# Deadweight Loss and Taxation

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Suppose that a market is described by the following supply and demand equations:

QS =2P,

QD =300−P.

(a) Solve for the equilibrium price and the equilibrium quantity.

(b) Suppose that a tax of T is placed on buyers, so the new demand equation is QD = 300 − (P + T). Solve for the new equilibrium. What happens to the price received by sellers, the price paid by buyers, and the quantity sold? Explain.

(c) Tax revenue is T · Q. Use your answer to part (b) to solve for tax revenue as a function of T. Graph this relationship for T between 0 and 300. Explain.

(d) Solve for deadweight loss as a function of T. Graph this relationship for T between 0 and 300. Explain.

(e) Use your results in (c) and (d) to plot the dead weight loss as a function of tax revenue.

(f) If the government doubles the tax on gasoline, can you be sure that revenue from the gasoline tax will rise? Can you be sure that the deadweight loss from the gasoline tax will rise? Explain.

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This solution uses equations and graphs to demonstrate how deadweight loss changes at different tax rates.

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(a) Solve for the equilibrium price and the equilibrium quantity.

Equilibrium occurs when QS= QD. Setting the given equations equal to each other, we find:

2P=300-P

3P = 300

P=100

So equilibrium price is 100. To find equilibrium quantity, we insert this value either of the other equations. For example, QD=300-100=200.

(b) Suppose that a tax of T is placed on buyers, so the new demand equation is QD = 300 − (P + T ). Solve for the new equilibrium. What happens to the price received by sellers, the price paid by buyers, and the quantity sold? Explain.

The new equilibrium is now found with this equation:

300 − (P + T) = 2P

300=2P+P+T

300 =3P+T

P = 100-1/3T

From this wee see that the equilibrium price will be less than 100. The consumers are not able to buy as much as before, due to the added cost of the tax. Producers must lower prices by 1/3T in order to sell as much as they did ...

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