The production function: efficiency and productivity

Explain what a firm's production function is and what the difference is between the short run and the long run. Use the example of a pizzeria with a single oven: which factor is fixed in the short run and which is variable?

A self-service laundromat has rented premises with $10$ washing machines, and uses detergent, water, electricity and the hours of an employee who cleans and serves customers.

1. a) Classify its factors as fixed and variable in the short run.
2. b) Explain which factors it could change in the long run and what that means for its production function.

A pizzeria has a single oven and hires cooks: with $1$ cook it bakes $10$ pizzas per hour; with $2$, $18$; with $3$, $24$; and with $4$, $28$. The figure represents the marginal product of each cook, that is, what each cook adds to production, but one of the bars is missing.

1. a) Calculate the marginal product of the third cook and draw the missing bar.
2. b) State the law of diminishing returns and explain, thinking about the oven, why it applies to this pizzeria.

You can do it on paper and send a photo with your answer.

A plant nursery employs $3$ workers who plant $90$ pots a day.

1. a) Calculate the average product per worker.
2. b) Explain in your own words the difference between total product, average product, and marginal product.

A food truck has a single griddle. With $1$ cook it prepares $40$ burgers per hour; with $2$, $90$; with $3$, $130$; and with $4$, $160$.

1. a) Calculate the marginal product of the second, third, and fourth cook.
2. b) State from which cook diminishing returns appear.
3. c) Explain, thinking about the griddle, why this happens.

A farmer manages a tomato farm with a fixed surface area and states: "Given that with $5$ farmhands we harvest $400\ \text{kg}$ per day ($80\ \text{kg}$ per farmhand), each worker generates an average of $\$160$, well exceeding their daily cost of $\$150$. Therefore, all $5$ are profitable and it would be advisable to hire a $6.^{\text{th}}$ if they contribute at least $76\ \text{kg}$." Daily production according to the farmhands hired is: $1$ farmhand: $100\ \text{kg}$; $2$: $190\ \text{kg}$; $3$: $270\ \text{kg}$; $4$: $340\ \text{kg}$; and $5$: $400\ \text{kg}$. Each kilogram of tomatoes sells on the market for $\$2$.

1. a) Prove by calculating the marginal product and the marginal value contributed by each worker why the farmer's reasoning is economically flawed, and determine the daily workforce that maximizes profit.
2. b) Explain what economic principle describes the behavior of marginal product and why this phenomenon inevitably occurs even if all farmhands possess the exact same skills and work with identical effort.
3. c) If a drought reduces the price of tomatoes to $1.70\ \text{\$/kg}$ and the farmhands agree on a wage increase up to $\$160$ per day, predict with reasoning the new optimal hiring level and calculate how the daily profit derived from labor changes compared to the initial situation.

An artisan pottery workshop has $3$ potter's wheels (fixed factor) and faces a daily fixed cost of $\$150$ for rent and facilities. Daily output according to the workers employed is: $1$ worker: $25$ pieces; $2$: $48$; $3$: $68$; $4$: $84$; $5$: $93$; and $6$: $98$. Each worker earns $\$120$ per day and each ceramic piece is sold for $\$18$.

1. a) The manager argues: "We should hire the 6th worker, since producing $98$ pieces per day generates total revenue ($\$1,764$) that comfortably exceeds total costs ($\$150 + 6 \times \$120 = \$870$), thus maximizing business volume." Refute their position using marginal analysis and calculate the number of workers that maximizes daily net profit, indicating its amount.
2. b) Determine beyond which number of workers diminishing marginal returns set in, and reason what change in the endowment of fixed factors would delay the saturation of installed capacity.
3. c) If the entry of new competitors pushes the selling price down to $\$10$ per piece, determine how many workers it would be advisable to hire and justify whether it is in the workshop's interest to continue operating in the short run or if it should shut down temporarily.

Explain the difference between technical efficiency and economic efficiency using this example: to pack $1,000$ jars of honey, technique A uses $3$ workers and $2$ machines, and technique B uses $2$ workers and $3$ machines. Can you tell whether either is technically inefficient? What piece of information is missing to know which is economically efficient?

To produce $100$ T-shirts per day, a workshop can use technique A ($4$ workers and $2$ machines) or technique B ($6$ workers and $1$ machine). Each worker costs $\$1,200$ per month and each machine costs $\$1,800$ per month.

1. a) Reason whether both techniques are technically efficient (does either one use more of all factors to produce the same output?).
2. b) Calculate the monthly cost of each technique and decide which one is economically efficient.

To produce $1,000$ yogurts a day, a dairy has three techniques: A ($5$ workers and $3$ machines), B ($4$ workers and $4$ machines) and C ($6$ workers and $3$ machines). Each worker costs $\$1,500$ a month and each machine, $\$900$.

1. a) Reason whether any technique is technically inefficient by comparing it with the others.
2. b) Calculate the monthly cost of the efficient techniques and decide which is economically efficient.
3. c) Explain what would happen to your decision if the cost of the machines rose sharply.

A hair salon employs $5$ hairdressers who serve $60$ haircuts a day.

1. a) Calculate labor productivity (haircuts per hairdresser per day).
2. b) Propose two ways of increasing that productivity without hiring more people.

A carpentry workshop produces $400$ chairs per month with $8$ workers. After training the workforce and reorganizing the workshop, it produces $450$ chairs per month with the same $8$ workers.

1. a) Calculate labor productivity (chairs per worker) before and after the change.
2. b) Calculate the percentage change in productivity and explain which two factors improved it.

A juice company employs $10$ workers who cost $\$15,000$ per month in total, and spends another $\$5,000$ per month on fruit, packaging and energy. It produces $2,000$ liters of juice which it sells for $\$12$ per liter.

1. a) Calculate the value of monthly production and the total cost of input factors.
2. b) Calculate the total factor productivity, $\text{PG} = \frac{\text{value of production}}{\text{cost of factors}}$, and interpret the result: how many dollars of value does it generate for every dollars spent?

A language academy has $6$ teachers who serve $480$ students a year. The following year, with $7$ teachers, it serves $630$ students.

1. a) Calculate labor productivity (students per teacher) for each year.
2. b) Calculate the percentage change in productivity.
3. c) Comment on whether the academy has improved and what other data you would ask for to judge it (quality of teaching, prices...).

A print shop produces $8,000$ posters a day, which it sells at $\$0.5$ each. Each day it pays $\$480$ in wages, $\$200$ in machine depreciation and $\$0.15$ per poster in paper and ink.

1. a) Calculate the value of daily production and the total cost of the factors.
2. b) Calculate the total factor productivity, $\text{PG} = \frac{\text{value of production}}{\text{cost of factors}}$, and interpret the result.

Company X produces $2,100$ parts a day with $10$ workers working $7$ hours; company Y produces $1,800$ parts with $12$ workers working $6$ hours.

1. a) Calculate productivity per hour worked for each company.
2. b) Explain why it is better to measure productivity per hour and not just per worker when workdays are different.

Explain what the division of labor is and what specialization consists of. Give the example of a sandwich assembly line (one person cuts the bread, another adds the ingredient, another wraps it) and point out two advantages and one disadvantage of organizing production in this way.

A neighborhood bakery buys a new kneading machine and, with the same workers and the same flour, kneads twice as many kilograms of bread a day.

1. a) Explain what has changed in its production function.
2. b) Define what is meant by technological progress and give another example from an industry you know.

A canning factory currently produces $10,000$ cans per day using $20$ workers on $8$-hour daily shifts. Management is considering an investment of $\$180,000$ in an automated line with an estimated useful life of $5$ years ($250$ working days per year). With this technology, production would increase to $16,000$ cans per day employing only $8$ workers. Each worker's wage is $12\ \text{\$/hour}$ and the daily maintenance and utility costs of the new machinery would amount to $\$160$ per day.

1. a) Calculate the productivity of the labor factor (in cans per person-hour) before and after the innovation, and explain how the production function shifts on a graph plotting output against the labor factor.
2. b) Evaluate whether the investment is economically efficient by calculating and comparing the direct cost per can (labor cost plus machinery cost per day) in both scenarios.
3. c) A labor union states: "Automation destroys net employment and harms general welfare." Use the impact of R&D&I to technically qualify this statement, explaining the medium- and long-term compensating mechanisms on overall productivity and employment in other economic sectors.

A workshop assembles bicycles with $6$ multi-skilled workers working individually; each one assembles a complete bicycle and finishes $4$ units in an $8$-hour shift. Management proposes reorganizing the workshop into an assembly line with $4$ sequential stages whose processing times per bicycle are:

• Stage 1 (frame and fork): $15$ minutes.

• Stage 2 (wheels and drivetrain): $35$ minutes.

• Stage 3 (brakes and cabling): $20$ minutes.

• Stage 4 (adjustment and quality control): $10$ minutes.

1. a) Identify the bottleneck of the proposed system. Distribute the $6$ workers among the $4$ stages to balance the line as best as possible, determine what the resulting maximum daily output of the workshop will be, and compare the average productivity per hour of work with that of the individual model.
2. b) Analyze two operational or organizational risks introduced by this specialization (such as lack of flexibility or job monotony) and design a work organization measure to mitigate them without reducing productivity.
3. c) Apply the underlying principle of the division of labor to international trade: explain why two countries with different opportunity costs in manufacturing components benefit mutually by specializing and trading, rather than attempting to manufacture complete bicycles under autarky.

A clothing company needs to make $10,000$ garments per month and is analyzing three available technologies:

• Technology A: $10$ workers and $6$ machines.

• Technology B: $25$ workers and $3$ machines.

• Technology C: $30$ workers and $4$ machines.

The rental cost of each machine is $\$3,000$ per month. The company can locate in the home country (where the monthly wage is $\$1,800$ per worker) or offshore to Country B (where the monthly wage is $\$500$ per worker). However, manufacturing in Country B adds transport and tariff costs of $\$1.20$ per garment, in addition to a carbon footprint levy of $\$2,000$ per month set by environmental regulations.

1. a) Analyze the technical efficiency of the three technologies and rule out with justification any that are technically inefficient.
2. b) Determine which technology is economically efficient and in which location it should be implemented, calculating the average total cost per garment for each viable option.
3. c) Using sensitivity analysis, calculate below what wage level in the home country offshoring would cease to be profitable. Point out a social or external cost that does not appear in the company's accounts but should be considered by society.

A supermarket serves $3,000$ customers per day over a $12$-hour operating day using $10$ traditional checkouts staffed by $10$ cashiers (at $12\ \text{\$/hour}$ each). The company decides to replace $6$ conventional checkouts with an area of $6$ self-checkout stations monitored by $1$ single supervisory employee. Each self-checkout terminal entails an equivalent daily cost (rental, maintenance, and depreciation) of $\$25$. To prevent delays and ensure customers do not experience undue lines, the supermarket adds $4$ daily hours of support staff for restocking and assistance (at $12\ \text{\$/hour}$).

1. a) Calculate the productivity of the labor factor (customers served per hour of checkout and assistance work) under the original model and under the new hybrid model.
2. b) Evaluate economic efficiency by calculating the total cost per customer served before and after the change, and quantify the daily net savings achieved by the supermarket.
3. c) Management commits to distributing the daily net productivity gain as follows: $40\ \%$ for business margin, $35\ \%$ to lower prices for customers, and $25\ \%$ for wage incentives for the remaining staff. Calculate the average price reduction per customer served (in dollars cents) and the new hourly wage of the store workers.

Two business models coexist in the bakery sector:

• Artisan Bakery: Employs $2$ master bakers earning $10\ \text{\$/hour}$ each in $8$-hour shifts. With a traditional oven (daily depreciation and utilities cost of $\$30$), they bake $320$ sourdough loaves per day, sold at $\$1.50$ each. The daily raw materials cost is $\$80$.

• Industrial Bakery: Employs $1$ line technician earning $15\ \text{\$/hour}$ in an $8$-hour shift. It has an automated line (daily depreciation, maintenance, and energy cost of $\$220$). It produces $1,200$ conventional loaves per day, sold at $\$0.60$ each. The daily raw materials cost is $\$240$.

1. a) Calculate for each model the physical labor productivity (loaves/man-hour) and productivity in terms of value added generated per hour of work (sales revenue minus raw material and depreciation costs, divided by total labor hours).
2. b) Even though the industrial bakery has more than seven times the physical productivity of the artisan one, analyze why the gap in value added per hour narrows dramatically. Explain what role capital intensity and product differentiation play in this outcome.
3. c) Translate these conclusions to the macroeconomic sphere: reason why economies with a higher endowment of capital and technology per worker can systematically pay higher real wages without losing competitiveness, and predict the consequences of raising wages by law in a sector without a parallel increase in its value added per employee.