# Government's intervention - price and quantity after taxation

Given that demand for a good X is equal to $q_D=393-2p$ and market supply is $q_S=p/4-12$. Find equilibrium price and quantity, consumer and producer surplus and draw a diagram illustrating the situation. Given that:

a) $T=2q$, every single item sold is taxed.

b) $T=20\% TR$ total revenue is taxed

Obviously i have calculated the equilibrium price and quantity before taxation that is $p=180,q=33$.But i have no idea how to caculate those two values after taxation.

• A tax does not shift the supply function in the way you are suggesting. In your expression, you subtract the tax $T_{ME}(p)$ from the supply function. I suppose you are thinking about subtracting the tax from the profit function of the supplier. Jan 9 '16 at 22:50
• Given supply and demand curves of the usual kind (supply increases and demand decreases as price increases), a tax should reduce the price received by suppliers, and in case (a) the price paid by consumers is just $2$ greater. If the equilibrium price is 180 before taxation, the consumer will pay less than 182 after taxation, definitely not more than 190. Jan 9 '16 at 22:53
• Indeed David, in fact, because demand is rather elastic relative to supply, the consumer ends up only paying about a quarter more. Jan 9 '16 at 23:01
• In the first comment, $T_{ME}(p)$ is the total revenue (money) collected by the government. Then the comment proposes the equation $q_D(p) = q_S(p) - T_{ME}(p)$ -- that is, something measured in number of goods is equal to something measured in number of goods minus some amount of money. The units aren't even compatible, so this cannot possibly be the correct equation. Jan 9 '16 at 23:03
• Regarding part (b), @mkropkowski, how do you measure total revenue? Is it the amount suppliers receive (with consumers paying an additional 20% directly to the government), or do the consumers only pay the suppliers, who must then turn over 20% of that amount to the government? There is a factor in the solution that is either $8/10$ or $10/12$ depending on which kind of tax you mean. Jan 9 '16 at 23:14

## 1 Answer

Suppose after the tax, $p$ is the price charged by the supplier. In the first case, the price paid by the consumer is $p+2$. To find the new equilibrium price (charged by the supplier), solve:

$$393-2(p+2) = \frac{p}{4} -12$$

The new price charged by the supplier is $178.22$ the consumer pays $\$180.22$. New quantity is$393-2(180.22)=32.56$. Notice that the tax collected is exactly$2q$. In the second case, if the supplier charges$p$, they receive only$80\%$of that price for each item sold. Thus, it is as if they receive$0.8p$when they charge$p$. We solve: $$393-2(p) = \frac{0.8p}{4} -12$$ The new price charged by the supplier is$\$184.091$ quantity sold is $393-2(184.091)=24.818$.

• For part (b) I wonder whether the tax is meant to be 20 percent of the total amount paid by consumers, or 20 percent of the amount received by suppliers. Corporate income tax might work in the first way; a typical sales tax would work in the second way (at least in the US). I find the problem ambiguous on that point. The parameterization of the equations in this answer is generally sound, however. Jan 9 '16 at 23:09
• Actually I find the cost to the consumer is 184.091 in one case and 183.396 in the other, including taxes. The supplier receives 147.273 and 152.83, respectively, not including taxes. For example, see wolframalpha.com/input/… Jan 9 '16 at 23:26
• Yes, I understand your point and I agree that it is a bit ambiguous. Jan 10 '16 at 0:08
• @DavidK i have posted the answer in a comments above. Jan 10 '16 at 10:49