# Solving Models: First-best and Second-best Solutions

^{1}(1) \(\max_{q,a\ge{0}}{P(q)}q-c(q,a)-te+\sigma q,\) (2) \(e=s(q,a).\) Under some additional assumptions on demand, we can use the following first-order conditions to investigate firm behavior: (3) \(P(q)+P'(q)q+\sigma = \frac{\partial c(q, a)}{\partial q}+\frac{\partial s(q,a)}{\partial q},\) (4) \(0 = \frac{\partial c (q,a)}{\partial a}+t\frac{\partial s(q,a)}{\partial a}.\) As shown in the preceding step, a first-best solution follows from \[\int_{0}^{q} P(\tilde{q})d\tilde{q}-c(q,a)-D(s(q,a)).\] This yields the following first-order conditions (5) \(P(q)=\frac{\partial c(q,a)}{\partial q}+D'(e)\frac{\partial s(q,a)}{\partial q},\) (6) \(0=\frac{\partial c(q,a)}{\partial a}+D'(e)\frac{\partial s(q,a)}{\partial a}.\) If we set \(t = D′(e)\) and \(\sigma = -P'(q)q,\) equations (3) and (5) as well as the equation (4) and (6) become identical. Thus with these policy settings, firms do exactly what is socially optimal. This is a

**first-best solution**and it will work, whenever we have a sufficient number of policy instruments to tackle our problem. In our case, we have two problem (emissions and market power) and solve them with two instruments (tax and subsidy). Note that, in this solution, we have derived the best possible state of the world from our welfare function (we differentiated the welfare function wrt q and a) and then used the policy variables to make the firm’s first-order conditions (equations (3)–(4)) equal to the first-order conditions derived from the welfare function (equations (5)–(6)). We have not optimized welfare wrt the policy instruments. Also note that, in the first-best solution, the tax is equal to the tax derived in the case with only one problem. The same holds for the subsidy: The above subsidy would be the correct subsidy, if we only had the problem of market power. This is a general result: In a first-best solution, every policy instrument is chosen in a way as if there would be only the problem addressed by this instrument. In many cases, however, we face less benign problems. Often, we do not have enough policy instruments to address all problems. Imagine, for example, that we would only have the emissions tax and could not use a subsidy. In this case, we could not make equations (3) and (5) as well as the equations (4) and (6) identical; the first-best solution is unattainable. Thus we have to look for a different way to tackle this problem. To this end, we rewrite the welfare function, taking into account that the values of q and a are chosen by the firm but are influenced by the tax: \[\int_{0}^{q(t)} P(\tilde{q})d\tilde{q}-c(q(t),a(t))-D(s(q(t),a(t))).\] We now differentiate this welfare function wrt to the tax \(t\) and set the result equal to zero: (7) \(0=P(q)\frac{\text{d}q(t)}{\text{d}t}-\frac{\partial c(q,a)}{\partial q}\frac{\text{d}q(t)}{\text{d}t}-\frac{\partial c(q,a)}{\partial a}\frac{\text{d}a(t)}{\text{d}t} -D'(e)\left(\begin{array}{c}\frac{\partial s(q,a)}{\partial q}\frac{\text{d}q(t)}{\text{d}t}+\frac{\partial s(q,a)}{\partial a}\frac{\text{d}a(t)}{\text{d}t} \end{array}\right).\) Using the firm’s first-order conditions (equations (3)–(4) with \(σ = 0\)), we get an optimal tax

^{2}(8) \(t=D'(e)+P'(q)\frac{q\frac{\text{d}q}{\text{d}t}}{\frac{\partial s(q,a)}{\partial q}\frac{\text{d}q}{\text{d}t}+\frac{\partial s(q,a)}{\partial a}\frac{\text{d}a}{\text{d}t}}.\) This is the

**second-best solution.**It achieves a lower level of welfare than the first-best solution, as we have to tackle two problems with one policy instrument, which makes a compromise necessary. In fact, the compromise can be clearly seen in equation (8): The optimal tax is a compromise between reducing emissions (first term on the right-hand side) and compensating the market power (second term on the right-hand side). The second term is usually negative (as \(P′(q) < 0\); an increasing quantity reduces the price), so that the second-best tax is smaller than the first-best tax. This is intuitive: As a monopolist will produce less than a producer under perfect competition, the second-best tax should be smaller than the first-best tax, otherwise we would reduce production too strongly. This implies that, in a second-best solution, we do not choose the policy instruments as if the other problems would not exist; the policy choice is a compromise between addressing different problems. Again, this is a general property of second-best solutions. Note that we derived the second-best solution in a totally different way than we calculated the first-best solution: We wrote social welfare as a function of the policy variable and then optimized directly with respect to the policy variable, using the firm’s first-order conditions as constraints. This is usually the best way to derive a second-best solution: Do not ask what the ideal state of the world might be (which is the first-best solution), but rather directly ask what is the best available policy.

^{1}As we have only one firm, we do not need to index all variables and functions.

^{2}To this end, rewrite the firm’s first-order conditions in a way that we can substitute them into equation (7), eg, solve the firm’s first-order conditions for \(\frac{\partial c(q,a)}{\partial q}\) and \(\frac{\partial c(q,a)}{\partial a}\).

#### Exploring Possible Futures: Modeling in Environmental and Energy Economics

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