Topic 8: Unconstrained Optimisation Reading: Jacques: Chapter 5, - - PowerPoint PPT Presentation

Topic 8: Unconstrained Optimisation

Reading: Jacques: Chapter 5, Section 5.4

• 1. Finding stationary points of functions of

several variables

• 2. Application I: Revenue, Costs and Profit

Stationary Points: Functions of one variable

• First-order condition (necessary condition):

( )

' = = x f dx dy

• Second-order conditions (sufficient conditions):

( )

x f y =

• Second-order conditions (sufficient conditions):

( ) 0

' '

2 2

< = x f dx y d

( ) 0

' '

2 2

> = x f dx y d

Maximum Minimum

Stationary Points: Functions of several variables

• The total differential of the function

( )

z x f y , =

• First order conditions (necessary conditions):

dz z y dx x y dy . . ∂ ∂ + ∂ ∂ =

• First order conditions (necessary conditions):

. . = ∂ ∂ + ∂ ∂ = dz z y dx x y dy

= = ∂ ∂

z

f z y = = ∂ ∂

x

f x y

and

Stationary Points: Functions of several variables

• Classifying stationary points:
• 3 possible turning points:
• Minimum: increase in each direction

Minimum

Y Z X B

Stationary Points: Functions of several variables

• Classifying stationary points:
• 3 possible turning points:
• Minimum: increase in each direction
• Maximum: decline in each direction

Maximum

Y A Z X

Stationary Points: Functions of several variables

• Classifying stationary points:
• 3 possible turning points:
• Minimum: increase in each direction
• Maximum: decline in each direction
• Saddle Points: increase in one direction, decline
• Saddle Points: increase in one direction, decline

in other direction

Saddle Point

Stationary Points: Functions of several variables

• Second order conditions (sufficient conditions):

– For minimum:

2 2 2 2 2 2 2 2 2 2

>         ∂ ∂ ∂ −         ∂ ∂         ∂ ∂ > ∂ ∂ > ∂ ∂ z x y z y . x y and z y , x y

– For maximum: – For maximum:

2 2 2 2 2 2 2 2 2 2

>         ∂ ∂ ∂ −         ∂ ∂         ∂ ∂ < ∂ ∂ < ∂ ∂ z x y z y . x y and z y , x y

– For saddle point:

2 2 2 2 2 2

<         ∂ ∂ ∂ −         ∂ ∂         ∂ ∂ z x y z y . x y

Stationary Points: Functions of several variables

Summary

• Total differential:

( )

z x f y , =

• First order conditions:

dz z y dx x y dy . . ∂ ∂ + ∂ ∂ =

= ∂ ∂ = ∂ ∂ ⇒ = z y and x y dy

2 2 2 2

y y y       ∂ ∂ ∂

• Second order conditions:

2 2 2 2

> ∇ > ∂ ∂ > ∂ ∂ and z y , x y

2 2 2 2

> ∇ < ∂ ∂ < ∂ ∂ and z y , x y < ∇

2 2 2 2 2 2

• y

y y x z x z       ∂ ∂ ∂ ∇ = −       ∂ ∂ ∂ ∂      

Minimum Maximum Saddle Point

Stationary Points: Functions of several variables

• Example:

Find and classify the stationary points of the function:

( )

2 3

3 , xz x x z x f + − =

( )

• Example:

The profit function of a firm producing 2 goods is given by

Application I: Revenue Costs and Profit

2 2 2 1 2 1 2 1

2 2 800 1000 Q Q Q Q Q Q − − − + = π

2 2 1 1 2 1

1 2

• Q

Q

• Example:

A firm charges different prices for its goods to its domestic and export markets. The inverse demand equation for the domestic market is given by

Application I: Revenue Costs and Profit

1 1

500 Q P − = and for the export market is given by

2 2

5 1 360 Q . P − =

Q TC 20 50000 + =

2 2

The total cost function is where

2 1

Q Q Q + =

What prices will the firm charge if it is to maximise its profits?

• Example:

A firms inverse demand function is given by

Application I: Revenue Costs and Profit

5 .

8 2 100 A Q P + − =

Where A is expenditure on advertising, P is the price of the product and Q is the quantity demanded. In order to the product and Q is the quantity demanded. In order to produce Q units of output it costs the firm

AQ Q Q TC + + =

2

4

(i) Find an expression for the firm’s profit in terms of A and Q (ii) Find the values of A and Q at which profits are maximised