~VP
f
~VH 5 ~V
P
f
~V
H
5 Q
f
; 10
~VP
d
~VH 5 ~V
P
d
~V
H
5 Q
d
. 11
Equations 10–11 are uncertainty analogues of Hotelling’s lemma and represent the foreign export and domestic supply functions. These supply functions hold for any
general unrestricted utility function. The exact functional form for the indirect expected utility function is unknown, but can be approximated by a second-order Taylor series
expansion around an expansion point Z thus
7
: V~P
f
, s
f
, P
d
, s
d
, w, H 5 V~Z 1
O
i51 6
V
i
~Z D
i
1 1
2
O
i51 6
O
j51 6
V
ij
~Z D
i
D
j
, 12
where V
1
5 V
P
f
, V
2
5 V
s
f
, V
3
5 V
P
d
, V
4
5 V
s
d
, V
5
5 V
w
, V
6
5 V
H
and V
ij
5 V
ji
are second partial derivatives.
8
All derivatives are evaluated at the expansion point Z, and D
i
represent deviations from Z. Differentiating equation 12 with respect to P
f
, P
d
, and H, and substituting into equations 10 and 11 yields the estimating equations:
Q
f
5
F
V
1
1
O
i51 6
V
1i
~D
i
G
Y
F
V
6
1
O
i51 6
V
6i
~D
i
G
; 13
Q
d
5
F
V
3
1 V
13
~D
1
1
O
i52 6
V
3i
~D
i
G
Y
F
V
6
1
O
i51 6
V
6i
~D
i
G
, 14
where all partial derivatives are evaluated at the expansion point, and the cross-equation restriction, V
31
5 V
13
is imposed in equation 14. For estimation purposes, an input price index w, rather than individual input prices, is used and the shift parameter, H, is set
equal to its initial value of zero.
9
The estimating equations above are homogeneous of degree zero in the parameters. To identify these parameters, V
H
5 V
6
is normalized to 1.
10
V. Hypothesis Testing
The first hypothesis to be tested is the symmetry restriction Q
f
P
d
5 Q
d
P
f
see Table 1, which is implied by both CARA and separability. The empirical restriction
corresponding to this is given by differentiating equation 10 with respect to P
d
, and equation 11 with respect to P
f
, and evaluating at the expansion point of the indirect expected utility function. Noting that at the expansion point V
H
5 V
6
5 1, the first parameter restriction for CARA and separability is given by:
V
61
5 ~ V
63
V
1
~V
3
, 15
7
Contrast the second-order Taylor series approximation of a general unrestricted utility function in this paper with the exponential utility function assumed by Chavas and Holt 1996, p. 331, equation 8.
8
Hlawitschka 1994 showed that for a wide family of utility functions, two-moment Taylor expansions provide excellent approximations to expected utility.
9
This implies that D
6
5 0. Thus, V
16
D
6
5 V
66
D
6
5 V
36
D
6
5 0.
10
This normalization implies that V
H
5 E[U9P] 5 1, at the point of expansion. This type of normalization is common in the empirical demand and production literature [see Christensen et al. 1975, p. 370].
320 S. Satyanarayan
which is imposable in equations 13 and 14. Next, note that Q 5 V
P
f
1 V
P
d
V
H
. The parameter restriction implied by the second symmetry condition see Table 1, evaluated
at the approximation point, is given by: V
11
5 ~V
13
1 V
33
2 V
63
~V
1
1 V
3
~V
15
2 V
65
V
1
~V
35
2 V
65
V
3
1
F
V
63
V
1
V
3
~V
1
1 V
3
2 V
13
G
, 16
where the prior restriction equation 15 is imposed in equation 16. If the restrictions given by equations 15 and 16 are rejected, then both CARA and separability must be
rejected. The comparative statics of the model provide one source of empirically refutable
symmetry restrictions. Additional symmetry restrictions can be easily derived by using the derivatives of the indirect expected utility function [Dalal 1994]. The indirect function
V corresponding to the separable utility function is given by:
V~P
f
, s
f
, P
d
, s
d
, w, H 5 P
f
Q
f
1 P
d
Q
d
2 C~w, Q 1 H 2 b~Q
f 2
s
f 2
1 Q
d 2
s
d 2
, 17
where s
f 2
and s
d 2
are the variance of the domestic and foreign price distributions. The envelope theorem applied to equation 17 immediately implies that V
P
f
5 Q
f
, V
P
d
5 Q
d
, V
s
f
5 22bQ
f 2
s
f
, V
s
d
5 22bQ
d 2
s
d
, V
w
5 2Cw 5 2C
w
, V
H
5 1, and Young’s theorem implies that:
~V
P
f
s
f
V
P
d
s
f
5 ~V
s
f
P
f
V
s
f
P
d
f
S
Q
f
s
f
Y
Q
d
s
f
D
5
S
Q
f
P
f
Y
Q
f
P
d
D
; 18
~V
P
f
s
d
V
P
d
s
d
5 ~V
s
d
P
f
V
s
d
P
d
f
S
Q
f
s
d
Y
Q
d
s
d
D
5
S
Q
d
P
f
Y
Q
d
P
d
D
; 19
V
Hu
5 V
u H
5 0 where u 5 ~P
f
, P
d
, s
f
, s
d
, w, H. 20
Now, equations 15 and 16 are valid restrictions for both CARA and the separable utility function. If these restrictions are rejected, then both CARA and separability must
be rejected. However, if these restrictions cannot be rejected, then CARA or separability are possible forms for the utility function. To distinguish separabilty from CARA, impose
the additional restrictions for separability given by equations 18, 19 and 20. If these restrictions are rejected,
11
the utility function is consistent with CARA but is not separable. However, if these restrictions cannot be rejected, then the utility function is
separable and does not display CARA.
12
VI. Data Description and Stationarity Properties of the Data
