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# How to build a theoretical model: Basic building blocks

Learn about the descriptive solution, the first-best solution and the second-best solution of solving a model.
6.2
We now come to the crucial step of modeling work, solving the model. This step is crucial, as it shows whether a model that we have developed is indeed useful, or whether it needs to be changed. Can the model be solved at all, and does a solution yield interesting insights? To see what solving the theoretical model means, let us have a look at a simple example. We consider a set of firms that produce the same good. Production leads to emissions, which can be reduced by producing less or by producing more carefully. A tax on emissions sets an incentive for emitting less. Using the Output Abatement Choice Model, we can describe this situation in the following way.
63.9
A first way to solve the model is to see what kind of firm behaviour it implies. That is, we use a model in the purely descriptive way. We want to know what happens if we set a certain tax level. To this end, we simply calculate the firm’s behaviour. We differentiate the profit function with regard to the firm’s decision variables, set the resulting equations equal to zero, and gain expressions for a candidate for an interior solution of the firm’s maximisation problem. Doing this for firm i leads to this. This calculation already reveals a lot of interesting information, if we use some assumptions on our cost and missions functions.
117.4
Let us assume that more production and more care lead to higher costs, and that production leads to higher emissions, but more care in production leads to less emissions. In this case, the first equation shows that a higher tax will lead to less production at each given market price. The second equation shows that a higher tax leads to more care being used in production. Thus taxing of firm leads to less production and a more careful production, and thus reduces emissions. If we introduce a demand function for the product that is downward sloping, we also get the information that the tax will increase the price of the product.
166.3
If we want to have even more information, we need to use specific functional forms for the cost emissions and demand function, which would give us equations for emissions, production, and the output price, as a function of the tax. Such a model could then be estimated. A second way to solve this model would be to ask, what is the best tax level that we could set? To answer this question, we have to use the welfare function that we already discussed in week two. In addition to calculating the firm’s behaviour, as above, we would use the welfare function to calculate the best possible outcome, that is, how much each firm should produce, and how carefully each firm should produce.
216.9
If we compare the equations describing the best possible outcome with the equations describing the firm’s behaviour, we see that those become equal if we set the tax equal to the marginal damage. This is a normative way of solving our model. We do not ask what happens, but we ask what we should do. We try to give policy advice. That is, we say how the tax should be set. Again, we would need specific functional forms and data to get numerical values for the best tax level. However, even without this additional information, our model tells us much. First, we know that the tax should be determined by the marginal damage.
267.8
Second, and this is the most important result, the model shows that we should use the same tax for all firms, even if they have vastly differing costs. If we extend the model to have different sectors with firms producing different goods, we would even see that all firms in all sectors should pay the same tax. This is a very strong result from a fairly simple model. Furthermore, it is a highly relevant result. The costs of climate policy could be reduced substantially if countries would be willing to implement this simple insight.

Solving the model is a crucial step, as it shows whether a model that we have developed is indeed useful, or whether it needs to be changed. Can the model be solved at all, and does a solution yield interesting insights?