Overview
This lecture analyzes consumer and producer surplus when a price ceiling is imposed, using linear supply and demand functions, and details calculations for resulting surpluses and deadweight loss.
Market Equilibrium Without Price Ceiling
- Supply function: ( S(x) = 31 + 0.5x ).
- Demand function: ( D(x) = 79 - x ).
- Set ( S(x) = D(x) ) to find equilibrium: ( 31 + 0.5x = 79 - x ).
- Solving for ( x ): Equilibrium quantity ( x_e = 32 ).
- Plug into either function for equilibrium price: ( P_e = 47 ).
Imposing a Price Ceiling
- Price ceiling set at ( P_c = 41 ), below equilibrium price (( 47 )).
- Price ceiling restricts the maximum allowable price in the market.
- New intersection with supply curve: set ( 41 = 31 + 0.5x ), solve for ( x ) gives ( x_c = 20 ).
- New market point due to ceiling: ( (20, 41) ).
Calculating Consumer Surplus with Price Ceiling
- Consumer surplus is area under demand curve above price ceiling up to ( x_c ).
- Integrate ( 79 - x ) from 0 to 20: result = 1380.
- Subtract revenue rectangle: ( 41 \times 20 = 820 ).
- Consumer surplus = ( 1380 - 820 = 560 ).
Calculating Producer Surplus with Price Ceiling
- Producer surplus is revenue rectangle area minus area under supply curve up to ( x_c ).
- Integrate ( 31 + 0.5x ) from 0 to 20: ( 31x + 0.25x^2 ) at 20 = 620.
- Producer surplus = ( 820 - 720 = 100 ).
Deadweight Loss Due to Price Ceiling
- Deadweight loss: Area between supply and demand curves from ( x_c = 20 ) to ( x_e = 32 ).
- Integrate ([79 - x] - [31 + 0.5x] = 48 - 1.5x) from 20 to 32.
- Calculated deadweight loss = 108.
Key Terms & Definitions
- Supply Function — Mathematical representation of quantity supplied at each price.
- Demand Function — Mathematical representation of quantity demanded at each price.
- Equilibrium — Point where supply equals demand.
- Price Ceiling — Legal maximum price allowed in the market.
- Consumer Surplus — Area between demand curve and price, up to quantity sold.
- Producer Surplus — Area between supply curve and price, up to quantity sold.
- Deadweight Loss — Lost total surplus due to market inefficiency, such as price ceilings.
Action Items / Next Steps
- Practice calculating surpluses and deadweight loss under other price controls.
- Review integration methods for calculating areas under curves.
- Prepare for homework on calculating market effects of other interventions.