2
Partial Derivatives
If you take the partial derivative of the Hicksian demand function
with respect to a price you are asking
how the demand changes as
you change a price whilst holding utility constant
.
123
2
(, , ,
.
.
. ,)
(
,
,
,...
, )
iN
i
j
ij
m
xppp p u
p
m
p
pp
∂
=
∂
∂
∂
=
∂∂
∂
This is the
substitution effect.
It is symmetric by Young’s Theorem:
(
)
(
)
()
22
p,
p,
p,
p,
j
i
ji
j
j
i i
x
u
xu mu mu
p
p
∂
∂
===
∂
∂
Partial Derivatives
E.g. the substitution effect on leisure induced by a change in the
price of goods, is the same as the substitution effect on goods
induced by a change in the wage.
(
)
(
)
(
)
p,
p,
p,
p,
p
p,
,
ii
j
j
jj
i
i
i
j
j
i
xm
x
pm
x
xu
p
p
p
m
=−
=
∂
∂
−
∂
∂
Alternative Notation
First-order partial derivatives can be written:
or
and
or
xy
ff
22 2
and the second-order derivatives are:
or
and
or
and
or
For a function
,
,
,... we sometimes
xx
yy
xy
ff f
f
xyx
y
fxxx
∂∂ ∂
∂∂∂
∂
121
11
3
2
1
3
write:
for
,
for
,
for
,
for
etc.