• Types of taxes
    • Quantity Taxes: a set tax levied on each unit of a good traded
    • Excise Tax: a tax levied on sellers
    • Sales Tax: a tax levied on buyers
  • The definition of equilibrium means that the market clears; that is, QS = QD, or D(p) = S(p)
  • A tax rate t makes the price paid by buyers Pb higher than the price received by sellers Ps: Pb - Ps = t
  • To solve for the market equilibrium after a tax is levied, we need two equations
    • Note that regardless of levying an excise or sales tax, the outcome is the same
      1. D(pb) = S(ps)
      2. pb - ps = t
  • The division of the tax burden between buys and sellers is known as the incidence of the tax
    • Note that the incidence of the tax is evenly split between buyers and sellers
    • Tax Graph
Solving for equilibrium price + quantityD(pb)=a−bpb,S(ps)=c+dpsTwo necessary equations: pb−ps=t,D(pb)=S(ps)pb=ps+t→a−b(ps+t)=c+dps→ps=a−c−btb+dpb=ps+t=→ps=a−c+dtb+dqt=D(pb)=S(ps)=a−bpb=ad+bc−bdtb+dNote that the equilibrium price and quantity are nearly the same as it is without taxes.As the tax, t, goes to 0, the tax equilibrium goes closer to the original equilibrium. The tax paid per unit by the buyers: pb−p∗=a−c+dtb+d−a−cb+d=dtb+dThe tax paid per unit by the sellers: p∗−ps=a−cb+d−a−c−btb+d=btb+dNote that the tax paid per unit by sellers and buyers is equal to t:btb+d+dtb+d=(b+d)tb+d=t\text{Solving for equilibrium price + quantity} \\ D(p_b) = a - bp_b, S(p_s) = c + dp_s \\ \text{Two necessary equations: } p_b - p_s = t, D(p_b) = S(p_s) \\ p_b = p_s + t \rightarrow a - b(p_s + t) = c + dp_s \rightarrow p_s = \frac{a - c - bt}{b + d} \\ p_b = p_s + t = \rightarrow p_s = \frac{a - c + dt}{b + d} \\ q^t = D(p_b) = S(p_s) = a - bp_b = \frac{ad + bc - bdt}{b + d} \\ \text{Note that the equilibrium price and quantity are nearly the same as it is without taxes.} \\ \text{As the tax, t, goes to 0, the tax equilibrium goes closer to the original equilibrium. } \\ \text{The tax paid per unit by the buyers: }\\ p_b - p^* = \frac{a - c + dt}{b + d} - \frac{a - c}{b + d} = \frac{dt}{b+d} \\ \text{The tax paid per unit by the sellers: }\\ p^* - p_s = \frac{a - c}{b + d} - \frac{a - c - bt}{b + d} = \frac{bt}{b+d} \\ \text{Note that the tax paid per unit by sellers and buyers is equal to t:} \\ \frac{bt}{b+d} + \frac{dt}{b+d} = \frac{(b+d)t}{b+d} = t
  • The total tax paid by buyers and sellers can be found by graphing the relationship between tax revenue and tax levied (AKA Laffer curve)
  • Laffer Curve Equation: T=tqt=tad+bc−bdtb+d\text{Equation: } T = tq^t = t \frac{ad + bc - bdt}{b + d}

Tax Incidence and Elasticities of Supply and Demand

Around p=p∗ the own-price elasticity of demand is approximately: εD≈△qq∗pb−p∗p∗→pb−p∗≈△q⋅p∗εD⋅q∗Around p=p∗ the own-price elasticity of supply is approximately: εS≈△qq∗ps−p∗p∗→ps−p∗≈△q⋅p∗εS⋅q∗\text{Around } p = p^* \text{ the own-price elasticity of demand is approximately: } \\ \varepsilon _D \approx \frac{\frac{\triangle q}{q^*}}{\frac{p_b - p^*}{p^*}} \rightarrow p_b - p^* \approx \frac{\triangle q \cdot p^*}{\varepsilon _D \cdot q^*} \\ \text{Around } p = p^* \text{ the own-price elasticity of supply is approximately: } \\ \varepsilon _S \approx \frac{\frac{\triangle q}{q^*}}{\frac{p_s - p^*}{p^*}} \rightarrow p_s - p^* \approx \frac{\triangle q \cdot p^*}{\varepsilon _S \cdot q^*} \\ Tax incidence (or the ratio of the tax burden on the buyers/sellers) can be written as:pb−p∗p∗−ps≈−εSεD\text{Tax incidence (or the ratio of the tax burden on the buyers/sellers) can be written as:}\\ \frac{p_b - p^*}{p^* - p_s} \approx - \frac{\varepsilon _S}{\varepsilon _D}
  • The fraction of a quantity tax, t, paid by buyers rises as supply becomes more elastic or as demand becomes less elastic

Elasticity Tax

  • Note that, with perfectly inelastic demand, buyers pay the entirety of the tax burden, and vice versa with supply. There is no deadweight loss because buyers are forced to buy and sellers are forced to sell at the new price
  • With perfectly elastic demand/supply, no trade occurs, as all buyers/sellers exit the market instantaneously. Thus, the deadweight loss is equal to the previous total surplus