# When solving for a new equilibrium price after the institution of a tax, I keep getting a price that is lower than my previous equil price. Why?

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.

• Hint: Consider that the tax is only paid by buyers. Sellers don't receive any additional revenue from the proportional tax. Oct 6 at 22:19

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 $$p_D = (1+t)p_S$$.