MA ECONOMICS ENTRANCE; DSE ; NUMERICALS TO DERIVE COST FUNCTIONS PRODUCTION FUNCTIONS
HC2 Cost Functions
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Transcript of HC2 Cost Functions
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Transport economics and managementCost functions
Eric Pels
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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
TEM32
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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38
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