Question

Frances sells pencils in the perfectly competitive pencil market. Her output per day and her total cost are shown in the following table: $$ \begin{array}{|c|c|} \hline \text { Output per Day } & \text { Total Cost } \\ \hline 0 & \$ 1.00 \\ \hline 1 & 2.50 \\ \hline 2 & 3.50 \\ \hline 3 & 4.20 \\ \hline 4 & 4.50 \\ \hline 5 & 5.20 \\ \hline 6 & 6.80 \\ \hline 7 & 8.70 \\ \hline 8 & 10.70 \\ \hline 9 & 13.00 \\ \hline \end{array} $$ a. If the current equilibrium price in the pencil market is \(\$ 1.80,\) how many pencils will Frances produce, what price will she charge, and how much profit (or loss) will she make? Draw a graph to illustrate your answer. Your graph should be clearly labeled and should include Frances's demand, \(A T C, A V C, M C,\) and \(M R\) curves; the price she is charging; the quantity she is producing; and the area representing her profit (or loss). b. Suppose the equilibrium price of pencils falls to \(\$ 1.00\). Now how many pencils will Frances produce, what price will she charge, and how much profit (or loss) will she make? Show your work. Draw a graph to illustrate this situation, using the instructions in part (a). c. Suppose the equilibrium price of pencils falls to \(\$ 0.25 .\) Now how many pencils will Frances produce, what price will she charge, and how much profit (or loss) will she make?

Short answer

a) When the price is $1.80, Frances must produce 4 pencils and will suffer a loss of $1.80. b) When the price falls to $1.00, she must produce 2 pencils and will incur a loss of $1.50. c) If the pencil price drops to $0.25, she should not produce any pencils, resulting in a loss of $1.00, which is her fixed cost.

Step-by-step solution

Calculate the ATC, AVC, and MC

First, calculate the Average Total Cost (ATC) by dividing the total cost by the quantity of pencils produced. Similarly, calculate the Average Variable Cost (AVC) and Marginal Cost (MC) by subtracting the cost of producing one less unit from the cost of producing the current quantity of units.

Find the best quantity to produce when price is $1.80

Considering the equilibrium price of $1.80, analyze the cost table and produce up to the quantity where the MC (Marginal Cost) does not exceed $1.80. The MC of producing the fifth pencil is higher than $1.80, hence Frances should produce and sell 4 pencils.

Calculate the profit/loss for price $1.80

Subtract the total cost of producing 4 pencils from the total revenue at a selling price of $1.80 each. So, the profit = (4 x $1.80) - $4.50 = $2.70 - $4.50 = -$1.80, indicating a loss of $1.80.

Determine quantity for price $1.00

Following the same process for the new equilibrium price of $1.00, Frances should produce up to the quantity where MC does not exceed $1.00. So, she should produce and sell 2 pencils.

Calculate the profit/loss for price $1.00

Again, subtract the total cost of producing 2 pencils from the total revenue at a selling price of $1.00 each. So, the profit = (2 x $1.00) - $3.50 = $2.00 - $3.50 = -$1.50, showing a loss of $1.50.

Determine quantity for price $0.25

For the new equilibrium price of $0.25, it seems that producing any pencils will result in a loss (since even the first pencil's MC is more than $0.25). Hence the quantity produced will be 0.

Calculate the profit/loss for the price $0.25

The revenue will be 0, and the loss equals to the fixed cost, which is $1.00.

Key concepts

Average Total Cost

To understand how Frances manages her pencil business in a perfectly competitive market, we first need to look at the Average Total Cost (ATC). ATC is calculated by dividing the total cost of production by the number of units produced. This gives us an idea of the cost per unit at different levels of output. In simpler terms, it shows how much each pencil costs to make on average.

In the provided table, if we consider producing 3 pencils, the total cost is $4.20. Hence, the ATC will be calculated as: \[ ATC = \frac{4.20}{3} = 1.40 \] Understanding ATC helps Frances make decisions about pricing and production by comparing it to market price. If her pencils sell for more than the ATC, she makes a profit. If less, she incurs a loss.

Marginal Cost

Marginal Cost (MC) is a key factor in determining how many pencils Frances should produce. It is the cost of producing an additional unit of output. Understanding MC helps in deciding where to stop production to maximize profit and minimize loss.

Frances looks at the difference in total cost when producing one more pencil. For example, producing one pencil costs \(2.50, and producing two costs \)3.50. Thus, MC of the second pencil is: \[ MC = 3.50 - 2.50 = 1.00 \] Frances should continue producing pencils as long as the MC is less than or equal to the market price. This ensures she is not selling a pencil for less than it costs to make.

Profit Maximization

Profit maximization is crucial for Frances while operating in a perfectly competitive market. She aims to produce the quantity where her marginal cost is just below or equal to the market price, maximizing her potential earnings.

By calculating the revenue from selling pencils and subtracting the total cost, Frances can determine profit or loss. For instance, at a price of $1.80 per pencil, producing 4 pencils leads her to a loss since total cost exceeds total revenue \[ Profit = (4 \times 1.80) - 4.50 = 7.20 - 4.50 = -1.80\]Thus, finding that optimal point where MC equals the market price ensures she does not make and sell excess units that would lead to unnecessary costs.

Equilibrium Price

Equilibrium price is where the quantity demanded by consumers matches the quantity supplied by producers. In perfect competition, like Frances's situation, this price also dictates how many units she should produce to stay in the market.

At each given price, Frances evaluates if it covers her costs. If the equilibrium price is lower than her total cost, she incurs a loss on each pencil sold. For example, with the equilibrium price at \(1.00, she produces where MC equals or is just below \)1.00, but still incurs a loss:\[ Profit = (2 \times 1.00) - 3.50 = 2.00 - 3.50 = -1.50 \]Equilibrium price offers clarity on production adjustments needed to achieve better efficiency and minimize losses in a competitive market.