# 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
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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