Effect of a tax on equilibrium

Statement

Demand $D(p)=120-2p$, supply $S(p)=3p-30$ (with $p$ in €). The government imposes a $5\ \text{€}$ per-unit tax on sellers, so supply becomes $S(p)=3(p-5)-30$. Find the new equilibrium price and the quantity traded.

Solution

The new supply is $S(p)=3(p-5)-30=3p-45$. At equilibrium: $120-2p=3p-45\iff 165=5p\iff p^{*}=33\ \text{€}.$ Quantity: $q^{*}=120-2\cdot 33=54$. (Without the tax the price was $30\ \text{€}$ with $q=60$: the tax raises the price and lowers the quantity.) $\boxed{p^{*}=33\ \text{€},\ q^{*}=54}$