# Step-by-Step Solution ## 1. **Given Log-Linear Demand Function** \[ \ln Q^d = 8.2 - 1.6 \ln P \] where - \(Q^d\): Quantity demanded - \(P\): Price ## 2. **Price Elasticity of Demand for Log-Linear Functions** For a demand function of the form \[ \ln Q^d = a + b \ln P \] the price elasticity of demand (\(E_d\)) is simply the coefficient \(b\) (i.e., the elasticity is **constant**): \[ E_d = \frac{\partial Q}{\partial P} \cdot \frac{P}{Q} \] \[ \frac{\partial \ln Q}{\partial \ln P} = b \implies E_d = b \] So, **the price elasticity of demand is \(-1.6\) at any price**, including $50. ## 3. **Check if Elasticity Changes with Price** In this log-linear form, the elasticity does **not** depend on the price; it's always \(-1.6\). ## 4. **Final Answer** \[ \boxed{-1.6} \] **At a price of $50, the predicted price elasticity of demand is \(-1.6\).** --- ### **Summary Table** | Quantity | Value | |----------|-------------------| | Coefficient of \(\ln P\) | \(-1.6\) | | Standard error | \(0.3\) | | 95% Confidence Interval | \([-2.2, -1.0]\) | | Predicted Elasticity | \(\boxed{-1.6}\) |

# Step-by-Step Calculation of Price Elasticity of Demand ## 1. **Understanding the Demand Function** The demand function given is: \[ \ln Q^d = 8.2 - 1.6 \ln P \] In this function: - \(Q^d\) represents the quantity demanded. - \(P\) is the price of the product. ## 2. **Identifying the Coefficient** From the demand function, the coefficient before \(\ln P\) is \(-1.6\). This coefficient directly represents the price elasticity of demand (\(E_d\)) in a log-linear model. ## 3. **Price Elasticity of Demand Formula** The formula for price elasticity of demand when using a log-linear function is: \[ E_d = \frac{\partial Q}{\partial P} \cdot \frac{P}{Q} \] However, in the log-linear form: \[ \frac{\partial \ln Q}{\partial \ln P} = b \] Here, \(b\) is the coefficient of \(\ln P\), which is \(-1.6\). Thus: \[ E_d = -1.6 \] ## 4. **Elasticity at a Specific Price** Since the elasticity derived from this function is constant, it remains \(-1.6\) regardless of the price level. Therefore, we can say: \[ E_d \text{ at } P = 50 \text{ is } -1.6. \] ## 5. **Final Answer** The predicted price elasticity of demand at a price of $50 is: \[ \boxed{-1.6} \] ### **Summary of Key Points** - The coefficient for \(\ln P\) is \(-1.6\), which indicates the price elasticity of demand. - The elasticity does not change with price in a log-linear demand function. - The final calculated price elasticity of demand at any price, including $50, is \(-1.6\).