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

a. The firm should produce 25 watches to maximize profit. b. The profit level will be $1050. c. The firm will produce a positive output at a minimum price of $0.

Step-by-step solution

Determine output level for profit maximization

The firm's revenue is given by \(p*q\), where \(p\) is the price and \(q\) is the quantity. Thus, when \(p = 100\), the revenue is \(100q\). The firm maximizes profit when marginal cost equals marginal revenue. Given that the marginal cost is \(4q\), setting this equal to marginal revenue (price), gives: \(4q = 100\). Solving for \(q\) results in \(q = 25\).

Compute profit level

The profit is given by revenue minus cost. Thus, substituting \(q = 25\) into \(C = 200 + 2q^2\) gives \(C = 200 + 2*625 = 200 + 1250 = 1450\). Thus, the profit is \(100*25 - 1450 = 2500 - 1450 = $1050\).

Determine minimum price for positive output

A firm will produce a positive output as long as the price is higher than the average cost. The average cost is \(C/q = (200 + 2q^2)/q\). Setting this equal to \(p\) and solving for \(p\) gives: \(p = 200/q + 2q\). As \(q\) approaches infinity, the term \(200/q\) approaches 0, leaving us with \(p = 2q\). Thus, the minimum price for a positive output is when \(p\) equals marginal cost, which is $0 in this case.

Key concepts

Cost of Production

Understanding the cost of production is fundamental for any manager in a competitive market, as it affects pricing, output, and overall profitability. In economical terms, it includes both fixed costs, which do not change regardless of the output level, and variable costs, which are dependent on the quantity of output produced.

For a watchmaking firm in our scenario, the cost of production formula is given by C = 200 + 2q^2. Here, 200 represents a fixed cost, which could be rent or the cost of machinery that stays constant regardless of how many watches are made. The term 2q^2 indicates a variable cost that increases with the square of the number of watches produced (q). This can reflect the cost of materials and labor, which increases as more watches are crafted.

By understanding this formula, the manager can calculate the total cost of producing a certain number of watches and make informed decisions about the level of production that will maximize the firm's profit.

Marginal Cost

Marginal cost is another pivotal concept in finding the optimal output for a company. It is defined as the increase in total cost that arises from producing one additional unit of a product. For our watchmaking firm, marginal cost is represented by the derivative of the total cost function with respect to output q, which is 4q.

As the firm increases its production of watches, each additional watch costs 4q to make. This cost plays a critical role in decision-making. The point at which the marginal cost of production equals the price of the product is typically where profit is maximized because it indicates that producing one more watch would cost just as much as the revenue it generates.

Marginal Revenue

Marginal revenue is the increase in revenue that comes from selling one more unit of the product. In competitive markets, the marginal revenue is simply the price of the good, assuming the firm is a price-taker -- which means the firm cannot influence the market price and must accept it as given.

For the watchmaking firm, since watches sell for \(100 each, the marginal revenue of selling an additional watch is also \)100. This is because selling one more or one fewer watch at the market price does not affect the price of the watches. In order to maximize profit, the manager needs to equate this marginal revenue with the marginal cost.

Average Cost

Average cost (AC), sometimes referred to as unit cost, is total cost divided by the number of units produced. It represents the cost per unit of output. When comparing to the price, if the average cost is below the price, the firm can make a profit.

In our case, the average cost function for the watchmaking firm is AC = C/q = (200 + 2q^2)/q. Simplifying this gives you an average cost per watch which varies depending on output level. The relationship between average cost and marginal cost is also important; if the marginal cost is below the average cost, then producing more will decrease the average cost and vice versa.

Output Level

The output level is the quantity of goods that a firm produces. Determining the right output level is critical to maximizing profits. In the case of our watchmaking firm, the output level for profit maximization is found when the marginal cost equals the marginal revenue, leading us to an output level of 25 watches.

Increasing or decreasing production incurs different costs and revenues, so establishing the output that aligns with the market price while covering production costs is key. It requires balancing the desire for profit with the reality of how market prices and cost structures constrain operations.