# Find dead weight loss and quantity produced from tax revenue

In this example, we will find what the quantity produced after the introduction of a tax and the associated dead weight loss (DWL).

Consider the example associated with the above graph. Suppose you have been told that the equilibrium quantity produced was 400 units and after introducing a tax the tax revenue is \$600 and the tax rate was \$3.  Solve for the new quantity produced and the dead weight loss.

Firstly, what we need to recognize is that the tax revenue is defined as:

Tax revenue = quantity sold x tax rate

We can now substitute the above values into our tax revenue equation and get:

\$600 = quantity sold x \$3

If we re-arrange the equation we can isolate for the quantity sold after the tax:

quantity sold = \$600/\$3 = 200

Therefore in our diagram above x = 200 which is the quantity sold after the introduction of the tax. Now that we have this value we can solve for the dead weight loss. The dead weight loss is the area shaded in the green - the triangle. Therefore, using our values, we get the area of the dead weight loss as follows:

(1÷2) * (Pd - Ps) * (Qe - Qn) = (1÷2) * (tax rate) * (400 - 200) = (1÷2) * (\$3) * (400 - 200) = 300

Pd = is the price that consumers would pay in this equilibrium

Ps = is the price that suppliers receive after the government has collected their tax

(Pd - Ps) = the tax rate. You should be able to see this as the sellers receive (Pd - tax rate) which re-arranged makes tax rate = (Pd - Ps). Intuitively, this also makes sense since the tax rate must equal what buyers pay minus how much sellers receive. It is the difference.

Qe = the quantity produced before the tax was introduced

Qn = the new quantity being sold and consumed after the tax.

We should be able to see that the dead weight loss which occurred because of the tax was \$300.
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