# Do the partial derivatives of the compensated demand have an interpretation?

When obtaining the Marshallian demand from a utility maximization problem we have a classification of normal goods, inferior goods, Giffen-goods, etc.. These are related to the first derivative of the Marshallian demand function with respect to prices or income.

My question is whether a similar classification exists for compensated demand functions of the expenditure minimization problem or whether the (same) goods classification can be derived via the compensated demand.

• I think you have to specify compensation in more detail. If it is a function of price and you take the derivative w.r.t. price the derivative of the Hicksian demand would be equal to the derivative of the Marshallian demand because of the envelope theorem. – Giskard Nov 23 '15 at 11:44
• By compensated demand I mean this: en.wikipedia.org/wiki/Hicksian_demand_function – HRSE Nov 23 '15 at 11:59
• I understood that. Does the price change from $p$ to $\hat{p}$ or is it just an infinitesimally small change? The derivative makes more sense in the second case, but then the demand functions coincide by virtue of the envelope theorem. – Giskard Nov 23 '15 at 16:17
• the derivative is meant. if by the envelope theorem one can prove that the derivatives are the same, then that is probably the answer. However, I am not sure whether this is true. For example, Marshallian demand for CD preferences would be $x_1=m/(2p_1)$ while the compensated demand depends on both $p_1$ and $p_2$. Thus, the cross price derivatives are for example not identical. – HRSE Nov 24 '15 at 2:00

To the question on whether the same classification can be achieved through the hicksian demand function: in general I believe the answer is no. To see this, relate the classifications to the Slutsky decomposition.

(Changes in) Hicksian demand is related to the substitution effect from the Slutsky decomposition. Any standard textbook should provide a reference to this.

1.Ordinary and Giffen goods: These refer to marshallian demand changes when prices change. This is what we call the Total Effect in the Slutsky decomposition.

1. Normal and inferior goods: These refer to marshallian demand changes when income changes. This is by definition the Income Effect in the Slutsky decomposition.

As you see, the substitution effect, which is represented by changes in hicksian demand, is not a part of any of these classifications you mentioned.