Profit Maximization for a Monopolist: A Comprehensive Guide

When a monopolist faces a demand curve and has a fixed supply, determining the optimal price and profit becomes a crucial economic question. This article will walk you through the steps to find the profit-maximizing price and the maximum profit for a monopolist with a fixed supply of 7 units. We will use a specific demand curve, Dp 10 - P, and see how it impacts the pricing and profit calculations.

Identifying the Demand Curve

The demand curve given is:

Dp 10 - P

This can be rearranged to find the price as a function of quantity:

P 10 - Q

Determining Total Revenue (TR)

Total revenue is calculated by multiplying the price by the quantity:

2

Determining Marginal Revenue (MR)

To find the marginal revenue, we take the derivative of the total revenue function with respect to quantity Q:

MR frac{dTR}{dQ} 10 - 2Q

Setting Quantity to Fixed Supply

Given the fixed supply of 7 units, we substitute Q 7 into the price equation:

P 10 - 7 3

Calculating Total Revenue (TR) at Q 7

Using the price found:

TR P × Q 3 × 7 21

Assuming Zero Total Cost (TC)

In the absence of specific cost information, we assume a total cost of zero for simplicity:

TC 0

Calculating Profit

Profit is calculated as the difference between total revenue and total cost:

Profit TR - TC 21 - 0 21

Summary

The profit-maximizing price for the monopolist is:

P 3

The maximum profit for the monopolist, given a fixed supply of 7 units, is:

Profit 21

If you have specific costs or further details, please provide them to refine this analysis!

Additional Considerations

The profit-maximizing price for a monopolist can vary based on the divisibility of units and the type of product:

Divisibility of Units: For units like oil or vehicles, the profit-maximizing price is more straightforward. For non-divisible units like perishable food or labor seats at a theatre, the profit maximization might differ due to storage and perishability constraints. Cost of Production: If the cost of production is greater than a certain threshold (e.g., 8), the company may be better off selling at that threshold to avoid losses, especially for storable goods. Accounting Practices: Different accounting practices (e.g., mark-to-market) can impact the actual profit reported. For storable goods like gold or silver, the company can store the inventory at cost and only sell at prices above the cost threshold.

For non-storable items, the optimal pricing strategy has already been calculated. By understanding these nuances, businesses can make more informed decisions on pricing and profit maximization.