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Chapter 1: Problem 30

Suppose the cost function associated with a product is \(C(x)=c x+F\) dollars and the revenue function is \(R(x)=\) \(s x\), where \(c\) denotes the unit cost of production, \(s\) the unit selling price, \(F\) the fixed cost incurred by the firm, and \(x\) the level of production and sales. Find the break-even quantity and the break-even revenue in terms of the constants \(c, s\), and \(F\), and interpret your results in economic terms.

Short Answer

Expert verified
The break-even quantity is \(x = \frac{F}{s - c}\) and the break-even revenue is \(R(x) = s(\frac{F}{s - c})\). This means that the company must produce and sell \(\frac{F}{s - c}\) units of the product to cover all costs without making a profit or incurring a loss. The break-even point is essential for businesses to determine the appropriate production levels to cover costs and achieve profits.

Step by step solution

01

Write down the given functions

The cost function and the revenue function given are: C(x) = cx + F R(x) = sx
02

Break-even quantity

To find the break-even quantity x, we need to solve the equation C(x) = R(x): cx + F = sx
03

Solve for x

Rearrange the equation and solve for x: \(x(s - c) = F\) Divide both sides by (s - c): \(x = \frac{F}{s - c}\) The break-even quantity is \(x = \frac{F}{s - c}\).
04

Break-even revenue

To find the break-even revenue, plug the break-even quantity x back into the revenue function R(x): \(R(x) = s(\frac{F}{s - c})\)
05

Interpret results in economic terms

The break-even quantity \(\frac{F}{s - c}\) represents the number of units that must be produced and sold to cover all the costs (variable and fixed) associated with the production. The break-even revenue \(s(\frac{F}{s - c})\) represents the total revenue earned at this break-even point, which is enough to cover all the costs without making a profit or incurring a loss. In this scenario, increasing the production and sales beyond the break-even quantity will result in making a profit, whereas producing and selling fewer units than the break-even quantity will result in a loss. The break-even point is an important indicator of the financial viability of a product and helps businesses to determine the appropriate production levels to cover costs and achieve profits.

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Key Concepts

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

Cost Function
The cost function is a mathematical representation that helps businesses understand how their costs change with different levels of production. In this problem, the cost function is given by the formula:\[ C(x) = cx + F \] This formula combines both variable and fixed costs associated with producing goods.
  • Variable Costs: Represented by the expression \( cx \), where \( c \) is the unit cost of production and \( x \) is the number of units produced. This part of the function changes as the level of production changes.
  • Fixed Costs: Represented by \( F \), a constant that does not change regardless of the production level. This is essential because fixed costs, like rent or salaries, remain constant even if no units are produced.
Thus, to manage production costs effectively, businesses need to not only control their variable costs but also cover their fixed costs through revenue.
Revenue Function
The revenue function describes how much money a company makes from selling its goods. Here, it's given by the formula \( R(x) = sx \), where:
  • \( s \) is the unit selling price, indicating the amount earned for each unit sold.
  • \( x \) is the number of units sold, which aligns with the production level in this context.
Revenue increases linearly with sales as long as the selling price per unit remains constant. In practical terms, understanding the revenue function helps businesses forecast earnings based on expected sales volumes and set sales targets to achieve financial goals. This straightforward formula also aids businesses in planning and making strategic decisions to maximize profits.
Fixed Costs
Fixed costs represent those expenses that do not fluctuate with the amount of goods produced or sold. In the given cost function \( C(x) = cx + F \), the \( F \) stands for these fixed costs. Examples of fixed costs include:
  • Rent or lease payments for facilities.
  • Salaries for permanent staff.
  • Insurance fees and property taxes.
Fixed costs are vital for businesses as they must be covered regardless of production or sales volume. Understanding fixed costs is crucial for a company to establish the baseline of revenue required to avoid losses, especially when determining the break-even point. This consideration ensures that the business remains viable even during lower sales periods.
Unit Cost of Production
The unit cost of production refers to the cost incurred by producing a single unit of a product. In the cost function \( C(x) = cx + F \), the \( c \) is the unit cost of production. This cost typically includes:
  • Direct materials needed for production.
  • Direct labor used in creating the product.
  • Variable overhead costs associated with production.
Managing the unit cost of production effectively allows a business to price its products competitively while ensuring profitability. When the unit cost is kept low, companies can either increase their profit margins or offer more competitive prices to attract customers. Tracking the unit cost helps businesses remain efficient and responsive to market changes and cost variations.

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Most popular questions from this chapter

Let \(L_{1}\) and \(L_{2}\) be two nonvertical straight lines in the plane with equations \(y=m_{1} x+b_{1}\) and \(y=m_{2} x+b_{2}\), respectively. Find conditions on \(m_{1}, m_{2}, b_{1}\), and \(b_{2}\) such that (a) \(L_{1}\) and \(L_{2}\) do not intersect, (b) \(L_{1}\) and \(L_{2}\) intersect at one and only one point, and (c) \(L_{1}\) and \(L_{2}\) intersect at infinitely many points.

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Find an equation of the line that satisfies the given condition. The line passing through \((-5,-4)\) and parallel to the line passing through \((-3,2)\) and \((6,8)\)

Moody's Corporation is the holding company for Moody's Investors Service, which has a \(40 \%\) share in the world credit-rating market. According to Company Reports, the total revenue (in billions of dollars) of the company is projected to be as follows \((x=0\) correspond to 2004\()\) : $$ \begin{array}{lccccc} \hline \text { Year } & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline \text { Revenue, } \boldsymbol{y} & 1.42 & 1.73 & 1.98 & 2.32 & 2.65 \\\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the results of part (a) to estimate the rate of change of the revenue of the company for the period in question. c. Use the result of part (a) to estimate the total revenue of the company in 2010 , assuming that the trend continues.

Find an equation of the line that satisfies the given condition. Given that the point \(P(2,-3)\) lies on the line \(-2 x+k y+\) \(10=0\), find \(k\)
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