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The price elasticity of demand, , captures the total effect of a price change. We can decompose the price elasticity of demand into two terms involving elasticities that capture the substitution and income effects. We measure the substitution effect using the *substitution elasticity of demand*,
*, which is the percentage that quantity demanded falls for a given percentage increase in price if we compensate the consumer to keep the consumer's utility constant. The income effect depends on the income elasticity,
, and the share of the budget spent on that good.

The relationship among the price elasticity of demand, , the substitution elasticity of demand, *, and the income elasticity of demand, , is the *Slutsky equation* (named after its discoverer, the Russian economist Eugene Slutsky):

= * + (-)

where is the budget share of this good: the amount spent on this good divided by the total budget.

If a consumer spends little on a good, a price change does not affect the consumer's total budget significantly. If the price of garlic triples, your purchasing power is hardly affected. The total effect, , for garlic hardly differs from the substitution effect, *, because the price change has little effect on income.

In Mimi's original equilibrium, *e*_{1}, where the price of beer was $12 and Mimi bought 26.7 gallons of beer per year, Mimi spent about three-quarters of her $419 budget on beer: 0.76 = (12 x 26.7)/419. Thus, Mimi's Slutsky equation is

-0.76 -0.09 - (0.76 x 0.88)

The total effect of a price change -- the price elasticity of demand -- is =

For a Giffen good to have an upward-sloping demand curve, must be positive. The substitution elasticity, *, is always negative: Consumers buy less of a good as its price goes up, holding utility constant. Thus, for a good to have an upward-sloping demand curve, the income effect, -, must be positive and large relative to the substitution effect. For the income effect to be positive, the good must be inferior, < 0. [Vandermeulen (1972) argues that both inferiority and satiability -- the consumer is near the point where more of this good is not better -- are necessary for a positive price effect.]

For the income effect, -, to be a large positive number, either the good is very inferior ( is a very negative number, which is not common) or the budget share, , is closer to one than to zero, or both. One reason why we don't see upward-sloping demand curves is that the goods on which consumers spend a large share of their budget, such as housing, are usually normal goods.

Vandermeulen, Daniel C., "Upward Sloping Demand Curves without the Giffen Paradox," *American Economic Review,* 62(3), June 1972:433-458.