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For Example:

$D(p) = 50 - p$

$S(p) = p/20$

The p* of this problem is $47.61$, however, when I add a .1 proportional tax I get an answer of $43.29$ which does not make sense because that price is lower than the previous equilibrium price.

I am using the equations:

$D(p) = 50 - (p + .1p) = (p + .1p)/20 = S(p)$

Am I dividing wrong? I suspect that I am getting a percentage change that is correct, it is just negative instead of positive. Any help would be appreciated.

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    $\begingroup$ Hint: Consider that the tax is only paid by buyers. Sellers don't receive any additional revenue from the proportional tax. $\endgroup$ Oct 6 at 22:19
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When solving this problem, it is important to notice that whenever you have a tax, there will be a difference between the price that the buyers pay and the price that the sellers will receive.

Let $p_D$ be the price that the buyers pay and let $p_S$ be the price that the sellers receive.

The price $p_D$ is the one that is relevant for the buyers, so it should enter the demand function, $D(p_D)$.

The selling price $p_S$ is the only price that is relevant for the sellers, so it is the price that should enter the supply function, $S(p_S)$.

At the moment you have two equations in three unknowns, the amount sold (or bought) $q$, the price $p_D$ and the price $p_S$. So you need an additional equation to solve the model.

The additional equation links the two prices $p_S$ and $p_D$. In particular, the difference between the two prices is given by the tax. If the seller receives $p_S$ then the buyer pays $p_D = (1+t)p_S$.

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