Chapter 1-6: Introduction to Taxation

Demand and market setup

• Demand: Qd = a - b Pd

• Supply: Qs = c + d Ps

• Per-unit tax on sellers: Pd = Ps + t

• Equilibrium condition: Qd = Qs with the relation Pd = Ps + t

Prices and quantities under a per-unit tax

• Solve for equilibrium with tax:

  • Seller price at equilibrium: P_s^* = rac{a - c - b t}{b + d}

  • Buyer price at equilibrium: Pd^* = Ps^* + t = rac{a - c + d t}{b + d}

  • Equilibrium quantity: Q^* = Qs(Ps^) = c + d Ps^ = a - b Pd^*

• Tax impact (t > 0):

  • Quantity falls: Q^* ext{ decreases}

  • Buyer price rises: P_d^* ext{ increases}

  • Seller price falls: P_s^* ext{ decreases relative to pre-tax}

Tax revenue and deadweight loss

• Government revenue: R = t \, Q^*

• Deadweight loss: DWL = frac{1}{2} t \, ig(Q^0 - Q^*ig) where Q^0 is the pre-tax quantity (t = 0)

• Social surplus changes: tax revenue increases but DWL reduces overall surplus

Incidence of taxation (who bears the burden)

• Burden depends on elasticities, not on who remits the tax

• Linear model intuition: if demand is more inelastic than supply, buyers bear more of the tax; if supply is more inelastic, sellers bear more

• In a simple linear setup (withDemand: Q = a - b Pd, Supply: Q = c + d Ps), the shares can be expressed (illustrative):

  • Share borne by buyers: rac{ ext{elasticity of supply}}{ ext{elasticity of demand} + ext{elasticity of supply}} = rac{d}{b + d}

  • Share borne by sellers: rac{b}{b + d}

• If the tax is imposed on buyers instead, the same incidence results; the labels change, but the economic burden and quantity effects are the same

Berkeley soda tax example (1 cent per ounce)

• Tax: t = 1 ext{ cent per ounce} = 0.01

• Observed price to consumers rose by about 0.4 ext{ cents} = 0.004 dollars

• Interpretation: price rise to consumers is smaller than the tax due to elasticity; government collects revenue R = t \, Q^*

• The tax reduces welfare via deadweight loss, with some of the tax collected as revenue

Quick algebraic reminder (from lecture quiz)

• For a simple linear model, the quick equilibrium result shown was:

  • Price: P^* = rac{A B}{1 + b}

  • Quantity: Q^* = rac{a}{1 + b}

• These reflect a specific linear-curve setup used for a rapid check; use the general framework above for your own derivations

Buyer vs. seller taxation (summary)

• If the tax is imposed on the buyer vs the seller, the same qualitative outcomes hold:

  • Equilibrium quantity decreases

  • The buyer’s price and the seller’s net price adjust so that the tax is shared (incidence)

  • Government revenue and deadweight loss are determined by the same tax and elasticities; who pays the government is a matter of administration, not of market outcome in the basic model

Practice note

• When solving equilibrium with a per-unit tax, use the relation Pd = Ps + t and set demand equal to supply: Qd(Pd) = Qs(Ps) to solve for Ps^, Pd^, and Q^*.