Under quantity regulation, the firm is forced to produce at Q = Q . But efficiency requires that for any value of v, the firm produce at Q*, where this is defined by the condition that actual marginal costs equal actual marginal benefits:

or:

Since Q* ф Q, there will be a deadweight loss from this policy; this is illustrated in Figure 4.6.2.

Applying the method in equation (4.19), we can see that for any value of v this loss is equal to:

The expected value of this is:

(which follows from the assumption that v has a zero mean). Now

a^{2} = — x 4 + — x 4 = 4 and so: ^{v} 2 2

Figure 4.6.2 The deadweight loss of quantity regulation under incomplete information: Social marginal benefits turn out to be less than expected

A Pigouvian tax

Under a Pigouvian tax, the firm faces a tax equal to the expected marginal benefit and marginal cost at Q = Q. This means that the tax is equal to

For any value of v, the firm will produce at Q' the point where actual marginal benefits equal the tax. This means that:

So:

But efficiency again requires that for any value of v, the firm produce at Q*, where Q * = ——^{C}° + ^{v}. Thus the difference between actual

C - B,

production and efficient production is:

On the other hand, the height of the deadweight loss triangle in this case is: