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       18.1   a.     P = 53  Q    PQ = 53Q  Q2

                    MR = 53  2Q = MC = 5

                     Q = 24, P = 29, π = (P  AC)  Q = 576

              b.    MC = P = 5      P = 5, Q = 48

              c.    Competitive Consumers' Surplus = 2(48)2 = 1152.

                    Under monopoly:




                    Notice that the sum of consumer surplus, profits, and Deadweight loss under
                    monopoly equals competitive consumer surplus.


       18.2   Market demand Q = 70  P, MR = 70  2Q.

              a.    AC = MC = 6. To maximize profits set MC = MR.
                     6 = 70  2Q
                    2Q = 64
                     Q = 32
                     P = 38
                     π = (P  AC)  Q = (38  6) 32 = 1024

              b.    TC = .25Q2  5Q + 300, MC = .5Q  5.    Set MC = MR
                    .5Q  5 = 70  2Q
                       2.5Q = 75
                          Q = 30
                          P = 40
                 π = TR  TC = (30)(40)  [.25(30)2  5(30) + 300]
                   = 1200  375 = 825.

       c.   TC = .0133Q3  5Q + 250.

            MC = .04Q2  5

            MC = MR  .04Q2  5 = 70  2Q

            or

                   .04Q2 + 2Q  75 = 0.

            Quadratic formula gives Q = 25.

            If Q = 25, P = 45

             TR = 1125

             TC = 332.8 (MC = 20)
              π = 792.2

                                                                     d.




18.3   a.   AC = MC = 10, Q = 60  P, MR = 60  2Q.
            For profit max., MC = MR       10 = 60  2Q
            2Q = 50 Q = 25 P = 35
             π = TR  TC = (25)(35)  (25)(10) = 625.

       b.   AC = MC = 10, Q = 45  .5P, MR = 90  4Q.
            For profit max., MC = MR     10 = 90  4Q
     80 = 4Q Q = 20 P = 50
     π = (20)(50)  (20)(10) = 800.

c.   AC = MC = 10, Q = 100  2P, MR = 50  Q.
     For profit max., MC = MR       10 = 50  Q
     Q = 40 P = 30.
     π = (40)(30)  (40)(10) = 800.

     Note: Here the inverse elasticity rule is clearly illustrated:

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     a              1(35/25) = 1.4                        .71 = (35  10)/35

     b              .5(50/20) = 1.25          .80 = (50  10)/50

     c              2(30/40) = 1.5                        .67 = (30  10)/30


                                                                            d.
            The supply curve for a monopoly is a single point, namely, that quantity-price
            combination for which MC = MR. Any attempt to connect equilibrium points
            (price-quantity points) on the market demand curves has little meaning and brings
            about a strange shape. One reason for this is that as the demand curve shifts, its
            elasticity (and its MR curve) usually changes bringing about widely varying price
            and quantity changes.




18.4   a.
       b.     No supply curve for monopoly; have to examine MR = MC intersection since a
              shift in demand is accompanied by a shift in MR curve. Case (1) and case (2)
              above show that P may rise or fall in response to an increase in demand.

       c.     Can use inverse elasticity rule to examine this

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              As e  toward 1, P  MR increases.

              Case 1 MC constant, MR constant

              If e  , P  MR , P .

              If e constant, P  MR constant, P constant.

              If e , P  MR , P .

              Case 2 MC falling, MR falls:

              If e , P  MR , MR , P may rise or fall.

              If e constant, P  MR constant. MR , P .

              If e , P  MR , MR , P .

              Case 3 MC rising, MR rising

              If e , P  MR , MR , P .

              If e constant, P  MR constant, MR , P .

              If e , P  MR , MR , P may rise or fall.


18.5   Q = (20  P)(1 + .1A  .01A2)

                                    Install Equation Editor and double -
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       Let K = 1 + 1A + .01A

       π = PQ  TC = (20P  P2)K  (200  10P)K  15  A
       Install Equation Editor and double -
       click here to view equation.           = (20  2P)K + 10K = 0.

       a.       20  2P = 10         P = 15 regardless of K or A
                If A = 0, Q = 5, TC = 65           π= 15  5  65 = 10

       b.       If P = 15,    = 75K  50K  15  A = 25K  15  A
                         = 10 + 1.5A  0.25A2
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                Q = 5(1 + .3  .09) = 5 1.21 = 6.05
                PQ = 90.75          TC = 60.5 + 15 + 3 = 78.5
               π = 12.25Can increase over the case A = 0.
18.6   A multiplant monopolist would still produce where MR = MC and would equalize MC
       among plants. This answer assumes the number of plants is fixed. If the number of
       plants is subject to choice by monopolist, this number should be chosen so that given the
       quantity to be produced, overall total costs are minimized.


                                                                   Install Equation Editor and double -
18.7   a.       Q1 = 55  P1       R1 = (55  Q1)Q1 = 55Q1        click here to view equation.


                MR1 = 55  2Q1 = 5            Q1 = 25, P1 = 30

                                     Install Equation Editor and double -
                Q2 = 70  2P2        click here to view equation.



                MR = 35  Q2 = 5        Q2 = 30, P2 = 20

                 π = (30  5) 25 + (20  5)  30

                   = 625 + 450 = 1075

       b.       Producer wants to maximize price differential in order to maximize profits but
                maximum price differential = $5.

                P1 = P2 + 5

                π = (P1  5)(55  P1) + (P2  5)(70  2P2)
              = π + λ (5  P1 + P2)



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                                                   60  2P1 = 4P2  80 and P1 = P2 + 5

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                                                   130 = 6P2    P2 = 21.66 P1 = 26.66

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                                                          Install Equation Editor and double -
       c.   P1 = P2 π = 140P  3P  625  2               click here to view equation.           = 140  6P = 0

                 Install Equation Editor and double -
            P=   click here to view equation.           = 23a    Q1 = 31b       Q2 = 23a         π = 1008a


                                                               Install Equation Editor and double -
       d.   If the firm adopts a linear tariff of the     form click here to view equation.           , it can
            maximize profit by setting m = 5,

            α1 = .5(55  5)(50) = 1250

            α2 = .5(35  5)(60) = 900

            and π = 2150.

            Notice that in this problem neither market can be uniquely identified as the "least
            willing" buyer so a solution similar to Example 18.5 is not possible. If the entry
            fee were constrained to be equal in the two markets, the firm could set m = 0, and
            charge a fee of 1225 (the most buyers in market 2 would pay). this would yield
            profits of 2450  125(5) = 1825 which is superior to profits yielded with T(Qi).


18.8   a.   For P.C.      MC = $10. For monopoly                MC = $12.

            QD = 1000  50P
     P.C.: P = MC = $10. Thus Q = 1000  50(10) = 500.

                               Install Equation Editor and double -
     Monopoly: P = 20  click here to view equation.                  Q, PQ = 20Q 
     Install Equation Editor and double -
     click here to view equation.         Q2

                                                    Install Equation Editor and double -
     Produce where MR = MC. MR = 20                click here to view equation.           Q = 12.

     Q = 200, P = $16

b.   See graph below.

     Loss of consumer surplus =
       Consumer surplus P.C.  Consumer surplus monopoly =
       2500  400 = 2100.
       c.




            Of this 2100 loss, 800 is a transfer into monopoly profit, 400 is a loss from
            increased costs under monopoly, and 900 is a "pure" deadweight loss.


18.9   a.   The government wishes the monopoly to expand output toward P = MC. A lump-
            sum subsidy will have no effect on the monopolist's profit maximizing choice, so
            this will not achieve the goal.

       b.   A subsidy per unit of output will effectively shift the MC curve downward. The
            figure illustrates this for the constant MC case.




       c.   A subsidy (t) must be chosen so that the monopoly chooses the socially optimal
            quantity, given t. Since social optimality requires P = MC and profit
                                                        Install Equation Editor and double -
            maximization requires that MR = MC  t =    click here to view equation.           ,
                                       Install Equation Editor and double -
                substitution yields    click here to view equation.           as was to be shown.
                Intuitively, the monopoly creates a gap between price and marginal cost and the
                optimal subsidy is chosen to equal that gap expressed as a ratio to price.


18.10 Since consumers only value X  Q, firms can be treated as selling that commodity (i.e.,
      batteries of a specific useful life). Firms seek to minimize the cost of producing X  Q for
      any level of that output. Setting up the Lagrangian,

                 = C(X)Q + λ (K  X  Q)

       yields the following first order conditions for a minimum:

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       Install Equation Editor and double -
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       Install Equation Editor and double -
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       Combining the first two shows that

                C(X)  C'(X)X = 0
       or

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                click here to view equation.           .


       Hence, the level of X chosen is independent of Q (and of market structure). The nature of
       the demand and cost functions here allow the durability decision to be separated from the
       output-pricing decision. (This may be the most general case for which such a result
       holds.)

				
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