Technology Adoption under Negative External Effects

In this section, we illustrate, by means of a simple example, that, for certain configurations of parameter values, quotas imposed on a monopolist may well be welfare reducing.

Suppose that g i ( x i ) = 2 d i x i , c _A (x) = ax ², c _B (x) = bx ², e(x) = θx ², and F _A = F _B , where d _i , a, b, and θ are all positive. These functional forms yield

and

where

The quantity of B-units x B * that maximizes the economy’s overall (conditional) welfare, i.e., such that

is given by

The monopolist’s profit maximizing A-output when technology B is not employed, i.e., the solution z _A to

is given by

The monopolist’s profit maximizing B-output when technology A is not employed, i.e., the solution z _B to

is given by^[10]

Conditions (9), (10), and (13) place the following constraints on the parameters.

1. Condition (9) can be expressed as

   (33) a < α 2 ( b + N θ ) .

2. Condition (10) is expressible as

   2 a x A ( x B / α ) + a < F B ( x B / α ) 2 + α 2 ( b + N θ ) .

   By (33), this inequality holds if 2 a x A ( x B / α ) < F B ( x B / α ) 2 , i.e., if

   2 a x A x B < α F B .

   This inequality does not need to hold for all (x _A , x _B ); it suffices to assume that it holds for any solution to the problem (21) (see footnote ^[5]). Because a solution ( x ̂ A , x ̂ B ) to (21) satisfies x ̂ A ≤ z A and x ̂ B ≤ z B , it suffices to assume that

   (34) 2 a z A z B < α F B .

3. Condition (13) can be expressed as

   a > α 2 ( b + 2 θ ) .

In sum, the following constraints guarantee that conditions (9), (10), and (13) are satisfied:

Using (30) and (32), we can rewrite the second equation as follows:

Next, we wish to choose a parameter configuration that will serve our purposes, i.e., one for which any quota yields a welfare loss. This configuration will be chosen in such a way that the conditions in (35)—and hence conditions (9), (10), and (13)—are satisfied. To this end, we seek parameter values for which

(see (32) and (28)). Arranging terms yields

Setting

and plugging this expression into the first equation in (35) yields

Consequently, under (38), (37) holds, and (35) becomes

Next, note that, because the second equation can be written as in (36), if

then 2az _A z _B < αF _B whenever α ²(b + 2θ) < a < α ²2(b + 2θ).

In sum, given (38) and (40), the following constraints guarantee that conditions (9), (10), and (13) are satisfied:

We claim that, for

and for a large enough within the bounds in (41), i.e., for

any binding quota z ̄ B results in a welfare loss. Note that (42) implies (38) and (40). Indeed, the second equation in (40) is expressible as F _B ≥ (d/2)^4/3, which is implied by the second equation in (42).

To see that, under (42) and (43), any binding quota z ̄ B results in a welfare loss, note first that, in the absence of quotas, the monopolist does not employ technology A, i.e., it sets its output at z _B (see (32)). To see this, it suffices to show that the monopolist’s profit at z _B exceeds the maximum profit under the adoption of both technologies.^[11] The monopolist’s profit at z _B is given by

where the inequality follows from (42).^[12] The monopolist’s maximum profit at a production plan that employs both technologies, i.e., a production plan (x _A , x _B ) with x _A > 0 and x _B > 0, is the monopolist’s profit at an interior solution to the problem (21); such a maximizer must satisfy the following first-order conditions:

Combining both equations gives

In our example, this equation becomes

Combining this equation with either (45) or (46) gives

The monopolist’s combined profit at (x _A , x _B ) is given by

Since

and since the last inequality holds by (42), it follows that Π(x _A , x _B ) < 0. Consequently, Π(z _B ) > 0 > Π(x _A , x _B ).

Thus, in the absence quotas, the monopolist does not adopt technology A and sets its B-output at z _B . In this scenario, there are three cases to consider:

1. Case 1. Under the quota, the monopolist does not employ technology A. This case is easy to handle, since any binding quota decreases the quantity of B-units traded in the market to a level below z _B (the profit maximizing B-output in the absence of quotas); since (37) holds (so that the efficient quantity of B-units, x B * , exceeds z _B ), the quota pushes market output further away from the efficient B-level, x B * ; this yields a welfare loss (since there is no A-output to replace the lost B-units).

2. Case 2. Under the quota, the monopolist switches to technology A. In this case, the monopolist produces and sells z _A (the profit maximizing A-output, see (30)) instead of z _B . This leads to a welfare loss. Indeed, the overall welfare at z _A is given by

   W ( z A ) = ∫ 0 z A G ̄ ( x ) d x − c A ( z A ) − F A = 2 d α z A − a z A 2 − F A ≈ d 4 / 3 2 − F A ,

   while the overall welfare at z _B can be expressed as

   W ( z B ) = ∫ 0 z B G ( x ) d x − N e ( z B ) − c B ( z A ) − F B = 2 d z B − ( b + N θ ) z B 2 − F B = 2 d z B − 2 ( b + 2 θ ) z B 2 − F B ≈ 3 2 5 / 3 d 4 / 3 − F B ≈ 0.94 d 4 / 3 − F B ;

   since F _A = F _B , it follows that W(z _B ) > W(z _A ). Intuitively, the profit maximizing quantity of A-units, z _A , is too low relative to the profit maximizing quantity of B-units, expressed in equivalent A-units, z _B /α. We know that replacing z _B B-units by the equivalent quantity of A-units, z _B /α, would lead to a welfare gain. This is, in fact, precisely our assumption from (9). While the quantity z _B /α of A-output need not coincide with the efficient quantity of A-output, x A * , which satisfies G ̄ x A * = c A ′ x A * , moving from z _B /α to x A * would bring about additional welfare gains. The monopolist, however, produces and sells too little A-output relative to x A * . Since, in our example, z _B /α happens to coincide with x A * , z _A is too low also in relation to z _B /α.

3. Case 3. It remains to consider the case when the monopolist adopts both technologies once the quota is implemented. This means that there is a solution (y _A , y _B ) to problem (22) such that y _A > 0 and y _B > 0. But since the feasible region for problem (22) is a subset of that for problem (21), it follows that the monopolist’s maximum profit for problem (22) is less than or equal to that for problem (21). Since, in our example, the maximum profit for problem (21) is negative (a fact that was established earlier), we see that the adoption of both technologies under the quota yields negative profits. The monopolist is better off by specializing in technology A, setting its A-output at z _A (see (30)), which yields a profit of

   Π ( z A ) = 7 d 4 / 3 16 − F A ≈ 0.44 d 4 / 3 − F A > 0 ,

   where the inequality follows from (42). Thus, the monopolist never adopts both technologies under the quota.

We have illustrated that, under certain configurations of parameter values, any quota imposed on a monopolist may well be welfare reducing. The same is true about Pigouvian taxes. Indeed, replacing quotas by any Pigouvian tax in the above example leads to a welfare loss.

To see this, consider a Pigouvian tax that levies $τ per unit of emissions/waste. As in Section 5.1, suppose that the monopolist’s B-waste is proportional to its B-output by a factor γ _B > 0, so that γ _B measures emissions/waste per unit of B-output. Under the tax, the monopolist solves the following problem:

In our example, an “unfettered” monopolist specializes in B-output, setting its level equal to z _B . Three case are possible under the Pigouvian tax τ:

1. Case 1. Under the tax, the monopolist does not employ technology A. In this case, there is no material difference between taxes and quotas when it comes to identifying the welfare effects of these policies: just as in the case of a binding quota, a binding tax decreases the quantity of B-units traded in the market to a level below z _B ; since (37) holds (so that the efficient quantity of B-units, x B * , exceeds z _B ), the tax pushes market output further away from the efficient B-level, x B * ; this yields a welfare loss (since there is no A-output to replace the lost B-units).

2. Case 2. Under the tax, the monopolist switches to technology A. In this case, the monopolist produces and sells z _A (the profit maximizing A-output, see (30)) instead of z _B , and we already know from the analysis of quotas that, in our example, this switch leads to a welfare loss.

3. Case 3. Under the tax, the monopolist adopts both technologies. This means that there is a solution (y _A , y _B ) to problem (47) such that y _A > 0 and y _B > 0. But, just as we showed that the maximum profit for problem (21) is negative, it can be shown that the maximum profit for problem (47) is negative. As in the quota case, the monopolist is better off by specializing in technology A, setting its A-output at z _A (see (30)), which yields a profit of

   Π ( z A ) = 7 d 4 / 3 16 − F A ≈ 0.44 d 4 / 3 − F A > 0 .

   Thus, in our example, the monopolist never adopts both technologies under the tax, just as it did not adopt both technologies under a quota.