Relationship between own and cross price elasticities for a compensated demand model.

In the context of a compensated demand model, which focuses on changes in quantity demanded while maintaining constant utility, the relationship between own and cross price elasticities is essential for understanding consumer behavior and the substitution effects between goods. Compensated demand functions adjust for the income effect by keeping the consumer's utility level constant, isolating the pure substitution effect.

Own price elasticity of demand measures the responsiveness of the quantity demanded of a good to changes in its own price, holding utility constant. Formally, it is defined as:

$\varepsilon_{ii} = \frac{\partial x_i}{\partial p_i} \cdot \frac{p_i}{x_i}$

where $x_i$ is the quantity demanded of good $i$, and $p_i$ is the price of good $i$. A negative own price elasticity indicates that as the price of the good increases, the quantity demanded decreases, consistent with the law of demand.

Cross price elasticity of demand measures the responsiveness of the quantity demanded of one good to changes in the price of another good. In the compensated framework, this is defined as:

$\varepsilon_{ij} = \frac{\partial x_i}{\partial p_j} \cdot \frac{p_j}{x_i}$

where $x_i$ is the quantity demanded of good $i$, and $p_j$ is the price of good $j$. The sign and magnitude of the cross price elasticity indicate the relationship between the goods. If $\varepsilon_{ij} > 0$, the goods are substitutes; if $\varepsilon_{ij} < 0$, they are complements.

In a compensated demand model, the Symmetry Property of the Slutsky equation provides a crucial link between own and cross price elasticities. The Slutsky equation decomposes the total effect of a price change into substitution and income effects. The compensated demand function focuses only on the substitution effect. The symmetry property states that the cross partial derivatives of the expenditure function (or the compensated demand functions) are equal, implying that:

$\frac{\partial x_i}{\partial p_j} = \frac{\partial x_j}{\partial p_i}$

This means that the cross price elasticity of demand for good $i$ with respect to the price of good $j$ is equal to the cross price elasticity of demand for good $j$ with respect to the price of good $i$, when compensated for constant utility:

$\varepsilon_{ij} = \varepsilon_{ji} \cdot \frac{x_j}{x_i} \cdot \frac{p_j}{p_i}$

This relationship highlights the mutual dependence and symmetry in consumer preferences regarding price changes and the substitution effect. For instance, if the price of good $j$ rises, the substitution effect would reflect how consumers would adjust their consumption of good $i$ in response, and vice versa. This symmetry helps in understanding and predicting the interplay between various goods in a market.