In the previous step, you have seen the main ideas that lead to the different building blocks of firm behaviour. Let us now go into somewhat more detail: let us investigate what a building block of firm behaviour has to do to fit into energy or environmental economic models and what kind of assumptions are typically used to achieve this.

As explained in the previous video, these building blocks describe the behaviour of some of the most important actors in our models: firms.

Firms are important because they produce goods, such as electricity, and decide how much to produce and what kind of technologies are used in the production. Furthermore, as a side effect of these decisions, firms determine a substantial part of the emissions in an economy, such as CO₂-emissions from electricity production. It is important to recognize that firms pursue their own objectives and might thus not decide in a way that is best for society. Thus, firm decisions should be modelled based on firm objectives.

To describe the behaviour of firms, you will need four items:

1. You have to determine which decisions the firms are able to make in your model: do they only decide how much to produce or also how to produce?

2. You have to describe the circumstances under which the firms make these decisions, such as market prices, like the electricity price, and policy measures, like a tax on CO₂-emissions.

3. You have to describe how firms decide: what is their objective when choosing among different options?

4. Finally, you have to include some technological constraints in your model: a firm using a coal-fired power plant can hardly decide to produce electricity without emissions.

Let us go through these items and investigate what the three building blocks discussed in the video imply.

Which decisions can your firms make? Answering this question determines which variables are chosen for the overall model according to the objectives of the modelled firms.

In reality, firms decide about how much and in which technology they invest, how their technologies are deployed in production, and what other inputs (such as labour, resources, materials) are used in addition. In our model, these decisions are usually described in much less detail.

The Emission Choice Model (ECM) lets firms decide only about how much they emit. Everything else, such as production, technology choice, use of labour, is not explicitly described in the model. This does not imply that these choices are not relevant nor implicitly accounted for. It is only not possible to calculate the results from the model. This model is thus of most use only if emissions are the central concern. Thus, if you want to describe the effects of climate policy on the emissions of the energy sector, it is the model to use. If you are concerned with more effects of climate policy, such as how much electricity will cost or how much unemployment the policy might generate, you will need more elaborate models.

The Output Abatement Choice Model (OACM) is one option to address more complex policy questions. This model lets firms decide how much and how carefully they produce. Thus, the OACM does not only track emissions (which are a result of the level of production and care in production) but also the output. You would, for example, be able to describe the effects of a climate policy on electricity production and the electricity price. However, you would still not know whether the policy generates unemployment.

Information about effects on the economy could be delivered by the Input Choice Model (ICM) that describes which factors of production, such as, labour, capital, fossil fuels, materials, are used, and to what extent, by a firm. This model can describe a process where a firm switches from a labour-intensive technology, such as coal, to a capital-intensive technology, such as PV, due to a CO₂ tax. It can thus describe, for example, employment effects of climate policies.

In each of these models, firms have to decide under given circumstances. For example, if you assume perfect competition, firms will treat all market prices as exogenously determined and only ask how much to produce under given prices and policies. Which prices and policies are relevant factors for the firms, depends on the model that you use.

In the ECM, your firm only chooses emissions. Thus the only relevant factor is the price of emissions, which could be set by a CO₂ tax. In the OACM, a firm chooses its output and (via the care used in production) also its emissions. Thus, it reacts to the output price (ie, the electricity price) and the price of emissions. Finally, in the ICM, a firm chooses how much to use each possible input and thereby (implicitly) also its level of production and its emissions. It thus reacts to all factor prices (such as wages, fuel prices, interest rates), the output price (eg, the electricity price), and the price of emissions.

You already see that the models do not only differ in their explanatory power but also in their complexity: the model that describes most (the ICM) also needs most information. Therefore, it is usually a good idea to choose the simplest possible model that can describe all effects that are relevant for your question.

The next choice that you must make is to determine what objective your firms follow in making their decisions. In this regard, the different building blocks are rather similar. They imply that the firm tries to maximise its benefit, which is usually measured as its profit. (In other models where the social planner is the main actor, the objective might be to maximise social welfare.)

In the ECM, the firm i (out of many firms) maximises the benefit of emitting (eg, the profit that can be attained if a certain level of emissions is allowed) minus the costs of emitting:

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(1)  $$\max \limits_{e_i \ \geq \ 0} \enspace {B}_i(e_i) - t \cdot e_i$$

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$$B_i(e_i)$$ describes the benefit of emitting and the costs of emitting are the tax $$t$$ times emissions $$e_i$$. As explained above, the firm chooses only emissions.

In the OACM, the firm j maximises its revenues (price, p, times produced quantity, $$q_i$$, minus the costs of production, $$c_i(q_i,a_i)$$. Here, the costs are composed of how much the firm produces, $$q_i$$, and the level of abatement, $$a_i$$, minus the costs of emissions (as above, but emissions are now also a function of production and abatement, $$e_i(q_i,a_i)$$):

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(2)  $$\max \limits_{q_i,a_i \ \geq \ 0} \enspace p \cdot q_i \ - \ c_i(q_i,a_i) - t \cdot e_i(q_i,a_i)$$

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This is simply a maximisation of profit (revenue minus costs).

Finally, in the ICM, our firm i again maximises its profit, but now costs are described in a different way accounting for the factors j needed for production:

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(3)  $$\max \limits_{q_i,a_i \ \geq \ 0} \enspace p \cdot f_i(x_{i,1}, x_{i,2}, \ldots, x_{i,m})-\sum_{j=1}^m W_j \cdot x_{i,j} - t \cdot e_i(x_{i,1}, x_{i,2}, \ldots, x_{i,m}) $$

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Costs are now described by factor costs, that is, factor price, $$w_j$$, times amount used of the factor, $$x_{i,j}$$, and revenues are now the output price times the amount produced, which is a function of all the inputs used ($$f_i(x_{i,1}, x_{i,2}, \ldots, x_{i,m})$$) Similarly, emissions are now a function of inputs ($$e_ix_{i,1}, x_{i,2}, \ldots, x_{i,m})$$). Again, equation (3) is simply a maximisation of profit.

Thus, although the models differ in how they describe firm behaviour, they are based on the same main assumption: firms pursue the objective of maximising their profit.

In pursuing this objective, the firms face technological constraints. Such constraints describe what is feasible: how much can emissions per unit of production be reduced? How costly is this in terms of the use of other factors of production? How much output can be generated for a given amount of inputs? Overall, the setup for all three models is a classical optimization formulation with objective and side constraints.

These constraints are described in different ways in the three model types. In the ECM, the function $$B_i(e_i)$$ is used to this end. This function describes the maximal profit a firm i can achieve for different levels of emissions. It thus describes the technological options for and the implied costs of reducing emissions. Take note that this function is thought to not only account for the direct costs of reducing emissions (such as using filters or scrubbers) but also for indirect costs, such as producing less and thus having less revenue.

In the OACM, technological constraints are included in the function $$c_i(q_,a_i)$$ that describes how much it costs to produce a given output, $$q_i$$, at a given level of care, $$a_i$$, and thus (implicitly), which technologies and factors are used to this end. Also, the function $$e_i(q_i,a_i)$$ is based on technological considerations: how strongly can emissions be reduced by balancing care against the amount of production?

Finally, the ICM describes technological constraints via a production function $$f_i(x_{i,1},x_{i,2}, \ldots ,x_{i,m})$$, which explains how much output a given combination of inputs yield, and the emission function $$e_i(x_{i,1},x_{i,2}, \ldots ,x_{i,m})$$, which describes the emissions resulting from the use of a given amount of inputs (such as fossil fuels).

In the upcoming examples of this chapter, you will see how such models can be filled with life and adjusted to real world data.