Exercises: Revenue and cost forecasting: the break-even point

Explain in your own words what the break-even point is, and what selling above or below it means.

The break-even formula is $q^* = \frac{CF}{p - CV_u}$. Identify what each term represents (CF, p, CV_u) and explain the meaning of the denominator, known as the unit contribution margin.

The figure represents total revenue (IT) and total costs (CT) of a business as a function of units sold, with fixed costs (CF) as a benchmark. Explain what the point marked with '?' represents and what occurs in the loss and profit regions.

A company has fixed costs of $\$120\,000$ per month, sells its product for $\$600$ per unit, and each unit costs $\$360$ to produce. a) Calculate the monthly break-even point using the formula $q^* = \frac{CF}{p - CV_u}$. b) Explain what this number means for the company.

The break-even point for a shop is 500 T-shirts per month. In June it sold 400 and in July 620. Indicate in which month it made a loss and in which a profit, reason why it is not necessary to know the exact amounts to know this, and explain what each T-shirt sold above 500 represents.

A bakery has a break-even point of 2 000 cakes per year. Analyse three levers to lower it — raising the price, reducing fixed costs, and reducing unit variable cost — explaining how each one acts on the formula and what risk using it entails.

A coffee shop is open 25 days a month with 12 four-seater tables and offers a single lunch shift. Its monthly fixed costs are $\$4,500$, the price per set menu is $\$12$, and the variable cost per menu is $\$7$. a) Calculate the break-even point in set menus per month and the average table occupancy rate needed to avoid incurring losses. b) If the demand forecast for the next month is $60\ \%$ of its maximum capacity, the business would incur losses. To prevent this, two alternatives are considered: 1) Offer a premium set menu at $\$15$ with a unit variable cost of $\$8.5$, maintaining fixed costs. 2) Reduce fixed costs by $20\ \%$ by renegotiating the rent, keeping the initial price and variable cost. Calculate the new break-even point for each alternative and justify which one allows the business to be profitable with the forecasted demand.

An entrepreneur plans to sell T-shirts with fixed costs of $\$5,000$ per year. He intends to set a selling price of $\$10$, even though his manufacturing variable cost is $\$11$ per T-shirt. In his business plan, he states: “Even though the unit margin is negative, if we reach a large enough sales volume, total revenues will exceed total costs and we will cover fixed costs.” a) Refute the entrepreneur's claim both quantitatively and conceptually: demonstrate using the profit function why it is mathematically impossible to reach the break-even point, and calculate the losses if he were to sell 5,000 T-shirts. b) To make the project viable, a mentor demands that the break-even point be set at a maximum of 1,000 T-shirts. If market constraints prevent the price from exceeding $\$14$, determine analytically: 1) The maximum allowable unit variable cost while maintaining fixed costs at $\$5,000$. 2) The percentage reduction in fixed costs required if the unit variable cost could only be lowered to $\$10.5$.

A food stall chain is hesitating between a city center location (fixed costs of $\$120\,000$ per month, price $\$180$, unit variable cost $\$80$, and 1.400 expected units) or a neighborhood location (fixed costs of $\$40\,000$, same price and variable cost, and 500 expected units). a) Calculate the break-even point and the expected profit of each option. b) Recommend one, also assessing the risk if actual sales fall 20% below expectations.