# Quantity regulation

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:

and so for any value of *v* the deadweight loss is:

The expected value of this is: