output supply in both markets, implying that domestic and foreign output are net complements. These results imply the following:
Lemma 1: Z
12
, 0 .0 if and only if S
Q
f
,P
d
5 S
Q
d
,P
f
, 0 .0 implies that Q
f
and Q
d
are net substitutes complements. The notion of output substitutabilitycomplementarity is useful in signing the compar-
ative statics results. These are summarized in Table 1 for a constant absolute risk aversion CARA utility function and a separable utility function.
6
The comparative statics results and the symmetry reciprocity results in Table 1 constitute refutable hypotheses which
are tested in the empirical section.
IV. Estimating Equations
The indirect expected utility function corresponding to equation 1 is given by VP
f
, s
f
, P
d
, s
d
, w, H, and the envelope theorem applied to it implies that:
6
A negative exponential utility function of the form UP 5 2e
2kP
exhibits CARA, where k is the coefficient of CARA. The separable utility function is given by UP 5 P 2 b[P 2 EP]
2
, where b . 0. This function has been especially useful in the theoretical and empirical analysis of uncertainty [Pope 1980;
Antonovitz and Roe 1986; Park and Antonovitz 1992a]. The derivation of the comparative statics results reported in Table 1 are available from the author on request.
Table 1. Comparative Statics Results
CARA CARA Complementarity
CARA Substitutability Separability
Symmetry Results
Q
f
P
f
.
Q
d
P
f
.
Q
d
P
f
,
Q
f
P
f
.
Q
f
P
d
5
Q
d
P
f
Q
d
P
d
.
Q
P
f
.
Q
f
P
d
,
Q
f
s
f
,
Q
w
i
Q
f
P
d
.
Q
f
P
d
Q
P
d
Q
P
f
5
Q
d
w
i
Q
f
w
i
Q
f
H
5
Q
P
d
.
Q
f
s
d
Q
d
H
5
Q
f
w
i
Q
f
w
i
Q
H
5
Q
d
w
i
Q
d
P
d
.
Q
s
f
,
Q
d
s
d
,
Q
f
s
d
,
Q
d
P
f
Q
d
s
f
Q
d
w
i
Note: The symmetry results are applicable under both CARA and Separability. Starred variables assume non-inferiority of inputs.
Econometric Tests of Decision Making 319
~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
