1. Unknown control costs1 ECON 4910 Spring 2007 Environmental Economics Lecture 10, Chapter 10 Lecturer: Finn R. Førsund

2. Unknown control costs 2 The regulator’s problems Purification cost functions unknown to the regulator What to do?  Form the expected cost function  Ask the polluters about their cost functions Assumption: only one polluter Decision of the regulator.  What policy instrument to choose

3. Unknown control costs 3 The regulator’s problem with uncertainty of the cost function The social problem The assumptions  One polluter  Only purification costs, output of polluter fixed  Damage function known  Standard curvature of functions

4. Unknown control costs 4 The solution to the regulator’s problem First-order condition Must have information to form the expected marginal cost function

5. Unknown control costs 5 Illustration of social optimum e D’(e) -E{c’(e)} Probability distribution e* -E{c’(e)} = D’(e)

6. Unknown control costs 6 The social solution with a specific distribution Simplifying assumption about the distribution function of the cost function  Only two types; high cost, c H (e) and low cost, c L (e)  Probability 0.5 for each type  Only assuming a single firm The expected marginal cost The social solution

7. Unknown control costs 7 Choice of policy instrument Emission tax The adjustment of the firm The firm knows its own type

8. Unknown control costs 8 Illustration of the simple distribution case e D’(e) -E{c’(e)} e* t -c H ’ -c L ’ tLtL tHtH e H (t) eHeH eLeL e L (t) Social loss if H Social loss if L

9. Unknown control costs 9 Regulator asks about cost type Assumptions  Emission tax is used as instrument  Regulator does not control if the information is correct, i.e. regulator does not measure actual emission  No ethical problems of the firm lying It will always minimise costs of the firm if it declares to be of the low cost type

10. Unknown control costs 10 Illustration of firm giving cost information e D’(e) -c H ’ -c L ’ tLtL tHtH eH*eH* eL*eL* Social loss if H and lying e H (L) e L (H)

11. Unknown control costs 11 Problems with the Kolstad model Nothing said about whether emission are measurable Assume that emissions are measurable:  When regulator is given the information about the cost function, then the regulator also knows the marginal cost function, and optimal emissions e L *, e H * can be calculated obeying

12. Unknown control costs 12 Problems with the Kolstad model, cont. When the firm reports its cost function then the Regulator knows the correct emission e j * and comparing this level and the choice of emission will tell the Regulator whether the firm is lying or not, thus inducing truth-telling. The possibility to observe emission, but not the cost function, is quite realistic. If emissions cannot be observed, then the firm may cheat about reported emission as well.

13. Unknown control costs 13 Kolstad: Introducing an incentive to tell the truth A high-cost firm can be compensated for telling the truth A reward for telling that the firm is high-cost- type  How to minimise the reward, R, and still make the firm willing to reveal that it is of the high-cost type If high-cost is true then

14. Unknown control costs 14 Incentive to tell the truth, cont. Setting the reward R such that it does not pay to lie when the true type is low-cost The range for R  Left-hand side positive, total cost telling the truth higher than cost when lying

15. Unknown control costs 15 The Kolstad analysis is not correct It is easy to demonstrate using diagrams that there is no amount of payment R that satisfies the equation above

16. Unknown control costs 16 Illustration of giving an incentive to the high-cost firm to tell the truth Difference in cost if H between lying and truth e D’(e) -c H ’ -c L ’ tLtL tHtH eH*eH* eL*eL* e H (L) e L (H)

17. Unknown control costs 17 Illustration of giving an incentive to the low-cost firm to tell the truth Difference in cost if L between truth and lying e D’(e) -c H ’ -c L ’ tLtL tHtH eH*eH* eL*eL* e H (L) e L (H)

18. Unknown control costs 18 An alternative model for asymmetric information Consider only two types of firms, high-cost H and low cost L Assume that a firm earns a profit π from its economic activity independent of type Emissions are measurable, but not cost functions The regulator offers contracts specifying permitted emission and a type-specific subsidy/tax - transfer

19. Unknown control costs 19 The contracts for the two cost types for emission quantity and tax Contract for low- cost type  π –T L *- c L (e L *) = 0 → T L * = π - c L (e L *) Contract for high-cost type  π –T H * - c H (e H *) = 0 → T H *= π - c H (e H *) Low-cost type choosing high-cost contract  π –T H * - c L (e H *) = π –T H * - c L (e H *) + c H (e H *) - c H (e H *) = {π –T H * - c H (e H *)} + c H (e H *) - c L (e H *) If high-cost gross profit is ≥ 0, then it pays the low-cost type to choose a high-type contract

20. Unknown control costs 20 The contracts for the two cost types, cont. Can it be profitable for the high-cost type to take a low-cost type contract  π –T L * - c H (e L *) = π –T L * - c H (e L *) + c L (e L *) - c L (e L *) = π – {T L * - c L (e L *)} - c H (e L *) + c L (e L *) = π- π - c H (e L *) + c L (e L *) < 0 It cannot be profitable for a high-cost type to take a low-cost type contract The problem for the regulator is the behaviour of the low-cost type taking the high-cost type contract

21. Unknown control costs 21 Contract to avoid low-cost firm taking a high-cost type contract The contracts should fulfil two objectives  Ensure participation of the firm, i.e. the gross profit must be non-negative  Give incentive to tell the truth about the cost type, i.e. transfer according to type must induce truth- telling

22. Unknown control costs 22 Contracts Participation:  U L = π -T L - c L (e L ) ≥ 0  U H = π -T H - c H (e H ) ≥ 0 The tax must not be so high that the firms close down The quantities to be permitted emitted in the contracts are endogenous and still not determined

23. Unknown control costs 23 Incentives Define gross profit (pure profit) when telling the truth as U j, j=L,H Incentives to tell the truth  U L ≥ π -T H - c L (e H )  U H ≥ π -T L - c H (e L ) It should give higher gross profit to tell the truth than to lie.

24. Unknown control costs 24 Combining participation and incentive The high-cost type will not take a low-cost- type of contract The tax-quantity pair in the contract can therefore be set such that gross profit is zero  U H = π -T H - c H (e H ) = 0 The incentive-condition for a high-cost type will not be a problem, i.e. fulfilled The participation condition for low-cost type is fulfilled when U H ≥ 0 implying U L > 0

25. Unknown control costs 25 The active participation- and incentive conditions The four original conditions are reduced to two  U H = 0  U L = U H + c H (e H ) - c L (e H ) = c H (e H ) - c L (e H ) The last condition was established with e H = e H * above

26. Unknown control costs 26 Determination of emission- and tax quantities of the contracts The objective function for the Regulator is  W = V + αU = T- D(e) + αU = π – c(e) – U – D(e) + αU = π – c(e)– D(e) + (1-α)U Assume that low-cost type appears with probability p and high-cost type with probability (1-p) The two possible objective functions are  W L = π – c L (e L )– D(e L ) + (1-α)U L  W H = π – c H (e H )– D(e H ) + (1-α)U H

27. Unknown control costs 27 Determination of emission- and tax quantities of the contracts, cont. Maximising the expected value of the objective function  E{W} = p(π – c L (e L ) – D(e L ) - (1-α)(c H (e H ) - c L (e H ) ) + (1-p)(π – c H (e H ) – D(e H ) ) (setting U H = 0 ) Differentiating:

28. Unknown control costs 28 Determination of emission- and tax quantities of the contracts, cont. The low-cost contract for emission is set by  This is the standard condition. The condition for emission allowed on the high-cost type contract is implying a higher emission than e H *.

29. Unknown control costs 29 The allowed emission for the high-cost type Rearranging the second condition for maximising the objective function High-cost- emission set such that marginal damage equals the sum of two terms, the marginal purification cost and a term reflecting the negative effect for the Regulator caused by the low-cost type trying to take a high-cost-type contract