# First-best or second-best: coping with realities

Do you remember our discussion of firm behaviour from last week? There, we have assumed that our firms behave competitively, that is, they take prices as given and do not try to manipulate markets to their advantage.

Unfortunately, this is a rather optimistic view of most energy markets. In particular, for electricity we often have one (or a few) big suppliers that are able to influence prices. In this step, we will use this example to highlight a crucial difference in designing energy and environmental policies, whether you aim for a first-best or a second-best outcome.

Let’s start with a first-best case. For this, take the equations that you have used in the last week to describe firm behaviour in the Output Abatement Choice Model (OACM) model:

(1) \(\max\limits_{q_i,a_i\geq0}\enspace{p\cdot q_i}-c_i(q_i,a_i)-t\cdot e_i(q_i,a_i)\)

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

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

Equation (1) represents each firm’s optimization problem and equations (2) and (3) are the necessary conditions for optimality resulting from this problem (excluding corner solutions).

These equations describe what a **firm** will do. However, how do you know whether this behaviour is good or bad from a **societal** perspective? In Week 2, we have introduced the concept of **social welfare** to this end. In the OACM model with \(n\) firms, social welfare can be written as

(4) \(\int_{0}^{\sum\limits_{i=1}^{n}{q_i} }{p(q)dq-\sum\limits_{i=1}^{n}{c_i(q_i,a_i)-D(\sum\limits_{i=1}^{n}{e_i(q_i,a_i)} )} }\)

The first term, \(\int_{0}^{\sum\limits_{i=1}^{n}{q_i} }{p(q)dq}\), describes the consumer rent; that is, the benefit that consumer have from consuming electricity (the area below the demand curve). The second term, \(\sum\limits_{i=1}^{n}{c_i(q_i,a_i)}\), equals the costs of producing electricity and the final term is the damage (measured by a damage function, \(D\)) caused by the sum of all emissions, \(\sum\limits_{i=1}^{n}{e_i(q_i,a_i)}\).

A benevolent dictator would maximize equation (4) with regard to output and abatement for all firms. Doing this for firm \(i\) (that is, differentiating with respect to \(q_i, a_i\) and setting the result equal to zero) leads to the following necessary conditions:

(5) \(p-\frac{∂c_i (q_i,a_i )}{∂q_i} -\frac{∂D (q_i,a_i )}{∂e_i (q_i ,a_i )}\cdot \frac{∂e_i (q_i ,a_i )}{∂q_i} =0\),

(6) \(-\frac{∂c_i (q_i,a_i )}{∂a_i} -\frac{∂D (q_i,a_i )}{∂e_i (q_i ,a_i )}\cdot \frac{∂e_i (q_i ,a_i )}{∂a_i} =0\)

Where \(D’\) is the derivative of the damage function with regard to emissions.

If you compare (2) and (5) as well as (3) and (6), you see that these equations become equal, if you set \(t =\frac{∂D (q_i,a_i )}{∂e_i (q_i ,a_i )}\). That is, there is a specific value of the emission tax that induces firms to produce exactly the right amount with exactly the right level of care (abatement). In fact, this is what is usually called a **Pigouvian tax**, which is a tax that internalizes the emission externality by making each firm pay the social costs of the next unit of emissions (ie, \(\frac{∂D (q_i,a_i )}{∂e_i (q_i ,a_i )}\) ) for all of its emissions. Note that this value is the same for all firms.

This calculation has several remarkable implications. First, you have two decision variables (production quantity and abatement) for each firm (altogether \(2 n\) variables) but a single policy instrument (a mandatory tax) suffices to set incentives so that each firm behaves in a socially optimal way. Second, it implies that all firms should pay the same tax for each unit of emissions, which is a pretty strong policy implication (in fact, all firms in all sectors of an economy that emit the same pollutant should pay the same tax).

These are rather strong results for a comparatively simple model. Furthermore, they nicely highlight what constitutes a **first-best solution**: you can achieve the best possible outcome, social welfare is maximized. In fact, by implementing a policy, you can align the interests of all firms in the economy with societal objectives. If the tax is chosen correctly (ie, \(t = \frac{∂D (q_i,a_i )}{∂e_i (q_i ,a_i )}\) ), the firms voluntarily do what is best for the society.

This holds in the benign situation where firms have no market power. Let us for a moment assume that this is not the case. Instead, assume that we only have a single firm, that is, a monopolist. What would this firm do?

A monopolist would maximize his profit taking into account the willingness of consumers to consume at different prices. Thus, we would integrate the inverse demand function, \(p(q)\), which describes the price consumers are willing to pay for a given amount, \(q\), into the monopolist’s decision problem (we drop the indices now, as there is only a single firm):

(7) \(\max\limits_{q,a\geq0}\enspace{p(q)\cdot q - c(q,a)-t\cdot e(q,a)}\)

(8) \(\frac{∂p(q)}{∂q}q+p(q)-\frac{∂c(q,a)}{∂q} -t\cdot \frac{∂e(q ,a)}{∂q} =0\)

(9) \(-\frac{∂c(q,a)}{∂a} -t\cdot\frac{∂e(q,a)}{∂a} =0\)

Again, you have described the maximization problem in equation (7), and then you derived the necessary conditions from it. The term \(\frac{∂p(q)}{∂q}\) denotes the derivative of the inverse demand function.

Comparing (2) and (8) as well as (3) and (9) shows that there is no value of the tax, \(t\), that makes these equations identical. You can either choose to have the right amount of output (ie, make (2) and (8) identical) or the right amount of abatement (ie, make (3) and (9) identical); achieving both simultaneously is impossible.

This is typical for a **second-best solution**: you cannot achieve the best possible outcome but have to choose among less desirable ones. The reason is that the monopolist already produces too little to keep prices up. Thus, taxing the monopolist with the Pigouvian tax, \(t = \frac{∂D (q_i,a_i )}{∂e_i (q_i ,a_i )}\), leads to a too strong drop in output. But a lower tax leads to too little abatement effort by the monopolist. Both of these outcomes reduce social welfare.

So, you have to compromise: setting a tax that is somewhat lower than the Pigouvian tax will usually be the best that you can do.

This is a typical situation in most **real world problems**. We are faced with more problems (market power and emissions, in your example) that we can cope with in terms of policy instruments (you only have a single instrument here). Therefore, we have to make a compromise and accept some welfare losses. It is still possible to increase welfare by introducing a tax but not up to the theoretical benchmark of the first-best solution.

Of course, this is a rather formal argument. What you should take out here is that there are two approaches to environmental and energy policy:

- You can try to achieve the best possible outcome. But to this end, you will typically need a rather complex policy (if you want, you might think about what additional policy instrument could be used to achieve the first-best outcome in the market power case). This is a first-best approach.
- Or, you can accept some welfare losses and make do with the insufficient policy instruments that you have. You still can do some good, but the scope is more limited. This is a second-best approach.

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