Derivation using Slutsky equation Agricultural household models
The profit effect
X a* = X a* ( pa , pm , w,Y * )
(
= X a* pa , pm , w,Y * ( pa , w, A)
)
How does the HH’s demand for the ag staple (Xa*) Δ
when the price of the ag staple (pa) s (assuming it is a
normal, non-Giffen good)? Does it  or  ?
&para; X a* &para; X a*
=
&para; pa &para; pa
(?)
(-)
+
Y *=const
&para; X a* &para;Y *
&times;
&para;Y * &para; pa
(+)
(+)
What would we have concluded if we had used a
standard consumer demand model for Xa?
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Agricultural household models
The profit effect (cont’d)
Using the Slutsky equation (see extra slides or Sadoulet &amp; de
Janvry 1995 for details), we can rewrite the expression as:
&para; X a* &para; X a*
=
&para; pa &para; pa
-X
*
&times;
a
U =const
&para; X a*
&para;Y *
*
+Q &times;
&para; X a*
&para;Y *
or, equivalently:
&para; X a* &para; X a*
=
&para; pa &para; pa
(-)
•
•
•
*
+ (Q - X
U =const
*
)&times;
a
&para; X a*
&para;Y *
(+)
Q*-Xa*?
What is
Let’s call this M*.
What is the the sign of M* (marketable surplus) for net buyers vs. net
sellers?
What can we say about the effect of an increase in pa on Xa* for net
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1
Slutsky equation: shows relationship b/w
Marshallian (M) &amp; Hicksian (M) demand functions
&para;x M &para;x H
&para;x M
Slutsky equation:
-x
=
&para;p
&para;p
&para;y
(1)
&para;x M
is holding y constant.
&para;p
&para;x H
H
H
is holding U constant.
x = x ( p,U ), so
&para;p
Recall x M = x M ( p, y), so
&para;x
&para;x
&para;x
-x
(2)
=
&para; p y=const. &para; p U =const. &para; y
Can rewrite Slutsky as
Apply Slutsky to get profit effect equation
in terms of marketable surplus (Q*-Xa*)
&middot;
&middot;
&middot;
Per Slutsky (2):
&para; X a* &para; X a*
=
&para; pa &para; pa
&para; X a*
&para; pa
Plug (4) into (3):
=
&para; pa
Y =const
&para; pa
=
- X a*
&para; X a*
&para;Y *
- X a*
U =const
(4)
&para; X a* &para; X a* &para;Y *
&para;Y *
+
&times;
(5)
&para;Y * &para; pa
*
&para;Y
&para;(p + wT ) &para;( paQ - wL + wT )
=
=
= Q* (6)
&para; pa
&para; pa
&para; pa
*
*
&middot;
Note that:
&middot;
Plug (6) into (5), rearrange terms, and simplify:
&para; X a*
&para; pa
&times;
(3)
&para;Y * &para; pa
U =const
&para; X a*
&para; pa
&para; X a* &para;Y *
Y *=const
&para; X a*
*
&para; X a*
+
=
Y * =const
&para; X a*
&para; pa
+ (Q* - X a* )
U =const
&para; X a*
&para;Y *
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