1 A Recursive Dynamic Computable General Equilibrium Model For ...

*
R_Area j = 1 −
+ ε j
(7)
C + R _ Re nt
0
where the sub-index j refers to each of the j observations available for each GTAP region: j
= 1, …, N, where N is the number of observations and equal to the number of crops times
the types of land suitability (with a maximum of 92 observations, 23 crops × 4 land
suitability classes); and ε j is the error term, distributed as a Normal N(0,σ 2 ) (as in non-linear
least squares). Then, the likelihood function is specified as:
L(
N
1
∏
j=
1 σ 2π
j
p
* 2
2 ⎢
0
j ⎥
α , b , C , p,
σ ) = e ⎣ ⎝
⎠⎦
(8)
0
leading to the the log-likelihood:
*
l(
α,
b
, C
0
, p,
σ
2
⎛
24
p
j
*
1 ⎡ ⎛ b ⎞⎤
− ⋅⎢R
_ Area −⎜α−
⎟⎥
2σ
⎜ C + R _ Rent
⎟
N ⎜
⎡ ⎛
⎞⎤
⎜ 1
1
b
*
∑ − log 2π
− log σ − R_Area − ⎢ ⎜ α − ⎟
⎜ 2
2 j
⎥
j 1
C + R_ Re nt p
⎜
2σ
⎣ ⎝ 0 j ⎠⎦
= ⎟ ⎟⎟⎟
) =
⎝
⎠
In our estimation we use the algorithms that GAUSS uses as default values: the
Broyden-Fletcher-Goldfarb-Shanno (BFGS) iteration procedure to obtain the direction δ,
and the cubic or quadratic method STEPBT to obtain the step length ρ. As initial values
for the parameters (α, b * , C 0, p) we assign respectively the values (1, 0.05, 0, 0.05). The α-
parameter is kept fixed at 1. We apply Weighted Maximum Likelihood estimation where
the weight for each observation is the value of R_Rentj, as in Tabeau et al. (2006), in order
to assign relatively more weight to those observations located at the ends in order to
improve the fit of the estimated function to the original data. Finally, among the different
alternatives to estimate the covariance matrix of the parameters (standard errors) we use
the method implemented by default, which uses the inverse of the Hessian (matrix of the
second derivatives of the log-likelihood function).
Writing the flexible non-linear expression for land supply in the GEMPACK model
code gives the following levels expression:
2
(9)
2⎞