Effect of a tax on the equilibrium

Problem

A tax of $T$ € per unit shifts the supply curve upwards by $T$. Show that, for linear demand and supply, the equilibrium moves towards higher prices and lower quantities, and compute the new $(p^*;\,q^*)$ as a function of $T$.

Solution

Let $D(p)=a-bp$ and $S(p)=cp-d$ with $a,b,c,d>0$. A tax of $T$ per unit means that producers, to supply a given quantity, need a price greater by $T$: the new supply is $S_T(p) = c(p-T)-d = cp - d - cT.$ New equilibrium $D(p)=S_T(p)$: $a-bp = cp-d-cT \iff a+d+cT = (b+c)\,p \iff \boxed{p^* = \frac{a+d+cT}{b+c}}.$ Without a tax ($T=0$) it would be $p_0=\dfrac{a+d}{b+c}$; hence the price increases by $\dfrac{cT}{b+c}>0$.

The equilibrium quantity: $q^* = a-bp^* = \frac{a(b+c)-b(a+d+cT)}{b+c} = \frac{ac-bd}{b+c} - \frac{bc}{b+c}\,T.$ Compared with the value without a tax $q_0=\dfrac{ac-bd}{b+c}$, the quantity decreases by $\dfrac{bc}{b+c}\,T>0$.

Conclusion: the tax raises the equilibrium price and lowers the quantity traded, as expected.

Links

Topics: Functions and properties
Concepts: Supply curve · Market equilibrium
Methods: Market equilibrium
Skills: Proving · Modelling · Solving systems
Exercise types: Modelling problem