1. **Problem statement:** We have two problems related to elasticity: a. Calculate the price elasticity of supply for the supply function $P = 5Q + 100$ at $Q=75$ and $P=150$. b. Calculate the price elasticity of demand for the demand function $Q_d = 100 - 2P^2$ at $P=5$ and $Q=68$. --- 2. **Elasticity formulas and rules:** - Price elasticity of supply (E_s) is given by: $$E_s = \frac{dQ}{dP} \times \frac{P}{Q}$$ - Price elasticity of demand (E_d) is given by: $$E_d = \frac{dQ_d}{dP} \times \frac{P}{Q_d}$$ - Important: For supply, $\frac{dQ}{dP}$ is the derivative of quantity with respect to price. - For demand, $\frac{dQ_d}{dP}$ is the derivative of demand quantity with respect to price. --- ### a. Elasticity of supply 3. Given $P = 5Q + 100$, solve for $Q$: $$Q = \frac{P - 100}{5}$$ 4. Differentiate $Q$ with respect to $P$: $$\frac{dQ}{dP} = \frac{1}{5}$$ 5. Calculate elasticity at $Q=75$, $P=150$: $$E_s = \frac{dQ}{dP} \times \frac{P}{Q} = \frac{1}{5} \times \frac{150}{75} = \frac{1}{5} \times 2 = \frac{2}{5} = 0.4$$ 6. **Interpretation:** An elasticity of 0.4 means supply is inelastic at this point; a 1% increase in price leads to a 0.4% increase in quantity supplied. --- ### b. Elasticity of demand 7. Given $Q_d = 100 - 2P^2$, differentiate with respect to $P$: $$\frac{dQ_d}{dP} = -4P$$ 8. Calculate derivative at $P=5$: $$\frac{dQ_d}{dP} = -4 \times 5 = -20$$ 9. Calculate elasticity at $P=5$, $Q_d=68$: $$E_d = \frac{dQ_d}{dP} \times \frac{P}{Q_d} = -20 \times \frac{5}{68} = -\frac{100}{68} = -\frac{50}{34} \approx -1.47$$ 10. **Interpretation:** An elasticity of approximately -1.47 means demand is elastic; a 1% increase in price causes about a 1.47% decrease in quantity demanded. --- **Final answers:** - Elasticity of supply at $Q=75$, $P=150$ is $0.4$ (inelastic supply). - Elasticity of demand at $P=5$, $Q=68$ is approximately $-1.47$ (elastic demand).