continuous_2nd_degree

# continuous_2nd_degree - Second-degree Price Discrimination...

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Second-degree Price Discrimination with a Continuum of Types Economics 201 b A monopolist wishes to practice second-degree price discrimination via quan- tity discounts when there is a continuum of types. Assumptions There is a continuum of types; specifically, a consumer’s type, θ , is an element of the real interval [ θ 0 , θ 1 ]. Marginal cost is a constant, c . Because marginal cost is a constant, there is no loss of generality in con- sidering a single consumer whose type is drawn from the distribution F ( · ) : [ θ 0 , θ 1 ]. 1 Assume F ( · ) is differentiable. Let f ( · ) be the derivative (the density func- tion). Assume f ( θ ) > 0 for all θ [ θ 0 , θ 1 ]. A type- θ consumer’s utility is b ( x, θ ) T , where x is consumption of the good in question and T is the amount paid to the firm (the price of a package with x units in it). A Spence-Mirrlees condition holds: 2 b ( x, θ ) ∂θ∂x > 0 . ( sm ) A consumer knows his type. The monopolist knows only that θ is drawn from [ θ 0 , θ 1 ] according to the distribution F ( · ). The monopolist also knows the functional form, b ( · , · ). Properties P1. ∂b/∂x > 0 for all x > 0. This follows because ∂b/∂x is the inverse demand curve and inverse demand curves are positive (at least over the relevant region of analysis). 1 As will be obvious later, we could multiply the expression for profits below by N , the population, without effecting the solution to the maximization problem. Copyright c 2005 Benjamin E. Hermalin. All rights reserved.

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Economics 201 b Page 2 P2. ∂b/∂θ > 0 for all x > 0. Proof: Let θ > θ . Observe 2 b ( x, θ ) b ( x, θ ) = x 0 ∂b ( z, θ ) ∂x dz x 0 ∂b ( z, θ ) ∂x dz = x 0 θ θ ∂b ( z, t ) ∂θ∂x dt dz > 0 . The last line follows because x > 0, θ > θ , and the quantity being inte- grated (the integrand) is positive by ( sm ). Notation The package intended for type θ has x ( θ ) units in it. The charge for an x ( θ )-unit package is T ( θ ). In equilibrium, a type- θ consumer purchases the package intended for him or her. Let U ( θ ) denote the equilibrium level he or she enjoys; that is, U ( θ ) b ( x ( θ ) , θ ) T ( θ ) . (1) A feasible price discrimination scheme is a pair of function x ( · ) and T ( · ) such that for every type θ , a type- θ consumer prefers an x ( θ )-unit package at price T ( θ ) to any other sized package and to not purchasing at all. 3 In other words, the scheme is feasible if it is incentive compatible and individually rational for each type to purchase the package intended for him or her. Analysis The monopolist wishes to design a feasible scheme that maximizes her profit, which is T ( θ ) cx ( θ ) if the customer proves to be type θ . This profit needs to be multiplied by the number (probability) of type- θ customers, which can be thought of as being f ( θ ). Hence, (expected) profit is Π = θ 1 θ 0 ( T ( θ ) cx ( θ ) ) f ( θ ) dθ . (2) Because the scheme must be feasible, it must satisfy both ( ir ) and ( ic ) con- straints; that is, respectively, consumers must participate and they must buy the appropriate package. The ( ir ) constraints are simply U ( θ ) 0 for all θ [ θ 0 , θ 1 ] . ( ir ) 2
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