1. 1 Topic 9 Constrained Optimization Jacques 5.5 and 5.6

2. 2 The behaviour of economic actors is often constrained by the economic resources they have at their disposal Examples: –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

3. 3

4. 4 Method 1: By Substitution

5. 5

6. 6

7. 7

8. 8 Method 2: By The Lagrange Multiplier

9. 9

10. 10

11. 11

12. 12

13. 13 Example 1 Question A consumers preferences can be represented by the Utility Function, U(x,y)=x.y. How much will the utility maximising consumer demand of goods x and y if they have an income of €100, the price of good x is €5 and the price of good y is €1?

14. 14 Solution Solution: (x*,y*) = (10,50) and U = 500

15. 15 Lagrangean Method:

16. 16 Lagrangean Method: