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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, e1, 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
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.