Optimal price skimming by a monopolist facing rational consumers.

*(English)*Zbl 0712.90007A problem of equilibrium pricing strategy over time for a monopoly seller of a new durable good is formulated as a noncooperative game between the monopoly and the consumers of the product, each of whom buys at most one unit, generating a value, (or revenue) v. The value v varies across consumers and the distribution of the valuations of the population (N) is uniform over an interval \([0,v^+]\equiv V\). So \(Nv/v^+\) is the number of consumers whose valuation is less than v. The constant (marginal) cost of production per unit is c and \(\delta\in (0,1)\) is a discount factor. The equilibrium concept is a subgame perfect Nash equilibrium defined as follows. A sequence of functions \(p_ t^*:\) \(V\to R_+\) and \(v_ t^*:\) \(R_+\times V\to V\) for each \(t=1,...,T\) is an equilibrium iff for any \(t\in \{1,...,T\}\) and \(v\in V\), \(p_ t^*\) and \(v_ t^*\), respectively, solve the problems A and B defined below.

Define \(V_ t^*:\) \(V\to V\) by \(V_ t^*(v)=v_ t^*(p_ t^*(v),v)\). For \(t\in \{1,...,T\}\) and \(\bar v\in V\), \(p_ t^*(\bar v)\) solves:

(A) \(_{p_ t}\sum^{T}_{s=t}\delta^{s-t}(p_ s-c)\) \((v_{s-1}- v_ s)\) \(N/v^+\) subject to \(v_{t-1}=\bar v\), \(v_ t=v_ t^*(p_ t,\bar v)\), \(v_{s+1}=V^*_{s+1}(v_ s)\) and \(p_{s+1}=p^*_{s+1}(v_ s)\) for \(s=t,...,T.\)

For \(t\in \{1,...,T\}\), \(\bar v\in V\), \(\bar p\geq 0\), \(v_ t^*(\bar p,\bar v)\) solves:

(B) Min v subject to \(v\in [0,\bar v]\), \(0\leq v-\bar p\geq \delta^ s[v- p_{t+s}]\) for \(s=1,...,T-t\), where \(p_{t+1}=p^*_{t+1}(v_ t^*(\bar p,\bar v))\), \(p_{t+1+s}=p^*_{t+1+s}(v_{t+s})\), \(v_{t+s}=V^*_{t+s}(v_{t+s-1})\) for \(s=1,...,T\) and \(v_ t=v_ t^*(\bar p,\bar v).\)

The objective function in (A) can be seen to be the discounted sum of profits. \(\delta^ s[v-p_{t+s}]\) is the profit of the consumer if he buys in period \(t+s\). (A) and (B) says that each agents policy function for each period is the optimal one for the restricted optimization problem for the rest of the horizon on the expectation that these policy functions will be adhered to on both sides. In this sense the equilibrium is a rational expectations one and subgame perfect.

Backward induction leads to the conjecture that the solution of (B) must satisfy \(v_ t^*(\bar p,\bar v)-\bar p=\delta [v_ t^*(\bar p,\bar v)-p^*_{t+1}(v_ t^*(\bar p,\bar v))]\) for \(t<T\), and is verified directly. This is then used to solve (A). For the last period T an explicit solution can be derived since there is no complicating factor of future periods. Backward induction then establishes that the equilibrium solutions can be described by certain explicit formulae (too numerous to reproduce here) which are defined recursively. Asymptotic forms for these formulae which approximate the solution for large T are also derived. The equilibrium solutions are then analyzed to obtain several properties. First, the equilibrium price sequence \(p_ t^*\) is strictly decreasing exhibiting intertemporal price discrimination. Second, that for \(t<T\), \(p_ t^*(v_{t-1})<(1/2)(v_{t-1}+c)\). The expression on the right is the profit maximizing price if the monopoly disregards future sales possibilities and maximizes profits for this single period given the consumers leftover in the market determined by \(v_{t-1}\). Third, a similar problem is analyzed but where the consumers are myopic in the sense that at any period consumers with valuation \(v\geq p_ t\), the announced price, buy the product. It is shown that here the equilibrium pricing policy \(\hat p_ t\) \((v_{t-1})>(1/2)(v^ t+c)\) for \(t<T\) and so, for each state \(v_{t-1}\), the myopic pricing is higher than the single period profit maximizing solution. Finally, some numerical examples are used to show that errors in pricing because of presuming myopic behaviour of consumers when actually they follow rational expectations can be substantial.

This reviewer found the formulation of the problem and its handling to be clever and the results interesting and believes it should be worth a quick scan for somebody interested in the literature on intertemporal pricing of durable goods and applied game theory. The notation and formal statements appear sometimes confused and misleading.

Define \(V_ t^*:\) \(V\to V\) by \(V_ t^*(v)=v_ t^*(p_ t^*(v),v)\). For \(t\in \{1,...,T\}\) and \(\bar v\in V\), \(p_ t^*(\bar v)\) solves:

(A) \(_{p_ t}\sum^{T}_{s=t}\delta^{s-t}(p_ s-c)\) \((v_{s-1}- v_ s)\) \(N/v^+\) subject to \(v_{t-1}=\bar v\), \(v_ t=v_ t^*(p_ t,\bar v)\), \(v_{s+1}=V^*_{s+1}(v_ s)\) and \(p_{s+1}=p^*_{s+1}(v_ s)\) for \(s=t,...,T.\)

For \(t\in \{1,...,T\}\), \(\bar v\in V\), \(\bar p\geq 0\), \(v_ t^*(\bar p,\bar v)\) solves:

(B) Min v subject to \(v\in [0,\bar v]\), \(0\leq v-\bar p\geq \delta^ s[v- p_{t+s}]\) for \(s=1,...,T-t\), where \(p_{t+1}=p^*_{t+1}(v_ t^*(\bar p,\bar v))\), \(p_{t+1+s}=p^*_{t+1+s}(v_{t+s})\), \(v_{t+s}=V^*_{t+s}(v_{t+s-1})\) for \(s=1,...,T\) and \(v_ t=v_ t^*(\bar p,\bar v).\)

The objective function in (A) can be seen to be the discounted sum of profits. \(\delta^ s[v-p_{t+s}]\) is the profit of the consumer if he buys in period \(t+s\). (A) and (B) says that each agents policy function for each period is the optimal one for the restricted optimization problem for the rest of the horizon on the expectation that these policy functions will be adhered to on both sides. In this sense the equilibrium is a rational expectations one and subgame perfect.

Backward induction leads to the conjecture that the solution of (B) must satisfy \(v_ t^*(\bar p,\bar v)-\bar p=\delta [v_ t^*(\bar p,\bar v)-p^*_{t+1}(v_ t^*(\bar p,\bar v))]\) for \(t<T\), and is verified directly. This is then used to solve (A). For the last period T an explicit solution can be derived since there is no complicating factor of future periods. Backward induction then establishes that the equilibrium solutions can be described by certain explicit formulae (too numerous to reproduce here) which are defined recursively. Asymptotic forms for these formulae which approximate the solution for large T are also derived. The equilibrium solutions are then analyzed to obtain several properties. First, the equilibrium price sequence \(p_ t^*\) is strictly decreasing exhibiting intertemporal price discrimination. Second, that for \(t<T\), \(p_ t^*(v_{t-1})<(1/2)(v_{t-1}+c)\). The expression on the right is the profit maximizing price if the monopoly disregards future sales possibilities and maximizes profits for this single period given the consumers leftover in the market determined by \(v_{t-1}\). Third, a similar problem is analyzed but where the consumers are myopic in the sense that at any period consumers with valuation \(v\geq p_ t\), the announced price, buy the product. It is shown that here the equilibrium pricing policy \(\hat p_ t\) \((v_{t-1})>(1/2)(v^ t+c)\) for \(t<T\) and so, for each state \(v_{t-1}\), the myopic pricing is higher than the single period profit maximizing solution. Finally, some numerical examples are used to show that errors in pricing because of presuming myopic behaviour of consumers when actually they follow rational expectations can be substantial.

This reviewer found the formulation of the problem and its handling to be clever and the results interesting and believes it should be worth a quick scan for somebody interested in the literature on intertemporal pricing of durable goods and applied game theory. The notation and formal statements appear sometimes confused and misleading.

Reviewer: S.Dasgupta

##### MSC:

91B24 | Microeconomic theory (price theory and economic markets) |

91A40 | Other game-theoretic models |

91B62 | Economic growth models |

91A10 | Noncooperative games |

91B26 | Auctions, bargaining, bidding and selling, and other market models |

91B50 | General equilibrium theory |

90C39 | Dynamic programming |