Why must the Slutsky matrix be symmetric and negative semidefinite?

Answer First

The Slutsky matrix must be symmetric because it is the Hessian of the expenditure function, which is twice‑differentiable and convex in prices. Hessians of convex functions are always symmetric.

It must be negative semidefinite because compensated demand satisfies the law of demand: holding utility constant, increasing a price cannot increase compensated demand.

• Symmetry → comes from Young’s Theorem (cross‑partial equality)
• Negative semidefinite → comes from convexity of expenditure and concavity of utility

Step-by-Step: Why These Properties Hold

Step 1: The Slutsky Matrix Comes From Hicksian Demand

Compensated demand is:

\[ h(p,u) = \nabla_p e(p,u) \]

So the Slutsky matrix is:

\[ S(p,u) = \nabla_p h(p,u) = \nabla^2_p e(p,u) \]

Step 2: Symmetry Comes From Cross‑Partial Equality

If the expenditure function is twice continuously differentiable:

\[ \frac{\partial^2 e}{\partial p_i \partial p_j} = \frac{\partial^2 e}{\partial p_j \partial p_i} \]

This is Young’s Theorem → the Hessian is symmetric.

Step 3: Negative Semidefinite Comes From Convexity

The expenditure function is concave in utility but convex in prices. Convexity implies:

\[ v^\top S v \le 0 \quad \text{for all } v \]

This is exactly negative semidefiniteness.

Step 4: Economic Interpretation

Symmetry → substitution effects are reciprocal. If increasing price of good i affects demand for j, then increasing price of j affects demand for i in the same compensated way.

Negative semidefinite → compensated demand obeys the law of demand.

Numerical Example

For a Cobb–Douglas utility:

\[ u(x,y) = x^\alpha y^{1-\alpha} \]

The Hicksian demands are:

\[ h_x = \alpha \frac{e}{p_x}, \quad h_y = (1-\alpha)\frac{e}{p_y} \]

Differentiating with respect to prices yields a Slutsky matrix:

\[ S = \begin{bmatrix} -\alpha \frac{e}{p_x^2} & 0 \\ 0 & -(1-\alpha)\frac{e}{p_y^2} \end{bmatrix} \]

This matrix is:

• symmetric (diagonal)
• negative semidefinite (all diagonal entries ≤ 0)