ECON

First-Order Condition (FOC) for interior optimum

All first-order partial derivatives must equal zero. (fx = 0 and fy = 0).

Sufficient condition for a 2-variable Local Maximum

Hessian determinant (H) > 0 and the first partial derivative (f_xx) < 0.

Sufficient condition for a 2-variable Local Minimum

Hessian determinant (H) > 0 and the first partial derivative (f_xx) > 0.

Saddle Point condition (2 variables)

The Hessian determinant (H) is strictly negative (H < 0).

3-Variable Local Maximum Hessian sign pattern

Principal minors must alternate in sign starting with negative: |H1| < 0, |H2| > 0, |H3| < 0.

3-Variable Local Minimum Hessian sign pattern

All principal minors must be strictly positive: |H1| > 0, |H2| > 0, |H3| > 0.

Young's Theorem (Symmetric Matrix)

The order of differentiation does not matter for continuous functions. Mixed partials are equal (f_xy = f_yx).

Envelope Theorem core intuition

The total derivative of the value function equals the direct partial derivative with respect to the parameter.

Envelope Theorem (Indirect effect)

At the optimum, small adjustments in choice variables have a zero first-order effect on the objective value.

Comparative Statics definition

Analyzing how an optimal choice changes when an exogenous parameter shifts, holding other variables constant.