The Slutsky Equation

The total effect of a price increase can thus be divided into a substitution and an income effect. In terms of the own price elasticity we can define the total effect as:

ΔQ^tot = ε . (Q₀/p₀) . Δp

the substitution effect is:

ΔQ^sub = ε^∗ . (Q₀/p₀) . Δp

and the income effect:

ΔQ^inc = η . (Q₀/Y₀) . ΔY

In general we can write:

ΔQ^tot = ΔQ^sub + ΔQ^inc

ε . (Q₀/p₀) . Δp = ε^∗ . (Q₀/p₀) . Δp + η . (Q₀/Y₀) . ΔY

If we use the Slutsky compensation criterion, that the consumer should be able to buy her initial bundle of consumption goods, x⁰, the income compensation can be written as:

ΔY = Δp . Q₀,

and we can rewrite the previous equation as,

ε .(Q₀/p₀) . Δp = ε^∗ . (Q₀/p₀) . Δp + η (Q₀/Y₀) . Δp . Q₀,

we now multiply and divided the last term on the right-hand side by p0 and after that we cancel all common terms:

ε . (Q₀/p₀) . Δp = ε^∗ . (Q₀/p₀) .  Δp + η . (Q₀/Y₀) . (p₀/p₀) .  Δp . Q₀,

ε = ε∗ + η .  [(Q₀ . p₀)/Y₀]

The very last term now shows the budget share of the good whose price has changed, or

θ = (Q₀ . p₀)/Y₀

Note that Q₀.p₀ gives the total expenditure on that particular good and Y₀ is total expenditure on all good, given that the consumer uses all her budget to buy goods. Hence, we can finally write the Slutsky equation in terms of elasticity,

ε = ε^∗ + η . θ

We should now remind ourselves that for a price increase the total effect is negative if the good is a normal good; the substitution effect is always negative (so the compensated price elasticity ε^∗ is always negative). The income effect is positive for a normal good, but in the case of a price increase the consumer’s real income decreases so the second term on the right-hand side is also negative (η < 0). If the good in question is an inferior good, the income effect is negative which means that demand for the good increases as the real income decreases and η > 0. In this case the sign of the total effect is unclear. This means that the uncompensated demand curve could be upward sloping (for a very inferior good). Such goods are called Giffin goods and could be a reality especially in poor countries their families are very dependent on a particular source of food for their survival (e.g. potatoes or maize). But, Gi?en goods must be considered to very unusual cases which goes against the “law of demand” (i.e., that the demand curves are downward sloping).

One more comment on the Slutsky equation is in order. In the derivation we assumed Slutsky compensation criterion instead of Hicks’. This was done to make the derivation easier, but it can be shown that for small price changes, the two compensation criteria coincide and we get the same final expression.

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