Lecture 5a UNSW 2009 Page 1. Mat hs Revision

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1 Lecture 5a UNSW 2009 Page 1 Mat hs Revision Fr om the document, MBAMat hs Unit 3 Functions & Int eres ts (sic) pages 46/47: Functions Quadratics. Aquadr atic function constains a var iable squared: y = ax 2 + bx + c The (single) maximum or minimum of y alw ays occur s at : x = b 2a We can see this by dif ferentiating y: dy dx zero tomaximize y,at x * = b/2a. = 2ax + b,which is Whet her y is a maximum or minimum depends on the sign of the coefficient a: when a is positive, y is a minimum, when a is negative, y is a maximum, and when a is zero, y is a linear function in x. >

2 Lecture 5a UNSW 2009 Page 2 Let s find the profit-maximizing values in L5. p.5 VonStackelberg: Reaction Function π C = (10 (Q S + Q C )) Q C 3Q C π C = Q 2 C + (7 Q S) Q C + 10 is a quadr atic in Q C, wit h a = 1, b = (7 Q S ),and c =10. π C max at Q * C = b 2a = 7 Q S 2 p.6 VonStackelberg: Solution π S = 1 2 Q2 S Q S is a quadr atic in Q S, wit h a = ½, b =3½, and c =0. π S max at Q * S = b 2a = p.8 Monopolis t: Solution π M = P M Q M 3Q M = Q 2 M + 7Q M is a quadr atic in Q M, wit h a = 1, b =7,and c =0. π M is max at Q * M =7/2.

3 Lecture 5a UNSW 2009 Page 3 p.11 Cour not: Reaction Function π S = (10 Q C Q S ) Q S 3Q S π S = Q 2 S + (7 Q C)Q S is a quadr atic in Q S, wit h a = 1, b = 7 Q C,and c =0. π S is a max at Q * S = 7 Q C 2. p.15 Imper fect Bertrand: Solution π p = P 2 p + ( P d ) P p 144 3P d is a quadr atic in P p, wit h a = 1, b = P d,and c = P d π p is max at P * p = P d p.26 Monopolis tic Car tel: Solution π M = (10 Q M ) Q M 1 Q M = Q 2 M + 9Q M is a quadr atic in Q M, wit h a = 1, b =9,and c =0. π M is max at Q * M = 9/2 =4½.

4 Lecture 5a UNSW 2009 Page 4 p.27 Cour not: Reaction Function π 2 = (10 y 2 y e 1 ) y 2 1 y 2 π 2 = y (9 y e 1 ) y 2 is a quadr atic in y 2, wit h a = 1, b = 9 y e 1,and c =0. π 2 is max at y * 2 = 1 2 (9 y e 1 ). p.28 VonStackelberg: Solution π 1 = (10 y 2 y 1 ) y 1 1 y 1 π 1 = ( (9 y 1) y 1 ) y 1 y 1 π 1 = 10y y y 2 1 y 2 1 y 1 π 1 = y y 2 1 is a quadr atic in y 1, wit h a = ½,b =4½, and c =0. π 1 is max at y * 1 =4½.

5 Lecture 5a UNSW 2009 Page 5 The Economics of Profit-Maximizing The profit-maximizing firmwill continue increasing its rat e of output until the revenue associated withselling another unit of output is equal toorless than the costofproducing that output, assuming rising costs. These two measures arecalled the Marginal Revenue and the Marginal Cost, respectivel y. The difference between Marginal Revenue and Marginal Cost is the Marginal Profit associated withthe lastunit of output produced and sold. In algebr a: M π = MR MC, where all three are functions of the level ofoutput Q (amongs t ot her things, such as the demand curve, orthe going market price P).

6 Lecture 5a UNSW 2009 Page 6 With rising costs, profit is maximized when Marginal Profit is no longer positive: when Tot alprofit no longer rises with the rat e of production. Or, when Mπ = MR MC = 0. If we have functions for MR and MC in ter ms of output Q, then we can deter mine theprofit-maximizing levelofoutput Q *. Marginal Revenue and Costare jus t theder ivatives of Tot al Revenue and Tot al Cos t wit h respect tooutput Q. TR is just P Q. Wit h alinear demand curve (say P = 10 Q), and a firmwit h some market power (which means can set its own price, subject todemand), MR can be calculat ed: TR = P Q = (10 Q) Q = 10Q Q 2.

7 Lecture 5a UNSW 2009 Page 7 Dif ferentiating with respect tooutput Q: = MR[Q] = 10 2Q dtr dq Let s say that it costs a constant amount (say, $3) toproduce and sell a unit of output. Then TC = 3Q,and MC =3. For this firm, choosing Q * to equat e MR and MC will result in the higes t profit : sol ve: 10 2Q = 3 to get Q * =3½. Fr om the demand function, with Q * =3½, the price P will be $6.50: the higher the output, the lowerthe price tosell all units.

8 Lecture 5a UNSW 2009 Page 8 Profit-Maximizing, Graphicall y We can plot the MC = $3/unit line and the demand line P = 10 Q. Wecan identify the Monopolist sprice & quantity (P M,Q M )and the price-t aker sprice & quantity (P C, Q C ). Monopolis ts s TR = P M Q M = (10 Q M ) Q M = 10Q M Q 2 M Dif ferentiating TR, weget MR = dtr = 10 2Q dq M M,which is exactl y twice as steep as the demand line, as shown below. Fr om above, profit-maximizing monopolistic output Q * M occur s when MR = MC,but here MC =$3. So Q * M from 10 2Q * M = 3,or Q * M = The diagram shows that the output Q * M =3½occur s where the red downwards-sloping MR line cuts the MC = $3 line, and reading up to the demand line gives P * M =$6.50.

9 Lecture 5a UNSW 2009 Page 9 Pr ice P $/unit MR Demand: P = 10 Q Monopol y (Q M =3.5, P M =$6.50) Pr ice-taking MC = AC (Q C=7, P C =$3) = Quantity Q

10 Lecture 5a UNSW 2009 Page 10 If the firm isbehaving as a price-t aker,then its Marginal Revenue is just the going price P. So it chooses its output whereits Marginal Revenue = $3, the Marginal Cost. The Marginal Cost =$3/unit is in effect the market Supply cur ve. The market quantity is Q C = 7,at P C =$3/unit, where Demand = the horizont al green Supplyline, as shown above. <