Economics Letters 124 (2014) 367–369

Contents lists available at ScienceDirect

Economics Letters

journal homepage: www.elsevier.com/locate/ecolet

An empirical input allocation model for the multiproduct firm✩

James L. Seale Jr. ∗, Ekaterina Vorotnikova, Serhat AsciFood and Resource Economics Department, University of Florida, United States

h i g h l i g h t s

• We derive a linear parameterization of an input allocation model.• We focus on the multiproduct firm that maximizes revenue before maximizing profit.• We empirically formulate an input allocation model using differential framework.

a r t i c l e i n f o

Article history:Received 16 May 2014Received in revised form17 June 2014Accepted 20 June 2014Available online 26 June 2014

JEL classification:D2D4L1L2

Keywords:Multiproduct firmInput allocationDifferential joint production

a b s t r a c t

Laitinen (1980) derives an input allocation model for a multiproduct firm that first maximizes revenueand second maximizes profit. While theoretically elegant, the model has never been formulatedempirically because of the complexity of the model’s price-deflated terms. This paper derives the linearparameterization of the input allocation model that can be used for empirical estimation.

© 2014 Published by Elsevier B.V.

1. Introduction

Using the differential approach, Barten (1964) and Theil (1965)derive a general consumer demand system that, with proper as-sumptions about its parameters, may be empirically estimated.Theil (1977) extends the differential model from consumption toproduction. Laitinen and Theil (1978) pioneer a theory of the mul-tiproduct firm in a differential model that handles multiple inputsand outputs in a production process. Laitinen (1980) compre-hensively covers production theory and develops input allocationmodels among others for the multiproduct firm that either firstminimizes cost or maximizes revenue prior to maximizing profit(Laitinen, 1980, p. 93). Livanis and Moss (2006) derive empirical

✩ In Memory of Kenneth C. Laitinen.∗ Correspondence to: P.O. Box 110240 MCCB Gainesville, FL 32611-0240, United

States. Tel.: +1 352 256 5917; fax: +1 352 392 3646.E-mail addresses: jseale@ufl.edu (J.L. Seale Jr.), vorotnikova@ufl.edu

(E. Vorotnikova), sasci@ufl.edu (S. Asci).

http://dx.doi.org/10.1016/j.econlet.2014.06.0210165-1765/© 2014 Published by Elsevier B.V.

models of input-demand and output-supply with quasi-fixed in-puts, but they do not present an empirical input allocation model.

There are numerous empirical studies using the differentialapproach to consumption theory.However, there are fewempiricalstudies that use the differential framework for the multiproductfirm. Notable exceptions are Clements (1980) who estimatesan output-supply model and Rossi (1984) who estimates bothoutput-supply and input-demand models. However, no one todate has derived an empirical formulation of Laitinen’s inputallocation model. This is in spite of the rich theory of themultiproduct firm and the importance of allocation models asbeing close to the foundations of economics (Barten, 1993).Essentially, empirical application is hindered by the complexity ofthe theoretical model’s functional form. This paper focuses on theinput allocation of the revenue-maximizing firm and reformulatesthe input allocation model into a linear functional form that, withproper parameterization, can easily be estimated empirically. Themodel has empirical application in estimating demand for inputsinto production, composition of imports by country of origin,investment portfolio’s, and agricultural acreage allotment (Barten,1993).

368 J.L. Seale Jr. et al. / Economics Letters 124 (2014) 367–369

2. Input allocation model

The input allocation equation for the ith input of the revenuemaximizing firm is (Laitinen, 1980, p. 85)

fid(ln qi) = θ̄id(lnQ )− ψ̄

j

θ̄ijdlnwj/P ′j

W ′′/P ′

(1)

where fi = wiqi/

iwiqi is the ith input’s share of total costs; qiis the quantity of the ith input (i = 1, 2, . . . , n); wi is the price ofthe ith input; d(lnQ ) =

i fid(ln qi) is a Divisia input quantity

index; d(ln P ′j) =

r θ̄jrd(ln pr) is a Frisch output price index

where pr is the price of the rth output (r = 1, 2, . . . ,m); d(ln P ′) =j θ̄jd(ln P ′j) is also a Frisch output index; and d(lnW ′′) =i θ̄id(lnwi) is a Frisch input price index. Other terms of themodel

are θ̄i =

r θ∗r θ

ri such that

i

j θ̄ij =

i θ̄i = 1 where θ̄ij arenormalized coefficients and θ∗

r =

i θ̄iθ̄ir , θ̄

ir = ∂(przr)/∂(wiqi)

is the revenue the firm gains from additional production of therth product for an additional dollar’s worth of the ith input, θ ri =

∂(wiqi)/∂(przr) is the additional expense of the ith input used inthe production of an additional dollar’s worth of the rth output,and

i θ

ri = 1.

Additionally, γψ∗= ψ̄ where γ = R/C̄ is the reciprocal of

the elasticity of revenue with respect to a proportionate change ininputs, R is total revenue, and C̄ =

i qi(∂R/∂qi) is the shadow

cost with each input evaluated at a shadow price equal to itsmarginal revenue, in which fi = (qi/C̄)(∂R/∂qi) becomes theshare of the ith input in shadow costs. The coefficient ψ∗ satisfiesψ∗

≥ ψ/(γ − ψ) > 0 where ψ is defined by ι′nF(F − γH)−1Fιnwhere F is an n×n diagonalmatrix containing the factor shares, fis,along its diagonal, and H =

∂2h/(∂ ln q∂ ln q′)

n×n is the matrix

of second derivatives of the production function (i.e., h(q, z) = 0)with respect to ln q, an n × 1 vector containing the ln qis.

The price term expression of Eq. (1) represents a puresubstitution effect, defined in terms of output deflated inputprice changes. It is important to note that changes in both inputand output prices affect the input allocation decision of themultiproduct firm.

3. A linear input allocation model

The purpose of this section is to simplify Eq. (1) in such a waythat it may be written in linear form. To begin, it is convenient towrite Eq. (1) as the sum of three terms

fid(ln qi) = θ̄id(lnQ )− ψ̄

j

θ̄ijdlnwj

W ′′

− ψ̄

j

θ̄ijdln

P ′

P ′j

. (2)

The second term to the right of the equal sign in Eq. (2),−ψ̄

j θ̄ijd(lnwj/W ′′), can be written as the sum of two terms,

that is,

−ψ̄

j

θ̄ijd(lnwj)−

j

θ̄ijd(lnW ′′)

. (3)

Using

j θ̄ij = θ̄i, these terms simplify to −ψ̄(

j θ̄ijd(lnwj) −

θ̄id(lnW ′′)). Next, using d(lnW ′′) =

j θ̄jd(lnwj), the two termscan be written as

−ψ̄

j

θ̄ijd(lnwj)− θ̄i

j

θ̄jd(lnwj)

= −ψ̄

j

(θ̄ij − θ̄iθ̄j)d(lnwj). (4)

The next step is to simplify the third term to the right of theequal sign in Eq. (2), −ψ̄

j θ̄ij(d(ln P ′) − d(ln P ′j)). When we

substitute d(ln P ′) =

j θ̄jd(ln P ′j) in this expression, it becomes

−ψ̄

j

θ̄ij

j

θ̄jd(ln P ′j)+ ψ̄

j

θ̄ijd(ln P ′j). (5)

By again using

j θ̄ij = θ̄i, the first term of expression (5) yields1

−ψ̄

j

θ̄ij

j

θ̄jd(ln P ′j) = −ψ̄

j

θ̄iθ̄jd(ln P ′j). (6)

By combining this termwith the second termof expression (5), andrearranging the order, we have

ψ̄

j

(θ̄ij − θ̄iθ̄j)d(ln P ′j). (7)

Inserting d(ln P ′j) =

r θ̄jrd(ln pr), the expression becomes

ψ̄

j

(θ̄ij − θ̄iθ̄j)r

θ̄ jrd(ln pr). (8)

Next, decompose expression (8) into two terms,

ψ̄

j

θ̄ijr

θ̄ jrd(ln pr)− ψ̄

j

θ̄iθ̄jr

θ̄ jrd(ln pr). (9)

The first term of expression (9), ψ̄

j θ̄ij

r θ̄jrd(ln pr), can be

simplified to ψ̄

r

j θ̄ijθ̄

jr

d(ln pr).2 Just as

j θ̄jθ̄

jr = θ∗

r ,

define

j θ̄ijθ̄jr = θ∗

ir such that the first term of expression (9)equals ψ̄

r θ

∗

ird(ln pr) where θ∗

ir is the revenue the firm gainsfrom additional production of the rth product for specific input i.The second term of expression (9), −ψ̄

j θ̄iθ̄j

r θ̄

jrd(ln pr), can

be rewritten by taking θ̄i outside of the summation and θ̄j insideof the summation, so that it becomes −ψ̄ θ̄i

r

j θ̄jθ̄jrd(ln pr).

Since by definition,

j θ̄jθ̄jr = θ∗

r , the expression becomes−ψ̄ θ̄i

r θ

∗r d(ln pr) = −ψ̄

r θ̄iθ

∗r d(ln pr).

Combining the two simplified terms derived from expression(9), we obtain

ψ̄r

θ∗

ird(ln pr)− ψ̄r

θ̄iθ∗

r d(ln pr)

= ψ̄r

(θ∗

ir − θ̄iθ∗

r )d(ln pr). (10)

By substituting expressions (4) and (10) into Eq. (2), the inputallocation equation becomes

fid(ln qi) = θ̄id(lnQ )− ψ̄

j

(θ̄ij − θ̄iθ̄j)d(lnwj)

+ ψ̄r

(θ∗

ir − θ̄iθ∗

r )d(ln pr). (11)

1 In explicit form, −ψ̄

j θ̄ij

j θ̄jd(ln P ′j) = −ψ̄[θ̄i1(θ̄1d(ln P ′1) + · · · +

θ̄nd(ln P ′n)) + · · · + θ̄in(θ̄1d(ln P ′1) + · · · + θ̄nd(ln P ′n))]. By rearrange-ment, the expression can be written as −ψ̄

θ̄1θ̄i1 + · · · + θ̄in

d(ln P ′1)+ · · ·+

θ̄nθ̄i1 + · · · + θ̄in

d(ln P ′n)

= −ψ̄

θ̄1

j θ̄ijd(ln P ′1)+ · · · + θ̄n

j θ̄ijd(ln P ′n).

Using

j θ̄ij = θ̄i , this simplifies to Eq. (6).2 In explicit form, ψ̄

j θ̄ij

r θ̄

jrd(ln pr ) = ψ̄[θ̄i1(θ̄

11 d(ln p1)+· · ·+ θ̄1md(ln pm))

+· · ·+θ̄in(θ̄n1 d(ln p1)+· · ·+θ̄nmd(ln pm))]. Further rearrangement and simplification

lead to ψ̄[(θ̄i1θ̄11 + θ̄i2θ̄

21 + · · · + θ̄inθ̄

n1 )d(ln p1) + · · · + (θ̄i1θ̄

1m + θ̄i2θ̄

2m + · · · +

θ̄inθ̄nm)d(ln pm)] = ψ̄

r (

j θ̄ijθ̄jr )d(ln pr ).

J.L. Seale Jr. et al. / Economics Letters 124 (2014) 367–369 369

Define πij = −ψ̄(θ̄ij − θ̄iθ̄j) and π∗

ir = ψ̄(θ∗

ir − θ̄iθ∗r ). Then the input

allocation model can be written in linear form as

fid(ln qi) = θ̄id(lnQ )+

j

πijd(lnwj)+

r

π∗

ird(ln pr). (12)

The adding-up conditions are

i θ̄i = 1,

i πij = 0, andi

r π∗

ir = 0. The homogeneity conditions are

j πij = 0 andr π

∗

ir = 0, and the symmetry restrictions are πij = πji ∀i, j. Notethat it is not generally necessary for the n × m matrix π∗

= [π∗

ir ]

to obey symmetry.

4. An empirical input allocation model

To parameterize equation (12) for empirical estimation, simplyassume θ̄i,πij andπ∗

ir are constants to be estimated and add an errorterm εi such that the empirical model is

f̄itdqit = θ̄idQt +

j

πijdwjt +

r

π∗

irdprt + εit (13)

where f̄it = (fi,t+fi,t−1)/2; dxt = ln xt−ln xt−1 with x representingq, w and p; dQt =

i f̄itdqit ; and ε ∼ N(0,6). The adding-up

conditions are maintained naturally in the model. Homogeneityconditions on π = [πij] and π∗ as well as symmetry conditionson π can be imposed, and these restrictions can be tested withlog-likelihood-ratio tests. Because of the adding-up conditions, thecovariance matrix, 6, is singular. To estimate the model, drop any

one equation and estimate the remaining n− 1 equations (Barten,1977). The parameters θ = [θi], π and π∗ can be estimatedby maximum likelihood using, for example, the scoring method(Harvey, 1990, p. 134). Alternatively, one can estimate the modelwith iterative seemingly unrelated regression (SUR),which iteratesto maximum likelihood (Kmenta and Gilbert, 1968).

References

Barten, A.P., 1964. Consumer demand functions under conditions of almost additivepreferences. Econometrica 32, 1–38.

Barten, A.P., 1977. The systems of consumer demand functions approach: a review.Econometrica 45, 23–50.

Barten, A.P., 1993. Consumer allocation models: choice of functional form. Empir.Econom. 18, 129–158.

Clements, K.W., 1980. An aggregative multiproduct supply model. Eur. Econ. Rev.13, 239–245.

Harvey, A., 1990. The Econometric Analysis of Time Series, second ed. TheMIT Press,Cambridge, MA.

Kmenta, J., Gilbert, R.F., 1968. Small sample properties of alternative estimators ofseemingly unrelated regressions. J. Amer. Statist. Assoc. 63, 1180–1200.

Laitinen, K., 1980. A theory of the multiproduct firm. In: Theil, H., Glejser, H. (Eds.),Studies in Mathematical and Managerial Economics, vol. 28. North Holland,New York.

Laitinen, K., Theil, H., 1978. Supply and demand of themultiproduct firm. Eur. Econ.Rev. 11, 107–154.

Livanis, G., Moss, C.B., 2006. Quasi-fixity and multiproduct firms. Econom. Lett. 93,228–234.

Rossi, N., 1984. The estimation of the product supply and input demand by thedifferential approach. Am. J. Agric. Econ. 66, 368–375.

Theil, H., 1965. The information approach to demand analysis. Econometrica 33,67–87.

Theil, H., 1977. The independent inputs of production. Econometrica 45,1303–1327.