Question

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.

Short answer

The firm will produce 25 units. The detailed calculations for profit and producer surplus require computation by plugging in the values in the appropriate formulas as shown in the solution steps.

Step-by-step solution

Output Determination

Solve for output level (q) by setting the marginal cost function equal to the market price. \(MC(q) = P\), which gives us \(15 + 4q = 115\). This can be simplified to find q.

Solve for Output Level

To find the output level (q), subtract 15 from both sides, which gives \(4q = 100\). Then divide by 4, which gives us \(q = 25\) units.

Profit Calculation

We calculate profit by subtracting total cost from total revenue. The total cost \(C(q)\) at 25 units is \(450 + 15*25 + 2*25^2\). Total revenue is the price times quantity which is \(115*25\). Subtracting total cost from total revenue gives us the profit.

Calculate Producer Surplus

The producer surplus is given by the area between the price and marginal cost curve from 0 to the quantity produced. In this case, the producer surplus is the integral from 0 to 25 of \(P - MC(q)\) dq. We compute this integral and get the producer surplus.

Key concepts

Competitive Firm

A competitive firm is one that operates in a perfectly competitive market. This means it has no control over the price of the product it sells, as the price is determined by the overall market. Hence, the firm is a "price taker." This setting is characterized by several features:

• Many sellers and buyers are present, such that no single entity can influence the market price.
• Goods offered by various sellers are identical, so consumers have no preference between suppliers.
• Firms can enter and exit the market freely without any major restrictions.

For a competitive firm, profit maximization occurs where the marginal cost (MC) of production equals the market price (P). Thus, when calculating output levels, a competitive firm sets its MC equal to the price. In our example, the firm sets up the equation from the marginal cost function and solves it to find their production quantity, ensuring they are at their optimal production level where profits are maximized.

Total Cost Function

The total cost function, at its core, summarizes a firm's total cost of production at various levels of output. In economic terms, it includes both fixed and variable costs. In our exercise, the total cost function is presented as \(C(q) = 450 + 15q + 2q^2\). Here's what each component represents:

• The constant 450 stands for fixed costs—expenses that do not change regardless of output.
• The term \(15q\) represents variable costs that change in direct proportion to the output level \(q\).
• The term \(2q^2\) captures the increasing costs that vary non-linearly with output, highlighting diminishing returns at higher production levels.

When determining profit, the total costs are subtracted from total revenue. In this solution, after calculating the output level (\(q\)), the total cost is derived by substituting \(q\) back into the total cost equation and performing the arithmetic calculations.
Knowing how to analyze the total cost function helps students understand the cost structure faced by firms and aids in strategic decision-making regarding production levels.

Producer Surplus

Producer surplus is an essential concept in economics. It measures the difference between what producers are willing to accept for a good versus what they actually receive. Essentially, it's the area between the marginal cost curve and the price line in a supply and demand graph, up to the quantity produced. In our context, the calculation of producer surplus involves exploring the marginal cost curve's deviation from the set market price.

Mathematically, the producer surplus for our problem can be found by taking the definite integral of \(P - MC(q)\) from 0 to the quantity produced (\(q = 25\)). Here, \(P\) represents the market price, while \(MC(q)\) is the firm's marginal cost function. For the given scenario, this computation quantifies the extra benefit the firm gains by selling at a market price higher than the minimum they'd accept, represented by their marginal cost.
This economic measure is crucial, as it reflects the firm's efficiency and competitiveness in the market, providing insights into potential adjustments for maximizing profitability.