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Chapter 8: Problem 7

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?

Short Answer

Expert verified
The variable cost is \(4q^2\), the fixed cost is 16, the average cost is \(4q + 16/q\), the average variable cost is \(4q\) and the average fixed cost is \(16/q\). The output that minimizes average cost is 2 units. The firm will produce positive output when the price is equal or higher than 8, have negative profits when the price is less than 12, and have positive profits when the price is greater than 12.

Step by step solution

01

Define the cost function

The cost function of the firm is given as \(C(q) = 4q^2 + 16\) where \(q\) represents the quantity of goods produced.
02

Find the variable cost and fixed cost

The variable cost is the cost that changes with the quantity produced. As seen in the cost function, the variable cost would be \(4q^2\). On the other hand, the fixed cost is cost which does not change with the quantity produced. In the cost function, the fixed cost would be 16.
03

Calculate mean costs

The Average Cost (AC) is computed as total cost divided by the quantity produced i.e. \(AC = C(q) / q = (4q^2 + 16) / q = 4q + 16/q\). The Average Variable Cost (AVC) is the variable cost divided by the quantity i.e. \(AVC = VC / q = 4q\). The Average Fixed Cost (AFC) is the fixed cost divided by the quantity i.e. \(AFC = FC / q = 16 / q\).
04

Show the cost curves on a graph

Plot average cost (AC), marginal cost (MC), and average variable cost (AVC) on the graph with quantity on the x-axis and cost on the y-axis.
05

Find minimum average cost output

To find the output level that minimizes AC we can take the derivative of AC with respect to q and set equal to zero. The derivative of \(4q + (16/q)\) is \(4 - (16/q^2)\). Setting this equal to zero we find \(q = 2\). So the output that minimizes average cost is 2 units.
06

Find the range of prices for positive output

The firm will produce positive output when the price is greater than or equal to the minimum point of the average variable cost. So, \(P >= 4q\), when \(q > 2\). Therefore, \(P >= 8\).
07

Find the range of prices for negative profit

The firm will face a negative profit when the price is less than the average cost. So, \(P < 4q + 16/q\), when \(q > 2\). Therefore, \(P < 8 + 8/2 = 12\).
08

Find the range of prices for positive profit

The firm will face a positive profit when the price is greater than average cost. So, \(P > 4q + 16/q\), when \(q = 2\). Therefore, \(P > 8 + 8/2 = 12\).

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Most popular questions from this chapter

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} \\] \(C(q)=722+\frac{q^{2}}{200} \quad\) 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. 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.

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