8. (LO4) Suppose the demand for a product is given by P=-0.8Q_(D)+150 and the supply for the same product is given by P=5.2Qs For both functions, D is the quantity and P is the price, in dollars Find the consumer surplus at the equilibrium price. a) 450 b) 250 c) 150 d) 550 e) 850

To find the consumer surplus at the equilibrium price, we need to follow these steps:<br /><br />1. **Find the equilibrium price and quantity:**<br /> - Set the demand equation equal to the supply equation to find the equilibrium quantity (\(Q_E\)):<br /> \[<br /> -0.8Q_D + 150 = 5.2Q_S<br /> \]<br /> Since \(Q_D = Q_S = Q_E\), we can rewrite it as:<br /> \[<br /> -0.8Q_E + 150 = 5.2Q_E<br /> \]<br /> Solving for \(Q_E\):<br /> \[<br /> 150 = 5.2Q_E + 0.8Q_E<br /> \]<br /> \[<br /> 150 = 6Q_E<br /> \]<br /> \[<br /> Q_E = \frac{150}{6} = 25<br /> \]<br /><br />2. **Find the equilibrium price (\(P_E\)) using the equilibrium quantity:**<br /> - Substitute \(Q_E = 25\) into either the demand or supply equation. Using the demand equation:<br /> \[<br /> P = -0.8Q_D + 150<br /> \]<br /> \[<br /> P = -0.8(25) + 150<br /> \]<br /> \[<br /> P = -20 + 150<br /> \]<br /> \[<br /> P = 130<br /> \]<br /><br />3. **Calculate the consumer surplus:**<br /> - Consumer surplus is the area of the triangle formed by the demand curve above the equilibrium price.<br /> - The formula for consumer surplus is:<br /> \[<br /> \text{Consumer Surplus} = \frac{1}{2} \times \text{Base} \times \text{Height}<br /> \]<br /> Here, the base is the equilibrium quantity (\(Q_E = 25\)) and the height is the difference between the maximum price consumers are willing to pay (intercept of the demand line) and the equilibrium price (\(P_E = 130\)):<br /> \[<br /> \text{Height} = 150 - 130 = 20<br /> \]<br /> Therefore,<br /> \[<br /> \text{Consumer Surplus} = \frac{1}{2} \times 25 \times 20 = \frac{1}{2} \times 500 = 250<br /> \]<br /><br />So, the consumer surplus at the equilibrium price is \(\$250\).<br /><br />The correct answer is:<br />b) \(\$250\)