constrained optimisation

Question 1

name three common examples of constrained optimisation problems ?

Answer 1

1. max utility ( budget constraint )
2. min cost ( min prod quota )
3. insurance max profit ( not scare away customers and moral constraint )

Question 2

what are binding and non-binding constraints ?

Answer 2

binding constraints fix the feasible region and non binding do not.

Question 3

what is the feasible region ?

Answer 3

area of all possible options

Question 4

where is the optimal choice when the feasible region is plotted ?

Answer 4

at the far most point on the binding constraint

Question 5

what is the form you put the main function and constraints in to be able to solve for the optimal x and y ?

Answer 5

the Lagrangian form where:

L(x,y) = F(x,y) + Lagrangian. Gk(x,y)

Question 6

how do you solve a Lagrangian issue ?

Answer 6

find the F.O.C ( all three or the first derivatives ) = 0

then put the x and y derivatives over each other to find one of the variables

then substitute into the Lagrangian derivative to find X* and Y*

Question 7

what do you do when you have two possible (x,y)’s ?

Answer 7

you must substitute them into the Lagrangian and check if they are positive to find the correct (X,Y)

Question 8

what are the local min/max or the global min/max ?

Answer 8

the local min or max is the min/max in a certain area

the global is over the whole function the absolute max/min

to find the global you can plug the local into the function to find the highest or lowest

Question 9

what is the hessian matrix ?

Answer 9

( d’‘z/dx’’ d’‘z/dydx )

d’‘z/dxdy d’‘z/dy’’

Question 10

using the determinant of the hessian and the leading diagonal how do you determine the nature of a point?

Answer 10

MIN ) + prod of lead diag, + determinant of H

MAX ) - prod of lead diag, + determinant of H

SADDLE ) - determinant of H

UNDETERMINED ) determinant = 0