Taxes:

A specific tax is a tax per unit sold. e.g. 2 cents per crate of oranges. Or $25/acre foot of water sold. An ad valorem tax is based on value. e.g. 8% of the value of the transaction, like a sales tax. We analyze a specific tax.

Consumers pay Pd per unit, which consists of the amount the producers receive Ps and the tax, t. That is Pd = Ps + t. Producers respond to what they receive, S(Ps) is their supply curve. Consumers purchase according to what it costs them, D(Pd) is their demand curve.

In algebraic form the model is S(Ps) = D(Pd) and Pd = Ps + t. One just solves for the prices Ps and Pd. The obvious solution technique is S(Ps) = D(Ps +t).

Now let D-1(Q) = Pd be the inverse demand curve, the function written the way we draw demand curves, p on the vertical axis, Q on the horizontal. Same for S-1(Q)= Ps. Equilibrium is D-1(Q) = Pd = Ps + t, so [D-1(Q) - t] = Ps = S-1(Q). Now the term in brackets [] is just the demand curve shifted downward by t; for every value of Q, find the associated value of p and subtract t from it!

The picture shows how to do this on a graph. D(Pd) is the demand curve. S is the supply curve. Their intersection gives the original, no tax, equilibrium. Call Pnotax the price to both consumers and producers in the no-tax equilibrium. Look at the dotted line labeled t. Its length is the amount of the tax. look at the bottom point of the dotted line. It gives the price to the producer--amount paid by the consumer less the governments tax take-- for the same quantity. Thus the lower curve is D(Ps), the demand seen by suppliers. The same construction can be done at any other quantity point, so D(Ps) is just D(Pd) shifted downward by t. The equilibrium quantity and price to producers with tax is found from the intersection of D(Ps) and S(Ps). The price to consumers is the price to producers plus t, so it is the price Pd at the top of the dotted line in the diagram.

One can just as well solve this problem by making the algebraic substitution in the other order.   Equilibrium is S-1(Q) = Ps = Pd - t, so S^-1(Q) + t =   Pd = D^-1(Q).  Now the supply curve from the consumer's point of view appears to have shifted up by t. 

Incidence:

Tax incidence compares the prices to consumers and producers after the tax has been imposed to the equilibrium that would exist if there were no tax.  In the diagram, S(Ps) = D(Pd) is the no tax equilibrium.

How much of the tax t is paid by consumers and how much by producers? The price change to consumers is Pd – Pnotax while the price change to producers is Ps – Pnotax. Problem set 2 explores this question of tax incidence in much greater detail.

The tax take:

If the equilbirum after tax quantity is Q*, then the government receives revenues of t Q*.  On the diagram, it is a rectangle.  (Note that the vertical axis is in $/unit and the horizontal axis is in units, so an area is in $).

The Environment:

Besides raising revenues, taxes are also good for changing behavior.  In the easy but unrealistic case where it takes one unit of clean air services (emitting one unit of pollution) to make one unit of output, a tax can be used to achieve the right level of output.  Suppose that each unit of clean air services used by the firm causes $t worth of damage to the environment and that the firm pays nothing for its pollution.   The firm will pollute too much (because supply shifts out when the cost of inputs, like clean air services go down).  By charging a tax of $t/unit the government can induce the firm to produce the correct output.  Of course, the government could also induce the correct output by simply requiring the firm to produce that output.  The firm always prefers the standard (the requiremnt to produce a specific output) over the tax.  Why.