2. Just-the-fax, max, inc. has determined that the demand for its fax machine is Q=3000-1.5P.

Question

A. Calculate the point price elasticity of demand when p=600.
B. At P=600, what is the firm's marginal revenue ?
C. Find the total revenue maximizing price &quantity for the firm.

Answer 1

To address the questions posed for Just-the-fax, max, inc., let's break each part down step-by-step:

### A. Calculate the Point Price Elasticity of Demand when P = 600

The demand function given is:
\[ Q = 3000 - 1.5P \]

The point price elasticity of demand (E) is calculated as:
\[ E = \left( \frac{dQ}{dP} \right) \cdot \left( \frac{P}{Q} \right) \]

First, find the derivative \( \frac{dQ}{dP} \):
\[ \frac{dQ}{dP} = -1.5 \]

Next, plug in \( P = 600 \) into the demand function to find the corresponding quantity (Q):
\[ Q = 3000 - 1.5(600) \]
\[ Q = 3000 - 900 \]
\[ Q = 2100 \]

Now, calculate the elasticity:
\[ E = (-1.5) \cdot \left( \frac{600}{2100} \right) \]
\[ E = -1.5 \cdot \frac{2}{7} \]
\[ E = -0.4286 \]

So, the point price elasticity of demand when \( P = 600 \) is approximately \(-0.4286\).

### B. At \( P = 600 \), What is the Firm's Marginal Revenue?

Marginal Revenue (MR) can be derived from the total revenue (TR) function. First, we need to find the TR function.

Total Revenue (TR) is:
\[ TR = P \cdot Q \]

From the demand function \( Q = 3000 - 1.5P \), we can derive the TR function in terms of P:
\[ TR = P \cdot (3000 - 1.5P) \]
\[ TR = 3000P - 1.5P^2 \]

To get Marginal Revenue (MR), we take the derivative of TR with respect to P:
\[ MR = \frac{d(TR)}{dP} \]
\[ MR = 3000 - 3P \]

Now plug in \( P = 600 \):
\[ MR = 3000 - 3(600) \]
\[ MR = 3000 - 1800 \]
\[ MR = 1200 \]

So, the firm's marginal revenue at \( P = 600 \) is 1200.

### C. Find the Total Revenue Maximizing Price and Quantity for the Firm

Total revenue is maximized when marginal revenue (MR) is zero. From the previous part, we know that:
\[ MR = 3000 - 3P \]

Set MR to 0 and solve for P:
\[ 0 = 3000 - 3P \]
\[ 3P = 3000 \]
\[ P = 1000 \]

Now, find the corresponding quantity (Q) using the demand function:
\[ Q = 3000 - 1.5(1000) \]
\[ Q = 3000 - 1500 \]
\[ Q = 1500 \]

So, the total revenue maximizing price is \( P = 1000 \) and the total revenue maximizing quantity is \( Q = 1500 \).

### Summary:
- **Point Price Elasticity of Demand at \( P = 600 \)**: \(-0.4286\)
- **Marginal Revenue at \( P = 600 \)**: 1200
- **Total Revenue Maximizing Price and Quantity**: \( P = 1000 \), \( Q = 1500 \)