ECON 000 Problem Set 7: Monopoly nd Gme Theory. () The monopolist will choose the production level tht mximizes its profits: The FOC of monopolist s problem is: So, the monopolist would set the quntity to mx [ bq] q cq q 0 [q] : (b + c)q = 0 q = (b + c) Subsequently, the monopolist will choose price tht clers the mrket t its optiml quntity: p = (b + c) b = b + c (b + c) (b) A price-tking firm with the sme cost function would produce t quntity tht MC(q c ) = p : ( q c = p c = + b ) ( c (b + c) = + b ) q c One cn note tht the price-tking firm with the sme cost function would produce more thn monopolist even when the mrket price is the sme s the one monopolist will choose to cler its mrket. (c) The demnd shift is clockwise rottion round the previous equilibrium. In competitive mrket supply function is independent of the demnd. Therefore, this type of demnd shift hs no effect on the equilibrium price nd quntity. Here, the new optiml price nd quntity re: q = (b + c) p = b + c (b + c) But we know the demnd is rotting round the previous equilibrium: [b + c] (b + c) = (b + c) b (b + c) = b + b + c This illustrtes tht the optiml quntity fter the chnge in demnd, q, is smller thn the optiml quntity before the chnge in demnd, q. On the other hnd, since both (q, p ) nd (q, p ) re on sme downwrd sloping demnd curve, we know the new optiml price, p, is higher thn the old price, p. Alterntively, one cn note tht the demnd elsticity hs risen t the old equilibrium quntity: ɛ d(q ) = b b ɛ d(q ) < ɛ d (q ) Using the Lerner index representtion of monopoly s profit mximizing quntity, the firm s optiml respond to the shift in demnd is to cut its production until the mrkup equls the new Lerner index: p(q ) MC(q ) p(q ) = ɛ d (q )
ECON 000. () The totl demnd tht the firm fces is: q tot (p) = 0 p, p > 0 + p, p < Since there is kink we hve to seprtely solve for the best price bove nd for the best price below. The FOC yields: p 5, p > = 0+, p < The corresponding quntity is: q tot 5, < 5 = 0+, > 0 3 Firm s profits in the two cses is given by: 5, p > π = (0+), p < The firm chooses to serve both mrkets if: (0 + ) ( ) > 5 > 0 =. This lso stisfies the condition > 0 3. For ny < the firm chooses to serve only the dult mrket. This lso stisfies the condition < 5. (b) If the firm cn set different prices for different segments of the popultion, it mximizes the profits for ech segment seprtely. The profit mximiztion for the dult segment is: The corresponding FOC is: The resulting equilibrium quntity is: mx (0 p ) p p [p ] : 0 p = 0 p = 5 q = 0 p = 5 The profit mximiztion for the student segment is: The corresponding FOC is: The resulting equilibrium quntity is: mx ( p s ) p s p s [p s ] : p s = 0 p s = q s = p s = (c) One cn see from prt b tht the firm s profits with third degree price discrimintion is: π d = 5 + = 00 + When the firm chrges the sme price for everyone its profits re: 5, > π nd = (0+), <
ECON 000 It is esy to see tht for <, third degree price discrimintion increses the profits of the firm. One cn see this is lso true when < : 00 + > (0 + ) 00 + > (0 + ) (0 ) > 0 The consumer surplus of dults nd students with third degree price discrimintion re: CS d = (0 p ) q = 5 CS s d = ( p s) q s = So, the totl consumer surplus with price discrimintion is: CS d = CS d + CS s d = 00 + The consumer surplus of dults nd students respectively when the firm chrges the sme price to everyone is: 5 CSnd, > = ( ) 0 0+ = (30 ) 3, < CS s nd = ( 0+ 0, > ) = (3 0) 3, < So consequently the combined CS without discrimintion is 5, > CS nd = (30 ) +(3 0) 3, < Third degree price discrimintion does not chnge the surplus of dults if > but decreses their surplus if < : (30 ) > 5 3 (30 ) > 00 < 0 which by ssumption is lwys true. The consumer surplus of students however lwys increses since when < (3 0) < ( )( 0) < 0 3 which by ssumption lwys holds. Finlly note tht the totl surplus without price discrimintion is CS nd + CS s nd = 5, > 0 0+000 3, < If < ā it is immedite tht consumer surplus is higher with price discrimintion. When > ā the totl consumer surplus is higher with price discrimintion if nd only if or equivlently if This holds if nd only if 0 0 + 000 3 < 00 + 0 0 + 000 < 00 +. 6( 0) < 0 which is impossible. Hence we see tht if > ā the totl consumer surplus is higher without price discrimintion. 3
ECON 000 3. Two-Prt Triff () Suppose tht the generl demnd form is given by p(q) = bq () Note tht b = in this cse. The profit mximiztion problem is given by mx π(q) = p(q)q c(q) = ( q)q q q () The FOC is thus Thus q + ( q) q = 0 (3) q = 3 () Substituting this bck into the demnd function, we cn get the equilibrium price s p = 3 (5) Equilibrium profit nd consumer surplus cn be clculted s π(q ) = p q c(q ) = 3 3 ( 3 ) = 6 (6) CS = ( p )q = (7) To clculte the ded weight loss, we use the fct tht under competitive equilibrium, we should hve price equls to mrginl cost. Thus we hve Substituting this into the consumer demnd function, we thus hve Then the ded weight loss is given by p = c (q) = q () q o = = po (9) DW L = [p c (q )](q o q ) = ( 3 3 )( 3 ) = 36 (0) (b) The optiml -prt triff is estblished by setting price (nd hence quntity) t the competitive equilibrium level p = p o = () nd chrging consumers n entry fee tht extrcts ll the consumer surplus Firm profit is thus F = CS o = () There is no DWL in prt triff. π(q ) = p q c(q ) + F = ( ) ( ) + = (3). Limited Functionlity Softwre
ECON 000 () Let u H (p, q) q q p be the utility of the high vlution consumer when she buys q nd pys p. Anlogously, let u L (p, q) q q p be the utility of the low vlution consumer when she buys q nd pys p. Since the firm cn perfectly discriminte, it will extrct ll consumer surplus. If we let u H (p H, ) = 0, we cn solve for p H = 3 while letting u L (p L, q L ) = 0 yields p L = q L q L. Since p L is the profit of the firm, we mximize p L by choosing q L = which implies tht p L =. (b) Now, the firm hs to set prices nd qulities beforehnd nd let consumers decide wht to buy. The firm s objective is to: mx p H,p L,q L αp H + ( α)p L subject to: (IC) Incentive Comptibility Constrint: u H (p H, ) u H (p L, q L ) (the high type wnts to buy high nd not low) (IR) Individul Rtionlity Constrint: u L (p L, q L ) 0 (the low type wnts to sty in the mrket no need to worry bout high type). Which yields the following Lgrngin: L = αp H + ( α)p L + λ[u H (p H, ) u H (p L, q L )] + µu L (p L, q L ). (c) You cn tke FOCs nd solve for prices, qulities nd Lgrnge multipliers. You will find tht ql = α α. Another wy to solve this is to remember tht the firm will extrct ll vlution from the low vlution consumer, while the high type my go wy with something. Tht mens tht u L (p L, q L ) = 0 which yields p L = q L ql. Also, note tht it mkes no sense for the firm to llow (IC) u H (p H, ) > u H (p L, q L ). It should be strict equlity. Then, using wht we know of p L we cn deduce tht p H = 3 q L. Using wht we know so fr, we cn plug ll this into the objective function to see tht the firm s problem becomes: mx ql α(3 q L ) + ( α)(q L ql ). Now the objective looks much more simple nd we cn tke FOC on q L to get q L = α α p H = + α α, p L = α ( α). (d) When α increses, the mrket gets more people with high vlution. Then, the firm will try to exploit them, not the minority of low types. In order to do this, p H must go up, but t the sme time, high types must not go nd buy low qulity. To mintin this Incentive Comptibility constrint, the firm mkes the low qulity so miserble tht they do not wnt to buy it. Imgine tht there re lots of rich people trying to fly. Then the irlines increse prices for first clss nd mke economy clss so miserble tht first clss pssengers would dred to go in or even look t. 5. Iterted Deletion of Dominted Strtegies () There re situtions, when n =, then A i = 0 () is the dominnt strtegy for individul i, s whtever the other plyer nnounces, 0 will be closer to of the men. When n >, there is no dominnt strtegy s for whtever nnouncement tht the individul mkes, I cn lwys find set of nnouncements tht other plyers mke such tht this individul nnouncement is no longer optiml. Consider if individul i mkes the nnouncement k > 0, A i = k (5) 5
ECON 000 (b) then I simply let the rest of the plyers nnounce 0. Then individul i would lose. Now suppose tht individul i nnounces A i = 0, then I let plyer j nnounces A j =, nd other plyer(s) nnounces A i,j = 00. Then plyer i would lose. But plyer i cn improve his pyoff by choosing the winning strtegy A i, which is given by the solution to the following problem: nd it lwys hs solution for ll A j [0, 00]. A i = rgmin Ai [0,00] A i n ( A j + A i ) (6) j i Hence we do not hve strtegy tht domintes ll other strtegy. A i = 00 (7) is clerly the dominted strtegy. If ll other plyers nnounce 00, then deviting to nnouncing something smller is clerly better s he will win the gme. If not ll plyers nnounce 00, then individul i would lose by nnouncing 00, but he cn improve his pyoff by nnouncing the winning pyoff A i (s solved in the previous prt). (c) After A i = 00 hs been removed, the sme logic lso pplies to the second lrgest nnouncement tht is A i = 99, nd it will be strictly dominted. (d) We cn keep deleting the strictly dominted strtegy nd we re left with the symmetric equilibrium of A i = 0 () for ll i. (e) It is Nsh equilibrium. Given for ll j i, plyer s pyoff is A j = 0 (9) f(x) = 00 n if A i = 0 0 if A i 0 (0) Hence it is optiml for plyer i not to devite. Thus it is Nsh Equilibrium. 6