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2.5

# Building blocks: Final steps

As the final steps in our modeling work, we want to discuss how to introduce different policy instruments in our models, how to introduce further complications, such as market power, and how to investigate how policies should be designed.

So far, we have always used the example of a tax on emissions in our models; for example, in the Output Abatement Choice Model:

(1)   $$\max_{q_{i},a_{i}\geq 0} {P\: q_{i} }-{c_{i}(q_{i}, a_{i})}-{t\: e_{i}},$$

(2)   $${e_{i}=s_{i}(q_{i},a_{i})}.$$

Here, $$t$$ is a tax that is levied on emissions $$e_{i}$$.

In theory, this policy is a simple and (usually) efficient policy. However in practice, it is often difficult to implement. One problem that is typically encountered is that it is necessary to monitor emissions in order to tax them. But monitoring emissions can be rather costly.

A simple alternative could be to use a tax on the final product instead of emissions. It is usually far easier to measure (and tax) the produced quantity than emissions. Even though this is hardly an optimal solution, such a tax might work: In absence of environmental policy, firms produce too much; a tax on the final product could reduce this “overproduction”.

In our model, we would simply use a tax, say $$\tilde{t}$$, levied on the produced quantity $$q_{i}$$, instead of emissions. Thus we would have $$\tilde{t} q_{i}$$ instead of $$t e_i$$ in equation (1).

The problem with this approach is that, although it reduces the produced quantity, it does not provide an incentive to produce more carefully; the firstorder condition with regard to care in production would be

$0=\frac{\partial c_{i}(q_{i},a_{i})}{\partial{a_{i}}}.$

This condition is independent of the tax, so that the policy does not influence $$a_{i}$$. Thus an output tax reduces emissions (by reducing production). But most likely it will be inefficient, as it cannot implement an optimal combination of producing less and producing more carefully.

Another option of environmental policy is emissions trading, which we already discussed last week. To model emissions trading, we could replace the tax in equation (1) by a price for emission permits (say $$p_{e}$$). This price would then result from emissions trading, that is, we have a market clearing condition for permits, where the sum of the number of permits available (say, firm $$i$$ receives $$z_{i}$$ permits) equals the number of emissions:

$\sum_{i=1}^{n} e_i^*= \sum_{i=1}^{n} z_{i}.$

This condition would define the permit price; the optimal emissions $$e_{i}^*$$ depend on the permit price $$p_{e}$$ and this permit price would adjust in a way so that total emissions (left-hand side) equal the total number of permits (right-hand side).

We could also model emission standards, where firms are required to reduce their emissions to a certain level. In this case, we would not have the costs of the environmental policy ($$t e_{i}$$) in equation (1). Rather, we would have an additional constraint, such as $$e_{i} ≤ \overline{e_i}$$, which describes the requirement that firm $$i$$ should not emit more than the standard $$\overline{e_i}$$ allows.

As an example of complications, we could introduce market power in our model. For example, we could consider a case where we have one firm that is a monopolist. In this case, the firm would not maximize its profit for a given product price $$P$$, but rather account for its power to influence this price. Thus we would replace $$P$$ in equation (1) by a function $$P(q_{i})$$ that describes the consumers’ marginal willingness to pay for the product (the inverse demand function). As you can easily see, this will lead to a different first-order condition for the firm, which will now produce less in order to keep the price high.

Finally, let us consider the problem of how a policy should be designed. What is the best value of an emissions tax or the best emission standard? To define this, we need a measure of how our policy performs. Usually, we try to find a policy that maximizes what we call social welfare, that is, the overall benefit that society gets from the firms’ and consumers’ activities.

How to define social welfare depends strongly on the building blocks used in our model. As discussed before, the Emission Choice Model is very simple, as it describes only emissions. We can thus account only for the benefits firms gain from emitting (their profits) and the damage caused by the total emissions, which is usually described by a damage function $$D$$ that depends on the sum of emissions. Social welfare is thus given by

$\sum_{i=1}^{n} B_{i}(e_{i})-D\left(\sum_{i=1}^{n}e_{i}\right).$

In case of the Output Abatement Choice Model, we also have to account for the welfare gained by consuming the product, which equals the area below the (inverse) demand curve. We thus have to account for this welfare, the costs of production, and the damage caused by emissions:

$\int_{0}^{Q=\sum_{i=1}^nq_i}P(\tilde{q}) d\tilde{q}-\sum_{i=1}^nc_i(q_i,a_i)-D\left(\sum_{i=1}^ns_i(q_1,a_i)\right).$

Finally, in case of the Input Choice Model, we have the same approach, but the costs of production are now described by the costs paid for the different inputs:

$\int_{0}^{Q=\sum_{i=1}^nf_i(x_{i,1},...,x_{i,m})}P(\tilde{q}) d\tilde{q}-\sum_{i=1}^n\sum_{j=1}^mw_jx_{i,j}-D\left(\sum_{i=1}^ns_i(x_{i,1},...,x_{i,m})\right).$

Note that, in all cases, the social welfare does not include the direct costs of the environmental policy. The reason is very simple: The taxes paid by the firms are costs to the firms but revenues for the tax-collecting authority. Thus they cancel out, when we derive social welfare.

We now have everything we need to build a model. You will be asked in the following steps to start building your own model. In Week 6, we will then discuss how to solve a model.