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Chapter 8: Problem 4

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

Expert verified
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

01

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.
02

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.
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Suppose the same firm's cost function is \(C(q)=4 q^{2}+16\) a. Find variable cost, fixed cost, average cost, average variable cost, and average fixed cost. (Hint: Marginal cost is given by \(\mathrm{MC}=8 q\).) b. Show the average cost, marginal cost, and average variable cost curves on a graph. c. Find the output that minimizes average cost. d. At what range of prices will the firm produce a positive output? e. At what range of prices will the firm earn a negative profit? f. At what range of prices will the firm earn a positive profit?

Suppose that a competitive firm has a total cost function \(C(q)=450+15 q+2 q^{2}\) and a marginal cost function \(M C(q)=15+4 q .\) If the market price is \(P=\$ 115\) per unit, find the level of output produced by the firm. Find the level of profit and the level of producer surplus.

Suppose you are given the following information about a particular industry: \\[ \begin{array}{ll} Q^{D}=6500-100 P & \text { Market demand } \\ Q^{S}=1200 P & \text { Market supply } \end{array} \\] \\[ C(q)=722+\frac{q^{2}}{200} \quad \text { Firm total cost function } \\] \\[ M C(q)=\frac{2 q}{200} \quad \text { Firm marginal cost function } \\] Assume that all firms are identical and that the market is characterized by perfect competition. a. Find the equilibrium price, the equilibrium quantity, the output supplied by the firm, and the profit of each firm. b. Would you expect to see entry into or exit from the industry in the long run? Explain. What effect will entry or exit have on market equilibrium? c. What is the lowest price at which each firm would sell its output in the long run? Is profit positive, negative, or zero at this price? Explain. d. What is the lowest price at which each firm would sell its output in the short run? Is profit positive, negative, or zero at this price? Explain.

Suppose that a competitive firm's marginal cost of producing output \(q\) is given by \(\mathrm{MC}(q)=3+2 q\). Assume that the market price of the firm's product is \(\$ 9\) a. What level of output will the firm produce? b. What is the firm's producer surplus? c. Suppose that the average variable cost of the firm is given by \(\mathrm{AVC}(q)=3+q .\) Suppose that the firm's fixed costs are known to be \(\$ 3 .\) Will the firm be earning a positive, negative, or zero profit in the short run?

A firm produces a product in a competitive industry and has a total cost function \(C=50+4 q+2 q^{2}\) and a marginal cost function \(\mathrm{MC}=4+4 q\). At the given market price of \(\$ 20,\) the firm is producing 5 units of output. Is the firm maximizing its profit? What quantity of output should the firm produce in the long run?
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