Lecture 3Production and Cost Function
EstimationBronwyn H. Hall
Economics 220C, UC BerkeleySpring 2005
Spring 2005 Economics 220C 2
Outline
• Production, Cost, and Profit functions– uses
• Data and estimation issues– Panel data– specification– Exit and selection
• parametric• Semi-parametric (Olley-Pakes)
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Why study them?• One piece of supply/demand framework
– Needed for any equilibrium computation– Form influences model (e.g. learning by doing,
networks)• Used to evaluate efficiency effects of policy
– Regulation - increasing returns, cost complementarities
– Mergers – cost reduction, synergies• Productivity analysis
– Impact of deregulation– Impact of public infrastructure– Impact of non-market production externalities
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Functions• Production
– Output = f(inputs, technical efficiency)• Cost
– Dual to production, assuming cost minimization given output
– Cost = f(output, prices, technical efficiency)• Profit
– Profit = Revenue - cost function = f(output, prices, technical efficiency)
– Similar to cost function, unless a demand model used to construct revenue function
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ProductionStart with Cobb-Douglas for firms (or plants)
indexed by i
Properties of estimator of (α,β) depend on the relationship between inputs and disturbance.
Why is this very simple form useful?– First order log-log approx., constant elasticity
• Identification of higher orders sometimes difficult– Corresponds to growth accounting framework– Easy to add additional inputs
or
where
i i i i
i i i i
i i
Q A L Kq a l ka
α β
α βµ ε
== + += +
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Drawbacks to Cobb-Douglas
• Elasticity of substitution always one• All tech change is neutral• Multiproduct firms – merger and antitrust
analysis– Cost synergies of interest– Explore subadditivity of cost function
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Some alternatives
flexible3--Generalized Leontieff (dual)
flexible35 (8 with t)
translog
σ=1/(1+ρ) for all inputs
23CES
1for all inputs
12Cobb-Douglas
Elasticity of substitution
# params if CRS, symmetry imposed
# params (2 inputs)
Functional form
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Variances in productivity
• Empirical facts1.Large variance in productivity ai across firms2.Productivities highly correlated over time (within firm)
• Suggests that input choices might depend on the disturbance
• Sources of dependence– True technology or management differences– Measurement error (inputs or outputs)– External factors (weather, strikes, breakdowns, etc.)
• How do input choices react to these shocks?
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Panel production function
• Now assume we have several periods of data for each firm– Add time dummies– Consider two types of transitory error
(transmitted and not transmitted)
where uit t it it it
it i it it
q l k ue
δ α βα η
= + + += + +
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Production function errorWhat’s in uit = αi+εit = αi+ηit+eit?
– αi = “permanent” differences in firm productivity (perhaps due to market power or varying product mix), known to firm when it chooses both variable and fixed inputs.
– ηit = transitory differences in firm productivity (due to demand or supply shocks), known to firm when it chooses variable inputs, but not fixed (capital) inputs.
– eit = transitory measurement error (the econometrician’s problem, but not the firm’s).
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Measurement problems• Production:
– Output usually sales (turnover or revenue) divided by a price index
• Most plants and firms have multiple output types• Same price for different firms with different product mix• For individual firms, reinterpret result as revenus productivity
– Labor input usually hours or person-years• No quality adjustment, although some exceptions
– Capital aggregates investment of different types at different times using simple depreciation models.
• Errors in quantity measurement usually mean errors in corresponding price (dual forms)
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Endogeneity
• If inputs respond to shocks (ηit or αi), OLS estimates will be biased– more serious for inputs that adjust quickly like labor
and materials• Some solutions
– Use panels and try to remove αi (more later)– Find instruments
• Lagged values of inputs problematic given serial correlation• Prices if you can find variance across firms unrelated to
disturbance
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ExampleSelected large U.S. manufacturing firms, 10 years
of data from 1986 to 1995.– y = log sales – output measure– l = log employment – labor measure– k = log gross P&E – capital measure
yit = λt + αkit + βlit + uitSubtracting labor from both sides of the eq provides an
easy test for scale economies:yit - lit = λt + α(kit – lit)+ (α+β-1)lit + uit
uit= αi+εit
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Log
sale
s
Selected U.S. Manufacturing Firms 1986-1995Log employment
-5 0 5
0
5
10
AmCyAmCyAmCyAmCyAmCyAmCyAmCyAmCy
B&J
B&J
B&JB&J
B&J
B&J
B&JB&J
B&J B&J
ChevChevChevChevChevChevChevChevChevChev
CokeCokeCoke
CokeCoke
Coke
Coke
CokeCoke
Coke
H-DH-D
H-D
H-D
H-D
H-D
H-DH-DH-DH-D
PlazaPlazaPlazaPlazaPlaza
Plaza
P&GP&GP&GP&GP&G
P&GP&G
P&GP&GP&G
StrikerStriker
StrikerStrikerStriker
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Log
sale
s
Selected U.S. Manufacturing Firms 1986-1995Log capital
-5 0 5 10
0
5
10
AmCyAmCyAmCyAmCyAmCyAmCy
AmCyAmCy
B&J
B&J
B&JB&J
B&J
B&J
B&JB&J
B&J B&J
ChevChevChev
ChevChevChevChevChev
ChevChev
CokeCokeCoke
CokeCoke
Coke
Coke
CokeCoke
Coke
H-DH-D
H-D
H-D
H-D
H-D
H-DH-DH-DH-D
PlazaPlaza
PlazaPlazaPlazaPlaza
P&GP&GP&GP&GP&G
P&GP&G
P&GP&GP&G
StrikerStrikerStrikerStrikerStriker
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Log
outp
ut-la
bor r
atio
Selected U.S. Manufacturing Firms 1986-1995Log capital-labor ratio
2 4 6 8
4
5
6
7
AmCy
AmCyAmCyAmCy
AmCyAmCy
AmCy
AmCy
B&J
B&J
B&J
B&J
B&J
B&J
B&J
B&J
B&J
B&J
Chev
Chev
Chev
Chev
ChevChevChevChev
ChevChev
CokeCokeCoke
Coke
Coke
Coke
Coke
CokeCokeCokeH-D
H-D
H-D
H-D
H-D
H-D
H-D
H-DH-DH-D
PlazaPlaza
PlazaPlaza
PlazaPlaza
P&GP&G
P&GP&GP&G
P&G
P&G
P&GP&G
P&G
Striker
Striker
Striker
StrikerStriker
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Panel data estimators
yit-yi,t-1 = (Xit –Xi,t-1)β + uit-ui,t-1
or ∆yit = ∆xitβ + ∆uit
First differences (FE)
yit =a + Xitβ + uit=a + Xitβ + αi+εit
Var(uit) = σα2+σε2Variance components (RE)
yit -yi = (Xit-Xi)β + (uit-ui)Within (FE)
yi = a + Xiβ + ui where i subscript denotes firm means
Between
yit = a + Xitβ + uitTotal
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OLS Production function estimates
.205.319.916.598.549R2
2.123 (<1.00)
0.104 (.000)
0.828 (.000)
--0.182 (.000)
Durbin-Watson
.132.411.150.293.329s.e.
0.176 (.018)
0.310 (.009)
0.237 (.015)
0.556 (.020)
0.525 (.008)
Log(K/L)
-.289 (.020)
-.016 (.005)
-.072 (.014)
-.018 (.007)
-.014 (.003)
LogL
First Diff.Var. Comp.
WithinBetweenTotalsEstimation method
Dep Var=Log(Sales/L) N=582 1986-1995 (T=10)
Var btwn=.086; Var within=.022; γ=0.975Hausman test: 88.1 with 2 df (p=.000)
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-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fixed Effects Distribution
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-1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Random Effect Distribution
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Production function summary
• Dynamics appear to be important• => endogeneity of inputs• Also want to consider selection• Next time
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The Dual• Assume cost minimization given output and prices w for labor and r
for capital (can be done for C-D, CES, translog, etc.)• E.g., Cobb-Douglas:
• Cobb-Douglas unit cost function with CRS:
• When can we use the Dual?– Firms face different prices (geography, taxes)– Firm does not choose output level or we have appropriate “demand
shifters” for instruments (or CRS)– All inputs can be varied costlessly or we incorporate adj costs (see
Nadiri, Prusa, Bernstein and co-authors)
1 ii i i ic w r q εα βµ
α β α β α β α β= + + + −
+ + + +
i i i i ic q w rµ α β ε− = + + + −
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Cost functions and input demands
• Deriving cost function assumed competitive factor markets, which implies factor demand equations
• Why not use them? E.g.,
where _ denotes coefficients to be estimated. This model has only one disturbance and is
overdetermined. So we will need to think about how to add more error.
1 1
_ _ _ _ _
_ _ _ _ _
i i i i i
i i i i i
i i i i i
c w r q
l w r qk w r q
α βµ εα β α β α β α β
µ εµ ε
= + + + −+ + + +
= + + + −= + + + −