Summary of Microeconomics Core Concepts and Problems

## Introduction Tax incidence is the study of how the burden of a tax is split between buyers and sellers. It asks who actually bears the economic cost of a tax once markets adjust: consumers through higher prices, producers through lower net receipts, or both. This material uses linear supply and demand examples to show how to compute equilibrium prices and quantities, measure welfare changes, and interpret the role of elasticities. > **Definition:** Tax incidence is the distribution of the economic burden of a tax between market participants after market adjustments. ## 1. Preliminaries: Market equilibrium without tax Consider a competitive market with linear demand and supply given by $$x_N = 150 - 50p$$ $$x_P = 60 + 40p$$ Here $x$ is quantity (liters of gasoline) and $p$ is the market price (per liter). Equilibrium solves $x_N = x_P$. Solve equilibrium: $$150 - 50p = 60 + 40p$$ Move terms: $$150 - 60 = 40p + 50p$$ $$90 = 90p$$ $$p^* = 1$$ Plug into either schedule to get quantity: $$x^* = 150 - 50\cdot 1 = 100$$ Consumer price and producer net price are both $p^* = 1$ before tax. ### Welfare before tax - Consumer surplus (CS): area between demand intercept and price. Demand intercept (price at $x=0$) solves $150 - 50p = 0$ so $p = 3$. Consumer surplus is a triangle with height $3 - 1 = 2$ and base $100$: $$CS = \tfrac{1}{2} \cdot 100 \cdot 2 = 100$$ - Producer surplus (PS): supply intercept (price at $x=0$) solves $60 + 40p = 0$ so $p = -\tfrac{3}{2}$, but easier is to compute PS as area between price and supply line at quantities. Compute the price on supply at $x=0$ is $p = -\tfrac{3}{2}$, so height from that intercept to market price is $1 - (-\tfrac{3}{2}) = \tfrac{5}{2}$. Base $100$: $$PS = \tfrac{1}{2} \cdot 100 \cdot \tfrac{5}{2} = \tfrac{100 \cdot 5}{4} = 125$$ - Total welfare (TW) before tax is $CS + PS = 100 + 125 = 225$. > **Note:** For linear supply with negative intercept, producer surplus can be computed via the triangular area between the supply curve and market price. ## 2. Price elasticities at the equilibrium Elasticity of demand at the equilibrium: elasticity formula for linear demand $x = a - bp$ is $\varepsilon_D = \dfrac{dx}{dp}\cdot\dfrac{p}{x} = -b\cdot\dfrac{p}{x}$. Here $b = 50$, $p = 1$, $x = 100$ so $$\varepsilon_D = -50 \cdot \dfrac{1}{100} = -0.5$$ Elasticity of supply for $x = 60 + 40p$ is $\varepsilon_S = \dfrac{dx}{dp}\cdot\dfrac{p}{x} = 40 \cdot \dfrac{1}{100} = 0.4$. Interpretation: demand is inelastic (absolute elasticity $0.5$) relative to supply (absolute elasticity $0.4$); both are inelastic, but demand is somewhat more elastic in absolute terms than supply (0.5 vs 0.4). ## 3. Introducing a specific tax: $t = 0.5$ per liter The tax of $t = 0.5$ is levied on producers (they remit it). A per-unit tax shifts the supply curve vertically upward by $t$ when plotted as a function of price received by buyers, or equivalently it shifts the price producers receive downward by $t$ relative to the buyer price. Let $p_b$ be the price paid by buyers (gross price) and $p_s$ the price received by sellers (net of tax). The supply schedule in terms of $p_s$ remains $$x = 60 + 40p_s$$ But buyers face $p_b = p_s + t$, so in terms of $p_b$ the supply becomes $$x = 60 + 40(p_b - t) = 60 + 40p_b - 40t$$ With $t = 0.5$ this is $$x = 60 + 40p_b - 20 = 40 + 40p_b$$ Equate demand and taxed supply to get new equilibrium quantity: $$150 - 50p_b = 40 + 40p_b$$ $$150 - 40 = 40p_b + 50p_b$$ $$110 = 90p_b$$ $$p_b = \dfrac{110}{90} = \dfrac{11}{9} \approx 1.222\ldots$$ Quantity after tax: $$x_{tax} = 150 - 50 \cdot \dfrac{11}{9} = 150 - \dfrac{550}{9} = \dfrac{1350 - 550}{9} = \dfrac{800}{9} \approx 88.888\ldots$$ Price received by sellers (net): $$p_s = p_b - t = \dfrac{11}{9} - \dfrac{1}{2} = \dfrac{22 - 9}{18} = \dfrac{13}{18} \approx 0.7222\ldots$$ Verify via supply: $x = 60 + 40\cdot \dfrac{13}{18} = 60 + \dfrac{520}{18} = \dfrac{1080 + 520}{18} = \dfrac{1600}{18}