Question

Sara is a dot.com entrepreneur who has established a Web site at which people can design and buy sweatshirts. Sara pays \(\$ 1,000\) a week for her Web server and Internet connection. The sweatshirts that her customers design are made to order by another firm, and Sara pays this firm \(\$ 20\) a sweatshirt. Sara has no other costs. The table sets out the demand schedule for Sara's sweatshirts. $$\begin{array}{cc} \begin{array}{c} \text { Price } \\ \text { [dollars per sweatshirt/ } \end{array} & \begin{array}{c} \text { Quantity demanded } \\ \text { (sweatshirts per week) } \end{array} \\ \hline 0 & 100 \\ 20 & 80 \\ 40 & 60 \\ 60 & 40 \\ 80 & 20 \\ 100 & 0 \end{array}$$ Calculate Sara's profit-maximizing output, price, and economic profit.

Short answer

Profit-maximizing output is 40 units at \(60\) dollars each, with an economic profit of \(600\) dollars.

Step-by-step solution

Understand the Cost and Demand Schedule

Familiarize with Sara's fixed and variable costs. Fixed cost: \(\$ 1000\) per week, variable cost: \(\$ 20\) per sweatshirt. Review the demand schedule provided.

Total Revenue Calculation

Calculate total revenue (TR) for each price-quantity pair using the formula: \[ TR = \text{Price} \times \text{Quantity Demanded} \ For example, if the price is \(20\) and quantity demanded is \(80\), then \ TR = 20 \times 80 = 1600. \] Calculate TR for all price points.

Total Cost Calculation

Calculate total cost (TC) using the formula: \[ TC = \text{Fixed Cost} + (\text{Variable Cost} \times \text{Quantity Demanded}) \ For example, if quantity demanded is \(80\), the total cost: \ TC = 1000 + (20 \times 80) = 2600. \] Calculate TC for all quantities.

Calculate Profit

Calculate profit using the formula: \[ \text{Profit} = \text{Total Revenue} - \text{Total Cost} \ For example, when TR is \(1600\) and TC is \(2600\): \ \text{Profit} = 1600 - 2600 = -1000 \] Compute the profit for all quantities.

Determine the Profit-Maximizing Output and Price

Review calculated profits for all price-quantity pairs to determine which one yields the highest profit.

Key concepts

fixed costs

Fixed costs are expenses that do not change with the level of output production. They remain constant regardless of how many units a company produces. For Sara, her fixed costs amount to \(\$ 1000\) per week, which she pays for her Web server and Internet connection. These costs are incurred even if she does not sell any sweatshirts.

variable costs

Variable costs change with the level of output produced. They vary depending on how many items are produced or services provided. For Sara, the variable cost is \(\$ 20\) per sweatshirt, which she pays to the firm that makes the sweatshirts. This means that if Sara sells more sweatshirts, her variable costs increase proportionally.

total revenue

Total revenue (TR) is the total amount of money a firm receives from selling its goods or services. It can be calculated using the formula: \[ TR = \text{Price} \times \text{Quantity Demanded} \] For instance, if Sara sets the price at \(\$ 20\) and sells \(80\) sweatshirts, her total revenue will be \(20 \times 80 = 1600\). Total revenue helps in understanding how well the business is performing in terms of sales.

total cost

Total cost (TC) is the sum of all costs incurred in producing a certain level of output. This includes both fixed and variable costs. The formula to calculate total cost is: \[ TC = \text{Fixed Cost} + (\text{Variable Cost} \times \text{Quantity Demanded}) \] For example, if Sara sells \(80\) sweatshirts, the total cost would be \(1000 + (20 \times 80) = 2600\). Understanding total cost is crucial as it helps in determining profitability.

demand schedule

A demand schedule is a table that shows the quantity of a good that consumers are willing to purchase at various prices. It helps in understanding how price changes affect the quantity demanded. In Sara's case, the demand schedule shows the relationship between different prices for her sweatshirts and the corresponding quantities demanded per week. For instance, at a price of \(\$ 20\), the quantity demanded is \(80\) sweatshirts, while at \(\$ 100\), no sweatshirts are demanded. This information is essential for making pricing and production decisions.