Suppose that for a product in a competitive market, the demand function is 

*p* = 1,400 − 3*x*

 and the supply function is 

*p* = 200 + 3*x*,

 where *x* is the number of units and *p* is in dollars. A firm's average cost function for this product is as follows.

*C*(*x*) = 

17,000

*x*

 + 60 + *x*

Find the maximum profit, in dollars. (*Hint*: First find the equilibrium price.)

To find the maximum profit, we will follow these steps:

1. **Find the equilibrium price and quantity** by setting the demand function equal to the supply function.
2. **Determine the profit** function using the equilibrium quantity, the average cost function, and the equilibrium price.
3. **Calculate the maximum profit**.

### Step 1: Find the Equilibrium Price and Quantity

Set the demand function equal to the supply function:

\[
1,400 - 3x = 200 + 3x
\]

Combine like terms:

\[
1,400 - 200 = 3x + 3x
\]

\[
1,200 = 6x
\]

\[
x = 200
\]

Now, substitute \( x = 200 \) back into either the demand or supply function to find the equilibrium price \( p \):

Using the demand function:

\[
p = 1,400 - 3(200) = 1,400 - 600 = 800
\]

Thus, the equilibrium price \( p \) is \( 800 \) dollars, and the equilibrium quantity \( x \) is \( 200 \).

### Step 2: Determine the Profit Function

The profit \( \Pi(x) \) is given by:

\[
\Pi(x) = R(x) - C(x)
\]

Where \( R(x) \) is total revenue and \( C(x) \) is total cost.

1. **Total Revenue \( R(x) \)**:

\[
R(x) = p \cdot x = 800x
\]

2. **Total Cost \( C(x) \)**:

From the average cost function given:

\[
C(x) = 17,000 \cdot x + 60 + x
\]

Thus, the total cost function becomes:

\[
C(x) = 17,001x + 60
\]

3. **Profit Function**:

Substituting \( R(x) \) and \( C(x) \) into the profit function:

\[
\Pi(x) = R(x) - C(x) = 800x - (17,001x + 60)
\]

\[
\Pi(x) = 800x - 17,001x - 60 = -16,201x - 60
\]

### Step 3: Calculate Maximum Profit

Since profit is linear in \( x \) and negative, maximum profit occurs at the equilibrium quantity:

\[
\Pi(200) = -16,201(200) - 60
\]

Calculating this:

\[
\Pi(200) = -3,240,200 - 60 = -3,240,260
\]

### Conclusion

The maximum profit, in dollars, is \(-3,240,260\).

### Questions to Explore Further

1. What are the implications of negative profit for this firm?
2. How would a change in demand or supply affect the equilibrium?
3. What strategies could the firm use to increase profitability?
4. How would fixed costs influence the average cost function?
5. What role does competition play in determining prices in this market?

### Tip

Consider analyzing the cost structure further to identify potential areas for cost reduction, which may improve profitability.

Suppose that for a product in a competitive market, the demand function is p = 1,400 − 3x and the supply function is p = 200 + 3x, where x is the number of units and p is in dollars. A firm's average cost function for this product is as follows. C(x) = 17,000 / x + 60 + x. Find the maximum profit, in dollars. (Hint: First find the equilibrium price.)

Calculate the maximum profit for a product in a competitive market with given demand, supply, and cost functions. This problem involves setting demand equal to supply to find equilibrium, then using profit maximization techniques, making it ideal for economics or business math students.

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