8/17/2019 c12sq.doc

1/4

Chapter 12: Price Discrimination

1. A price-discriminating monopolist faces the following inverse demand functions:In Market One it is P1 = !-"1 where P1 is the price charged in Market 1 and "1 is the #uantit$ demanded inMarket one. In Market %wo it is P = 1&-1.&" where P is the price charged in Market and " is the#uantit$ demanded in Market %wo. Marginal cost is constant at '&. (ind the profit-ma)imi*ing #uantit$and price charged in each market. +alculate profit in each market and ,oint profit.hat would this firms price/ #uantit$ and profit 0e if it were constrained to charge the same price to allconsumers 2how this outcome on a graph.

. A price-discriminating monopolist faces the following inverse demand functions:In Market One it is P1 = 3!-"1  and in Market %wo it is P = 4!-" . Marginal cost is constant at '1!.+onsumers in market two can resell the good to consumers in market one at a cost of '5 per unit.(ind the profit-ma)imi*ing #uantit$ and price charged in each market su0,ect to the resale constraint.

6. +onsider the following inverse demand function: P = 1! 7 ". Marginal cost is constant at '.a. (ind the %8 and M8 functions under perfect-price discrimination. hat is profit-ma)imi*ing #uantit$hat is profit 0. %he perfect-price discrimination outcome can 0e attained 0$ an all-or-nothing packaging scheme.9)plain how.

5. +onsider the following inverse demand function: P =5!!-". Marginal and average cost are constant at'!!.a. (ind the monopol$ #uantit$/ price and profit. raw a graph illustrating this outcome. 0. If this firm emplo$s a two-part tariff scheme/ what will the user charge for this good 0e 9)plain. ;oure)planation should include a calculation of the profit-ma)imi*ing and 0>.

&. a. hat conditions make price discrimination possi0le 0. A monopol$ can sell its good in the ?2/ where the elasticit$ of demand is -/ and in 2outh @orea/ wherethe elasticit$ of demand is -5. Its marginal cost is '1!. At what price does the monopol$ sell its good ineach countr$ if resales are not possi0le

ANSWERS

1. %he firm ma)imi*es profit 0$ producing where M8 =M+ in 9A+ market.(irst/ find the M8 function for each market: %8 1=!"1-"1

 which gives us M8 1 = !-"1. (or markettwo: %8 =1&"-1.&"

 which gives us M8  = 1&-6".2econd/ set M8 e#ual to M+ and solve for #uantities.Market One: !-"1 = & which gives us Q1 = 7.5/ and Market %wo: 1&-6" = &/ which gives us Q2= 3.33.%hird/ plugging these #uantities into the demand function gives us prices. P1 = !-B.& = 12.5  and P = 1&-1.&6.66>= 1!.

π1= 1.&-&>B.&> = 5".25. π=1!-&>6.66> = 1"."7. π ,oint = '&4.&C'14.4B=72.#2.

(irst/ we need to find the market demand curve 0$ hori*ontall$ adding the individual demand curves. %odo this we must solve for #uantit$ for each 0u$er: "1=!-P and " = 1!-D6P. Ee)t/ " = "1C " =!-P>C1!-D6P>=6!-&D6>P. After doing this/ we re-solve for P so that P = 13-6D&>".

%his allows us to write the market M8 e#uation as M8 = 13-4D&>". N$%E: Fiven the constraint that the price must 0e the same for 0oth groups of 0u$ers/ $ou +AEEO% get the market M8 curve 0e hori*ontall$summing the individual M8 curvesGdoing this implicitl$ assumes that the M8 in each market will 0e thesame at the profit-ma)imi*ing solution. Hut this would not 0e true if the price is the same in each market.%hat is wh$ $ou must first derive the market demand curve and then get the M8 curve from that.>

2econd/ set M8 = M+ and solve for ". 13-4D&>"=&/ and so Q 1!.&33. 

%hird/ Plug " = 1!.366 into market demand curve to get price. P =13-6D&>1!.366> ≅ 11.5!.

(ourth/ find profit. Profit = 11.&!-&>1!.366> = 7!.'2.

8/17/2019 c12sq.doc

2/4

. 1st. 2et up the agrangian.

= P1"1 C P" 7 M+1"1-M+" C λJP1-P-tK where t = cost of reselling good.

= 3!-"1>"1> C = 4!-">"> - 1!"1 7 1!" C λJ3!-"1> 7 4!-">-5K

  = B!"1 - "1 C = &!"-"

  C λJ14-"1 C"K

nd. (ind the first order conditions.

i. ∂D∂"1 = B! 7 "1 -λ = !.   λ = B! 7 "1 . ii. ∂D∂" = &! 7"  C λ = !.   λ = -&! C "

iii. ∂D∂λ = 14-"1 C " = !   "1 = 14 C ".

6rd. i and ii impl$: B! 7 "1 = -&!C " or "1 = 4! - ".?sing this along with iii/ gives us 14 C " = 4! - ".   " = 55 or " = .

5th. Eow that we have "/ we can find "1/ P1 and P."1 = 14 C " = 63. P1 = 3!-"1 = 3!- 63 = '5. P = 4!-" = 4!-= '63.

6.a. ?nder perfect price discrimination/ total revenue e#uals the entire under the demand curve up to the#uantit$ sold/ so %8 = ∫1!-" d" = 1!" 7!.&" . M8 = d%8Dd" = 1! 7 ". 2etting M8 = M+  1! 7 " = / or " =3. Profit is %8 7 %+ = 1!3> 7!.&3> 7 >3> = '6.

 0. 8e#uire the consumer to purchase the good in a package of 3 units or not at all. %he all-or-nothing pricee#uals average revenue at a #uantit$ of 3 units A8 = 1! 7 !.&" = 1!- !.&3> = '4.

2o the price of a package of 3 units is '4L3= '53.Profit = '53 - '14 = '6.

5a. M8 = 5!!-5". a. 5!!-5"=!! which gives us " = &!. (or price/ P=5!!-&!>='6!!/ and for profitwe have 6!!-!!>&!>='&/!!!. 0. %he firm should set the user charge P> e#ual to M+ which is !!> and charge an entrance fee e#ual tothe consumer surplus that would e)ist at P=!!.Profit = entrance feeCP-A+>">= !.&5!!-!!>1!!> C !!-!!>1!!>= '1!/!!!.c.

2ingle-P monopol$ %wo-Part %ariff Monopol$ +hange+onsumer 2urplus !.&5!!-6!!>&!>='/&!! ! -/&!!Producer 2urplus 6!!-!!>&!>= '&/!!! '1!/!!! C&/!!!2ocial %otal 'B/&!! '1!/!!! C/&!!

%he social total is increased under a two-part tariff. In fact/ there is no deadweight loss under a two-parttariff for identical consumers>.

8/17/2019 c12sq.doc

3/4

&. a. (irst/ the firm must have some degree of market power/ 0. 8esale of the good must 0e difficult/ andc. %he firm must 0e a0le to identif$ or the consumers have to implicitl$ identif$ themselves> the differentgroups of 0u$ers that it wants to price discriminate against. 0. P?2 = M+DJ1C 1De?2>K = '1!DJ!.&K = 2! and P2@  = '1!DJ!.3!K = 13.33 

"uestion :2ame price to all consumers.  '!

P=!-" for P'1&>

'1&  P=13-6D&>" for PN'1&>  '1  P='11.&!

mkt.  '1! M8=13-4D&>"

  '& M+  (R m)t

  & 1!.366 "

 Eote: Hecause there is a kink in the market demand curve/ there will 0e a discontinuit$ in the market M8

curve.

"uestion 6.' per unit Perfect Price iscrimination

  1!

  Pall-or-nothing = '4

' M+

= M8 A8 

  3 "

8/17/2019 c12sq.doc

4/4