💰 Specific Taxes with Linear Functions
Effect on supply: A specific (per-unit) tax of $t per unit shifts the supply curve vertically upward by $t. The new supply function becomes: $$Q_s = c + d(P - t)$$ which simplifies to: $$Q_s = (c - dt) + dP$$ The slope is unchanged but the Q-intercept falls by $dt$.
Calculating the new equilibrium
Set the original demand equal to the new (taxed) supply and solve for the new $P^*_{\text{consumer}}$ (price consumers pay). The price producers receive is: $P_{\text{producer}} = P_{\text{consumer}} - t$.
Worked example: Demand: $Q_d = 100 - 5P$. Supply: $Q_s = -20 + 4P$. Tax: $t = $3. \n New supply: $Q_s = -20 + 4(P - 3) = -32 + 4P$. \n Equilibrium: $100 - 5P = -32 + 4P$ → $132 = 9P$ → $P_c = $14.67$ \n $P_p = 14.67 - 3 = $11.67$ \n $Q^* = 100 - 5(14.67) = 26.7$ units \n Tax revenue = $3 × 26.7 = $80.00
Tax incidence depends on relative elasticities. Consumer burden = $P_c - P_{\text{original}}$. Producer burden = $P_{\text{original}} - P_p$. The more inelastic side bears the greater burden.
🎁 Subsidies and Welfare Calculations
Effect of a subsidy
A specific subsidy of $s per unit shifts the supply curve vertically downward by $s. The new supply function: $$Q_s = c + d(P + s) = (c + ds) + dP$$ The Q-intercept rises by $ds$.
Welfare calculations
- Consumer surplus (CS) = area of triangle above P and below demand curve = $\frac{1}{2} \times Q^ \times (P_{\text{max}} - P^*)$.
- Producer surplus (PS) = area of triangle below P and above supply curve = $\frac{1}{2} \times Q^ \times (P^* - P_{\text{min}})$.
- Tax revenue = tax × Q* (a rectangle).
- Deadweight loss (DWL) = $\frac{1}{2} \times t \times \Delta Q$ where $\Delta Q$ is the reduction in quantity.
- Government expenditure on subsidy = subsidy × new Q*.
In IB exams, you may be asked to 'calculate the deadweight loss' or 'calculate the change in consumer surplus' using the linear functions. Draw the diagram first, then use area formulas (triangles and rectangles).