Lecture 12: Sensitivity Examples (Shadow Price Interpreted)

1. Lecture 12: Sensitivity Examples (Shadow Price Interpreted) AGEC 352 Spring 2012 – February 29 R. Keeney

2. Shadow Price signs • Signs on shadow prices differ whether the inequality constraint is ≤ or ≥. • They also differ for maximization and minimization problems.

3. Less than (<=) case • A boundary that is <= (upper bound) • We use +1 definition of shadow price • The +1 will always ‘relax’ the upper bound • A decision maker facing a less restrictive choice set • Can be better off (binding constraint) • Can be unaffected (slack constraint) • Better off depends on max vs. min

4. Great than (>=) case • A boundary that is >= (lower bound) • We use +1 definition of shadow price • The +1 will always ‘tighten’ a lower bound • A decision maker facing a more restrictive choice set • Can be worse off (binding constraint) • Can be unaffected (slack constraint) • Better off depends on max vs. min

5. Example (Upper/Max) • Upper bound • Maximization • Land available to plant • Shadow price = the change in returns generated by a +1 to the land constraint • Shadow price = Maximum rent that can be paid • Use extra profits from additional resources to acquire the resource

6. Example (Upper/Min) • Upper bound • Minimization • Fertilizer mix phosphate limit • Shadow price = the change in costs from a 1 unit increase in the phos limit • Shadow price = discount the mixer could offer to the buyer to expand the phos limit • Pass some of cost savings to buyer

7. Example (Lower/Max) • Lower bound • Maximization • Every 10 acres of corn planted requires 1 acre left fallow (set aside) • Shadow price = change in profits from increasing set-aside by 1 • Shadow price = payment farmer must receive to participate

8. Example (Lower/Min) • Lower bound • Minimization • Calcium requirement in a daily diet • Shadow price = change in cost of requiring an extra unit of calcium • Shadow price = maximum price that can be paid per unit of non-food calcium supplement

9. Lab Assignment Problem • 4 Fertilizers (see lab 5 for fertilizer info) • Different compositions of nitrogen, potash, and phosphate • Meet an order (at minimum cost) by mixing the four fertilizers that has: • Exactly 1000 units of fertilizer • At least 20% (by weight) nitrogen • At least 30% (by weight) potash • At most 8% (by weight) phosphate

10. Shadow Prices in Fert. Problem

11. Interpretation of Potash • Potash constraint • Required to have a minimum amount of potash in the fertilizer mix • Increasing the RHS of the potash constraint makes the problem more restrictive, higher percentage of potash required • Shadow price is positive because costs will increase with the increase of RHS • Interpret this as the amount we would be willing to pay to avoid having the RHS increase • Also, the discount we could offer for a mix that had 0.1% less potash content

12. Interpretation Phosphate • Phosphate constraint • Upper limit on the phosphate content • Increasing the RHS of the phosphate constraint makes the problem less restrictive, higher percentage of phosphate allowed • Shadow price is negative because costs will decrease with the increase of RHS • Interpret this as the amount we would be willing to pay to relax the RHS by one unit • Also, the markup we should charge if someone required 0.1% less phosphate in their fertilizer mix

13. Interpretation in general • Always should be in context of the problem • Signs are actually trivial if you understand the problem (better off/worse off) • Does an increase in the RHS improve or worsen the objective? • If it improves, then we know the willingness to pay for increasing the RHS • If it worsens, then we know the willingness to pay to avoid having the RHS increase

14. Advanced Analysis: Which constraint is the most costly? • Recall the cereal problem from lecture • Two cereals mixed to meet minimum requirements on thiamine, niacin, and calcium

15. Rather than comparing units, we want to compare % of RHS • 1 mg of thiamine and 1 mg of niacin are not directly comparable • % increases in the RHS of constraints are however

16. Ranking the constraints • Thiamine was the most costly constraint to meet • We would have judged this the same just comparing shadow prices, but that could be misleading • Similar to elasticity interpretations • Elasticity of demand for food versus cars • Requires that you understand the problem and interpretation to make the comparisons

17. Fertilizer Problem • Consider • Is total comparable to others? • How to deal with positive vs negative shadow prices? • Compare relaxations of constraints…

18. Common percentage and direction (of objective variable) • Cost saving, 1% change in K • Total cost reduces $30.00 • Cost saving, 1% change in P • Total cost reduces by $11.20

19. Planting Problem • Shadow price for land is 2X labor • 1 unit of land is usually worth more than a unit of labor • Compare them as 1% increase in our resource base (labor > land > allot)