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Suppose we have the following system of equations:

$Q_s=-20+3P$
$Q_d=-220-5P$
$Q_s=Q_d$

Say we want to find the tax burden of the consumer, the tax burden of the firm, and the total revenue generated for the government for some excise tax t.

Do we do this by looking at the elasticity of each the supplier and consumer?

The Elasticity of Q with respect to P can be calculated by:

$\eta_Q,_P = P/Q*dQ/dP$

With this we see that the elasticity of supply is 3 and the elasticity of demand is -5

Can we use these to find the tax burden? And how do we calculate the government tax revenue in terms of t for the government?

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is this a homework question, by any chance? –  EnergyNumbers Nov 2 '11 at 7:41
    
Nope, I'm studying for a test tomorow. I missed a lecture and my prof was going over problems such as these. I don't care about the specific solution, I just want to know the method used :) –  fdart17 Nov 2 '11 at 7:57
    
in that case, the approach will be the same - it will be more helpful for you to get hints on how to solve it, than just given the solution - after all, this question with those exact coefficients is unlikely to be coming up on the test! –  EnergyNumbers Nov 2 '11 at 8:04
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2 Answers

You've got three equations: $$(1)Q_s=-20+3P$$ $$(2)Q_d=-220-5P$$ $$(3)Q_s=Q_d$$

You can substitute the first and second equations, into the third, like so:

$$-20 + 3P = -220 - 5P$$ All I've done there, is taken the value of Qs from equation 1, and substituted into equation 3. Similarly, I've taken the value of Qd from equation 2, and substituted into equation 3. We can do that, because the supplier and consumer see the same price.

Solving that in the usual way would give you the equilibrium price, for the untaxed scenario. And once you've got that, you can get the equilibrium quantity, too.

Now, what happens with tax t? Is it still the case that supplier and consumer see the same price? If not, can you express the price that one sees, in terms of the price the other sees, and the tax?

Having gone through that, you can then revise either equation 1 or equation 2. You can then substitute equations 1 and 2 into equation 3, and solve as before, to get the new equilibrium price. And once you've got that, you can put that into either equation 1 or 2, to get the new equilibrium quantity. And from there, you can calculate the tax yield.

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Another approach to find the equilibrium with tax is to modify equations $(1)$ and $(2)$ in Energy Numbers' solution to:

$(1)Q_s=-20+2P_s$

$(2)Q_d=-220-5P_d$

(where $P_s$ is the price the producer gets and $P_d$ is the price the consumer pays).

Equation $(3)$ remains the same, but now you add another equation, (4), which reads:

$(4)P_{d}=P_{s}+t$

Instead of now looking for an equilibrium price, you're looking for an equilibrium price "wedge" given by equation (4).

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