lect441-81

lect441-81 - Advanced Microeconomics Leonardo Felli EC441...

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Advanced Microeconomics Leonardo Felli EC441: Room D.106, Z.332, D.109 Lecture 8 bis: 24 November 2004
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Leonardo Felli Lecture 8 bis: 24 November 2004 Monopoly Consider now the pricing behavior of a profit maximizing monopolist : a firm that is the only producer of a commodity. Let the aggregate demand of this commodity at price p be x ( p ) assumed to be continuous, strictly decreasing and such that x ( p ) > 0 . Assume that there exists a price ¯ p < + such that x ( p ) = 0 for every p ¯ p . Assume that the monopolist knows x ( p ) and is endowed with a technology characterized by the cost function c ( q ) . Slide 1
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Leonardo Felli Lecture 8 bis: 24 November 2004 The monopolist’s problem is then: max p p x ( p ) - c ( x ( p )) Equivalent formulation in terms of quantity choice q is derived using the inverse demand function P ( · ) = x - 1 ( · ) : max q 0 P ( q ) q - c ( q ) We focus on this (equivalent) formulation and assume that: P ( · ) and c ( · ) are twice continuously differentiable, P (0) > c (0) and there exists a unique output q c such that P ( q c ) = c ( q c ) . Slide 2
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Leonardo Felli Lecture 8 bis: 24 November 2004 The solution to the monopolist’s problem q m satisfies the following necessary first order conditions : P ( q m ) q m + P ( q m ) c ( q m ) with equality if q m > 0 The left-hand-side is known as the marginal revenue and it is equal to the deriva- tive of the revenue function R ( q ) = P ( q ) q . The right-hand-side is the familiar marginal cost . Slide 3
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Leonardo Felli Lecture 8 bis: 24 November 2004 Since P (0) > c (0) the necessary first order conditions can only be satisfied at q m > 0 . Therefore the monopolist’s optimal quantity choice is the one that sets marginal revenue equal to marginal cost : P ( q m ) q m + P ( q m ) = c ( q m ) In the typical case P ( q ) < 0 we obtain that: P ( q m ) > c ( q m ) The price under monopoly exceeds marginal cost . Slide 4
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Leonardo Felli Lecture 8 bis: 24 November 2004 Correspondingly: q m < q c A reduction in the quantity sold by the monopolist allows him to increase the price charged on the remaining sales. The effect on profits is captured by the term P ( q m ) q m . The welfare loss, known as the deadweight loss of monopoly is measured by the change in surplus: q c q m [ P ( s ) - c ( s )] ds > 0 Slide 5
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Leonardo Felli Lecture 8 bis: 24 November 2004 Notice that the deadweight loss is absent in the special case of a perfectly elastic demand : P ( q ) = 0 for all q .
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