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ch7

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									       Chapter 8 and 9
Cost Theory and Applications
1. Relevant Costs- costs which vary over alternatives of a
   decision
2. Sunk costs- costs incurred regardless of alternative action.
   Also, cost of purchased resources with no opportunity
   value
3. Incremental cost- change in cost with a change in activity
   level
   · Proper measure (long run v. short run) must be geared to
   the duration of the planning horizon.
   · long run- all inputs are variable (i.e., the flow of resources
   per time period can be changed).
   · short run- the flow of one or more resources are fixed per
   period of time.
4. Alternative use or opportunity cost basis for valuation
4. Traceable v. Nontraceable costs-
   Two types of nontraceable costs
    a. joint costs- costs incurred in the production of two or more
       types of output which are produced in fixed proportion.
         1. passengers above deck and freight below deck on an
           aircraft. Direct aircraft flying expenses are joint between
           passenger services above deck and freight services below
           deck.
         2. Marginal costs may not be precisely determined
    b. common costs- costs incurred in the production of two or
       more types of output in which the outputs can be varied
       seperately.
         1. Refined petroleum products from crude oil
         2. Marginal cost can be precisely determined
5. Implicit costs
Cost Functions
1. See derivation of cost function in Production Theory notes
   Example- If Q = K.5L.5 and PK = PL = 1
    Long Run Total Cost = 2 Q
    If K = 1, short run total costs = 1 + 1 Q2
    If K = 4, short run total costs = 4 + 1/4 Q2
Appendix
        Q = a Lb1 K b2
        C = CL L + CK K


C = CL[b1/(b1+b2)] CK [b1/(b1+b2)]
     [(b2/b1)[b1/(b1+b2)]+(b2/b1)[-b2/(b1+b2)]] (Q/a )[1/(b1+b2)]
    COST FUNCTION FOR COBB-DOUGLAS
  PRODUCTION FUNCTION: Q = X.5 Y.5|PX=PY=1

$/PERIOD                     SHORT RUN
 40
          SHORT RUN          TOTAL COST
 35                          |Y=4
          TOTALCOST |Y = 1                 LONG RUN
 30
                                           TOTAL COST
 25
 20
 15
 10
  5
  0
      0          5           10       15         20
                     QUANTITY / PERIOD
         MARGINAL, AVERAGE TOTAL AND
        AVERAGE VARIABLE COST FUNCTIONS

9
8
                  SMC|Y=1
                                SAVC|Y=1
7
6
5               SAC|Y=1
4
                                            SAC|Y=4
3
2
                                            LMC=LAC
1
0
                 AFC|Y=1
    0       2       4       6        8     10    12
2. Long and Short Run Cost Concepts-
  a. long run: all inputs can be varied
        Returns to Scale v. long run costs
  b. Short run: certain inputs are fixed per time period
    Average fixed cost = AFC = total fixed cost/Q
    Average variable cost = SAVC = total variable cost/Q
    Average total cost = SAC = total cost/Q
    Incremental cost = marginal cost = SMC = d total cost/dQ
3. Cost Elasticity = dC/dQ Q/C = Marginal cost/ average cost
    a. if cost elasticity < 1, economies of scale in LR
                               economies of utilization in SR
    b. if cost elasticity > 1, diseconomies of scale in LR
                               diseconomies of utilization in SR
    c. If long run cost elasticity = short run cost elasticity,
         firm has efficient size plant for that output
    d. Railroad Example:
                         Long Run      Short Run
         Small Railroads .70             .67
         Large Railroads .99             .77
       Small roads have too little output
       Large roads have excess capacity
         Unit Cost vs. Cost Elasticity

$/Q     SRAC1
8
7
6                                     SRMC2
                    SRMC1
5                                         SRAC2
4                                         LRAC
3
2
1
0
    0           5           10   15           20
                    QUANTITY
Factors Producing Scale Economies
•   Specialization of Labor
•   Technological factors
•   Quantity discounts
•   Lower cost of capital
•   Principle of massed reserves
•   Principle of multiples
Breakeven Analysis:
  Used to examine the profitability of new product lines

 Assume a linear total cost function and a linear total
 revenue curve (completely elastic demand curve):
      Total Revenues = Total Costs
                   P Q = F + V Q implies
      Solving for Q:
      Qbreakeven = F/(P-V)
                 = (Fixed Cost)/(Unit Profit Contribution)
Example: P(Price) = $2 per unit,
         F(Fixed Cost) = $40,000,
         and V (Variable Cost per unit) = $1.20 per unit

Qbreakeven = F/(P-V) = 40,000/(2-1.2)= 50,000
                    Breakeven Analysis
$ of Revenue and Cost
400
                                                 Total Revenue
350

300
                                                         Total Cost
250                                          Profit
200

150

100

 50
                                                        Output
  0
      0   20   40    60   80   100   120   140    160   180   200
                    QBE
                    Operating Leverage-
   extent to which fixed production facilities are used in
       the operation to lower cost and increase risk

1. Degree of Operating Leverage (DOL) = the percentage
   change in profits with a percentage change in output
   A measure of risk of a more capital intensive production process
2. DOL =     total profit contribution/total profits
       =     Q (P - V) / [Q (P - V) - F]

3. Example: P = 2, F = 40,000, V = 1.2, then at Q = 100,000
  DOL = 100,000(2-1.2)/[100,000 (2-1.2) - 40,000]
       = 80,000/40,000 = 2
           Degree of Operating Leverage

400
                                                 Total Revenue
350

300
                                                        Total Cost
250                                          Profit
                          DOL = 2
200

150

100

 50

  0
      0   20   40    60   80   100   120   140    160   180   200
                    QBE
 Degree of Operating Leverage Comparison
     of Two Cost Functions at Q = 100
400
                                                 Total Revenue
350
                                                         DOL = 2
300
                                                        Total Cost=
250                                          Profit     40 + 1.2 Q
200                                                      Total Cost=
150                                                      100 + .6 Q
100                                               DOL= (2-.6)100/40
 50                                                  = 3.5
  0
      0   20   40    60   80   100   120   140    160   180   200
                    QBE
Empirical Cost Estimati
Approaches
 A. Accounting
 B. Statistical
 C. Engineering
 D. Survey or Survivor Tech
      A. Accounting: may involve the following

1. separate fixed and variable cost components
2. assignment of variable portion to output measures,
   input measures, quality measures, etc.
3. obtain unit costs by dividing the cost assigned to any
   category by the number of units
4. To estimate the cost for a particular product or service,
   multiple the unit costs by their respective number of units
   output, input, etc
                      Accounting Costing
                                              All Costs   100

                             variable costs               Fixed Costs         Costs Unrelated to
                                                                                 cost study
                            80                       10                       10
    Output #1                 Output #2                   Output #3


   Units of #1   10           Units of #2     5           Units of #3     3

    Costs #1    40             Costs #2     25                Costs #3   15

Cost per unit of #1       Costs per unit of #2      Costs Per unit of #3

     40/10=4                   25/5=5                     15/3=5
Cost = F + v1 X1 + v2 X2 + v3 X3
           10         4              5                    5
                 Statistical
A. Long Run (Planning) v. Short Run (Operating)
 Cross sectional data v. Time series data
  Cross sectional: data gathered on a number of
 individuals at approximately the same point in
 time
 Time Series: data gathered on a single individual
 at different points in time
 Long Run and Short Run Costs
30

25       Chrysler
                           Ford
20

15
                                  199
10             1994               4
 5

 0
     0          5     10    15          20
                 B. Requirements
1. Output Matching - example of deferred
   maintenance in RR
2. Uniform production with a time period
3. no technological change- might add a time
   variable to the regression equation
4. no changes in factor prices or inflation
   1. deflate by a price index
   2. reconstruct costs based on future prices and
      historic input and output levels
   3. include factor prices in the cost function
                 Output Matching
     Railroad Maintenance of Way and Structures ($000)
12

10

 8

 6

 4

 2

 0
     0       2        4       6        8        10       12

                           Gross Ton Miles (000,000)
          Nonuniform Production
         If 1/2 month at Q=2 and 1/2 month at Q=8,
30
              Theoretical Cost = 10
25
              Actual Cost = 16
20

15

10

 5

 0
     0            2             4            6       8
        With Technical Change
We can add a time term to the Cost Function

  ln Cost = b0 + b1 ln Q + b2 t + e
  Q = output
  t = time
  b2 = the percentage change in cost per year
      Adjust for Factor Prices
• Deflate by a price index
  – Cost / CPI = b0 + b1 Q
• Include factor prices in the Cost Function
  – ln Cost = b0 + b1 ln Q + b2 ln PL
                  + b3 ln PF + b4 ln PK + e
• Reconstitute Costs based on future prices
      Reconstituted Costs
          Q= L.5 K.5
PL   PK   Labor   Cap.   Out.   Cost   Rev.
10   10   10      10     10     200    210
12   12   8       8      8      192    168
20   10   8       16     11.3   320    240
18   9    5       10     7.1    180    150
15   10   6       12     8.5    210    180
12   9    9       12     10.4   216    216
      Recorded and Revised Costs
350
              Recorded Costs vs. Output
300
250       Reconstituted Cost vs. Output
200
150
100
 50
  0
      0        2        4        6        8   10   12
Biases with Cross Sectional Data
16
                 A cross section of plants but each one is of a
14               different vintage. So the decrease in costs may
12               be due to the technology of the plant rather than
                 to economies of scale.
10
 8
 6
 4
 2
 0
     0   1   2   3     4    5     6    7    8     9    10   11   12
C. Common Functional Forms for statistical
  estimation
   1. linear       TC = a + b Q
   2. quadratic    TC = a + b Q + c Q2
   3. Cubic        TC = a + b Q + c Q2 + d Q3
   4. log linear log TC = a + b log Q
   5. log quadratic log TC = a + b log Q + c (log Q)2
   6. Translog     log TC = a + b log Q + c (log Q)2
                    + S di log wi + S S eij log wi log wj
                   + S fi log wi log Q
      where wi is the price of factor i (labor, capital, etc.)
D. Long Run Cost estimation: use cross-
 sectional data

Empirical results of earlier studies- L shaped
 cost functions
     L Shaped Average Cost Curve
 $/Q
12

10
                         Minimum Efficient Size

 8

 6

 4

 2

 0
                                                    Q
     0   1   2   3   4     5   6   7   8   9   10 11 12
           Survivor Technique
Example- steel ingots open hearth Bessimer process
 Firm Size Percent of Industry Capacity Number of Firms
 % ind cap     1930 1938 1951           1930 1938 1951
Very small:
  < .5%         7        6      5        39     29   22
Small:
.5-2.5%       19        13     14        18     13   13
Medium:
2.5-25%       35        45     46         6      7    7
Large:
> 25%         39       36      35         1      1    1
           Engineering Technique
Example: Oil Pipeline
   Throughput = f(diameter of pipe, horsepower of
  engines driving fluids, number of pumping
  stations)
  T2.735 = H D4.735/.01046 or T = k H.37D1.73
 where
 T = throughput
 H = horsepower
 D = diameter of pipe
          Oil Pipeline Long-Run and Short-Run Average Costs

      5

      4
                  Long Run Ave. Cost
      3
                          Short Run Diam. = 10
$/Q




      2
                           Short Run Diam. = 20
      1

      0
                               11

                                    13

                                         15

                                              17

                                                   19

                                                        21
          1

              3

                  5

                      7

                           9




                                Q
          Breakeven Analysis Exam Problems

1. The Ajax Company estimates its fixed cost at $500,000
   and its average variable cost at $2.00 per unit. Ajax
   sells its product at a price of $4.00 per unit.
   a. What is Ajax's current break even output level?
        Q = 500000/(4-2)
   b. What is the firm's cost elasticity at Q = 500,000?
        CE = 2 (500,000)/[2 (500,000) + 500,000]
c. At Q = 500,000, what is the firm's degree of
   operating leverage? Interpret it.
   DOL=(4-2)(500,000)/[(4-2)(500,000)-500,000]

d. What price would yield an average profit
   contribution of 40 percent?
  .4=(P - 2)/P or P=2/.6
2. The Baker Company estimates its fixed cost at
   $1,000,000 and its average variable cost at $4.00
   per unit. The firm's goal is to sell 500,000 units.
   a. What is Baker's break even price?
Profits = (P-4)500,000 - 1000000 = 0
P=6

  b. What is its cost elasticity at Q = 500,000?
Cost Elas. = 4(500,000)/[1,000,000+4(500,000)]
            = 0.67
  c. What price must Baker charge if its average
  profit contribution is to be 60 percent?
(P-4)/P = .6
P = 10

  d. At Q = 500,000 and price = $8, what is Baker's
  degree of operating leverage? Interpret it.
DOL = (8-4)500,000/((8-4)500,000-1000000)
    =2

								
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