Twopart tariffs set by a riskaverse monopolist

Authors: Xiangkang Yin

Publish Date: 2012/05/26

Volume: 109, Issue: 2, Pages: 175-192

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Abstract

This paper revisits the classical issues of twopart 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 Marginalcost 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 aversionX Yin thanks Buly Cardak Baibing Li YewKwang Ng David Prentice Robert Waschik and seminar participants at the University of Strasbourg for fruitful discussions and constructive comments A particular thank to Benjamin Hermalin for his constructive and detailed comments and suggestionsDefine certainty equivalent profit pi E of a random profit pi by Vpi Eequiv EVpi 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 Vcdot is monotonically increasing Second certainty equivalent profit is not greater than expected profit ie pi Ele barpi because of concavity of Vcdot and they are equal if and only if profit is deterministic Define a general nonlinear pricing scheme as transferring a total amount of tq dollars from a buyer to the seller which is differentiable on qin 0+infty except at a finite number of points Facing tq the ex post demand qvarepsilon is thus implicitly determined by condition u qq varepsilon =tprime q The optimal nonlinear pricing is a problem of choosing quantity allocation qvarepsilon and monetary transfer tau varepsilon equiv tqvarepsilon to maximize the seller’s expected utility subject to buyers’ participation constraint Similar to 4 this participation constraint can be written as Euqvarepsilon varepsilon tau varepsilon =0 which implies that barpi =Euqvarepsilon varepsilon cqvarepsilon Hence the maximal expected profit barpi is obtained when u qqvarepsilon varepsilon =c Because the unit price of the optimal twopart tariff is equal to marginal cost it also results in barpi Moreover it generates a certainty equivalent profit equal to barpi because barpi is deterministic Thus to the seller a general nonlinear pricing scheme cannot be better than the optimal twopart tariff

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