Lecture 3 Production and Cost Function Estimationbhhall/e220c/e220c_S05_Lec3.pdf · Lecture 3 Production and Cost Function Estimation ... Spring 2005 Economics 220C 16 Log output-labor

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)


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


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


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

== + += +


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


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


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?


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

= + + += + +


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).


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)


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


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


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


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


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


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


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)


-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


-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


Production function summary

• Dynamics appear to be important• => endogeneity of inputs• Also want to consider selection• Next time


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− = + + + −


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

= + + + −+ + + +

= + + + −= + + + −

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Lecture 3 Production and Cost Function Estimationbhhall/e220c/e220c_S05_Lec3.pdf · Lecture 3 Production and Cost Function Estimation ... Spring 2005 Economics 220C 16 Log output-labor