. (i) Show marginal-cost pricing and the optimal lump-sum fee equal to the expected surplus of clients.

Explicitly considering $\pi$ as a function of $q$ and applying the mean value theorem to $V^{\prime }(\cdot )$ we obtain $V^{\prime }(\pi (q))=V^{\prime }(\pi (\bar{q}))+V^{\prime \prime }(\pi (q^\# ))(p-c)(q-\bar{q})$, where $q^\# \in (q,\;\bar{q})$. Applying it to (3) yields

$$\begin{aligned} 0&= E[{V^{\prime }(\pi )(q-\bar{q})}]+(p-c)E[ {V^{\prime }(\pi )q_p } ] \\&= (p-c)\{ {E[ {V^{\prime \prime }(\pi (q^\# ))(q-\bar{q})^2} ]+E[ {V^{\prime }(\pi (q))q_p }]}\}. \end{aligned}$$

Because $V^{\prime }(\cdot )>0, V^{\prime \prime }(\cdot )<0,$ and $q_p <0,$ we have $E[{V^{\prime \prime }(\pi (q^\# ))(q-q)^2}]<0$ and $E[ {V^{\prime }(\pi (q)q_p }]<0$, which implies $p=c$. The result of lump-sum fee equal to the expected surplus of clients is obvious by (4).

(ii) Show the optimal two-part tariff maximizes the seller’s expected utility even when the seller can choose more general schedules of nonlinear pricing.

Define certainty equivalent profit $\pi ^E$ of a random profit $\pi$ by $V(\pi ^E)\equiv E[V(\pi )],$ which means that a seller views receiving a certain amount of profit $\pi ^E$ as equivalent to receiving the random profit $\pi$. Two observations should be brought to notice. First, a profit maximizing the seller’s expected utility must also maximizes its certainty equivalent profit because $V(\cdot )$ is monotonically increasing. Second, certainty equivalent profit is not greater than expected profit (i.e., $\pi ^E\le \bar{\pi })$ because of concavity of $V(\cdot )$ and they are equal if and only if profit is deterministic. Define a general nonlinear pricing scheme as transferring a total amount of $t(q)$ dollars from a buyer to the seller, which is differentiable on $q\in [0,\;+\infty )$ except at a finite number of points. Facing $t(q)$, the ex post demand, $q(\varepsilon ),$ is thus implicitly determined by condition $u_{q}(q, \varepsilon )=t^{\prime }(q)$. The optimal nonlinear pricing is a problem of choosing quantity allocation $q(\varepsilon )$ and monetary transfer $\tau (\varepsilon )\equiv t(q(\varepsilon ))$ to maximize the seller’s expected utility, subject to buyers’ participation constraint. Similar to (4), this participation constraint can be written as $E[u(q(\varepsilon ),\varepsilon )-\tau (\varepsilon )]=0,$ which implies that $\bar{\pi }=E[u(q(\varepsilon ),\varepsilon )-cq(\varepsilon )]$. Hence, the maximal expected profit $\bar{\pi }^*$ is obtained when $u_{q}(q(\varepsilon ), \varepsilon )=c$. Because the unit price of the optimal two-part tariff is equal to marginal cost, it also results in $\bar{\pi }^{*}$. Moreover, it generates a certainty equivalent profit equal to $\bar{\pi }^{*}$ because $\bar{\pi }^{*}$ is deterministic. Thus, to the seller a general nonlinear pricing scheme cannot be better than the optimal two-part tariff.

(iii) Show the uniqueness of the optimal pricing.

To see the uniqueness, let us consider a pricing scheme whose marginal price is not equal to marginal cost in a small area around point $q^{\# }$, i.e., $t^{\prime }(q^{\# })\ne c$. Then, the corresponding profit is random and $\pi ^E<\bar{\pi }^{*}$, which implies that the certainty equivalent profit is not maximized and so does the expected utility of the seller. In other words, for scheme $t(q)$ leading to a deterministic profit, it must satisfy $t^{\prime }(q)=c$ for all $q,$ which unavoidably results in a two-part tariff.

Proof of Proposition 4

Let $P$ be the optimal unit price set by a less-risk-averse seller. Define $\pi (P,c)\equiv (P-c)Q(P)+s_1 (P)$. It decreases in $c$ so that $\pi (P,c)>\pi (P,P)$ if $c<P$ but $\pi (P,c)<\pi (P,P)$ if $c>P$. Recalling that $V(\pi )$ increases in $\pi$ and $\varphi ^{\prime }(V)$ decreases in $V$, we have $\varphi ^{\prime }(V(\pi ))$ decreases in $\pi$. Evaluating the derivative of $E[\varphi (V(\pi ))]$ with respect to $p$ at $p=P$, we obtain that

$$\begin{aligned}&E[{\varphi ^{\prime }(V(\pi (P,c)))V^{\prime }(\pi (P,c))(P-c)q^{\prime }}] \\&\quad =E_{c>P} [ {\varphi ^{\prime }(V(\pi (P,c)))V^{\prime }(\pi (P,c))(P-c)q^{\prime }}]\\&\qquad +E_{c<P} [{\varphi ^{\prime }(V(\pi (P,c)))V^{\prime }(\pi (P,c))(P-c)q^{\prime }}] \\&\quad >E_{c>P} [ {\varphi ^{\prime }(V(\pi (P,P)))V^{\prime }(\pi (P,c))(P-c)q^{\prime }}]\\&\qquad +E_{c<P} [ {\varphi ^{\prime }(V(\pi (P,P)))V^{\prime }(\pi (P,c))(P-c)q^{\prime }}] \\&\quad =\varphi ^{\prime }(V(\pi (P,P)))E[ {V^{\prime }(\pi (P,c))(P-c)q^{\prime }}]=0. \end{aligned}$$

Since the derivative is positive and the expected utility is concave in $p$, it is immediate that a more-risk-averse seller sets a unit price higher than $P$. That the entry fee increases in risk aversion can be obtained by noting that ${de} / {dp}=-q_1 <0.$

Proof of Proposition 7

Part (i) The proof is the same as Proposition 1 and is omitted.

Part (ii) Let the buyers’ constant coefficient of absolute risk aversion be $\gamma$. Their participation constraint can be written as

$$\begin{aligned} E[\exp (-\gamma \{u(q(p,\varepsilon ),\varepsilon )-pq(p,\varepsilon )-e)\})]=1. \end{aligned}$$

Clearly, it determines a lump-sum fee $e(p, \gamma )$ given $p$ and $\gamma$. It can be shown further that $e_{p\gamma }(p, \gamma )>0$ [(see the proof of Lemma 2 in Png and Wang (2010)]. Define $\bar{V}(p,\gamma )\equiv E[V(\pi (p, e(p,\gamma ), \varepsilon )]$ where $\pi (p, e(p, \gamma ), \varepsilon ) = e(p, \gamma )+(p-c)q(p, \varepsilon )$. Let $p(\gamma )$ denote the optimal unit price when buyers are of type $\gamma$. Then $\bar{V}(p(\gamma ),\gamma )\equiv E[V(\pi (p(\gamma ),e(p(\gamma ),\gamma ),\varepsilon )]$ is the maximal expected utility of the seller and

$$\begin{aligned} \bar{V}_{p\gamma } (p(\gamma ),\gamma )&= E[V^{\prime \prime }(\cdot )\{(q+(p-c)q_p +e_p \}e_\gamma ]+E[V^{\prime }(\cdot )e_{p\gamma } ] \\&= -e_\gamma rE[V^{\prime }(\cdot )\{(q+(p-c)q_p +e_p \}]+e_{p\gamma } E[V^{\prime }(\cdot )]>0 , \end{aligned}$$

where $r$ is the seller’s constant coefficient of absolute risk aversion. In reaching the inequality, the first order conditions (8) and (9) are applied.

Suppose $p(\gamma )$ decreases on $[\gamma _{1}, \gamma _{3})$. Because $p(\cdot )$ and $\bar{V}_{p\gamma } (\cdot ,\cdot )$ are continuous functions, $\bar{V}_{p\gamma } (p,\gamma )>0$ for all ($p, \gamma$) in a small area around ($p(\gamma _{1}), \gamma _{1})$. Obviously, we can find a $\gamma _2 \in [\gamma _1 ,\;\gamma _3 )$ which is sufficiently close to $\gamma _{1}$ so that ($p(\gamma _{2}), \gamma _{2})$ falls in this small area. Now, consider the difference function $\bar{V}(p,\gamma _2 )-\bar{V}(p,\gamma _1 ),$ which increases in $p$ because of $\bar{V}_{p\gamma } (p,\gamma )>0$. As $p(\gamma )$ decreases there are $p(\gamma _{2})<p (\gamma _{1})$ and

$$\begin{aligned} \bar{V}(p(\gamma _2 ),\gamma _2 )-\bar{V}(p(\gamma _2 ),\gamma _1 )<\bar{V}(p(\gamma _1 ),\gamma _2 )-\bar{V}(p(\gamma _1 ),\gamma _1 ). \end{aligned}$$

Rearrange it to obtain

$$\begin{aligned} \bar{V}(p(\gamma _2 ),\gamma _2 )-\bar{V}(p(\gamma _1 ),\gamma _2 )<\bar{V}(p(\gamma _2 ),\gamma _1 )-\bar{V}(p(\gamma _1 ),\gamma _1 )<0, \end{aligned}$$

which means $p(\gamma _{2})$ is not the optimal unit price and we obtain a contraction. This contradiction implies that $p(\gamma )$ must increase in $p$ rather than decrease.

Proof of $\pi _{p\varepsilon } > 0$ for additive and multiplicative demand shock

Because client heterogeneity does not change the result, we focus here on the case of homogeneous but risk-averse clients. For demand $q(p, \varepsilon )=q(p)+\varepsilon ,$ there is $\pi =s(p)+(p-c)(q(p)+\varepsilon )$. Then, $\pi _{p\varepsilon }=1>0$. For demand $q(p, \varepsilon )=\varepsilon q(p),$ there are $\pi =s(p)+(p-c)q^{\prime }(p)$ and $\pi _{p\varepsilon }=q(p)+(p-c)q^{\prime }(p)$. Let $p^{n}$ be the optimal unit price set by a risk-neutral seller. We now argue that $\pi _{p\varepsilon }>0$ on $p \in [c, p^{n}]$. For a risk-neutral seller, the first-order condition implies that $\bar{\varepsilon }[q(p^n)+(p^n-c)q^{\prime }(p^n)]={E[U^{\prime }(\cdot )\varepsilon q(p)]} /{E[U^{\prime }(\cdot )]}$. Thus, $\pi _{p\varepsilon }(p^{n}, \varepsilon )={E[U^{\prime }(\cdot )\varepsilon q(p)]}/{\{\bar{\varepsilon }E[U^{\prime }(\cdot )]}\}>0$. Noting $\bar{\varepsilon }[q(p^n)+(p^n-c)q^{\prime }(p^n)]$ is the first derivative of the expected profit when the seller sets linear price, it is a decreasing function of $p$ since the expected profit is assumed to be concave. Thus, $\pi _{p\varepsilon }(p, \varepsilon )>\pi _{p\varepsilon }(p^{n}, \varepsilon )>0$ for $p\in [c,\;p^n]$. Finally, because optimal price declines as the seller becomes more risk averse, we have no need to consider the situation where $p>p^{n}$.