# Building blocks: Firm behaviour

- 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).

^{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.

#### Exploring Possible Futures: Modeling in Environmental and Energy Economics

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