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					Econ 201 Problem Set on Chapter 7

1. Complete the following table (round each answer to the nearest whole number):
                                Total     Variable     Fixed   Marginal    Average      Avg.         Avg. Fixed
                                                                                        Var.
                     Output     Cost        Cost        Cost     Cost        Cost       Cost           Cost
                        0
                        1                                           5
                        2                                                     30
                        3                                                                13
                        4       105                                                                    10
                        5                   110
                        6                                         50

    Solution:
                                Total     Variable     Fixed   Marginal    Average      Avg.         Avg. Fixed
                                                                                        Var.
                     Output     Cost        Cost        Cost     Cost        Cost       Cost           Cost
                        0        40           0          40       –           –          –             –
                        1        45           5          40        5          45          5            40
                        2        60          20          40       15          30         10            20
                        3        79          40          40       19          26         13            13
                        4       105          65          40       26          26         16            10
                        5       150         110          40       45          30         22             8
                        6       200         160          40       50          33         27             7


2.Trisha believes the production of a dress requires 4 labor hours and 2 machine hours to produce.
                If Trisha decides to operate in the short run, she must spend $500 to lease her
                business space. Also, a labor hour costs $15 and a machine hours costs $35.
                What is Trisha's cost of production as a function of dresses produced?

    Solution: Since the production of a dress requires spending $60 for labor and $70 for
              machine hours, Trisha's cost function is: C  q   130q  500.



3. A firm's total cost function is given by the equation:
                              TC = 4000 + 5Q + 10Q2.
                 (1) Write an expression for each of the following cost concepts:
                a.     Total Fixed Cost
                b.     Average Fixed Cost
                c.     Total Variable Cost
                d.     Average Variable Cost
                e.     Average Total Cost
                f.     Marginal Cost
             (2) Determine the quantity that minimizes average total cost. Demonstrate that
                 the predicted relationship between marginal cost and average cost holds.

Solution:
PART (1)
       a.
             TFC  4000


       b.
                     4000
             AFC 
                      Q
       c.
             TVC  TC  TFC
             TVC  5Q  10Q2


       d.
                     TVC 5Q  10Q 2
             AVC                   5  10 Q
                      Q      Q


       e.
                     TC 4000  5Q  10Q2
             ATC      
                     Q          Q


        f.
             MC  5  20Q
PART (2)
      ATC is minimized where MC is equal to ATC.
      Equating MC to ATC
                  4000  5Q  10Q 2
                                     5  20Q
                          Q
                  4000  5Q  102  5Q  20Q 2
                  4000  10Q 2
                  Q 2  400
                  Q  20


      ATC is minimized at 20 units of output. Up to 20, ATC falls, while beyond 20
      ATC rises.
      MC should be less than ATC for any quantity less than 20.
      For example, let Q = 10:
                  MC = 5 + 20(10) = 205
                          4000  510   10 10 
                                                2
                  ATC                              505
                                   10
      MC is indeed less than ATC for quantities smaller than 20.
      MC should exceed ATC for any quantity greater than 20.
      For example, let Q = 25:
                  MC = 5 + 20(25) = 505
                          4000  525   10 25 
                                                 2
                  ATC                              415
                                    25
      MC is indeed greater than ATC for quantities greater than 20.
4.Davy Metal Company produces brass fittings. Davy's engineers estimate the production
              function represented below as relevant for their long-run capital-labor decisions.
                             Q = 500L0.6K0.8,
                 where Q = annual output measured in pounds,
                 L = labor measured in person hours,
                 K = capital measured in machine hours.
                 The marginal products of labor and capital are:
                             MPL = 300L-0.4K0.8      MPK = 400L0.6K-0.2
                 Davy's employees are relatively highly skilled and earn $15 per hour. The firm
                 estimates a rental charge of $50 per hour on capital. Davy forecasts annual costs
                 of $500,000 per year, measured in real dollars.
                 a.   Determine the firm's optimal capital-labor ratio, given the information
                      above.
                 b.   How much capital and labor should the firm employ, given the $500,000
                      budget? Calculate the firm's output

Solution:
            a.
                                             K 0.8
                 MPL  300L0.4 K 0.8  300
                                             L0.4
                                             L0.6
                 MPK  400L0.6 K 0.2    400 0.2
                                             K
                              K 0.8
                          300 0.4       0.8  0.2
                 MRTS        L  0.75 K K
                              L0.6     L0.4 L0.6
                          400 0.2
                              K
                              K
                 MRTS  0.75
                              L
                           w 15
                 Equate to  .
                           r 50
                             K 15
                         0.75 
                             L 50
                             K
                         0.75  0.3
                             L
                 K
                    0.4; K = 0.4L
                 L
            b.
                             C = 500,000
                             C = wL + rK
                             500,000 = 15L + 50K
                 K = 0.4L from optimal ratio
                             500,000 = 15L + 50(0.4L)
                             500,000 = 15L + 20L
                             500,000 = 35L
                                 L = 14,285.71 or 14,286 hours
                 Substitute to solve for K.
                             500,000 = 15(14286) + 50K
                             500,000 = 214,290 + 50K
                             285,710 = 50K
                                 K = 5714.20
                                 or K = 5714
                             Q = 500(14,286)0.6(5,714)0.8
                             Q = 157,568,191

5.A paper company dumps nondegradable waste into a river that flows by the firm's plant. The
              firm estimates its production function to be:
                             Q = 6KW,
                 where Q = annual paper production measured in pounds, K = machine hours of
                 capital, and W = gallons of polluted water dumped into the river per year. The
                 marginal products of capital and labor are given as follows:
                             MPK = 6W            MPW = 6K
                 The firm currently faces no environmental regulation in dumping waste into the
                 river. Without regulation, it costs the firm $7.50 per gallon dumped. The firm
                 estimates a $30 per hour rental rate on capital. The operating budget for capital
                 and waste water is $300,000 per year.
                 a.   Determine the firm's optimal ratio of waste water to capital.
                 b.   Given the firm's $300,000 budget, how much capital and waste water should
                      the firm employ? How much output will the firm produce?

Solution:
            a.
                             MPW = 6K
                             MPK = 6W
                                       6K K
                             MRTS       
                                       6W W
                 Rate of water charge to price of capital:
                              PW 7.5
                                     .25
                              PU   30
                 Equating MRTS to ratio of input prices
                K
                   0.25, K = 0.25W
                W
b.
                C = PWW + PKK
                300,000 = 7.50W + 30K
     recall K = 0.25W
                300,000 = 7.5W + 30(0.25W)
                300,000 = 7.5W + 7.5W
                    W = 20,000 gallons
                K = 0.25W
                K = 0.25(20,000)
                    K = 5000
                Q = 6(5000)(20,000)
                    Q = 600,000,000

				
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