# The firm model: the mathematics

It is, again, time to have a look at our model in somewhat more detail. This model includes several new concepts and is based on the model paradigms learned in this week.

In all models before, we have used a modeling paradigm that is typical for bottom-up energy economic models: we have focused the model on a description of technologies and this description was set up in a way that makes it easy to connect the model to technological data. In particular, we have assumed that technologies have variable costs that are proportional to the number of units produced. The so-called marginal costs (the additional costs of producing an additional unit of output) are constant for each technology in all previous models. This makes it easy to connect the model to data.

However, using marginal costs renders optimal behaviour in such models very simple. Imagine a firm that aims to maximize its profit. For each produced unit, the firm gets a price, *p*. Each unit that it produces has constant marginal costs, *mc*. As these values never change, regardless how much the firm produces, the firm will produce as much as it can, whenever *p>mc* (in this case, the firm makes a profit (*p-mc*) for each produced unit). In contrast, as soon as *p<mc*, the firm will not produce anything, as it would lose money for each produced unit. Thus at the boundary, where *p=mc*, the firm will switch from producing nothing to full production for a very slight change of the price, *p*.

This is hardly a realistic model of behaviour. Furthermore, most production technologies provide firms with some flexibility that the firm can use to adjust to market and policy conditions. For example, a coal-fired power plant can be run at different loads, some of which have higher technical efficiency than others and thus lead to less CO_{2}-emissions per unit of output. Or, for wind power, there are more and less suitable sites. Assuming that the best sites will be used first (as they generate the highest profit), each new power plant will either be more costly to build or less efficient in operation.

From now on, we will try to include such aspects in our model: the model will focus on firm behaviour and thus we will give firms a bit more leeway in using their technologies. The price of this is that the model becomes more abstract, needs assumptions that are only approximations of real technologies and uses variables that are less intuitive, and it will be harder to match the model to data. This new model paradigm is typical for environmental economic models or top-down energy economic models.

Let’s start by looking at the objective of a firm. The typical assumption is that the firm wants to maximize its profit. To this end, it can change two variables: how much it produces (output, *q*) and how cleanly it produces (abatement of emissions, *a*). We thus use one of the building blocks that you already know: the Output Abatement Choice Model (OACM).

In this model a firm, *i*, maximizes its profit:

Here, *p* is the market price for electricity and \(s_i\) is a subsidy, so that \((p+s_i) \cdot q_i\) is the revenue of firm, *i*. The term \(c_i(q_i,a_i)\) describes production and abatement costs. The term \(t \cdot e_i(q_i, a_i)\) denotes possible payments for emissions, due to an emissions tax, *t*. Finally, \(t_i^{nuc}\) is a security deposit for the risk induced by nuclear power.

Production and abatement costs are defined by \(c_i(q_i,a_i)=\alpha_i \cdot q_i + \frac{\beta_i}{2} \cdot q_i^2+\gamma_i \cdot q_i \cdot a_i\)

The parameters \(\alpha_i, \beta_i, \gamma_i\) are non-negative. So, costs increase more than proportionally with production and the costs of reducing emissions (via abatement, \(a_i\)) are proportional to the production level.

Emissions are given by \(e_i(q_i,a_i) = q_i \cdot \left(\overline{e}_i + \frac{\delta_i}{a_i}\right)\). Thus, emissions increase proportionally with production. For very high values of \(a_i\), emissions go to the lower boundary, \(\overline{e}_i\), which is the technically minimal level of emissions. For low values of \(a_i\), emissions become much higher.

As explained a few steps ago, a model needs as many equations as endogenous variables to calculate the optimal solution. The optimality conditions are again derived by differentiating the profit with regard to \(q_i\) and \(a_i\), setting the results equal to zero and solving for \(q_i\) and \(a_i\). This leads to

(1) \(a_i = \sqrt{\frac{t \cdot \delta_i}{\gamma_i}}\)

(2) \(q_i = \frac {(p+s_i)-\alpha_i - t \ \cdot \ \overline{e}_i - 2\sqrt{t \ \cdot \ \gamma_i \ \cdot \ \delta_i}}{\beta_i}\)

For \(a_i\), this approach always works because the above equations ensure that for all finite levels of the emission tax, \(t \geq 0\), the optimal level of abatement is between zero and infinity (we always have an interior optimum with regard to abatement).

This is not the case for \(q_i\), as equation (2) shows. Depending on the price, \(p\), the emission tax, \(t\), and the parameters, the optimal production level, \(q_i\), could be negative (in which case it is optimal to produce nothing, as negative production is not feasible). Thus, optimal production needs to be written as \(q_i = Max \left( \frac{(p+s_i)-a_i-t\cdot \overline{e}_i-2\sqrt{t \cdot \gamma_i \cdot \delta_i}}{\beta_i},0\right)\).

This holds for each firm, \(i\). In our model, we assume that there is one firm using each technology, that is, one firm using coal, one firm using gas, one firm using a nuclear power plant, and one firm each for PV and wind power. All these firms are assumed to face perfect competition; that is, no firm’s decision influences on the market price \(p\) for electricity into account.

Finally, we assume that demand is given by a function \(d= \overline{Q}-\eta \cdot p\) with \(\overline{Q}, \eta \gt 0\). Thus, demand is highest for \(p=0\) (it takes on the value \(\overline{Q}\)) and becomes smaller with an increasing price. Remember from your general economics course that demand slopes downward – here, \(\eta\) is an elasticity of demand that determines the slope of the line.

In the model exercise you were able to use the sliders to set the price \(p\), the tax \(t\), the deposit \(t_i^{nuc}\), and the subsidies \(s_{PV}\) and \(s_{Wind}\). Only PV and wind power get subsidies, for all other firms \(s_i\) equals zero. Furthermore, only the firm using nuclear power faces a security deposit \(t_i^{nuc}\), this value is zero for all other firms. CO_{2}-emissions occur only for gas-fired and coal-fired power plants, so that \(\overline{e}_i=\gamma_i=\delta_i=0\), for all other firms.

Based on your inputs the model has calculated the optimal production, \(q_i\), and abatement, \(a_i\), decisions of all firms as well as the resulting emissions.

Note that this model is clearly an approximation. In contrast to the previous models, it does not include investment costs. Furthermore, emissions would go to infinity if \(a_i\) becomes zero. However, the model provides a much more detailed picture of how a firm reacts to different policy and market signals.

Although, the model is more aggregated in its technical detail it can still be connected to real world data, albeit somewhat more loosely than the previous models. To match the above model with your desired real world example you will need to capture the cost parameters \(\alpha_i,\beta_i,\gamma_i\) of the production and abatement costs function and \(\overline{e}_i,\delta_i\) of the emission function. Finally, you will need to define the demand parameters, \(\overline{Q},\eta\).

For our model we aimed to link the parameter assumptions to the first model exercises of our example country. Thus, the overall demand should be again in the range of 150 TWh (terawatt hours), the costs for coal, gas, and nuclear should be oriented at the costs figures of the last models, and the renewable production should account for investment costs.

The \(\alpha_i\) of the cost function is basically the lowest marginal cost level achievable. We fix this to the same cost level of the respective power plant type in our energy system models for coal, gas, and nuclear. The increase in costs can then be justified with two effects: the usage of less efficient power plants and/or the need to build new power plants which then requires investment costs in addition. As we have used variable and investment costs in our first model exercises we use the investment cost level as proxy to define the increase parameter, \(\beta_i\). Namely, we assume that for a yearly output of 75 TWh (half the total demand), completely new plants would need to be constructed leading to a price level equal to the levelized costs of generation.

For renewable generation we define a slightly different cost structure justification. We expect a decrease in output per unit the more wind and PV plants are installed as the good sites are used first. To capture this dynamic we again use the investment costs as proxy. Assuming a decrease in the average utilization for increasing capacity additions, we can obtain a cost function by dividing the investment costs per unit with the expected output per unit. For wind, we assume that the average output decreases from 3000 to 750 full load hours, and for PV from 2000 to 500 full load hours.

A similar logic is used to define the abatement cost structures. Assuming the coal and gas firm can decide about the efficiency of their plant they can influence the emissions they have with each unit of output. However, a higher efficiency comes with the cost burden of installing those efficiency improvements. We fix the minimum achievable emission level by defining the maximum achievable efficiency level (50% for coal and 70% for gas) and the current emissions by using relatively low efficiencies (30% and 40%, respectively). We now assume that if a plant wants to obtain a reduction towards 90% of the maximum achievable reduction it will need to invest in a completely new set of power plants. This, again, defines the increase factor, \(\gamma_i\).

Finally, we calibrate the demand function to achieve consumption of 150 TWh with a market price of ca. 35 dollars per MWh (megawatt hour).

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