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Lecture 3Production and Cost Function

EstimationBronwyn H. Hall

Economics 220C, UC Berkeley

Spring 2005

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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, costcomplementarities

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 givenoutput

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(,) depend on therelationship 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 K

q a l k

a

=

= + +

= +

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Some alternatives

flexible3--Generalized

Leontieff (dual)

flexible35

(8 with t)

translog

=1/(1+)for all inputs

23CES

1

for 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 facts

1.Large variance in productivity aiacross firms

2.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 u

it t it it it

it i it it

q l k u

e

= + + +

= + +

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Production function errorWhats in uit= i+it= i+it+eit?

i= permanent differences in firm productivity(perhaps due to market power or varying productmix), known to firm when it chooses both variable andfixed inputs.

it= transitory differences in firm productivity (due todemand or supply shocks), known to firm when itchooses variable inputs, but not fixed (capital) inputs.

eit= transitory measurement error (the

econometricians problem, but not the firms).

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Measurement problems Production:

Output usually sales (turnover or revenue) divided bya 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 atdifferent times using simple depreciation models.

Errors in quantity measurement usually meanerrors in corresponding price (dual forms)

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Endogeneity If inputs respond to shocks (itori), OLS

estimates will be biased more serious for inputs that adjust quickly like labor

and materials

Some solutions Use panels and try to remove i(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= t+ kit+ lit+ uitSubtracting labor from both sides of the eq provides an

easy test for scale economies:

yit- lit = t+ (kit lit)+ (+-1)lit+ uituit= i+it

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Log

sales

Selected U.S. Manufacturing Firms 1986-1995Log employment

-5 0 50

5

10

AmCyAmCyAmCyAmCyAmCyAmCyAmCyAmCy

B&J

B&J

B&JB&J

B&J

B&J

B&J

B&J

B&JB&J

Chev

ChevChevChev

ChevChev

ChevChevChevChev

CokeCokeCoke

CokeCoke

Coke

Coke

Coke

Coke

Coke

H-DH-D

H-D

H-D

H-D

H-D

H-D

H-DH-DH-D

Plaza

PlazaPlazaPlazaPlaza

Plaza

P&GP&G

P&GP&GP&GP&GP&G

P&GP&GP&G

StrikerStriker

StrikerStrikerStriker

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Log

sales

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&J

B&J

B&JB&J

ChevChevChev

ChevChevChevChevChev

ChevChev

CokeCokeCoke

CokeCoke

Coke

Coke

Coke

Coke

Coke

H-D

H-D

H-DH-D

H-D

H-D

H-DH-DH-D

H-D

Plaza

PlazaPlazaPlazaPlaza

Plaza

P&GP&G

P&GP&GP&G

P&G

P&G

P&GP&GP&G

Striker

StrikerStrikerStrikerStriker

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Log

output-laborra

tio

Selected U.S. Manufacturing Firms 1986-1995Log capital-labor ratio

2 4 6 8

4

5

6

7

AmCy

AmCy

AmCyAmCy

AmCyAmCy

AmCyAmCy

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

CokeCoke

Coke

Coke

Coke

Coke

Coke

CokeCokeCokeH-D

H-D

H-D

H-D

H-D

H-D

H-D

H-D

H-DH-D

Plaza Plaza

PlazaPlaza

Plaza

Plaza

P&GP&G

P&GP&G

P&GP&G

P&G

P&GP&G

P&G

Striker

Striker

Striker

Striker

Striker

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Panel data estimators

yit-yi,t-1 = (XitXi,t-1) + uit-ui,t-1

or yit= xit + uit

First differences

(FE)

yit=a + Xit + uit=a + Xit + i+it

Var(uit) = 2

+2

Variance

components (RE)

yit-yi= (Xit-Xi) + (uit-ui)Within (FE)

yi= a + Xi + uiwhere isubscript denotes firm means

Between

yit= a + Xit + uitTotal

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OLS Production function estimates

.205.319.916.598.549R2

2.123(

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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 demandshifters 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 i c w r q

= + + +

+ + + +

i i i i i c q w r = + + +

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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 isoverdetermined. So we will need to think about how toadd 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 q k w r q

= + + +

+ + + +

= + + +

= + + +

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