Decomposing the Slutsky Decomposition for the First Time in a Century

Abstract

<p>The Slutsky decomposition is a mathematical formula which has been used for a very long time in economics to analyze how the demand for a good changes when its price goes up. Such a change in the demand is called the price effect. Most people will expect a decrease in the demand in response to a rise in the price. In other words, they will have a downward sloping demand curve in mind. Indeed they are right in usual cases.In terms of economicsit is said that the price effect is usually negative. But why? The Slutsky decomposition gives a correct answer to this question by decomposing the price effect into the substitution effect and the income effect. It is already known that the substitution effect is always negative, while the income effect is also negative if a good under considerationis “normal.”Since the sum of the two effects equals the price effect, it can be concluded that a demand for the good decreases when its price goes up, or the demand curve is downward sloping, in a “normal” situation. The Slutsky decomposition is so elegant and powerful that there would not be any economist who studies consumer behavior without mentioning it. Then, is there other decomposition than the Slutsky? This paperintroduces a new one which decomposes the price effect into the unit-elasticity effect and the ratio effect. The unit-elasticity effect means by how much the demand decreases in response to a rise in the price with the expenditure on it as fixed. The ratio effect means by how much the demand changes due to a change in the ratio of the expenditure on it to total income. The former effect is always negative, but the latter effect may be positive even in a “normal” situation. It is shown that anew decomposition is obtained by decomposing the Slutsky decomposition.</p>