Cost Theory and Applications

Topics to be covered

• Relevant Cost

• Cost functions and cost elasticity

• Cost estimation

• Economies of Scope

• Learning Curve

1. Relevant Costs- costs that vary with alternatives

2. Sunk costs- costs incurred regardless of alternative. Costs that cannot be recovered (no opportunity value)

3. Incremental cost- costs that change with a change in activity level (output level, investment level, etc.)

· Proper cost measure (long run v. short run) must be tied to the duration of the planning horizon.

· long run- all inputs are variable.

· short run- one or more inputs are fixed.

Consider variable, incremental and marginal costsIgnore sunk costsVariable costs depends on planning horizon

Rail vs Motor Carrier

AFC

AVCAVC

AFC

Price

Rail Motor

Minimum Prices should be based on average variable or marginal costs. Of course, actual prices can be higher. In 1980, Congress defined relevant cost as variable costs.

Example of Defining Relevant Costs

What costs are relevant in multi product firms?

Proposed Madison-Chicago train service?

– Chicago-Madison Train

• Variable cost

• Fixed Cost

• Hiawatha Line- common line costs

– Midwest territory- common territory costs

• Amtrak System- common system costs

All services on a train should cover their direct costs + common train costs.All trains on a line should cover their direct costs + common line costsAll lines in a territory should cover their direct costs + common territory costsAll territories in the system should cover their direct cost + common system costs.

Trains

Lines

Territories

Deriving cost functionsSuppose Price of labor = $40 and Price of capital =

$10/250hpShort Run: Y=1 (HP=250), vary X Q STC Avg Cost 1 10(1) + 40(1) = 50 50 2 10(1) + 40(2) = 90 45 4 10(1) + 40(3) = 130 32 6 10(1) + 40(4) = 170 28 16 10(1) + 40(5) = 210 13 29 10(1) + 40(6) = 250 9 44 10(1) + 40(7) = 290 7 50 10(1) + 40(8) = 330 7 55 10(1) + 40(9) = 370 7Long Run Suppose Optimal X = Y 1 10(1) + 40(1) = 50 50 6 10(2) + 40(2) = 100 17 29 10(3) + 40(3) = 150 5 55 10(4) + 40(4) = 200 4 61 10(5) + 40(5) = 250 4 63 10(6) + 40(6) = 300 5 64 10(7) + 40(7) = 350 5 65 10(8) + 40(8) = 400 6

LRTCSRTC

1 2 4 16 1 6 29 55Q Q

Total Cost $/period Total Cost $/period

Short and Long Run Total Costs

SRTC1

SRTC2LRTC

Quantity

$/Yr

LAC=LRTC/QLMC = dLRTC/dQAFC=TFC/QSAC = SRTC/QSAVC= TVC/QSMC= dSRTC/dQ

TFC

TVC

Unit Cost Curves

$/Q

Quantity

SAV1 SMC1SMC2

SAC2

LAC

LMC

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

SMCSMC

SAC

SAC

LAC=LMC

Small RR Large RR

Unit Cost vs. Cost Elasticity

0

1

2

3

4

5

6

7

8

0 5 10 15 20

SRMC2

SRAC2SRMC1

SRAC1

LRAC

$/Q

QUANTITY

Empirical Cost EstimationApproaches

A. Accounting: may involve the following

1. separate fixed and variable costs

2. assign variable portion to output measures, 3. calculate unit costs by dividing assigned cost by output units

4. For a particular product or service, multiple the unit costs by the respective number of units output for that product

B. Engineering

C. Survey or Survivor Techniques

D. Statistical

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

Statistical A. Long Run v. Short Run

Cross sectional v. Time series data

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

Long Run and Short Run Costs

0

5

10

15

20

25

30

0 5 10 15 20

ChryslerFord

200020001998 1998

$/Q

Output (Q)

Technical ChangeAdd 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

Production with Two Outputs--Economies of Scope

• Examples:– Chicken farm--poultry and eggs– Automobile company--cars and trucks– University--Teaching and research

Production with Two Outputs--Economies of Scope

• The degree of economies of scope measures the savings in cost can be written:

– C(Q1) is the cost of producing Q1 alone

– C(Q2) is the cost of producing Q2 alone

– C(Q1Q2) is the joint cost of producing both products

)(

)()()C( SC

2,1

2,121

QQC

QQCQCQ

Production with Two Outputs--Economies of Scope

• Interpretation:– If SC > 0 -- Economies of scope– If SC < 0 -- Diseconomies of scope

Example: Economies of Scopein the Trucking Industry

• Issues– TruckLoad versus Less-than-TruckLoad– Direct versus indirect routing– Length of haul

Example: Economies of Scopein the Trucking Industry

• Empirical Findings– Results

• SC = 1.576 for reasonably large firm

• SC = 0.104 for very large firms

– Interpretation• Combining partial loads at an intermediate location

lowers cost management difficulties with very large firms.

Many network industries exhibit economies of scope, density or scale

From Shapiro and Varian, Information Rules One of the most fundamental features of information goods is that their cost of production is dominated by “first copy costs.” In the language of economics, the fixed costs of production are large, but the variable cost of reproduction are small. This cost structure leads to substantial economies of scale: the more you produce, the lower your average cost of production. But there is more to it than just economies of scale, the fixed costs and the variable costs of producing information have a special structure.

The dominant component of the fixed costs of producing information are sunk cost, cost that are not recoverable if production is halted. … Sunk costs generally have to be paid up front, before commencing production.

The variable cost of information production also have an unusual structure: the cost of producing an additional copy typically does not increase, even if a great many copies are made.

These cost structures characterize the airline industry and others but the information goods is an extreme example.

Lessons in Pricing: Don’t get greedy.Play tough

Airline Industry Hubs with Density Economies

LTL truckers tried it but foundit provided too slow servicein regional markets

Old Route Structures Hub and Spoke

Network Effects

• Demand Based

• Value depends upon how many others in the group use the product.