6. Given the demand function as Q_(1)=b_(0)-b_(1)P Derive the Average Revenue and Marginal Revenue functions and compare the two (5 marks)

To derive the Average Revenue (AR) and Marginal Revenue (MR) functions from the given demand function \( Q_1 = b_0 - b_1P \), we need to follow these steps:<br /><br />1. **Determine the Total Revenue (TR) function:**<br /> - Total Revenue is calculated as the product of price (P) and quantity (Q).<br /> - Given \( Q_1 = b_0 - b_1P \), the Total Revenue function is:<br /> \[<br /> TR = P \times Q_1 = P \times (b_0 - b_1P)<br /> \]<br /> - Simplifying this, we get:<br /> \[<br /> TR = b_0P - b_1P^2<br /> \]<br /><br />2. **Derive the Average Revenue (AR) function:**<br /> - Average Revenue is the Total Revenue divided by the quantity.<br /> - Using the Total Revenue function \( TR = b_0P - b_1P^2 \), we have:<br /> \[<br /> AR = \frac{TR}{Q_1} = \frac{b_0P - b_1P^2}{b_0 - b_1P}<br /> \]<br /><br />3. **Derive the Marginal Revenue (MR) function:**<br /> - Marginal Revenue is the additional revenue from selling one more unit of the good.<br /> - To find MR, we take the derivative of the Total Revenue function with respect to quantity (Q).<br /> \[<br /> MR = \frac{d(TR)}{dQ_1}<br /> \]<br /> - Given \( TR = b_0P - b_1P^2 \), we need to express \( P \) in terms of \( Q_1 \). From the demand function \( Q_1 = b_0 - b_1P \), we solve for \( P \):<br /> \[<br /> P = \frac{b_0 - Q_1}{b_1}<br /> \]<br /> - Substitute \( P \) into the Total Revenue function:<br /> \[<br /> TR = b_0 \left(\frac{b_0 - Q_1}{b_1}\right) - b_1 \left(\frac{b_0 - Q_1}{b_1}\right)^2<br /> \]<br /> - Simplify the expression:<br /> \[<br /> TR = \frac{b_0^2 - b_0Q_1}{b_1} - \frac{b_1(b_0 - Q_1)^2}{b_1^2}<br /> \]<br /> \[<br /> TR = \frac{b_0^2 - b_0Q_1 - b_1(b_0 - Q_1)^2}{b_1}<br /> \]<br /> - Differentiate \( TR \) with respect to \( Q_1 \) to find \( MR \):<br /> \[<br /> MR = \frac{d}{dQ_1} \left( \frac{b_0^2 - b_0Q_1 - b_1(b_0 - Q_1)^2}{b_1} \right)<br /> \]<br /> \[<br /> MR = \frac{-b_0 - 2b_1(b_0 - Q_1)}{b_1}<br /> \]<br /> \[<br /> MR = \frac{-b_0 - 2b_1b_0 + 2b_1Q_1}{b_1}<br /> \]<br /> \[<br /> MR = \frac{2b_1Q_1 - 3b_0}{b_1}<br /> \]<br /> \[<br /> MR = \frac{2Q_1}{b_1} - 3<br /> \]<br /><br />4. **Compare Average Revenue (AR) and Marginal Revenue (MR):**<br /> - The Average Revenue function is:<br /> \[<br /> AR = \frac{b_0P - b_1P^2}{b_0 - b_1P}<br /> \]<br /> - The Marginal Revenue function is:<br /> \[<br /> MR = \frac{2Q_1}{b_1} - 3<br /> \]<br /> - To compare the two, we can express \( P \) in terms of \( Q_1 \) from the demand function:<br /> \[<br /> P = \frac{b_0 - Q_1}{b_1}<br /> \]<br /> - Substitute \( P \) into the AR function:<br /> \[<br /> AR = \frac{b_0 \left(\frac{b_0 - Q_1}{b_1}\right) - b_1 \left(\frac{b_0 - Q_1}{b_1}\right)^2}{b_0 - b_1 \left(\frac{b_0 - Q