4 market outcome is e cient Third degree price discrimination price depends on

4 market outcome is e cient third degree price

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price discrimination (Figure 14.4), market outcome is e¢ cient! Third-degree price discrimination: price depends on the identity but not on quantity. Separate the market into di/erent segments. Example 14.5: p 1 = 24 ² q 1 and p 2 = 12 ² 0 . 5 q 2 , MC = 6 . max [( 24 ² q 1 ) q 1 + ( 12 ² 0 . 5 q 2 ) q 2 ] ² 6 ( q 1 + q 2 ) . Second-degree price discrimination: price depends on quantity but not on the identity. Q. Wen (UW) Econ 400 University of Washington 33 / 52
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Chapter 15 Imperfect Competition Many ±rms, compete in either price or quantity, simultaneously or sequentially. Bertrand model : simultaneous price-setting/price-competition game. Two ±rms, 1 and 2, each has a constant MC c > 0 . Products of the two ±rms are homogenous /identical. Firm i chooses price p i ³ 0, for i = 1 and 2. Market demand function: q = a ² bp . Because of homogenous products , ±rm 1°s pro±t is not continuous: π 1 ( p 1 , p 2 ) = 8 > < > : ( a ² bp 1 ) ( p 1 ² c ) if p 1 < p 2 , 1 2 ( a ² bp 1 ) ( p 1 ² c ) if p 1 = p 2 , 0 if p 1 > p 2 . Q. Wen (UW) Econ 400 University of Washington 34 / 52
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Chapter 15 Imperfect Competition Bertrand model : simultaneous price-setting game. Firm 2°s pro±t function is not continuous as well: π 2 ( p 1 , p 2 ) = 8 > < > : ( a ² bp 2 ) ( p 2 ² c ) if p 2 < p 1 , 1 2 ( a ² bp 2 ) ( p 2 ² c ) if p 2 = p 1 , 0 if p 2 > p 1 . Because discontinuous payo/ functions, a ±rm does not always have a best response . However, if p 2 = c , then any p 1 ³ c is a best response of ±rm 1. There is a Nash equilibrium where p 1 = p 2 = c , which is the same as the competitive equilibrium outcome. Problem 15.4: Bertrand model with product di/erentiation . Demand for ±rm 1°s product is q 1 = 1 ² p 1 + bp 2 with b < 2, For simplicity, assume production cost is 0. Q. Wen (UW) Econ 400 University of Washington 35 / 52
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Chapter 15 Imperfect Competition Example 15.5 Hotelling°s Beach Model Horizontal product di/erentiation Consumers are distributed uniformly on [ 0 , 2 ] . Two ±rms, ±rm A is located at 0 and ±rm B is located at 2. A consumer°s transportation cost is 1 4 ° ( distance) 2 . Consumer x is indi/erence to buy from ±rm A or ±rm B if p A + 1 4 x 2 | {z } cost to buy from A = p B + 1 4 ( 2 ² x ) 2 | {z } cost to buy from B See Figure 15.5 Solve x = 1 + p B ² p A . Q. Wen (UW) Econ 400 University of Washington 36 / 52
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Chapter 15 Imperfect Competition Example 15.5 Hotelling°s Beach Model Demand function for ±rm A : q A ( p A , p B ) = x = 1 ² p A + p B . Demand function for ±rm B : q B ( p A , p B ) = 2 ² x = 1 + p A ² p B . If two ±rms chooses prices simultaneously (Bertrand competition) max p A ( 1 ² p A + p B ) p A max p B ( 1 + p A ² p B ) p B BR A ( p B ) = 1 + p B 2 BR B ( p A ) = 1 + p A 2 Nash equilibrium: p ± A = p ± B = 1 . Firms°pro±t: π ± A = π ± B = ( 1 ² 1 + 1 ) ° 1 = 1 . Q. Wen (UW) Econ 400 University of Washington 37 / 52
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Chapter 15 Imperfect Competition Cournot model: simultaneous quantity-setting game There are 2 ±rms, each has a constant MC c ³ 0. Denote ±rm i °s quantity output by q i ³ 0. Let the inverse market demand function be p = a ² bQ = a ² b ( q 1 + q 2 ) . Firm i °s pro±t/payo/ function is π i ( q i , q j ) = [ a ² b ( q 1 + q 2 )] q i ² cq i , which is continuous. To ±nd ±rm i °s best response function, max q i [ a ² b ( q 1 + q 2 )] q i ² cq i . Find ±rm i °s best response function: BR i ( q j ) = a ² c 2 b ² 1 2 q j . Q. Wen (UW) Econ 400 University of Washington 38 / 52
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Chapter 15 Imperfect Competition Cournot model: simultaneous quantity-setting game To ±nd a Nash equilibrium ( q ± 1 , q ± 2 ) , set q 1 = BR 1 ( q 2 ) and q 2
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