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Is the cross price elasticity of demand between two goods always lower in magnitude than the price elasticities of demand of each good?

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Not necessarily.

For example if the utility function is $U=\min\{x,y\}$, then the demand function for x is given by $D=\frac{M}{P_x+P_y}$.

The own price elasticity of demand for x is: $-\frac{M}{(P_x+P_y)^2} \frac{P_x}{x}$.

The cross price elasticity of demand for x is: $-\frac{M}{(P_x+P_y)^2} \frac{P_y}{x}$.

If these elasticities are evaluated at $P_x=P_y=p$, then for any value of x, you have the same elasticities.

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