# Building blocks: Firm behaviour

You have seen that environmental and energy economic models consist of different types of building blocks and we have discussed the first and most important type of building block: Firm behaviour.

Modeling firm behaviour is most important, because many parts of environmental or energy policy aim at firms: For example, the European Emissions Trading System aims at reducing firms’ greenhouse gas emissions or many parts of energy policy aim at altering firms’ investment behaviour.

Let us go into somewhat more detail regarding the three building blocks for firm behaviour from Xepapadeas (1997) that we have already discussed. All three building blocks model firms as maximizing their profit. In case of the Emission Choice Model, a firm’s profit is described by a function \(B_{i}(e_{i})\), which denotes the maximal profit firm \(i\) can make for a given level of emissions \(e_{i}\). In addition, we have to account for the costs induced by environmental policy, which could, for example, be given by a tax paid for remaining emissions: \(t e_{i}\).

Thus the model for firm \(i\) can be written as:

(1) \(\max_{e_{i} \geq 0} B_{i}(e_{i})-t e_{i}\)

Note that there are no constraints regarding the number of firms and that firms can differ substantially from each other; each firm can be described by a different function \(B_{i}(e_{i})\).

To make sure that the model works, we often use assumptions regarding the function \(B_{i}(e_{i})\). For example, we will typically assume:

- that \(B_{i}(e_{i})\) is differentiable
- that, at least for low values of \(e_{i}\), \(B_{i}(e_{i})\) is increasing in \(e_{i}\) (emitting more reduces production costs) and that, for \(e_{i}\) close to zero, the slope of \(B_{i}(e_{i})\) is strictly greater than the tax \(t\)
- that \(B_{i}(e_{i})\) is strictly concave, implying that it is less costly to reduce emissions by one unit, if a firm emits much (and thus has many options to reduce its emissions) than if the firm has already reduced its emissions substantially (all cheap options to reduce emissions are already used, only the expensive options remain)
- that \(B_{i}(e_{i})\) is bounded above (even if a firm can emit as much as it wants, it receives a finite profit).

Under these assumptions, the model yields a unique optimal emission level \(e_{i}^*\) for each tax \(t\) and each firm \(i\). This optimal tax level can be calculated by differentiating the model with respect to \(e_{i}\) and setting the result equal to zero (ie, by deriving the first-order condition), which yields:

(2) \(B_{i}'(e_{i}^*)=t\)

If we use a specific functional form for \(B_{i}(e_{i})\), we can solve this equation and derive an explicit solution for \(e_i^*\). In many cases however, equation (2) is already sufficient to gain interesting insights into a problem. For example, it shows that if we use the same tax \(t\) for all firms, all firms will have equal marginal profits \((B_i^{'})\), which is a necessary condition for the policy to be economically efficient.

A similar approach can be used with the Output Abatement Choice Model, which we described by:

(3) \(\max_{q_{i},a_{i}\geq0}Pq_{i}-c_{i}(q_{i},a_{i})-t e_{i},\)

(4) \({e_{i}=s_{i}(q_{i},a_{i})}.\)

Here, the firm is described as maximizing its profit by choosing its quantity of production \(q_{i}\) and the care used in production (\(a_{i}\)). Emissions result from these choices according to a function \(si(q_{i},a_{i})\). Using specific assumptions on this function as well as on the cost function \(ci(q_{i}, a_{i})\),^{1} we can ensure that there is a unique optimal level of production and care that can be calculated from the first-order conditions:

(5) \(P = \frac{\partial c_{i}(q_{i},a_{i})}{\partial{q_{i}}} +t \frac{\partial s_{i}(q_{i},a_{i})} {\partial{q_{i}}},\)

(6) \(0= \frac{\partial c_{i}(q_{i},a_{i})}{\partial{a_{i}}} +t \frac{\partial s_{i}(q_{i},a_{i})} {\partial{a_{i}}}.\)

These equations show that the tax will reduce a firms output (by equation (5), it works like a reduction of the price \(P\)) and increase the care used in production (see equation (6)).

Finally, a similar approach can be used in case of the Input Choice Model, where we need to use the firms’ first-order conditions with respect to all input choices \(x_{i,1}, x_{i,2}, . . . , x_{i,m}\).

To sum up: In all models, we will need assumptions (usually, differentiability and a convexity/concavity assumption) to make sure that the problem can be solved (that is, that there exists an optimum). Then, we use the first-order conditions to derive this optimum. Thereby, we get one first-order condition for each decision variable of each firm. In the Emission Choice Model, firms choose only their emissions, thus we get one condition per firm. In the Output Abatement Choice Model, firms choose production quantity and care used in production. Thus we get two conditions per firm. In the Input Choice Model, we have as many conditions per firm as we have inputs that the firms choose.

Thus you see these models differ substantially in complexity. As we argued before: A model can become rather complex very easily. Thus always use the simplest model that can describe your problem.

^{1} Most importantly, we need to assume (a) differentiability of both functions, (b) that \(c_{i}(q_{i},a_{i})\) is increasing wrt both arguments, (c) that \(s_{i}(q_{i},a_{i})\) is increasing in \(q_{i}\) and decreasing in \(a_{i}\), and (d) that both functions are strictly convex wrt \((q_{i},a_{i})\) and (e) bounded below by zero.

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