HC2 Cost Functions

7/27/2019 HC2 Cost Functions

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Transport economics and managementCost functions

Eric Pels

[email protected]


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TEM2

Volkskrant, 21/2/2013

Privatiseren spoor is kapitaalvernietiging op kosten vanbelastingbetaler

Rail privatisation is capital destruction at the expense of the taxpayer.

EU wants to privatise EU rail market by 2019; members have topartition national network and tender the parts regularly.

Members have to facilitate equipment of winning companies.

Cost aspects?


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TEM3

This lecture

Cost functions.

Theory of cost.

Scale effects

Empirics

Examples

Learning objective:

Understand assumptions underlying cost functions

Understand how cost functions are applied in transport related

studies


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TEM4

Introduction

Factors of production for bus company:

Land (raw materials) e.g. fuel Labor e.g. drivers

Capital (man-made resources) e.g. buses Machines

Computer systems

Financial capital

Entrepreneurship e.g. ownership/

management?

Risk-taking; organization of other factors


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TEM5

Cost function

Choose production factors so that costs are minimized

Produce output Q (e.g. Q=L

K

) Use e.g. production factors:

Labour L at price w

Capital K at price r

Minimize: C=w*L+r*K Subject to: target level Q can be produced from (K,L)

Cost function C=C(Q,r,k): mimimum cost of producing Q

given input prices, using optimal levels of (K,L).


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Cost function

TEM6 capital

labour

Iso-cost: L=(C-r*K)/w

Production: Q=f(L,K)

Isoquant: L=f(Q,K)

L*

K*

Same cost, lower output


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Cost function

Cost minimization: slope iso-cost = slope isoquant

r/w=L/K

Can be solved explicitly when f(K,L) is specified.

C=C(Q,w,r)

Increasing in Q

Non-decreasing in w,r

C(Q,x*w,x*r)=x*C(Q,w,r)

Application of C=C(Q,w,r) (implicitly) assumes cost

minimization!


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Costs

Fixed costs

Fixed costs for transport company?

Variable costs

Outsourcing transforms fixed into variable costs.

Marginal costs: change in TC (VC) resulting from a unit

change in output.

TC/Q; TC=total cost, Q=output


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Economies of scale

Cost function: C(Q,w,r)

Average cost (AC): C/Q

Marginal cost (MC): C/Q

Elasticity of cost with respect to output:

(C/Q)*Q/C = (C/Q)/(C/Q) = MC/AC

MC < AC: economies of scale or increasing returns MC > AC: diseconomies of scale or decreasing returns

MC=AC: No economies of scale (constant returns to scale)


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Average costs

Output


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Sources of economies of scale

Technical economies of scale

E.g. aircraft size.

Managerial economies of scale Producers with good reputation attract expensive but efficient

management. Fewer workers necessary, large output.

Marketing economies of scale

Large producers can negotiate favorable contracts with suppliers Marketing effort spread over large output

Financial economies of scale

Large producers perceived to have lower risk


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Sources of diseconomies of scale

Red tape

Paper work and coordination effort increases with size ofproducer.

Communication problems

Chain of command becomes longer as producer grows.


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Economies of size in (transport) networks

Transport companies have networks

Measures of size of company

Outputs (passengers, seats, tons of freight, passengerkilometers,

seatkilometers, tonkilometers, trainkilometers etc.)

Network size (e.g. points served)

Two measures are used in the empiricalliterature to

analyze economies of size for network companies: economies of scale

economies of density


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Economies of density

Economies of density:

average costs are reduced when output is increased by

using existing capital more extensively (M+G, p. 76).

the reciprocal of the elasticity of total cost with respect to

output, with all other variables (including points served,

average load factor and input prices) held fixed (Gillen et

al., 1990, Caves et al., 1984).


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Economies of density

Empirical literature may be confusing

Economies of density estimated using (C/Q)*Q/C = MC/AC

Theoretical definition of economies of scale

but this uses the elasticity of cost with respect to output!

Economies of scale focuses on physical network size (e.g.

number of points served in network)

1, where y

Q

C Q MC RTD

Q C AC


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Economies of scale

In applied transport literature, economies of scale focuses

on physical network size

defined as the reciprocal of the sum of the cost elasticities

of the output and points served, with all other variables,

including average load factor, held fixed.

Easy interpretation: When we increase the number ofpassengers and destinations in our network, the average

cost per passenger decreases.

1 , where (P points served)p

Q p

C PRTSP C


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But

Can we keep average load factor constant?

B

C

A

150

pax

150

pax

300 pax, 3 stations

B

C

A

150

pax

100

pax

150

pax

400 pax, 4 stations

D

+1

station


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Applications of cost functions

Description of production technology (economics)

E.g. scale economies Efficiency analysis

Cost minimization

Firm with lowest cost is peer.

Technological change

Add trend variable t to C; C/t is technological change


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TEM19

Specifications

Cobb-Douglas specification:

C=Qwr

lnC=ln+lnQ+lnw+lnr

Economies of scale parameter:

Marginal cost: C/Q=Q-1

w

r

Average cost: C/Q=Q-1wr

MC/AC=C/Q*Q/C= (lnC/lnQ=)

1: diseconomies of scale


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TEM20

Specifications

Short-run Cobb-Douglas specification:

Assume capital is fixed:

lnC=ln+lnQ+lnw+lnK

Amount of fixed capital is explanatory variable! (Ratherthan price of capital)


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TEM21

Specifications

Translog specification:

lnC=0+*(lnQ-lnQ*)+*(lnw-lnw*)+*(lnr-lnr*)+

(1/2)**(lnQ-lnQ*)*(lnQ-lnQ*)+

(1/2)**(lnw-lnw*)*(lnw-lnw*)+

(1/2)**(lnr-lnr*

)*(lnr-lnr*

)+*(lnQ-lnQ*)*(lnw-lnw*)+*(lnQ-lnQ*)*(lnr-lnr*)

Often used in literature.

Standardization.


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TEM22

Specifications

Translog specification:

lnC/lnQ= +2**(lnQ-lnQ*)+*(lnw-lnw*)+*(lnr-lnr*)

Flexible functional form: no a-priori restrictions on

parameters economies of scale dependent on output level


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TEM23

Estimation

Cobb-Douglas: OLS

Translog: OLS or more complicated (SUR)


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TEM24

Example

The cost of air service fragmentation (Tolofari,

Ashford and Caves, 1995) Different airports around London

Capacity problem: congestion

Other airports have surplus capacity

Cost implication of moving flights from largeairport to small


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TEM25

Example

Sample:

7 airports controlled by BAA 1 company; same accounting principles

12 years (1975/76-1986/87): trend variable

necessary

Translog

Standardization: mean


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TEM26

Example

Variables:

Output: WLU ( person = 100kg freight)

Input prices: Labour: labour cost / #employees

Equipment: equipment cost / net value of airport property(value depreciation sales of asset)

Residual: (operational cost labour equipment) / net valueof airport property

Operating characteristics: passengers per ATM, %international traffic, capital stock, capacity utilisation

Heathrow dummy


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TEM27

Example of results table

coefficient Variable Parameter

value

t-value

y Output 0.4459 11.09

LP Labour 0.4984 34.36

EP Equipment 0.1543 31.177

RP Residual

factors

0.3474 35.3970

YY *output2 0.2153 8.1634

50 other coefficients

R2adjusted: 0.99


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TEM28

Example

Some results

: Economies of scale for average airport

Airport specific values: LHR: 0.47; LGW: 0.51; STAN:0.27

What is the cost implication of moving flights from large

airport (LHR) to small (STAN or LGW)?

20.2153

ln ... 0.4459 ln ln 7.2922 ln ln 7.2922 ...2

vC y y

7.2922

ln0.4459

ln

v

y

C

y


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TEM29

Efficiency analysis, benchmarking

Cost minimization: minimize cost to produce given

output level.

Benchmarking: comparing business (performance

metrics) to industry bests.

Various tools

Partial indicators (e.g. labour productivity)

Frontier analysis

Cost frontier

Production frontier


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TEM30

Applied literature, scale effects, summary.

Economies of

scale (density)

Focus Most popular

specification

Remarks

Rail Yes U.S. (freight)Europe

(passengers)

Translog Popular topicPolicy oriented

Airlines Yes U.S. Translog Deregulation

Buses/Urban

transport

Yes Asia Translog Often public

Motor

carriers/trucking

companies

Mixed results U.S. Translog Small

companies

eos?

Airports Yes U.K. Translog 1 study


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TEM31

Rail

Economies of density found in literature

What is a potential effect of partitioning a rail network

on cost per passenger?


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Regression - OLS

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ln(C)

ln(X)

observations

regression line:lnC=K+a*ln(X)errors (v)

Determine K and

a such that (v)2

is minimized


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Regression - COLS

TEM33

ln(C)

ln(X)

Corrected OLS; error term gives deviation from

minimum cost.

OLS curve

COLS curve

Most efficient observation (firm)


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TEM34

Frontier analysis

OLS:

ln(C)=K+a*ln(X)+v

v has normal distribution

Stochastic frontier:

ln(C)=K+a*ln(X)+u+v

v has normal distribution

u is non-negative error termv+u: compound error term (composed error model)


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TEM35

Frontier analysis

ln(C)=K+a*ln(X)+u+v

C=K*Xa*eU+V

Efficiency coefficient:

EC = CMIN/C = K*Xa*ev/K*Xa*eu+v

=e-u

Same output can be obtained at fraction EC of costs

Estimation of cost frontier:

Maximum likelihood (freeware: FRONTIER)

Interpretation: coefficients as with OLS; efficiency coefficients


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TEM36

Example

Efficiency of European railways (Cantos and Maudos,2001)

Cost inefficiency of regulated railway companies inEurope, 1970-1990 1991: change in accounting principles

Method: translog, stochastic frontier

Data: 16 companies, 21 years

Some missing values


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TEM37

Example

Variables:

Outputs: passengerkilometers, tonkilometers

Price of labour

Price of energy

Price of materials

Average levels of cost efficiency

1970 1975 1980 1985 1990 1970-

1990

NS 0.95 0.91 0.94 0.93 0.93 0.92

Average 0.88 0.87 0.89 0.90 0.87 0.87


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Summary

Cost function: assumes cost minimization

Economies of scale/density: average costsdecrease as output increases

Applications: Cost function estimation

Scale effects important in transport sector

Pricing, mergers

Benchmarking

Documents

HC2 Cost Functions