Suppose that preferences are concave. Is it still the case that the substitution effect is negat

step 1 Yes, it is still negative. According to the rule of economics, when preferences are concave, it

Suppose that preferences are concave. Is it still the case...

Key Concepts

Substitution Effect

The substitution effect represents the change in consumption that results solely from a change in relative prices, holding the consumer's overall level of satisfaction (utility) constant. It captures the consumer’s tendency to substitute away from a good that has become relatively more expensive toward one that is relatively cheaper. This effect is typically negative, meaning that if the price of a good increases, the compensated demand for that good decreases.

Concave Preferences

Concave preferences are represented by utility functions that are concave, embodying diminishing marginal utility and risk aversion. Although concavity is a stronger condition than quasi?concavity (which ensures convex upper contour sets), concave preferences still guarantee that the consumer’s choice problem is well-behaved, even if potential issues such as non-uniqueness of optimal bundles might arise. This property is crucial in analyzing how changes in prices affect choice.

Expenditure Function and Duality

The expenditure function gives the minimal amount of money a consumer needs to achieve a given level of utility at prevailing prices. Through duality theory, the compensated (Hicksian) demand can be derived as the gradient of this function with respect to prices (as per Shephard’s lemma). Because the expenditure function is concave in prices, its derivative is decreasing, a fact that mathematically underpins the negative nature of the substitution effect.

Slutsky Decomposition

The Slutsky decomposition breaks down the total change in demand due to a price change into two parts: the substitution effect and the income effect. The substitution effect isolates the impact of the change in relative prices by holding utility constant, while the income effect accounts for the change in purchasing power. Under standard assumptions, including the properties of concave preferences, the substitution component is negative, confirming that higher prices lead to lower compensated demand for the good.

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