?
=
Τ
𝜕? 𝜕 ?
? 𝑈=𝑈
∗
+ (
Τ
𝜕? 𝜕𝐼)(
Τ
𝜕𝐼 𝜕?
?
)
(A4.17)
The first term on the right side of equation (A4.17) is the substitution effect
(because utility is fixed); the second term is the income effect (because income
increases).
From the consumer’s budget constraint,
, we know by
differentiation that
𝐼 = ?
?
? + ?
?
?
Τ
𝜕𝐼 𝜕?
?
= ?
(A4.18)

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59 of 50
It is customary to write the income effect as negative (reflecting a
loss of purchasing power) rather than as a positive. Equation
(A4.17) then appears as follows:
(A4.19)
In this new form, called the
Slutsky equation
, the first term represents the
substitution effect
: the change in demand for good X obtained by keeping utility
fixed. The second term is the
income effect
: the change in purchasing power
resulting from the price change times the change in demand resulting from
a change in purchasing power.
Τ
?? ??
?
=
Τ
𝜕? 𝜕 ?
? 𝑈=𝑈
∗
− ?(
Τ
𝜕? 𝜕𝐼)
●
Slutsky equation
Formula for decomposing the effects of a price change into
substitution and income effects.