Finding max/min under constraint The behaviour of economic actors - - PowerPoint PPT Presentation

Topic 9: Constrained Optimisation Reading: Jacques: Chapter 5.5, 5.6

1. Finding max/min under constraint:

• Substitution Method
• Lagrange Multiplier Method

2. Application I: Utility 3. Application II: Production

Finding max/min under constraint

• The behaviour of economic actors is often constrained by the

economic resources they have at their disposal

• Example:

– Individuals maximising utility will be subject to a budget constraint – Firms maximising output will be subject to a cost constraint

• The function we want to maximise is called the objective

function

• The restriction is called the constraint
• Two methods:

– Substitution Method – Lagrange Multiplier Method

Finding max/min under constraint

• Example:

Find the minimum value of the objective function

2 2

2 z x y   

subject to the constraint

1 2   x z

Substitution Method

Finding max/min under constraint

STEP 1: Use the constraint to express z in terms of x STEP 2: Substitute expression for z into the objective function STEP3: Find the value of x that maximises or minimises the objective function STEP 4: Substitute this value into constraint to find corresponding value of z STEP 5: Substitute values for x and z to find optimal value of y (if required)

• The Substitution Method:

   

z x g M z x f y , subject to , Optimise  

Finding max/min under constraint

• Example:

Find the minimum value of the objective function

xz z x y 2 4 4

2 2

  

subject to the constraint

x z   8

Finding max/min under constraint

STEP 1: Define a new function

• The Lagrange Multiplier Method:

   

z x g M z x f y , subject to , Optimise  

     

z x g M z x f L , ,    

STEP 2: Find all first order partial derivatives STEP 3: Solve the system of equations:

, ,           L z L x L

STEP 4: Substitute values for x and z to find optimal value of y (if required)

Finding max/min under constraint

• Example:

Use the Lagrange Multiplier method to find the optimal value

• f the objective function

x xz x y 12 3

2

  

subject to the constraint

6 3 2   z x

Application I: Production

• Production can be described by:

 

L K f Q , 

K r

• f

price the is

L w

• f

price the is

costs d constraine is M

• The constrained optimisation problem can be described as:

 

wL rK M L K f    subject to , Q Optimise

Figure 5.14

Application I: Production

• Example:

A firm has a Cobb-Douglas production function:

 

5 . 5 .

10 , K L L K f Q  

What combination of K and L will be chosen to maximise output if the firm’s total cost of production is constrained to €100, the price of labour is €4 and the price of capital is €10?

Application II: Utility

• Assume and individual’s utility can be described

as:

 

2 1, x

x f U 

1 1

• f

price the is x p

2 2

• f

price the is x p

income consumer is M

• The constrained optimisation problem can be

described as:

 

2 2 1 1 2 1

subject to , Optimise x p x p M x x f U   

Application II: Utility

• Example:

Consumers’ preferences can be represented by the utility function

 

2 1 2 1,

x x x x f U  

How much will the utility maximising consumer demand of good 1 and good 2 if they have an income of €100, the price

• f good 1 is €5 and the price of good 2 is €1