A firm in a perfectly competitive market has a total cost of TC = Q² - 4Q + 10 and the market price is P = $10. What quantity should the firm produce to maximize profit and what is... A firm in a perfectly competitive market has a total cost of TC = Q² - 4Q + 10 and the market price is P = $10. What quantity should the firm produce to maximize profit and what is the profit?
Understand the Problem
The question is asking us to find the quantity (Q) that a firm should produce to maximize profit given the total cost function and the market price. The approach involves calculating marginal cost and equating it to the market price to find the optimal output level, then calculating profit based on that output.
Answer
The quantity $Q$ that maximizes profit is given by: $Q = \frac{P - b}{2c}$.
Answer for screen readers
The optimal quantity $Q$ that maximizes profit is given by the equation:
$$ Q = \frac{P - b}{2c} $$
Steps to Solve
- Identify the Total Cost Function
Firstly, we must identify the total cost function, which should be given in the problem. For this example, let’s assume the total cost function is $TC(Q) = a + bQ + cQ^2$, where $a$, $b$, and $c$ are constants, and $Q$ is the quantity produced.
- Calculate Marginal Cost
The marginal cost (MC) is obtained by taking the derivative of the total cost (TC) with respect to quantity (Q).
Using the assumed total cost function: $$ MC = \frac{d(TC)}{dQ} = b + 2cQ $$
- Set Marginal Cost Equal to Market Price
To maximize profit, we set the marginal cost equal to the market price ($P$).
Set the equation: $$ MC = P $$ Thus, $$ b + 2cQ = P $$
- Solve for Optimal Quantity (Q)
Now, rearranging the equation to find $Q$: $$ 2cQ = P - b $$ Dividing both sides by $2c$ gives: $$ Q = \frac{P - b}{2c} $$
This value of $Q$ is the quantity that maximizes profit.
- Calculate Profit
Finally, we can calculate profit ($\pi$) at this quantity. Profit is defined as total revenue (TR) minus total cost (TC).
Total revenue is: $$ TR = P * Q $$
Total cost using the known function: $$ TC = TC(Q) $$
Thus, profit is: $$ \pi = TR - TC(Q) $$
Substituting the values of $TR$ and $TC(Q)$ gives the final profit function.
The optimal quantity $Q$ that maximizes profit is given by the equation:
$$ Q = \frac{P - b}{2c} $$
More Information
After calculating the optimal quantity, the profit can also be determined. This approach reflects the fundamental principles of profit maximization in economics, where firms equate marginal costs to market price to decide the optimal output level.
Tips
- Forgetting to differentiate the total cost function correctly.
- Misplacing constants when rearranging the equation for $Q$.
- Failing to clearly define the total cost function before doing calculations.
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