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# Derivation using Slutsky equation

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```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  ?
¶ X a* ¶ X a*
=
¶ pa ¶ pa
(?)
(-)
+
Y *=const
¶ X a* ¶Y *
×
¶Y * ¶ pa
(+)
(+)
What would we have concluded if we had used a
standard consumer demand model for Xa?
1
Agricultural household models
The profit effect (cont’d)
Using the Slutsky equation (see extra slides or Sadoulet & de
Janvry 1995 for details), we can rewrite the expression as:
¶ X a* ¶ X a*
=
¶ pa ¶ pa
-X
*
×
a
U =const
¶ X a*
¶Y *
*
+Q ×
¶ X a*
¶Y *
or, equivalently:
¶ X a* ¶ X a*
=
¶ pa ¶ pa
(-)
•
•
•
*
+ (Q - X
U =const
*
)×
a
¶ X a*
¶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
buyers vs. net sellers?
2
1
Slutsky equation: shows relationship b/w
Marshallian (M) & Hicksian (M) demand functions
¶x M ¶x H
¶x M
Slutsky equation:
-x
=
¶p
¶p
¶y
(1)
¶x M
is holding y constant.
¶p
¶x H
H
H
is holding U constant.
x = x ( p,U ), so
¶p
Recall x M = x M ( p, y), so
¶x
¶x
¶x
-x
(2)
=
¶ p y=const. ¶ p U =const. ¶ y
Can rewrite Slutsky as
Apply Slutsky to get profit effect equation
in terms of marketable surplus (Q*-Xa*)
·
·
·
We already showed that:
Per Slutsky (2):
¶ X a* ¶ X a*
=
¶ pa ¶ pa
¶ X a*
¶ pa
Plug (4) into (3):
=
¶ pa
Y =const
¶ pa
=
- X a*
¶ X a*
¶Y *
- X a*
U =const
(4)
¶ X a* ¶ X a* ¶Y *
¶Y *
+
×
(5)
¶Y * ¶ pa
*
¶Y
¶(p + wT ) ¶( paQ - wL + wT )
=
=
= Q* (6)
¶ pa
¶ pa
¶ pa
*
*
·
Note that:
·
Plug (6) into (5), rearrange terms, and simplify:
¶ X a*
¶ pa
×
(3)
¶Y * ¶ pa
U =const
¶ X a*
¶ pa
¶ X a* ¶Y *
Y *=const
¶ X a*
*
¶ X a*
+
=
Y * =const
¶ X a*
¶ pa
+ (Q* - X a* )
U =const
¶ X a*
¶Y *
4
2
```