Production costs and profit

Explain the difference between fixed costs and variable costs and write down how to calculate a firm's total cost. Give two examples of each type of cost in a café.

Classify these bakery costs as fixed or variable and justify each decision: shop rent, flour, the administrator's salary (the same payroll every month), yeast, and the wage of an assistant hired by the hour according to demand.

Classify these gym costs as fixed or variable and justify each decision: the rent for the premises, the receptionist's fixed salary, the towels washed for each member who trains, the monthly fee for the management software and the drinks restocked in the vending machine.

The figure shows the average cost (AC) and marginal cost (MC) curves of a firm, with their characteristic shape. The point where they intersect is marked with a question mark.

1. a) What does the marked point represent for the firm? Think about what happens to average cost at that level of production.
2. b) Reason why the marginal cost curve intersects the average cost curve precisely at its minimum, using the grade analogy: if your next mark is lower than your average, the average goes down.

A churro stall spends $60\ \text{€}$ producing $100$ servings of churros.

1. a) Calculate the average cost per serving.
2. b) If preparing serving number $101$ costs it $0,40\ \text{€}$, explain what that figure is (marginal cost) and how it differs from average cost.

A T-shirt company has recorded its total costs: producing $0$ units costs $100\ \text{€}$ (rent is paid regardless); $1$ unit, $140\ \text{€}$; $2$ units, $170\ \text{€}$; $3$ units, $210\ \text{€}$; and $4$ units, $260\ \text{€}$.

1. a) What is the firm's fixed cost? And what is the variable cost of producing $3$ units?
2. b) Calculate the marginal cost of the second, third and fourth units.
3. c) Calculate the average cost of producing $4$ units and explain how the marginal cost is evolving and why.

A fruit shop takes in $3.000\ \text{€}$ a month from sales and spends $2.200\ \text{€}$ on stock, rent and wages.

1. a) Calculate its monthly profit and write down the formula you used.
2. b) Explain, with this example, the difference between revenue and profit.

Explain in your own words what a firm's break-even point is and why it is a figure every entrepreneur wants to know before opening. Use the example of a café that knows how many coffees it has to sell a day to avoid losing money.

An empanada bakery has fixed costs of $1.500\ \text{€}$ a month and a variable cost of $3\ \text{€}$ per empanada. It sells them for $8\ \text{€}$ each.

1. a) Calculate the total cost and the profit for a month in which it sells $400$ empanadas.
2. b) Calculate the break-even point, $q^{*} = \frac{CF}{p - CVMe}$, and interpret the result.
3. c) Reason what would happen to the break-even point if the rent rose by $300\ \text{€}$ a month.

A coach company has sold $30$ of the $40$ seats on a trip at $20\ \text{€}$ each. A group of $5$ students asks for tickets at $12\ \text{€}$; the additional cost of each passenger (extra fuel, insurance) is $6\ \text{€}$.

1. a) Reason whether it is worth accepting the group by comparing the additional revenue and cost.
2. b) Explain why the company can charge different prices for the same trip without losing money, as long as the price exceeds a certain threshold.

The commercial director of a backpack factory proposes accepting an extraordinary order of $200$ units for a conference at a special price of $22\ \text{€}$ per backpack. He presents the following argument to management: "Our average cost when manufacturing our usual output of $1.000$ backpacks is $18\ \text{€}$, and we sell them in shops at $25\ \text{€}$. Since they pay us $22\ \text{€}$, we will make a clean $4\ \text{€}$ per backpack above average cost, providing $800\ \text{€}$ of additional net profit." However, the plant operates at the limit of its standard capacity: to manufacture those $200$ extra backpacks, overtime night shifts will be required, which will increase total variable costs from $12.000\ \text{€}$ to $17.000\ \text{€}$, while fixed costs remain at $6.000\ \text{€}$.

1. a) Refute the commercial director's argument using precise calculations: determine the marginal cost per unit of the extraordinary order and calculate the real impact that accepting the offer would have on the firm's overall profit.
2. b) Explain the conceptual error made by the commercial director by making production decisions based on average cost instead of marginal cost.
3. c) Determine the minimum unit price at which the company could accept the order of the $200$ backpacks without reducing its prior profit, and state the marginalist rule for production optimisation.

A $20$-room rural hotel (maximum monthly capacity: $600$ room-nights) incurs fixed costs of $4.200\ \text{€}$ per month and a variable cost of $12\ \text{€}$ per occupied room each night (laundry, breakfast, utilities). Its standard rate is $50\ \text{€}$ per night. In a low-season month, confirmed bookings from private guests are only $100$ room-nights.

A sports federation proposes booking an additional $120$ room-nights that month for a tournament, but offering a discounted rate of $25\ \text{€}$ per night. The hotel's investment partner rejects the offer, arguing: "If we accept, our average cost per room sold will be over $31\ \text{€}$; selling at $25\ \text{€}$ would make us lose money on every room and worsen our accounts."

1. a) Refute the investment partner's analysis with data: calculate the hotel's monthly profit with and without the agreement with the federation, and show which cost measure he should have considered to make the correct decision.
2. b) Now assume that, to accommodate the entire federation ($120$ room-nights), the hotel had to cancel $30$ of the bookings from private guests who were paying the standard rate of $50\ \text{€}$. Determine whether in this new scenario it would be advisable to accept the deal, quantifying the opportunity cost.
3. c) Design two conditions or clauses that the hotel should include in the negotiation with the federation to prevent regular customers from finding out about the discounted rate or demanding the same price (price discrimination without damaging brand image).

The owner of a traditional bakery states: "If I invest in a continuous industrial kneading line, I will automatically reduce my unit costs and be able to sell much cheaper than the competition, securing the future of the business." Currently, he makes $1.000$ baguettes a day at an average cost of $0,80\ \text{€}$ (with daily fixed costs of $500\ \text{€}$ and a variable cost of $0,30\ \text{€}$ per baguette). The new industrial technology would raise daily fixed costs to $1.400\ \text{€}$, but it would reduce the variable cost to just $0,15\ \text{€}$ per baguette. However, a market study estimates that maximum demand in his delivery area will not exceed $2.000$ baguettes per day at a price of $0,70\ \text{€}$.

1. a) Numerically demonstrate whether the owner's statement is correct for his potential market: calculate the average cost per baguette with the new machinery if he produces and sells $2.000$ baguettes a day, and compare the daily economic profit of both options.
2. b) Calculate the exact daily production volume at which the industrial technology would become cheaper per unit than the traditional one (break-even point / point of indifference) and explain how this result relates to the minimum efficient scale.
3. c) Refute the owner's reasoning by analyzing two strategic risks or diseconomies of scale that threaten a small business that over-expands its technical capacity beyond the real size of its market.

A beach bar operates under a municipal concession. Its unavoidable annual fixed costs are $36.000\ \text{€}$ (municipal fees, insurance, depreciation, and basic maintenance), which amounts to $3.000\ \text{€}$ per month, whether it opens or stays closed. During the $7$ months of high season (April to October), the business generates total revenue of $140.000\ \text{€}$ with cumulative variable costs of $63.000\ \text{€}$.

For the $5$ months of low season (November to March), management is debating what to do. It has three options:

1. Close down completely: it generates neither revenue nor operating variable costs, but preparing and sealing the facilities costs $1.000\ \text{€}$ in total.

2. Open only on weekends: it estimates generating $3.200\ \text{€}$ per month with variable costs of $2.600\ \text{€}$ per month.

3. Lease operations to a third party in winter: an event organiser offers to pay a total fixed rent of $2.000\ \text{€}$ to use the premises for the $5$ months, with this operator assuming all variable and maintenance costs.

1. a) Calculate the consolidated economic result for the whole year under each of the three options and determine which one maximises the concessionaire's annual profit.
2. b) Technically justify the shutdown rule: why is it economically rational to maintain activity in winter under Option 2 even though monthly revenue ($3.200\ \text{€}$) is lower than total monthly cost ($5.600\ \text{€}$)?
3. c) Predict what would have to happen in the long run (when the municipal concession expires) for it to be worthwhile for the owner to renew the business, distinguishing the short-run decision logic from that of the long run.

Competition authorities are reviewing the acquisition of a logistics operator by a large e-commerce chain. Prior to the transaction, the company sold $10$ million items per year at an average cost of $14\ \text{€}$ ($6\ \text{€}$ of fixed costs and $8\ \text{€}$ of variable costs). With the acquisition, the company doubles its capacity to $20$ million items per year: its overall fixed costs rise to $100$ million euros per year, but its variable costs drop to $3\ \text{€}$ per item thanks to full automation.

Following the investment, the company sets its selling price at $7,50\ \text{€}$ per unit across its entire catalogue. Its competitors, whose average cost is no lower than $11\ \text{€}$, report that the large chain is engaging in predatory pricing to drive them out of the market and monopolise the sector.

1. a) Demonstrate quantitatively whether the firm enjoys economies of scale: calculate its unit average cost when operating at $100\ \%$ of its new capacity ($20$ million units) and when a drop in demand reduces its volume to $12$ million.
2. b) Evaluate the predatory pricing accusation: justify whether selling at $7,50\ \text{€}$ constitutes selling below cost (internal dumping) under both output levels and analyse what incentives the firm will have regarding prices once the competition has shut down.
3. c) Draft a reasoned opinion as a Competition Commission analyst: should the transaction be approved unconditionally, prohibited, or made subject to specific remedies (for example, granting competitors access to the logistics network)? Argue your position by weighing productive efficiency gains against monopolisation risks.

A financial analyst with a net salary of $36.000\ \text{€}$ per year decides to quit her job to found her own sustainability consultancy. To start the project, she contributes $40.000\ \text{€}$ from her personal savings, which until then had been generating an annual return of $5\ \%$ in guaranteed investment funds. In addition, she uses commercial premises she owns as the company headquarters, which she previously rented out to an academy for $900\ \text{€}$ per month.

At the close of her first year of activity, the company's accounts show the following data:

• Total revenue from consulting services provided: $105.000\ \text{€}$.

• Explicit deductible expenses (utilities, software licenses, self-employment social security contributions, marketing, and an administrative assistant's salary): $58.000\ \text{€}$.

A relative tells her: «The business has been an extraordinary success: you have made a clean $47.000\ \text{€}$ in your first year, much more than you were earning in your old job».

1. a) Refute the relative's statement by calculating the accounting profit and the economic profit of the first year separately, accurately identifying and quantifying each of the implicit or opportunity costs.
2. b) Interpret the meaning of the economic profit obtained: determine whether the entrepreneur has created or destroyed economic value compared to her starting situation, and explain what rational decision she should make in the medium term if the figures remain unchanged.
3. c) Design a feasibility analysis: calculate the exact annual turnover the consultancy would need to reach to cover all its opportunity costs (zero economic profit or normal profit) and explain why in economics a zero economic profit is considered a satisfactory equilibrium situation.