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A manufacturer of automobiles is planning a new model and wants to determine the responsiveness of demand in various scenarios. The demand function for the new model is given by the following function:

\[ Q = 30000 - 3P + 2000\ln(PA) + Y \]

Where:
- \( Q \) is the quantity sold of the new model,
- \( P \) is the price for the new model,
- \( PA \) is the price of the competitor’s model, and
- \( Y \) is the annual income of a typical purchaser.

The new model price is planned to be £20,000, the competitor is charging £25,000, and the annual income of a typical purchaser is £30,000.

Answer :

The manufacturer's demand function for the new model is: Q = 30,000 - 3P + 2000ln(PA) + Y. Given P = £20,000, PA = £25,000, and Y = £30,000, we can calculate the demand (Q).

Step 1: Plug in the values into the demand function.
Q = 30,000 - 3(20,000) + 2000ln(25,000) + 30,000

Step 2: Simplify the equation.
Q = 30,000 - 60,000 + 2000ln(25,000) + 30,000

Step 3: Calculate 2000ln(25,000).
2000ln(25,000) ≈ 23,766

Step 4: Add the remaining numbers.
Q = -30,000 + 23,766 + 30,000

Step 5: Calculate Q.
Q ≈ 23,766

Approximately 23,766 units of the new model will be sold given the provided values for P, PA, and Y.

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