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					Readings



                          Readings
   Baye 6th edition or 7th edition, Chapter 5




                    BA 445 Lesson I.7 Minimizing Cost   1
Overview



                 Overview




           BA 445 Lesson I.7 Minimizing Cost   2
Overview


   Productivity Measures including total product and marginal product help firms and
   charities find the right allocation of inputs to produce output cheaply. They
   determine the value of buying additional inputs.

   Cost Minimization by firms and charities predicts effects of a crackdown on
   immigration, of outsourcing jobs, of minimum wage legislation, and of
   encouraging handicapped employment.

   Cost Measures including total cost and marginal cost are alternatives to
   productivity measures to help firms chose output to maximize profit. They also
   determine when it is best to shut down production.




                               BA 445 Lesson I.7 Minimizing Cost                       3
Productivity Measures



                Productivity Measures




                        BA 445 Lesson I.7 Minimizing Cost   4
Productivity Measures


   Overview

   Productivity Measures including total product and marginal
   product help firms and charities find the right allocation of
   inputs to produce output cheaply. They determine the
   value of buying additional inputs.




                        BA 445 Lesson I.7 Minimizing Cost          5
Productivity Measures


   Everyone Should Minimize Cost
   •   All firms in all market environments
           Monopoly (1 firm with restricted entry)
           Duopoly (2 firms with restricted entry)
           Oligopoly (a few firms with restricted entry)
           Monopolistic competition (a few firms but with free entry)
           Perfect Competition (many firms with free entry)
   •   Charities
           Toys for Tots should minimize the cost of getting toys to tots
              • They should often solicit cash over donated toys or labor. (Instead
                of having 1000 people each buying 1 toy for $20, people could each
                donate $20 and Toys for Tots could buy more than 1000 toys for
                $20,000 by getting a quantity discount.)
   •   Donors
           Perfectly altruistic people helping kids (not feeling warm and fuzzy)
            should donate money rather than toys or labor, and so minimize their
            own cost of giving. (see
            http://faculty.pepperdine.edu/jburke2/giving.pdf)


                         BA 445 Lesson I.7 Minimizing Cost                            6
Productivity Measures


   • Production Function (allows detailed analysis)
       Q = F(K,L)

            • Q is quantity of output produced.
            • K is capital input.
                – Capital is everything that is not labor.
                – Capital can be disaggregated to, say, machines and
                   land.
                – “Capital” is often rented (used for a time) but not
                   consumed in production.
            • L is labor input.
                – Labor is always rented and not consumed (except for
                   Cannibalism, which is frowned upon in polite society).
            • F is a function relating inputs to a single output.
          F(K,L) is the maximum amount of output that can be
           produced with K units of capital and L units of labor.
            • There is thus an assumption of a type of efficiency implicit in
              the definition of the production function.
                        BA 445 Lesson I.7 Minimizing Cost                       7
Productivity Measures


   Short-Run vs. Long-Run Decisions
   • Some capital or labor is fixed in the short-run.
       Office space may not be adjustable in the short-run.

       Labor with contracts is not completely adjustable in

        the short-run.
          • It takes time to terminate labor.
   • Unless stated otherwise in a homework or exam
     problem, just consider the case where labor is variable in
     both the short-run and long-run, and capital is fixed in
     the short-run but flexible in the long-run.




                        BA 445 Lesson I.7 Minimizing Cost         8
Productivity Measures


   Production Function Algebraic Forms
   • Linear production function: inputs are perfect substitutes.
       Machines and unskilled labor as alternatives on an assembly
        line. Other examples?
                             Q  F K , L   aK  bL
   • Leontief production function: inputs are used in fixed proportions.
       Example: K = an artificial heart, and L = surgeon’s labor.

                             Q  F K , L   min bK , cL
   • Cobb-Douglas production function: inputs have a degree of
     substitutability.
       Example: Producing houses with K = wood and L = labor in U.S.
        vs. India. --- Other examples where labor can substitute for
        materials?
                               Q  F K , L   K L
                                                 a b



                        BA 445 Lesson I.7 Minimizing Cost                  9
Productivity Measures


   • Total Product (TP): maximum output produced with
     given amounts of inputs.
   • Example: Cobb-Douglas Production Function:
     Q = F(K,L) = K.5 L.5
          K is fixed at 16 units in the short-run.
          Short-run Cobb-Douglass production function:
            Q = (16).5 L.5 = 4 L.5
          Total Product when 100 units of labor are used?
            Q = 4 (100).5 = 4(10) = 40 units




                        BA 445 Lesson I.7 Minimizing Cost    10
Productivity Measures


   Average Product of an Input: measures output produced
   per unit of input. (It is a common measure of productivity,
   but it is less useful than marginal product, explained later.)
          Average Product of Labor: APL = Q/L.
            • Measures the output of an “average” worker.
            • Example: Q = F(K,L) = K.5 L.5
                – If the inputs are K = 16 and L = 16, then the average product of
                  labor is APL = [(16) 0.5(16)0.5]/16 = 1.
          Average Product of Capital: APK = Q/K.
            • Measures the output of an “average” unit of capital.
            • Example: Q = F(K,L) = K.5 L.5
                – If the inputs are K = 16 and L = 16, then the average product of
                  capital is APK = [(16)0.5(16)0.5]/16 = 1.




                        BA 445 Lesson I.7 Minimizing Cost                            11
Productivity Measures


   Marginal Product on an Input: change in total output attributable to
   the last unit of an input.
         Marginal Product of Labor: MPL = DQ/DL
            • Measures the output produced by the marginal (the last)
              worker.
            • Slope of the short-run production function (with respect to
              labor).
            • Example: Q = F(K,L) = K.5 L.5
                 – If the inputs are K = 16 and L = 16, then the marginal
                   product of labor is MPL = .5(16) -0.5(16)0.5 = .5.
         Marginal Product of Capital: MPK = DQ/DK
            • Measures the output produced by the last unit of capital.
            • When capital is allowed to vary in the short run, MPK is the
              slope of the production function (with respect to capital).
   • As we will see, MP is the essential measure of input productivity for
      profit maximization and cost minimization.

                        BA 445 Lesson I.7 Minimizing Cost                    12
Productivity Measures


   Increasing, Diminishing and Negative Marginal
   Productivity

                    Increasing     Diminishing   Negative
                Q    Marginal       Marginal     Marginal
                    Productivity   Productivity Productivity




                                                        Q=F(K,L)



                                                             AP
                                                            L
                                                   MP
                        BA 445 Lesson I.7 Minimizing Cost          13
Productivity Measures


   Guiding the Production Process
   In terms of the production function and marginal product, profit
   maximization requires two areas of management:
   • Use the best technology so that, for any input levels (K,L) of capital
       and labor, the firm produces maximal output, F(K,L).
   • Employ the right level of inputs
         In the case where output sells at a competitive price p, to
          maximize profit a manager will vary inputs (if possible) and
          should
            • hire labor until the value of the marginal product of labor
              VMPL = P x MPL equals the wage, VMPL = w.
                – For example, if employing an extra hour of labor earns
                   $12 more revenue and the wage is $8, you should hire
                   that hour.
                – Equilibrium (no more hires, and no less) requires VMPL =
                   w.
            • hire capital until the value of marginal product of capital
              VMPK = P x MPK equals the rental rate, VMPK = r.
                        BA 445 Lesson I.7 Minimizing Cost                     14
Cost Minimization



                    Cost Minimization




                    BA 445 Lesson I.7 Minimizing Cost   15
Cost Minimization


   Overview

   Cost Minimization by firms and charities predicts effects of
   a crackdown on immigration, of outsourcing jobs, of
   minimum wage legislation, and of encouraging
   handicapped employment.




                    BA 445 Lesson I.7 Minimizing Cost             16
Cost Minimization


   Guiding the Production Process
   In terms of the production function and marginal product,
   profit maximization requires two areas of management:
   • Use the best technology so that, for any input levels
      (K,L) of capital and labor, the firm produces maximal
      output, F(K,L).
   • Employ the right level of inputs.




                    BA 445 Lesson I.7 Minimizing Cost          17
Cost Minimization


   Employing the right level of inputs is simplest in the case
   where output sells at a competitive market price p and the
   inputs are chosen to maximize profit a manager will vary
   inputs (if possible) and should
      • hire labor until the value of the marginal product of
        labor              VMPL = P x MPL equals the wage,
        VMPL = w.
               – For example, if employing an extra hour of
                 labor earns $12 more revenue and the wage is
                 $8, you should hire that hour.
               – Equilibrium (no more hires, and no less)
                 requires VMPL = w.
      • hire capital until the value of marginal product of
        capital             VMPK = P x MPK equals the rental
        rate, VMPK = r.

                    BA 445 Lesson I.7 Minimizing Cost            18
Cost Minimization


   Employing the right level of inputs is harder in the case
   where either firms are in non-competitive markets or
   charities are not choosing inputs to maximize profit. In
   those cases, the manager or charity director has two
   choices to make.

   1) Choose the right level of output. We leave aside that
   decision for now.

   2) Choose the right level of inputs that minimize the cost of
   producing the chosen output.




                    BA 445 Lesson I.7 Minimizing Cost              19
Cost Minimization


   Cost Minimization
   To minimize cost, the marginal product per dollar spent
   should be equal for all inputs:
                MPL/w = MPK/r

   • For example, if using an extra dollar to employ 1/w more
     hours of labor yields 12 more units of output (MPL/w =
     12) and if using an extra dollar to employ 1/r more hours
     of labor yields 10 more units of output (MPK/r = 10) , you
     should hire more labor and less capital until you restore
     the cost-minimization equality
                 MPL/w = MPK/r


                    BA 445 Lesson I.7 Minimizing Cost             20
Cost Minimization


   Optimal Input Substitution
       The general case of substitution (holding output constant) when
       there may be more than two production inputs is like the substitution
       effect for consumption goods (holding purchasing power constant).

   Decomposing the Effects of a Rental Rate Decrease in
   Input X
      In the input substitution effect (holding output constant), the rental
       rate decrease in Input X makes the quantity demanded for Input X
       increase, and makes demand
           decrease for Input Y if the two inputs are substitutes (Wood and
            Labor in building houses).
               If there are only two inputs (that is, if the production process
                uses only two inputs, like Capital and Labor), then they must
                be substitutes.
           increase for Input Y if the two are complements (Manuel Labor
            and Shovels).


                         BA 445 Lesson I.7 Minimizing Cost                         21
Cost Minimization


   Example 1: Effects of a Crackdown on Illegal Labor
   Step 1: There are various ways to crack down on illegal labor:
        • decrease the number of illegal aliens successfully crossing the
        boarder
        • decrease the supply of illegal aliens wanting to cross the boarder
          by increasing penalties if they are caught, increase the penalties
          on U.S. producers that employ illegal labor
        • increase the likelihood of catching and penalizing a U.S. employer
          of illegal labor.
   Any one of those crackdowns increases the rental rate of illegal labor.
   Step 2: The input substitution effect increases the demand for
   substitutes (like unskilled domestic labor) and decreases the demand
   for complements (like shovels and interpreters).




                       BA 445 Lesson I.7 Minimizing Cost                       22
Cost Minimization


   Example 2: Effects of Outsourcing Jobs to Other Countries
   Step 1: Outsourcing jobs to India is caused by a decrease
   in the rental rate of Indian labor. (For example, the rental
   rate includes the cost of phone and internet service so that
   Indians can provide customer service in America.)
   Step 2: The input substitution effect of a decrease in the
   rental rate of Indian labor is a decreases the demand for
   substitutes and increases the demand for complements:
       • Demand decreases for domestic substitute labor
           • Customer service, computer programming,
           radiologists.
       • Demand increases for domestic complement labor
           • Managers.


                    BA 445 Lesson I.7 Minimizing Cost             23
Cost Minimization


   Example 3: Minimum Wages
   The input substitution effect of any input cost change is
   always a change in the quantity of input demanded in the
   opposite direction.
      • Increasing minimum wages decreases the quantity
      demanded for unskilled labor, which is often replaced by
      machines (like in McDonalds).




                    BA 445 Lesson I.7 Minimizing Cost            24
Cost Minimization


   Example 4: Jobs for Handicapped Workers
   The input substitution effect of any input cost change is always a
   change in the quantity of input demanded in the opposite direction.
      • Decreasing the cost of hiring handicapped workers increases the
      quantity demanded for handicapped workers. But is that good?
      • Thinking like a perfect altruist, ask: what is best for a handicapped
      person? Consider the following example:
          • Suppose handicapped Mr. H values his time at $9 per hour.
          • Suppose In-N-Out and all other fast-food places value Mr. H’s
          time at $7 per hour up to 40 hours per week.
          • Will Mr. H have a job?
          • Suppose the government subsidizes Mr. H’s employment $3
          per hour. Will Mr. H now have a job?
          • Can you reallocate the subsidy money to make Mr. H
          happier?




                       BA 445 Lesson I.7 Minimizing Cost                        25
Cost Measures



                Cost Measures




                BA 445 Lesson I.7 Minimizing Cost   26
Cost Measures


   Overview

   Cost Measures including total cost and marginal cost are
   alternatives to productivity measures to help firms chose
   output to maximize profit. They also determine when it is
   best to shut down production.




                   BA 445 Lesson I.7 Minimizing Cost           27
Cost Measures


   Preview
   • Cost functions are an alternative to production functions
     in formulating production technology.
      • Bad: They do not contain as much information as
         production functions.
      • Good: Cost functions contain enough information to
         help determine profit-maximizing output under
         various market conditions (monopoly, oligopoly, …)
         discussed later.
      • Good: Cost functions help accounting for cost,
         revenue, and profit.




                   BA 445 Lesson I.7 Minimizing Cost             28
Cost Measures


   • Types of Costs
         Short-Run
           • Fixed costs (FC) is a fixed amount owed for any positive
             output.
                – If capital is fixed in the short-run, then FC equals the cost of
                  capital.
           • Sunk costs is the part of fixed cost that is owed even if output
             is zero.
                – For example, you lease a railroad car for $10,000 for a month,
                  but can recoup $6,000 if you do not use the car.
                – FC = $10,000, and sunk cost = $4,000.
           • Short-run total costs (C)
           • Short-run variable costs (VC). Defined by VC = C – FC.
         Long-Run
           • All costs are variable
           • No fixed costs
                        BA 445 Lesson I.7 Minimizing Cost                            29
Cost Measures


   C(Q): Minimum total cost of
   producing alternative levels of
   output:                       $
      C(Q) = VC(Q) + FC                                C(Q) = VC + FC

   VC(Q): Costs that vary with
                                                                 VC(Q)
   output.

   FC: Costs that do not vary
   with output, as long as output
   is positive.                                                    FC


   Sunk Cost: Cost if output is
   zero.                             0                              Q


                   BA 445 Lesson I.7 Minimizing Cost                     30
Cost Measures


   FC: Costs that do not
   change as output changes.

   Since sunk cost is forever   $
   lost after it has been paid,                            C(Q) = VC + FC
   decision makers should
   ignore sunk costs to                                              VC(Q)
   maximize profit or minimize
   losses.
       • Being rational in your
      personal life is a challenge.
      The extra $120,000 I paid for                                    FC
      my house in 2007 (compared
      with prices 1 year later) is like
      a sunk cost.
                                                                        Q
                       BA 445 Lesson I.7 Minimizing Cost                    31
Cost Measures


   Marginal Cost MC = DC/DQ
   • Marginal cost curves often initially
   decrease with output Q because of
   specialization of labor.
         • Example: Workers at the Malibu
         subway work best when there are        $                 MC
         several customers (Q midsize)
         because the workers specialize.
   • Marginal cost curves eventually
   increase with output when inputs crowd.
         •Example: Workers at the Malibu
         subway work become less
         productive and costs increase if Q
         is so large that workers are
                                                               Decreasing
         crowded.
                                                               productivity
   • Unless a homework or exam problem
                                                               of inputs
   explicitly states otherwise, draw the
                                                               because of
   marginal cost curves U-shaped, as on
                                                               crowding
   the right.
                                              Specializatio
                                              n of Labor

                                                                              Q
                           BA 445 Lesson I.7 Minimizing Cost                      32
Cost Measures


   Average Total Cost
      ATC = AVC + AFC
      ATC(Q) = C(Q)/Q
                                          $
                                                              MC   ATC
   dATC/dQ = (dC/dQ)/Q – C/Q2
            = (MC-ATC)/Q

   From that derivative calculation,
   • MC > ATC implies ATC is increasing
   • MC < ATC implies ATC is decreasing

   (There is an analogy to your
   GPA: If your next grade > GPA,
   your GPA is increasing.)

   Thus, ATC is decreasing until it
   intersects MC, then it is
   increasing.                                                     Q
                          BA 445 Lesson I.7 Minimizing Cost              33
Cost Measures


   Average Variable Cost
      AVC = VC(Q)/Q

   dAVC/dQ = (dVC/dQ)/Q –          $
                                                           MC   ATC
   VC/Q2
           = (MC-AVC)/Q                                          AVC

   From that derivative calculation,
   • MC > AVC implies AVC is
   increasing
   • MC < AVC implies AVC is
   decreasing




                                                                Q
                       BA 445 Lesson I.7 Minimizing Cost               34
Cost Measures


   Summary
   Average Total Cost
      ATC = AVC + AFC
      ATC = C(Q)/Q            $
                                                       MC   ATC
                                                             AVC
   Average Variable Cost
      AVC = VC(Q)/Q

   Average Fixed Cost
      AFC = FC/Q = ATC-AVC

   Marginal Cost                                            AFC
     MC = DC/DQ

                                                            Q
                   BA 445 Lesson I.7 Minimizing Cost               35
Cost Measures


   Recovering Fixed Cost
                                Q0(ATC-AVC)
                                                         MC
           $                    = Q0 AFC                     ATC
                                = Q0(FC/ Q0)                 AVC
                                = FC

          ATC
   AFC            Fixed Cost
          AVC


                                 Q0                            Q


                     BA 445 Lesson I.7 Minimizing Cost              36
Cost Measures


   Recovering Variable Cost

                          Q0AVC                          MC
       $
                                                               ATC
                          = Q0[VC(Q0)/ Q0]
                                                               AVC
                          = VC(Q0)




      AVC
                Variable Cost                 Minimum of AVC

                                Q0                              Q

                      BA 445 Lesson I.7 Minimizing Cost              37
Cost Measures


   Recovering Total Cost
                              Q0ATC
                                                      MC
      $
                              = Q0[C(Q0)/ Q0]                 ATC

                              = C(Q0)                         AVC


    ATC

                Total Cost                          Minimum of ATC




                             Q0                                 Q

                      BA 445 Lesson I.7 Minimizing Cost              38
Cost Measures


   Cubic Cost Function
   • C(Q) = f + a Q + b Q2 + cQ3
   • Marginal Cost?
       Calculus:

         MC(Q) = dC/dQ = a + 2bQ + 3cQ2




                 BA 445 Lesson I.7 Minimizing Cost   39
Cost Measures


   Quadratic Cost Function
         Total Cost: C(Q) = 10 + Q + Q2
         Variable cost function:
                    VC(Q) = Q + Q2
         Variable cost of producing 2 units:
                    VC(2) = 2 + (2)2 = 6
         Fixed costs (all of which are sunk):
                    FC = 10
         Marginal cost function:
                    MC(Q) = 1 + 2Q
         Marginal cost of producing 2 units:
                    MC(2) = 1 + 2(2) = 5




                       BA 445 Lesson I.7 Minimizing Cost   40
Summary



                Summary




          BA 445 Lesson I.7 Minimizing Cost   41
Summary


  Summary of Choosing the Right Level of Inputs
  1) For managers in perfectly-competitive, for-profit
     industries, to maximize profits (including minimize
     costs), must use inputs such that the value of marginal
     of each input equals the price the firm must pay to
     employ the input.
     • For example, w = VMPL = P x MPL
  2) For any industry and for charities, to minimize cost of
     producing any level of output, the marginal product per
     dollar spent should equal for all inputs:
     • For example, MPL/w = MPK/r
   Note: Equation 2) does not determine the level of output, and
    Equation 1) does determine output but only in the special case of
    perfectly-competitive, for-profit industries.

                     BA 445 Lesson I.7 Minimizing Cost                  42
Review Questions


   Review Questions
      You should try to answer some of the review
     questions (see the online syllabus) before the next
     class.
      You will not turn in your answers, but students may
     request to discuss their answers to begin the next class.
      Your upcoming Exam 1 and cumulative Final Exam
     will contain some similar questions, so you should
     eventually consider every review question before taking
     your exams.




                   BA 445 Lesson I.7 Minimizing Cost             43
BA 445                                       Managerial Economics



         End of Lesson I.7




         BA 445 Lesson I.7 Minimizing Cost                     44

				
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