1 Optimization

2 General Problem

3 One Independent Variable x y (Local) maximum Slope = 0

4 One Independent Variable (cont’d) slope=0 is necessary but not sufficient condition for a maximum Minimum, f’(x)=0 too f’(x)=0 too, but it is a point of inflexion

5 One Independent Variable x y (Local) maximum Slope = 0 x f’(x) Slope of f’(x) is zero x0x0 x0x0

6 One Independent Variable

7 Two Independent Variables x x* z* y A

8 Two Independent Variable

9 Constrained Optimization f(x,z)=25, better but not feasible F(x,z)=0 f(x,z)=20 x z contours

10 Constrained Optimization (cont’d)

11 Constrained Optimization (cont’d)

12 Lagrange’s method Lagange’s method introduces a third variable called the Lagrangian multiplier (λ) that allows us to treat the problem as if an unconstrained one.

13 Lagrange’s method (cont’d)

14 Lagrange’s method

15 The Farmer’s Problem

16 The Farmer’s Problem (cont’d)

17 Envelope theorem

18 Envelope theorem (Cont’d) The total derivative of the maximum value of y with respect to α is just equal to the partial derivative of y with respect to α, evaluated at the optimal choice of the x i ’s. This is called envelope theorem!

19 Envelope theorem (cont’d)

20 Envelope theorem (cont’d)