Case #$2$

From Harshbarger & Reynolds

Operating Leverage and Business Risk

Once you have determined the break$-$even point for your product, you can use it to examine the effects of increasing or decreasing the role of fixed costs in your operating structure. The extent to which a business uses fixed costs $($compared to variable costs$)$ in its operations is referred to as operating leverage. The greater the use of operating leverage $($fixed costs, often associated with fixed assets$),$ the larger the increase in profit as sales rise, and the larger the increase in loss as sales fall $($Source: www$.$toolkit.cch$.$com$/$text$/$P$06\_7540.$asp$).$ The higher the break$-$even quantity for your product, the greater your business risk.

To see how operating leverage is related to business risk, complete the following.

$1.$ If x is the number of units of a product that is sold and the price is $p per unit, write the function that gives the revenue R for the product.

$2.($a$)$ If the fixed cost for production of this product is $$10,000$ and the variable cost is $$100$ per unit, find the function that gives the total cost C for the product.

$($b$)$ What type of function is this?

$3.$ Write an equation containing only the variables p and x that describes when the break$-$even point occurs.

$4.($a$)$ Solve the equation from Question $3$ for x$,$ so that x is written as a function of p$.$

$($b$)$ What type of function is this?

$($c$)$ What is the domain of this function?$($d$)$ What is the domain of this function in the context of this application?

$5.($a$)$ Graph the function from Question $4$ for the domain of the problem.

$($b$)$ Does the function increase or decrease as p increases?

$6.$ Suppose that the company has a monopoly for this product so that it can set the price as it chooses. What are the benefit and the danger to the company of pricing its product at $$1100$ per unit?

$7.$ If the company has a monopoly for this product so that it can set the price, what are the benefit and the danger to the company of pricing its product at $$101$ per unit?

$8.$ Suppose the company currently is labor$-$intensive, so that the monthly fixed cost is $$10,000$ and the variable cost is $$100$ per unit, but installing modern equipment will reduce the variable cost to $$50$ per unit while increasing the monthly fixed cost to $$30,000.$ Also suppose the selling price per unit is P $=$ $$200.$

$($a$)$ Which of these operations will have the highest operating leverage? Explain.

$($b$)$ Which of these operations will have the highest business risk? Explain.

$($c$)$ What is the relation between high operating leverage and risk? Discuss the benefit and danger of the second operating plan to justify your answer.