Duality methods for incomplete systems of consumer demand equations are adapted to the dual structure of variable cost functions in joint production. This allows the identification of necessary and sufficient restrictions on technology and cost so that the conditional factor demands can be written as functions of input prices, fixed inputs, and cost. These are observable when the variable inputs are chosen and committed to production, hence the identified restrictions allow ex ante conditional demands to be studied using observable data. This class of production technologies is consistent with all von Neumann-Morgenstern utility functions when ex post production is uncertain.Joint production, variable cost, duality theory

Jeffrey LaFrance

Rulon Pope

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
                                                              
 
    
 
Working Paper Series
WP 2009-02 
School of Economic Sciences
Duality Theory for Variable 
Costs in Joint Production* 
 
By 
 
Jeffrey T. LaFrance and Rulon D. Pope 
 
 
 
 
 
 
January 2009 
 
 
  1 
Duality Theory for Variable Costs in Joint Production 
  Jeffrey T. LaFrance   Rulon D. Pope 
 Washington  State  University  Brigham Young University 
Abstract: Duality methods for incomplete systems of consumer demand equations are 
adapted to the dual structure of variable cost functions in joint production. This allows 
the identification of necessary and sufficient restrictions on technology and cost so that 
the conditional factor demands can be written as functions of input prices, fixed inputs, 
and cost. These are observable when the variable inputs are chosen and committed to 
production, hence the identified restrictions allow ex ante conditional demands to be 
studied using observable data. This class of production technologies is consistent with all 
von Neumann-Morgenstern utility functions when ex post production is uncertain. 
Key Words:    Joint production, variable cost, duality theory 
JEL Classification:    C3, D2, D8 
 
Please address correspondence to: 
Professor Jeffrey T. LaFrance 
School of Economic Sciences 
101 Hulbert Hall, PO Box 646210 
Washington State University 
Pullman, WA 99164-6210 
(509)-334-6079 
jtlafrance@wsu.edu  2 
Duality Theory for Variable Costs in Joint Production 
Analysis of multi-product behavior of firms is common in agricultural economics. Tech-
niques of analysis might be based on the distance or production functions, or profit, reve-
nue, or cost functions (Färe and Primont 1995; Just, Zilberman, and Hochman 1988; 
Shumway 1983, Lopez 1983; Akridge and Hertel 1986). There is a large literature on 
functional structure and duality that helps guide empirical formulations and testing based 
on concepts of non-jointness and separability (Lau 1972, 1978; Blackorby, Primont and 
Russell 1977; Chambers 1984). For example, separability in some partition of inputs or 
outputs often results in separability in a similar partition of prices so long as aggregator 
functions are homothetic (e.g., Blackorby, Primont and Russell 1977; Lau 1978). This 
allows a researcher to test hypotheses about the structure of technology using cost or 
profit functions (Shumway 1983). Similarly, the implications of non-jointness often re-
duce to some form of additivity (Hall 1973; Kohli 1983). Such restrictions on technology 
guide empiricists as they think about aggregation based on functional structure. 
In this short paper an issue of functional structure is considered which is somewhat 
non-standard but useful to empirical work. The question considered is: when can conven-
tional short-run cost minimizing factor demands a)  (,,) = xX w y z  be written as b) 
(,,) c = xX wz  , where  X  and  X   are vector valued functions, w the corresponding vector 
of input prices, z is a vector of fixed inputs, y is a vector of outputs, and c is cost? More 
precisely, what restrictions on technology, and hence costs, imply that the conditional 
factor demands can be written as functions of input prices, fixed inputs, and cost rather 
than the more standard representation in a)?  3 
Interest in answering this question comes from two sources. First, by analogy with 
Gorman’s theory of exact aggregation, if there is cost heterogeneity, it will be natural to 
think of conditional input demands as dependent on c just as consumer demands depend 
on income or expenditure. The second reason is more involved. There is a fairly large 
literature which proposes solutions to the specification of ex ante cost functions when 
output is uncertain under potentially risk-averse behavior (e.g., Pope and Chavas 1994; 
Pope and Just 1998; Chambers and Quiggin 2000; Chavas 2008). The essential problem 
is that if inputs are applied ex ante under stochastic production, then the outputs in a) 
can’t be observed. One approach is to make the assumptions required such that the ex 
ante cost function exists in an empirically convenient form. For example, given random 
supply shocks  i ε of the form 
  [ ] (,, ) , (,, ) |,, 0 , 1 , , , i i ii ii y yyH E H i n εε =+ = = yz yz xyz "  (1) 
the existence of a transformation function,  (,,) 0 F ≤ xyz , defined over variable inputs, x, 
planned outputs,  , y  and fixed inputs, z, then the reasoning in Pope and Chavas (1994) 
implies the existence of a cost function in which  y  replaces  y . That is, minimizing the 
variable cost of planned output yields  { } (,,) m i n : (,,) 0 , cC F = ≡≤ ≥
x
wyz wx xyz x 0
T  for 
all von Neumann-Morgenstern utility functions in both static and dynamic environments. 
The conditional factor demands,  (,,) X wyz, will continue to depend on the unobservable 
variables,  y . However, these input demand functions only depend on (,,) c wz , all of 
which are observable, when a) reduces to b). Thus, the restrictions we seek are those that 
allow ex ante conditional demands to be studied using only observable variables.   4 
To simplify notation by letting y now denote planned output, our main result on the 
dual restriction that is necessary and sufficient for  (,,) X wyz to be written as  ( , , ) c X wz   
i s  (,(,) ,) (,(,) ,) cC F θ θ =⇔ w y zz x y zz. That is, outputs must be weakly separable from 
variable inputs in the joint production technology, or equivalently, outputs must be 
weakly separable from variable input prices in the cost function. Though this result is 
somewhat restrictive in outputs,
1 it is fairly flexible in  and . xz  
Duality and the Main Result 
The neoclassical model of conditional demands for variable inputs with joint production, 
fixed inputs, and production uncertainty is 
  { } (,,) a r g m i n : (,,) 0 , , F =≤ ≥ Xwyz wx xyz x 0
T  (2) 
where 
x n
++ ∈⊆ x \ X  is an nx–vector of variable inputs, 
x n
++ ∈⊆ w \ W  is an nx–vector of 
input prices, 
y n
++ ∈⊆ y \ Y  is an ny–vector of outputs, 
z n
++ ∈⊆ z \ Z  is an nz–vector of 
fixed inputs,  :, , FF
∞ ××→ ∈ \ XYZ C is the transformation function that defines the 
boundary of a closed, convex production possibilities set with free disposal in the inputs 
and the outputs,  : , ××→ X W Y ZX  ,
∞ ∈ X C  is the nx–vector of variable input demand 
functions, and  (,,) (,,) , C ≡ w y zw x w y z
T  :, C ++ × ×→ \ WYZ  , C
∞ ∈C  is the variable 
                                                 
1 Among other things it implies that marginal rates of product transformation are independent of the vari-
able inputs and factor intensities.  5 
cost function.
2 By Hotelling’s/Shephard’s Lemma, we have 
  1 ( , , ) ( ) ( / ,..., / ) ,
x wn CC w C w =∇ ≡ ∂ ∂ ∂ Xwyz w , y , z
T  (3) 
where 
T
 denotes vector/matrix transposition. Note that  X is positively homogeneous of 
degree zero in w. Integrating with respect to w to the variable cost function, we obtain  
 (,,) (,,,(,) ) , cC C θ =≡ w y zw y z y z   (4) 
where  : θ ×→ \ YZ  is the constant of integration. In the present case, this means that θ  
is constant with respect to w. The structure of θ cannot be identified from the variable 
input demands, and captures the structure of the joint production process relating to the 
fixed inputs and the outputs that is separable from the variable inputs.
3 
Under standard and well-known conditions, the variable cost function is strictly de-
creasing in z, strictly increasing in y, and convex in (,) yz. We are free to choose the sign 
of θ  so that, without loss of generality  0. C θ ∂ ∂>   
Because C   is strictly increasing in θ, it has a unique inverse,  (,,,) c θ γ = wyz , where 
: γ ++ ××× → \\ WYZ  is the inverse of C   with respect to θ. The function  ( , , , ) c γ w y z  
is a quasi-indirect production function, analogous to the quasi-indirect utility function of 
                                                 
2 The paper focuses on interior solutions and smooth functions. The results can be extended in the standard 
way to corner solutions by a continuous extension of F or C to the boundary of the strictly positive orthant 
in (x,y,z) or (w,y,z) space (see, e.g., Blackorby, Primont, and Russell 1977). Also, smoothness can be re-
laxed to twice continuous differentiability with no change in the arguments that follow. 
3 We elucidate this point further in what follows.  6 
consumer theory (Hausman 1981; Epstein 1982; LaFrance 1985, 1986, 1990, 2004; La-
France and Hanemann 1989; von Haefen 2002). For all interior, feasible (,) ∈× yz YZ , γ 
is strictly increasing in c, strictly decreasing and quasi-convex in w, and 0° homogeneous 
in (w,c). Two identities are simple implications of the inverse function theorem, 
 (,,,(,,,) ) , cC c γ ≡ wyz wyz   (5) 
a n d  (,,,(,,,) ) . C θ γθ ≡ wyz wyz   (6) 
This construction lets one write the conditional demands for the variable inputs as 
 (,,,) . Cc =∇ ≡ w xG w y z   (7) 
One question of particular interest – answered below – is, “What is the necessary and suf-
ficient condition for the conditional demands in a general model of joint production, (7), 
to reduce to  ( , , )? c = xX wz  ” 
Before addressing this question, we complete the development of the duality of vari-
able cost functions in joint production. Define the quasi-production function by 
  { }
(, )
(,,) m i n (,,,) : , , 0 .
c
cc c υγ ≡≤ ≥ ≥
w
xyz wyz wx w 0
T  (8) 
The name quasi-production function indicates that  ( , , ) υ x y z  only reveals that part of the 
structure of the joint production process associated with how the variable inputs interact 
directly with the fixed inputs and the outputs. As before, this is analogous to the situation 
where one only recovers that part of the structure of a direct utility function associated 
with the market demands for a subset of consumption goods.  
From the identity  ( , ) ( , , , ( , , , ( , ))) C θγ θ ≡ y zw y zw y z y z  , we have   7 
  {}
(, )
(,) (,,, (,,,(,) ) )
min ( , , , ): , , 0
(,,) ,
c
C
cc c
θγ θ
γ
υ
≡
≥≤ ≥ ≥
≡
w
yz wyz wyz yz
wyz wx w
xyz
0

T  (9) 
for all interior feasible ( , , ) . ∈ ×× xyz X Y Z  The inequality follows from the fact that 
(,) θ y z  is feasible but is not necessarily optimal in the minimization problem. Because 
(,,) 0 F = xyz  defines the boundary of the production possibility set, and since F is 
strictly increasing in  y  and strictly decreasing in z, ( , ) ( , , ) θ υ = y zx y z  is logically 
equivalent to  ( , , ) 0 F = xyz . That is, the quasi-production function is defined equiva-
lently as the unique solution with respect to θ  of the implicit function,
4 
 (,,) (,,,(,) ) 0 FF θ ≡ = xyz xyz yz  . (10) 
What class of variable cost functions generates conditional input demand equations in 
the form,  
 (,,) ? c = xX wz   (11) 
We can now prove the following: 
Proposition: The variable input demand equations have the structure (11) if and 
only if the variable cost function has the weakly separable structure 
                                                 
4 The existence of  : θ ×→ \ YZ  is not an issue here. For example, one could always define the function 
(,,,(,) ) (,,) (,) , FF θθ ≡+ x y z y zx y z y z   with  (,) 0 θ ≡ yz , and all of the properties listed above are met. The 
issue is when θ  is the only way that y enters the joint production technology.  8 
 ( , , ) ( , ( , ), ), cC C θ =≡ w y zw y zz   (12) 
and the variable cost function has the weakly separable structure (12) if and only 
if the joint production transformation function has the weakly separable structure, 
 (,,) (,(,) ,) . FF θ ≡ x y zx y zz   (13) 
Proof: First, differentiating (12) with respect to w, Shephard’s Lemma implies, 
 . C =∇ w x   (14) 
C   is strictly monotonic in and has a unique inverse with respect to θ, say  ( , , ) c θ γ = wz  . 
Substituting this into (14) obtains 
 (,(,,) ,) (,,) . Cc c γ =∇ ≡ w xw w z z X w z     (15) 
Second, and conversely, integrating (15) with respect to w returns a variable cost 
function with the separable structure in (12), where  (,) θ yz is again the constant of inte-
gration for the system of partial differential equations. 
Third, if the representation of technology has the separable structure in (13), it fol-
lows that 
  { } argmin : ( , , ( , )) 0, ( , , ( , )). F θθ ≤≥ ≡ wx xz y zxX w z y z 0 ~~ T  (16) 
This implies that the variable cost function has the separable structure 
  (,(,) ,) (,(,) ,) . C θθ ≡ w X wy z z wy z z   T  (17) 
So far we have shown that  ( , ( , ), ) 0 F θ ≤ xy z z   (,(,) ,) θ ⇒ X w y zz   (,(,) ,) . cC θ ⇔= w y zz   
T o  s h o w  t h a t   (,(,) ,) (,(,) ,) cC θθ =⇒ = w y zz x X w y zz   , one can proceed in one of two 
ways. The way first is to note that, given monotonicity of C   in θ  and the smoothness  9 
assumption, one can apply the results of Primont and Sawyer (1993) to recover the tech-
nically efficient representation of technology.  
Perhaps a more direct and illuminating approach to establish that (12) implies the 
weakly separable production technology in the Proposition is to use the concept of the 
quasi-production function which satisfies 
  { }
(, )
(,) m i n (,,) : , , 0 .
c
cc c υγ ≡≤ ≥ ≥
w
xz wz wx w 0 ~~
T  (18) 
By the same logic that leads to (9) above,  ( , ) ( , , ( , , ( , ))) ( , ) C θγ θ υ ≡≥ y zx z x z y zx z    for 
all interior feasible( , , )∈×× xyz X Y Z , with the boundary of the closed, convex feasible 
production possibilities set defined by equality on the far right. The marginal rates of 
transformation between outputs are therefore independent of the variable inputs,  
 
(,)
0, , , .
(,)
i
kj
y
ij k
xy
θ
θ
⎛⎞ ∂∂ ∂
=∀ ⎜⎟ ⎜⎟ ∂∂ ∂ ⎝⎠
yz
yz
 (19) 
Hence, y is weakly separable from x in the transformation function (Goldman and Uzawa 
1964, Lemma 1). Therefore, since  ( , , ) 0 F = xyz  defines the boundary of the production 
possibility set and F is strictly increasing in y, it follows that  ( , ) ( , ) υ θ = xz y z   is equiva-
lent to ( , ( , ), ) 0 F θ = xy z z  .   
Conclusions 
An empirically important question concerns when cost-minimizing input demands can be 
stated in terms of empirically observable ex ante data: costs, input prices, and fixed or 
quasi-fixed inputs. We conclude that separability of expected output from variable inputs 
must occur in technology and similarly separability of expected or planned outputs from  10 
input prices must occur in the cost function. If these restrictions are deemed too strong, 
then alternative approaches to cost function formulation must be pursued.   11 
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Duality Theory for Variable Costs in Joint Production

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Duality Theory for Variable Costs in Joint Production

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