3.4

## University of Basel

We have now discussed models of firm behaviour in detail. To close this discussion, it is helpful to make some general observations.

First, you have seen that there are variables in the model that are explained by part of the model. For example, the Output Abatement Choice Model (OACM) explains the level of production and the emissions of a firm by assuming that the firm chooses these variables in a way that maximizes the firm’s profit.

As presented the last week, the variables explained in such a way in a model are called endogenous variables; variables that are calculated using the model. As you will see, once we add additional model structures, additional variables will be explained in this way. For example, the electricity price might be exogenous to a firm in the OACM but could be explained by a market model, and thus, electricity prices would be an endogenous variable in a market model.

In contrast, we have the exogenous variables like the carbon tax in the preceding models. This tax will be set by the government and, as long as you do not aim at explaining how a government sets this tax with the model, it will not be calculated from the model.

One rule that is useful in modeling is that the number of equations for solving the model has to be as least as large as the number of endogenous variables: we need at least one equation to calculate each of these variables. This is an important check for each model that you build: do you have enough equations to explain every endogenous variable of your model?

Let us perform this check for the OACM model. Here, you have two endogenous variables (output and abatement) for each firm. Thus for $n$ firms, we have $2 n$ endogenous variables. In terms of equations, the model is an optimization model with two decision variables (output and abatement) in one objective function (profit) for each firm:

(1) $p \cdot q_i - c_i(q_i,a_i) - t \cdot e_i(q_i,a_i)$

As is usual in optimization, the optimality conditions are obtained by differentiating the objective of each firm i with regard to the decision variables of this firm (that is, $q_i$ and $a_i$) and setting the result equal to zero:

(2) $p - \frac{\partial c_i(q_i,a_i)}{\partial q_i} - t \cdot \frac{\partial e_i(q_i,a_i)}{\partial q_i} = 0$

(3) $- \frac{\partial c_i(q_i,a_i)}{\partial a_i} - t \cdot \frac{\partial e_i(q_i,a_i)}{\partial a_i} = 0$

Those are the two equations you will need to solve the model.

Equations (2) and (3) are two equations for each firm, $i$, so that we have $2 n$ equations for $n$ firms. Thus, you have $2 n$ equations for $2 n$ endogenous variables, which is the minimal necessary number. This works as long as the model does not get corner solutions (that is, as long as the values of $q_i$ and $a_i$ implied by these equations are strictly greater than zero).

A second general observation that can be made at this point of the course is that models should be built so that they are fit for a given purpose. It is not useful at all to always use the most general model available, nor is it useful to build models that are not sufficiently detailed to answer your questions. As Albert Einstein once said: ‘Everything should be made as simple as possible but not simpler.’

So again, you always need to think about ‘what is the problem you want to answer with the model?’ before starting to build a model. This will help to identify which variables have to be explained by the model (ie, which variables have to be endogenous) and which are exogenous variables.

Once you know this, you can build a model that is as simple as possible while still being able to answer your questions. To this end, you can ask which actors would choose the variables that you want to calculate. These actors will then be modeled by building blocks that are able to explain the variables of interest. This provides the core of your model.

The next step is then to add some additional building blocks to connect the actors in your model and to fill some gaps. We will discuss this in the following steps.