Introduction to Quantitative Economics by kmb15358

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									                                               Topic 10

                                          Integration Solutions




1. A profit maximising firm has MR  34  3Q and MC  Q  10Q  26 .
                                                                           2



i) How much will the profit maximising firm produce (i.e. where MR = MC)?

Step 1: set MR=MC and find output that maximises profit, q*
       Q 2  10Q  26  34  3Q
       Q 2  7Q  8  0


                                                                   b  b 2  4a c 
                                                           Q
                                                                          2a 
Solve the quadratic for value of Q using formula                                          :


       a=1, b=-7, c=-8

            7  49  41 8 7  9 so
       Q                     
                  21           2
     Q  1 (inadmissible) or Q  8

     Thus 8 units produced by profit max firm


     ii)     Find expressions for total revenue TR and total Cost TC, and hence profits.

 Integrate MR and MC to find TR & TC, and thus profits

                                              TR  TC
TR   MR.dQ   34  3Q dQ  34Q  Q 2  c
                                     3
                                     2
In this case, the constant of integration c  0 , since the firm makes no revenue when Q=0
                      3 2
So   TR  34Q          Q
                      2
                                   1
                                                 
TC   MC.dQ   Q 2  10Q  26 dQ  Q 3  5Q 2  26Q  F
                                    3
F = the constant of integration = Fixed Costs



                                                                                              1
 = TR - TC
                  3         1
  34Q  Q 2  Q 3  5Q 2  26Q  F
                  2         3

                 1         7
         Q 3  Q 2  8Q  F
                 3         2

iii) What level of profits will the firm make when producing the profit maximising level
of output (found in part (i))

substitute in q* to TR and TC to get profit max values when producing q*

Substituting in Q  8 for profit max.

                8  8  88  170 2  224  64  117 1  F
                 1 3 7 2
                 3      2                  3                  3


 iv) Find the level of Fixed Costs at which the firm will make zero profits.
  Set profit =0 (thus TR – TC = 0), & solve for F


      Setting     0 , gives               1
                                     0  117  F
                                            3
                                               1
      Thus, value of F at  = 0 is   F  117
                                               3




2. A firm which has NO fixed costs has MC and MR given as follows:

                  MC=2Q2 – 6Q + 6;

                  MR = 22 – 2Q;

        Find total profit for profit maximising firm when MR=MC?

Solution:
1) Find profit max output Q where MR = MC




                                                                                           2
                  22 – 2Q = 2Q2 – 6Q + 6
                  gives Q2 – 2Q – 8 = 0

Solve quadratic for Q, by using formula, or

         (Q - 4)(Q + 8) = 0 so Q = +4 or Q =-2

so Q = +4     (since Q=-2 inadmissable)


2) Find TR and TC
TR( Q )   22  2Q dQ
 TR( Q )  22 dQ  2 QdQ

 TR( Q )  22Q  Q 2  c
TR = c when Q=0; but TR = 0 when Q = 0; so therefore c = 0

so TR = 22Q – Q2

MC = f (Q) = 2Q2 – 6Q + 6
              
TC ( Q )   2Q 2  6Q  6 dQ 
 TC ( Q )  2 Q 2dQ  6 QdQ 6  dQ
              2 3
 TC ( Q )      Q  3Q 2  6Q  F
              3
                                                            2 3
F = Fixed Cost = 0        (from question) so…. TC ( Q )      Q  3Q 2  6Q
                                                            3

3. Find profit = TR-TC, by substituting in value of q* when MR = MC

Profit = TR – TC
TR if q*=4:            22(4) - 42 = 88-16 = 72
                       2
TC if q* =4:             /3 (4)3 – 3(4)2 + 6(4)
                                          = 2/3(64) – 48 + 24
                                 = 182/3
so total profit when producing at MR=MC at q*=4 is

TR – TC = 72 - 182/3 = 53 1/3


3.    The inverse demand and supply functions for a good are, respectively:

        pD  100  1 q            and   pS  20  q
                      2



                                                                              3
      (a)    Find the market equilibrium values of p and q.

      For market equilibrium, pD  100  2 q  pS  20  q
                                         1


      So 120 = 3/2 q

      Hence 3q = 240 and so q = 80

      Since q = 80 and p = 100 – ½ (80) = 60

      Equilibrium q = 80 and p = 60



(b) Sketch the diagram and highlight Consumer and Producer Surplus at equilibrium

      Demand curve: pD  100  1 q so intercepts: at q = 0, p = 100 so (0,100) and at p =
                                                 2
0 , q = 50 so (50, 0)

      Supply curve: pS  20  q , so intercept: at q = 0, p = - 20



               P                          CS S
                          100              14



            p*=60

      PS
                                                     D
                    0
                                       q* = 80       50
               -20  Consumer Surplus
                                                                  Q




      (c) Find the consumers’ surplus when the market is in equilibrium.

Difference between value to consumers and to the market…. Area above price line and under
Demand curve


                                                                                            4
CS   Dq dq  p * q *
         q*

         0



                                            80
     Consumer surplus (CS)                  (100  2 q )dq  pq
                                                     1

                                            0


 100q  4 q 2 
          1            100q  4 q 2 
                                1             80  60  8000  1600  4800  1600
               q 80                q 0



d) Find the producers’ surplus when the market is in equilibrium

Producer Surplus
Difference between market value and total cost to producers… area below price line and
above Supply curve



PS  p * q *   S q .dq
                           q*

                           0


PS  60 80            20  q dQ
                      80

                  0


              
PS  4800   20 q  1 q 2
                     2                80
                                       0


PS  4800   20 (80 )  1 (6400 )   20 0  1 0
                          2                       2


PS  4800  (1600  3200)  4800  1600  3200  3200


(e) What is the measure of Total Surplus?

Total Surplus = CS + PS = 1600 + 3200 = 4800




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