1. Frank Cowell: Lecture Examples Example – single technique z1z1 z2z2 0 1 z1z1  3 z1z1 z2z2 0 3 z2z2  1 8 Oct 2015 1

2. Frank Cowell: Lecture Examples Example – two techniques z1z1 z2z2 0 3 1 z1z1 z2 z2   3 1 8 Oct 2015 2

3. Frank Cowell: Lecture Examples Example – multiple techniques z1z1 z2z2 0 3 1 3 1 z2 z2  z1z1  8 Oct 2015 3

4. Frank Cowell: Lecture Examples Example: 4 8 Oct 2015 z2z2 z1z1 Use spreadsheet to find (z 1, z 2 ) such that log 2 = 0.25 log z 1 + 0.75log z 2 ) Plot on graph Z(2) = {z:  (z)  2}

5. Frank Cowell: Lecture Examples Example 5 8 Oct 2015 z2z2 z1z1 Isoquant q = 2 (as before) Isoquant q = 1 Isoquant q = 3 Equation of isoquant Homotheticity Check HD 1 from original equation double inputs → double output

6. Frank Cowell: Lecture Examples Example 6 8 Oct 2015 Production function Keep input 2 constant Marginal product of good 1

7. Frank Cowell: Lecture Examples 7 8 Oct 2015

8. Frank Cowell: Lecture Examples Example – cost-min, single technique z1z1 z2z2 0 1 z1z1  3 z1z1 z2z2 0 3 z2z2  1 8 Oct 2015 8

9. Frank Cowell: Lecture Examples Example – cost-min, two techniques z1z1 z2z2 0 3 1 z1z1 z2 z2   3 1 8 Oct 2015 9

10. Frank Cowell: Lecture Examples Example 10 8 Oct 2015 z2z2 z1z1 Isoquant (as before) does not touch either axis Constraint set for given q Cost minimisation must have interior solution

11. Frank Cowell: Lecture Examples Lagrangean for cost minimisation Necessary and sufficient for minimum: Evaluate first-order conditions Example 11 8 Oct 2015 z2z2 z1z1 z*

12. Frank Cowell: Lecture Examples First-order conditions for cost-min: Rearrange the first two of these: Substitute back into the third FOC: Rearrange to get the optimised Lagrange multiplier Example 12 8 Oct 2015

13. Frank Cowell: Lecture Examples From first-order conditions: Rearrange to get cost-min inputs: By definition minimised cost is: In this case the expression just becomes * So cost function is Example 13 8 Oct 2015

14. Frank Cowell: Lecture Examples 8 Oct 2015 14

15. Frank Cowell: Lecture Examples Example – input demand with two techniques z1z1 z2z2 0 3 1 z1z1 z2 z2   3 1 7 Oct 2015 15 z1z1 3 1 w1w1 1

16. Frank Cowell: Lecture Examples First-order conditions for cost-min: Rearrange the first two of these: Substitute back into the third FOC: Rearrange to get the optimised Lagrange multiplier Example 16 8 Oct 2015

17. Frank Cowell: Lecture Examples From last lecture, cost function is Differentiate w.r.t. w 1 and w 2 Slope of conditional demand functions Example 8 Oct 2015 17