Question

Suppose you are the manager of a watchmaking firm operating in a competitive market. Your cost of production is given by \(C=200+2 q^{2}\), where \(q\) is the level of output and \(C\) is total cost. (The marginal cost of production is \(4 q ;\) the fixed cost is \(\$ 200 .\) ) a. If the price of watches is \(\$ 100,\) how many watches should you produce to maximize profit? b. What will the profit level be? c. At what minimum price will the firm produce a positive output?

Short answer

To maximize the profit, around 12 watches should be produced. The maximum profit given this output would be approximately $400. The firm would need to price the watches at a minimum of $200 to produce a positive output.

Step-by-step solution

Calculation of quantity to maximize profit

Start by setting up the profit equation, which is profit = total revenue - total cost. Since total revenue is output (q) times price (P), and total cost is given by the problem as \(C=200+2q^2\), the profit equation becomes profit = qP - (200+2q^2). Given that the price of watches is $100, substitute P=100 to find q. Taking derivative of the profit equation with respect to q, setting it to 0 and solve for q to get the quantity to maximize profit.

Calculation of maximum profit level

Once you have found the value for q that maximizes profit, substitute this value back into the profit equation to find the maximum profit. This is done by plugging the optimal q into the profit = qP - (200+2q^2) and evaluating.

Determination of minimum price for positive output

To produce a positive output, the total revenue should exceed the fixed cost of producing the good. Thus, set the total revenue (qP) equal to the fixed cost (200) and solve for the price P, assuming the quantity produced q is at least 1.

Key concepts

Marginal Cost

Understanding the marginal cost is essential in determining how production affects costs incrementally. Marginal Cost (MC) represents the additional cost incurred when producing one more unit of a good. In this scenario, the marginal cost is given by the function \(MC = 4q\). This means, for every watch produced, the cost increases by 4 times the quantity.

• Marginal cost is crucial for decision-making to ensure costs don't outweigh revenues.
• It helps determine the optimal production level to maximize profit.
• In a competitive market, firms often equate marginal cost with market price to find the ideal output level.

Understanding marginal cost is vital as it guides decisions on how much more of a product to make or when to stop increasing output.

Profit Maximization

Profit maximization is the core goal for any competitive firm. It's about adjusting production to achieve the highest possible profit. Profit is calculated as total revenue minus total cost, where total revenue is the price times quantity (\(qP\)), and total cost includes fixed and variable costs (\(C=200+2q^2\)).
To find the profit-maximizing output level:

• Set the derivative of the profit equation with respect to quantity to zero.
• This involves finding where marginal cost equals marginal revenue.
• In this exercise, with prices set at $100 per watch, setting \(4q = 100\) leads to the optimal production level.

By solving these, firms can find the exact quantity that lets them price their way into maximum profit territory, making resource allocation highly efficient.

Competitive Market

Operating in a competitive market means that firms are price takers. They cannot influence the price of their products due to the large number of sellers offering similar goods. This assumption affects strategy significantly.

• Competitive markets drive firms to produce efficiently to stay profitable.
• Prices are determined collectively by market demand and supply.
• Firms must focus on minimizing cost to improve profit margins.

Being in a competitive market means a firm needs to ensure its total revenue is sufficient to cover both fixed and variable costs. The minimum price for producing a positive output, in this case, is found by ensuring \(qP\) covers at least the fixed cost of 200 units, driving the production of watches.