Supplementary material to
Carpentier A, Letort E (2013) Multicrop Production Models with Multinomial Logit Acreage Shares.
Environmental and Resource Economics (doi: 10.1007/s10640-013-9748-6)
Appendix A. MNL acreage share functions, proof of Proposition 1
In the standard MNL case, the producer’s problem is given by:
max s  0
(A.1)

K

s  k  A   k 1 ck sk  a 1  k 1 sk ln sk
k 1 k
K
K


s.t.  k 1 sk  1 .
K
Note that we consider the non-negativity constraint s  0 , and not sk  0 for k  1,..., K . The result stating that
lims
k
0
sk ln sk  0 leads to the usual convention 0  ln 0  0 and thus to the extension of the domain of the
entropy function to values of s with null elements. The (modified) Lagrangian function associated to the considered
maximization problem is defined by:
(A.2)

 
L(s,  )   k 1 sk  k  A   k 1 ck sk  a 1  k 1 sk ln sk  
K
K
K
K

s 1 .
k 1 k
It leads to the following FOCs:
(A.3)
L(s,  )
  k  ck  a 1  (ln sk  1)    0 and
sk
With lim s
k
(A.4)
 0

K
s 1  0 .
k 1 k
a 1 ln sk   we know that corner solutions in sk at 0 cannot occur. Equation (A.3) leads to:
sk  exp  a  ( k  ck )  exp  (a  1)  .
Equation (A.3) and

K

s  1 yield exp  (a  1)   
m 1 m
K
k 1
exp  a  ( m  cm )  

1
and, as a result:
(A.5)
sk 
exp  a  ( k  ck ) 

K
m 1
exp  a  ( m  cm ) 
.
Note finally that the optimal level of  is given by S* (π)  ln  k 1 exp  a( k  ck )  .
K
Appendix B. Nested MNL acreage share functions, proof of Proposition 2
In the nested MNL case, the producer’s program is given by:
(B.1)
maxs0

Q
1
s 

s
mB ( ) m /


 ( m  cm   1 ln sm / )  a 1  ln s  A s.t.  k 1 sk  1
K
with s   mB ( ) sm and sm /  sm s 1 . The corresponding Lagrangian function is defined by:
(B.2)
L(s,  )   1 s 
Q

s
mB ( ) m /
The first order conditions for the crop k in nest
(B.3)

 ( m  cm   1 ln sm/ )  a 1  ln s  A   

K
are given by:
L(s,  )
  k  ck  a 1  (ln s  1)   1 ln sk /    0 and
sk

K
s 1  0 .
k 1 k
Equations (B.3) leads to:
(B.4)
 1 ln sk /  a 1 ln s   k  ck  a 1  
Equation (B.4) and sk /  sk s 1 allow then showing that:
(B.5)
sk  exp    ( k  ck )   exp     (a 1   )   exp  (1   a 1 )  ln s  .
Use of the definition s   mB ( ) sm leads to:
(B.6)

s 1 .
k 1 k
s  exp    (a1   )   exp  (1   a1 )  ln s    mB( ) exp    ( m  cm ) 
and, as result, to:
sk 
(B.7)
exp    ( k  ck ) 

mB (
exp    ( m  cm ) 
)
 s and sk / 

exp    ( k  ck ) 
mB ( )
exp    ( m  cm ) 
.
Equation (B.7) allows showing that:
s  exp  (1  a )    mB ( ) exp    ( m  cm ) 


(B.8)
With

(B.9)
s  1 we obtain exp(1  a )  
1
Q
s 

exp    ( m  cm ) 
1   mB ( )

Q

exp    ( m  cm )  
 mB ( )


a  1
a  1
and finally:
a  1

exp    ( m  cm )  
1   mB ( )

Q
a  1
.
This implies that:
(B.10)
sk 


exp    ( m  cm )  
 mB ( )

exp    ( k  ck ) 
mB ( )
exp    ( m  cm ) 
if crop k belongs to nest

Q
1
a  1

exp    ( m  cm )  
 mB ( )

a  1
.
Corally 2 presents an alternative parameterization of CN (s) .

Corollary 2. For any level of d  (dk ; k  1,..., K ) , there exist A and ω  (k ; k  1,..., K ) , with ωι  1 and
ω  0 , ensuring that:
CN (s)  A  
Q
1

s dk  a 1 
kB ( ) k
Q
1
 s (ln s  ln  )  a
where k /  k  1 and    kB ( )  for k  B( ) and
1
s

 1,...,Q , and:
s (ln sk /  ln k /
kB ( ) k /

k exp   n ( k  d k ) 
sk ( π) 
 mB (n) m exp  n ( m  dm ) 
2
(n ) n

( n  a )
( ) 
1
Q
2

 exp   n ( m  d m )  
 mB ( n ) m

(  a)
a n1

 exp   ( m  d m )  
 mB ( ) m

a  1
for k  B(n) and n  1,..., Q .
Proof. The formulas for CN (s) and sk (π) in this corollary are obtained from equations (8) and (9) with ω  s(d)
and A  A   *N (d) .
Appendix C. Berry’s (1994) device, presentation
Let nest Q be composed of a single crop which is crop K, i.e. the “outside crop”. We have parameter Q  a in this
case. We thus have:
(C.1)
sQ 
exp  a  ( K  cK ) 
exp  a  ( K  cK )   

exp    ( m  cm )  
1   mB ( )

Q 1
implying that for any crop k  K belonging to nest
(C.2)
a  1
 sK
we have:


sk exp    ( k  ck )    mB ( ) exp    ( m  cm ) 

sK
exp  a  ( K  cK ) 
a  1 1
.
Using the fact that ln sk /    ( k  ck )  ln  mB ( ) exp    ( m  cm )  , we finally obtain:
(C.3)
ln sk  ln sK  a  [( k  ck )  ( K  cK )]  (1  a  1 )  ln sk / ,
i.e. an estimating equation which is empirically tractable as long as the endogeneity of ln sk / is accounted for.
Appendix D. Crop production model, case with sequential input decisions
Let consider a generic yield function defined by:
(D.1)
y  a  (1/ 2)  (b  x)Γ 1 (b  x) with a  E[a] .
Note first that equation (D.1) is equivalent to the usual quadratic yield function:
y    xβ  (1/ 2)  xΩx with   a  (1/ 2)  bΓ 1b , β  Γ 1b and Ω   Γ 1 .
The aim of this appendix is to demonstrate that the yield supply, input demand and gross margin functions given in
the main text are fairly general.
We consider cases where a and b are random and where farmers sequentially choose their inputs. The
considered crop is sold at the known price p and inputs are purchased at the known price w. The input use vector
is decomposed into two sub-vectors, i.e. x  (x0 , x1 ) . This decomposition corresponds to that of b, i.e. b  (b0 , b1 ) ,
which accounts for the fact that the realization of b0 is observed before that of b1 . The price vector w is
decomposed accordingly as w  (w0 , w1 ) . We will also use the following decomposition of Γ and of G  Γ1 :
(D.2a)
Γ
Γ   00
 Γ01
Γ 01 
G
and G   00

Γ11 
 G 01
G 01 
G11 
as well as the matrix block inversion results for Γ  G 1 stating that:
(D.2b)
1
1
1
1
1
and Γ11  G11
.
Γ00  (G 00  G 01G11
G 01 ) 1 , Γ 01  Γ 00G 01G11
 G11
G 01Γ 00G 01G11
We use here the assumption stating that Γ is positive definite.
Let now assume that x 0 is chosen before b1 is observed, implying that b1 is random at the time x 0 is chosen
(the realization of a is only observed at the end of the production process). In this case the expected crop gross
margin is maximized by backward induction, i.e. x 0 is optimally chosen, accounting for the fact that x1 will be
optimally chosen.
In the second stage of the optimization process b is known and the level of x 0 is fixed. The famer’s yield
expectation is then given by (A.1) with b known and he solves the problem:
(D.3a)
max x1  0  p   a  (1/ 2)  (b  x)G (b  x)   xw .
The solution in x1 to this problem, x1* ( p, w1 ; x0 ) , is the solution in x1 to:
(D.3b)
max x1  0  p   a  (1/ 2)  (b1  x1 )G11 (b1  x1 )  (b0  x0 )G 01 (b1  x1 )   x1w1 
and is given by:
(D.4)
1
1
x1* ( p, w1 ; x0 )  b1  G11
G 01 (b 0  x 0 )  G11
w1 p 1
since it is the solution to the first order equation G11  b1  x1* ( p, w1 ; x0 )   G 01 (b 0  x0 )  w1  0 . We assume here
that the solution in x1 to problem (D.3b) is interior.
In the first stage of the optimization process, the farmer chooses x 0 according to the realization of b0 . The
realization of b1 is unknown but the farmer knows that his optimal use of x1 depends on b1 according to
x1* ( p, w1 ; x0 ) . This implies that, at the time he chooses x 0 , the farmer’s yield expectation is given by:
(D.5a)
1
a  (1/ 2)  w1G11w1 p 2  (1/ 2)  (b 0  x0 )(G 00  G 01G11
G 01 )(b 0  x 0 ) ,
and his expected expenditure of x1 is given by:
(D.5b)
1
1
w1 E[b1 ]  w1G11
G 01 (b 0  x0 )  w1G11
w1 p 1 .
This implies that the famer’s first stage optimization problem is defined as:
(D.6a)
1
 p  a  (1/ 2)  w1G11w1 p 2  (1/ 2)  (b0  x0 )(G 00  G 01G11
G 01 )(b0  x0 ) 


max x0 0 

1
w1  E[b1 ]  G11 G 01 (b0  x0 )   w 0 x0




1
1
 G 00  G 01G11
G 01 . This yields:
where we recognize Γ00
(D.6b)
1
1
1
x*0 ( p, w )  b 0  (G 00  G 01G11
G 01 ) 1 G 01G11
w1 p 1  (G 00  G 01G11
G 01 ) 1 w 0 p 1
and, using equations (D.2):
(D.7a)
x*0 ( p, w )  b 0  Γ01w1 p 1  Γ00 w 0 p 1 . We assume that the solution in x 0 to problem (D.6a) is interior.
The substitution of x*0 ( p, w ) for x 0 in equation (D.4) and use of equations (D.2) then lead to:
(D.7b)
x1* ( p, w )  x1*  p, w1 ; x*0 ( p, w )   b1  Γ01w 0 p 1  Γ11w1 p 1 .
We thus recover the formula provided in the text x* ( p, w )  b  Γwp 1 .
Since this result holds for any partitions of b and x, it also holds for any sequence of input use decisions related
to the corresponding sequence of realization of the elements of b. It suffices to consider the relevant backward
recursion process.
Appendix E. GMM estimators of θ 0s and of θ 0n
The variables ln skit for k  1, 2 are the only endogenous explanatory variables of the considered models. These
variables can be instrumented by using first step estimates of the terms ln skit (θ 0n ) for k  1, 2 where:
(E.1)
skit (θ0n ) 

exp  a0 (1  0 ) 1 (δk ,0 zkit  ψ k ,0 z its ) 
exp  a0 (1  0 ) 1 (δ ,0 zit  ψ  ,0 z its ) 
1,2
.
The rational for this instrument choice is that ln skit (θ 0n ) is an approximation to the conditional expectation
s
apart) in the equation of
E[ln skit | z it ] which is the best instrumental variable for ln skit (heteroskedasticity of  kit
ln( skit s3it1 ) (Chamberlain 1987). Note that z2,it can be used as an instrument for ln s1it in the equation of ln( s1it s3it1 )

, just as z1,it
can be used as an instrument for ln s2it in the equation of ln( s2it s3it1 ) . This is useful to define simple
consistent estimators of θ 0n .
In both equations systems the parameter vector δ 0 can be identified by the yield supply sub-systems only. This
observation lies at the root of our “sequential” identification strategy of the interest parameters θ 0s and θ 0n . Our
estimators of θ 0s and of θ 0n are defined such that (i) δ 0 is (mostly) estimated by the information content of the
yield equation sub-system of the models and while (ii) κ 0s and κ 0n are (mostly) estimated by the information
content of the acreage sub-system of the models. I.e. we refrain from taking advantage of the fact that δ 0 is
shared by the yield and acreage equation subsystems of the models. We do not consider orthogonality conditions
derived from the acreage share equations which could supplement the identification of δ 0 (and that of κ 0s and κ 0n
due to over-identifying restrictions). This additional moment conditions would increase the asymptotic efficiency
of our estimators (assuming that the models are correctly specified). But our “sequential” identification strategy
ensures that the model parameters are mainly identified by their “natural” identification sources.
We base the GMM estimators of the model parameters on two subsets of moment conditions. The first one is
common to both models:
(E.2)
E[z
 ]  0 for k  1, 2,3 .
y
y
kit kit
It ensures identification of δ 0 . The second one depends on the considered model. Given that δ 0 is identified by
the moment conditions (E.2), the subset of moment conditions
(E.3)


s

 E[(δk ,0 z kit  δ3,0 z 3it ) kit ]  0 for k  1, 2

E[z its  kits ]  0 for k  1, 2


ensure the identification of κ 0s in the SMNL multicrop model (30) while conditions
(E.4)
 E[(δk ,0 z kit  δ3,0 z 3it ) kits ]  0 for k  1, 2

E[z its  kits ]  0 for k  1, 2


E[ln skit (θ0n ) kits ]  0 for k  1, 2

ensure that of κ 0n in the NMNL multicrop model (31). GMM estimators of θ 0s and of θ 0n based on these moment
conditions are easily constructed since the instruments (δk ,0 zkit  δ3,0 z3it ) and ln skit (θ 0n ) can be replaced by
consistent estimates without any (asymptotic) efficiency loss.
Note that the considered moment conditions almost just-identify the interest parameters. There is only one
over-identifying restriction for estimating θ 0s ( a0 is shared by two equations), and there are only two overidentifying restrictions for estimating θ 0n ( a0 and 0 , or 0 , are shared by two equations). Parameters κ 0s and
κ 0n are mainly identified by the information content of the acreage share equation sub-systems while δ 0 is mostly
estimated by that of the yield supply equation sub-system, according to a “sequential” identification strategy.
Estimates of the terms (δk ,0 zkit  δ3,0 z3it ) are implicitly “supplied” to the acreage share equation sub-systems. The
moment conditions derived from these acreage share equation sub-systems are mostly devoted to the estimation
of the parameters κ 0s and κ 0n , provided that δ 0 is mostly estimated by the yield supply equation sub-system.
The GMM estimate of θ 0s is obtained in two steps:
(1) Compute the GMM estimator of δ 0 , δ , based on (E.2).
(2) Compute the GMM estimator of θ 0s , θˆ s , based on (E.2) and (E.3) while substituting (δk zkit  δ3 z 3it ) for
(δk ,0 zkit  δ3,0 z3it ) .
The GMM estimate of θ 0n is obtained in three steps:
(1) Compute the GMM estimator of δ 0 , δ , based on (E.2).
(2) Compute the GMM estimator of θ 0n , θ n , based on (E.2) and (E.4) replacing (δk ,0 zkit  δ3,0 z3it ) by

(δk z kit  δ3 z 3it ) , ln s1it (θ 0n ) by z2it , and ln s2it (θ0n ) by z1it
.
(3) Compute the GMM estimator of θ 0n , θˆ n , based on (E.2) and (E.4) while replacing (δk ,0 zkit  δ3,0 z3it ) by
(δk z kit  δ3 z 3it ) , and ln skit (θ 0n ) by ln skit (θ n ) .
GMM estimators of the parameters of the “land use” versions of the MNL acreage share MNL are easily
constructed by adapting the procedures leading to θˆ s and θˆ n . Let define z ckit  (zkit , z3it , zit ) and:
(E.5)
skit (λ 0n ) 

exp  (1  0 ) 1 ( ηk ,0 zkit  τ k ,0 z its ) 
1,2
exp  (1  0 ) 1 ( η ,0 zkit  τ  ,0 z its ) 
n
s
for k  1, 2 , as well as cit  (z1it , z2it , z3it , zit ) , λ 0s  (η1,0 , η2,0 , η3,0 , τ1,0 , τ 2,0 ) and λ 0  (λ 0 , 0 ) . The GMM estimate
s
of λ 0 , λˆ s , is obtained in one step:
s
(1) Compute the GMM estimator of λ 0 based on E[z ckit  kits ]  0 for k  1, 2 .
n
The GMM estimate of λ 0 , λˆ n , is obtained in two steps:
n
(1) Compute the GMM estimator of λ 0 , λ n , based on E[cit  kits ]  0 for k  1, 2 .
n
(2) Compute the GMM estimator of λ 0 , λˆ n , based on E[z ckit  kits ]  0 and E[ln skit (λ 0n ) kits ]  0 for k  1, 2 ,
while replacing ln skit (λ 0n ) by ln skit (λ n ) .