Partial equilibrium q analysis: y Monopoly Lectures in Microeconomic Theory Fall 2010, Part 17 07.07.2010 G.B. Asheim, ECON4230-35, #17 Pricemaking 1 price ( y, c) p ( y ) y cy a by y cy Inverse demand fn.: p ( y ) a by c y y 07.07.2010 G.B. Asheim, ECON4230-35, #17 quantity 2 1 Profit maximization in special cases P ( y, c) p ( y ) y cy Special case 1: a by c y max a by c y y FOC : a 2by c 0 p ( ym ) ac 2b p ( ym ) cy FOC : 1 1b p c p ( ym ) ym Special case 2: y Ap b 07.07.2010 (a c) 2 ( y, c) 4b Y ym ( y, c) A y 1 b ac 2 1 1b c 1 b1 G.B. Asheim, ECON4230-35, #17 3 General analysis ( y ) p( y ) y c( y ) FOC : y p ( y ) p( y ) y c( y ) 0 p ( y ) c( y ) y 1 p( y ) p( y) p( y) where h 1 p( y) i the is h elasticity l i i off demand. d d p( y ) y 2 SOC : 2 p( y ) p( y ) y c( y ) 0 y 2 07.07.2010 G.B. Asheim, ECON4230-35, #17 4 2 Comparative statics c( y ) cy SOC p( y ) 0 2 p( y ) p( y ) y 0 Special p case 1: ( y, c) p ( y ) y cy ( y, c) 0 ac y p( ym ) 2 ( y, c) ( y, c) dy dc 0 dp b y dcy 0 dc 2b dy 1 0 Special case 2: c d y y dc d dc 2 p( y ) p( y ) y p ( ym ) 1 b1 p( y ) dp dp dy 1 dp 0 1 dc dy dc 2 p( y ) p( y ) y dc 1 1b 2 2 2 2 2 2 07.07.2010 5 G.B. Asheim, ECON4230-35, #17 Welfare and output Welfare as a function of output: p W ( x) u ( x) c( x) Welfare maximization: u ( x0 ) p ( x0 ) c( x0 ) c((x) p ( xm ) Monopoly output satisfies: p( x0 ) p(x) p( xm ) p( xm ) xm c( xm ) W ( xm ) u ( xm ) c( xm ) p( xm ) xm u( xm ) xm 0 Monopolist’s gain is smaller than consumers’ loss. 07.07.2010 G.B. Asheim, ECON4230-35, #17 u(x) xm x0 x Deadweight loss 6 3 Price discrimination Monopolist’s dilemma: A higher quantity leads to a lower price price. price p( y) p ( y ) The monopolist can get out of this dilemma by • sorting consumers • charging different prices t different to diff t consumers c This requires that the monopolist can sort, and that consumers cannot resale. 07.07.2010 G.B. Asheim, ECON4230-35, #17 y y quantity How can the monopolist sort? 7 Types of price discrimination First-degree price discrimination (Also called perfect discrimination) “Special “Sp i l price pri fforr you”” Price = maximal willingness-to-pay for each unit. Second-degree price discrimination Price differs according to consumed quantity (or quality), but not across consumers. Ex: Full price/disc. tickets for t transportation t ti Third-degree price discrimination Price differs across consumers, but does not depend on consumed quantity. Ex: Different Ticket price depends on age, etc. price dom. and abroad 07.07.2010 G.B. Asheim, ECON4230-35, #17 8 4 1st-degr. price discr. p Maximization of total surplus c((x) u ( x0 ) c( x0 ) No surplus to consumers: p( x0 ) p(x) x0 u(x) u ( x0 ) u (0) p ( x)dx 0 0 x Whole surplus to x0 monopolist. Why is first-degree price discrimination difficult to implement for the monopolist? 07.07.2010 9 G.B. Asheim, ECON4230-35, #17 Model with two consumers p Low demand consumer (L) High demand consumer (H) Assumptions: H has higher total willingness-to-pay uH (x) u H ( x) u H (0) u L ( x) u L (0) 0 H has higher marginal willingness-to-pay uL (x) x 07.07.2010 G.B. Asheim, ECON4230-35, #17 u H ( x) u L ( x) 0 10 5 2nd-degr. price discr. (self-selection) Each consumer is offered a pair of total payment and quantity: (ri , xi ) p xL rL u L ( x)dx uH (x) Assume no costs. xH rH rL u H ( x)dx xL 0 Ensures participation and self-selection uL (x) How to determine xL and xH ? No distortion at the top: p u( xH ) 0 MC xL xL xH 07.07.2010 x L’s quantity is distorted: u( xL ) 0 MC Why? 11 G.B. Asheim, ECON4230-35, #17 3rd degr. price discr. (segmentation) The monopolist is able to treat the two consumers as separate markets. p uH ((xx) uL (x) p pH M Monopoly l price i and d quantity i iin each h market. k Higher elasticity leads to lower price. What are the welfare effects of requiring the same price in both markets? 33rdd degr. d price i distr. di t is i welfare lf improving only if it leads to a higher quantity. pL xL xH 07.07.2010 Assume no costs. x x G.B. Asheim, ECON4230-35, #17 x 12 6