Unconstrained Optimization in Mathematical Economics, Lecture notes of Mathematics

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ECON 3031 Mathematical EconomicsECON 3031 Mathematical Economics

I I

Unconstrained Optimization Unconstrained Optimization

See p. 692-693. Solve the problems on p. 694.

Suppose that Q=f(K,L) is a production function with K as the

cprapital input and L as the labor input. The price per unit of

output is P, the cost (or rental) per unit of capital is r, the wage

rate is w. The constants p, r, and w are all positive. The profit π

from producing and selling F(K,L) units is then given by the

function

P ( K L , ) = pF K L ( , ) - rK - wL

Example

Sufficient Conditions for a Max.and Min.

Suppose that (x

0

,y

0

) is a stationary point for C

2

function f(x,y) in a

convex set S.

(a) If for all (x,y) in S,

2

11 22 11 22 12

f ( , x y ) 0, f ( , x y ) 0, f ( , x y f ) ( , x y ) ( f ( , x y )) 0

then (x 0

,y 0

) is a maximum point for f(x,y) in S.

(b) If for all (x,y) in S,

2

11 22 11 22 12

f ( , x y ) 0, f ( , x y ) 0, f ( , x y f ) ( , x y ) ( f ( , x y )) 0

then (x ,y ) is a minimum point for f(x,y) in S.