This equation tells us something about the optimal path of the shadow price.

Transcription

1 PROBLEM 1 A transboundary environmental problem can be a problem where the damage from emissions (or more likely the stock of pollutants) in one country depends on the total level of emissions (from all countries), i.e. a uniformly mixing pollutant, where only the sum of emissions matter. A very relevant example is the emission of greenhouse gasses and global warming. It can also be a problem of nonuniformly mixing pollutants, where emissions in one country end up in a receptor in another country (or several other countries). An example of this is acid rain (Great Britain uses coal, acid rain destroys fishing in Norway). What they have in common is that the problem extends across national borders. In the non-uniformly mixing case, the main problem is that one country has only the benefits from emitting and the other has all the damages (in the most extreme case). The first country will not have any incentive to cut back on emissions unless they are influenced by the other country (either financially or morally). International treatises may be a solution, where the polluter can compensate the damages or the other country can pay to avoid the damages, as in the Coase theorem. A big problem here is deciding who has to pay whom (which in Coase depends on property rights, a topic that is much more complicated internationally). In most cases it is not so black and white, and many countries can both pollute and have damages, with several pollutants and receptors (one pollutant can affect several receptors, and one receptor can be affected by several pollutants). This does not make the problem any easier, since all countries may be affected differently and have different benefits, an agreement can be very difficult to reach (an international agreement has to be entered voluntarily by all members, and no one will agree to something they consider to be bad for their country, even if it may be good for everyone seen as one). In the uniformly mixing case, there are also problems with forming international agreements. All countries gain when one country abates (i.e. cuts down emissions) but that country will not abate unless its own reduced damages are larger than the cost of abating. This means that even though the total reduction in damages greatly exceeds the private abatement cost, no country will abate on their own initiative. An international agreement would appear to be able to sort this out, if the total benefit is large. The optimal situation is where the marginal cost of abating is equal to the sum of all damages (from every single country affected by the emissions). Since this sum is equal for all, the countries will adapt to these damages, and will end up in a least-cost situation (where marginal abatement costs are equal for all, i.e. abatement will be greater in the countries where it is cheapest). The problem with this is, as in the non-uniformly mixing case, that though it is good collectively, it may not be good for the individual members. One solution to this is to offer side payments, i.e. that those who benefit the most compensate those who "suffer", since no one will agree unless it is in their own best interest. Another problem with the uniformly mixing case is the free-rider problem. When total abatement is what matters, and there are many polluters/potential abaters, the contribution from any single country has very little effect on the total. It may therefore be tempting to free-ride on the effort of the other countries, enjoy the reduced damages while not paying any of the costs. If no one else abates then one country abating would do close to nothing to reduce damages, and when many abate one country not abating has close to no bearing on the resulting reduction in damages. Individually, no country then has any incentive to join an agreement. If an agreement is put in place, there is a stability condition for the coalition to hold. The internal stability condition says that no one in the agreement should have any incentive to leave the coalition, and the external stability condition says that no one outside the coalition should have any incentive to join. If these are satisfied, we have a stable coalition. Unfortunately, most research up till now points in the direction of very small stable coalitions (about 3 with identical, quadratic cost functions).

2 PROBLEM 2 a) In the optimal solution the marginal productivity of x must equal - µ, where µ is the shadowprice of emissions, which captures the effect of the growth in the stock of carbon (following from the emissions produced when using the input x) on the social welfare. Social welfare decreases when the stock increases, so µ is negative. That means that both sides of equation (i) are positive. In the optimal solution the marginal productivity of y must equal the marginal cost of producing y. This is a basic economic result. We know from the problem text that the marginal cost is positive (increasing cost fn.). This equation tells us something about the optimal path of the shadow price. This is the transversality condition, and will be used to arrive at a solution to the optimal time path of the shadow price (or rather, the tax, see b). b) From (i) we see that in optimum, the marginal productivity of x must equal - µ. In order to achieve this, it is possible to set a tax equal to - µ (see equations below). Then the sector will produce using the right amount of x, given by this equation (which they would not do otherwise, because the shadow price is not a price they would have to pay, and so they would not include it in their private optimization problem). This optimal tax depends on the damage parameter k and the two rates r and δ(see eq. (vi)). r is the discount rate and δ is the depreciation rate of the stock (the rate at which the stock "disappears" naturally, i.e. takes another (less harmful) form). If the damage is large and the sum of the rates small, the damage will dominate the equation and the tax will be high, and vice versa. c) If the tax rate is lower than the optimal tax given in b), we need a second best solution where a subsidy on y motivates a correct input-mix. Since x is "free" without the tax, and y has a production cost, the sector will prefer to use x (assuming x and y are perfect substitutes). The tax therefore decreases the use of x by putting a cost on it (internalizing the harmful effect of x on the environment). When the tax is not high enough, the sector will use too much x (and too little y). It can then be optimal for the government to put a subsidy on the use of y to encourage the sector to use more of this input (and less of x). This subsidy is positive in the case where the tax is lower than optimal, and negative if the tax is higher than optimal. 2. Social welfare at time t: ( ) ( ) «benefit» of emissions Damage from emissions a) Maximize the present value of the social welfare at time t: ( ( ) ( )

3 s.t. Need to set up the Hamiltonian: ( ) ( ) ( ) (i) (ii) (iii) (iv) b) Set a tax: Then from (iii): (v) Using rules for differentiation ( ) From (iv) we have that: (drop the minus below, doesn t make a difference)

4 ( ( )) ( ) then we must have Is the optimal tax per unit of x. Alternative: I know that ( ) since and we know that: (iv) : a rise in k (which is a rise in damages) will increase the optimal tax : a rise in r or (or both) will decrease the optimal tax

5 A rise in r means a higher discount rate, more concern for the present, less for the future. A rise in means the stock depreciates faster; ie. less damage through a lower stock c) Private optimization problem: FOC: tax subsidy Optimal (first best) if and (answer from a) and b)). is the bytteforhold (red.anm. rate of substitution) in first best when need: which, when solving for s, gives: ( ) > 0 k > 0, c > 0 and if