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 launderette 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 $160\ \text{€}$, well exceeding their daily cost of $150\ \text{€}$. 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 $2\ \text{€}$.

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 $160\ \text{€}$ 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 $3$ potters' wheels (fixed factor) and faces a daily fixed cost of $150\ \text{€}$ 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 $120\ \text{€}$ per day and each ceramic piece is sold for $18\ \text{€}$.

1. a) The manager argues: "We should hire the 6th worker, since producing $98$ pieces per day generates total revenue ($1.764\ \text{€}$) that comfortably exceeds total costs ($150\ \text{€} + 6 \times 120\ \text{€} = 870\ \text{€}$), thus maximising business volume." Refute their position using marginal analysis and calculate the number of workers that maximises 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 $10\ \text{€}$ 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 $1.200\ \text{€}$ per month and each machine costs $1.800\ \text{€}$ 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 $1.000$ yoghurts 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 $1.500\ \text{€}$ a month and each machine, $900\ \text{€}$.

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 labour 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 reorganising the workshop, it produces $450$ chairs per month with the same $8$ workers.

1. a) Calculate labour 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 $15.000\ \text{€}$ per month in total, and spends another $5.000\ \text{€}$ per month on fruit, packaging and energy. It produces $2.000$ litres of juice which it sells for $12\ \text{€}$ per litre.

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 euros of value does it generate for every euro spent?

Explain what the division of labour is and what specialisation 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 organising production in this way.

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

1. a) Explain what has changed in its production function.
2. b) Define what is meant by technical progress and give another example from a sector 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 $180.000\ \text{€}$ 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 $160\ \text{€}$ per day.

1. a) Calculate the productivity of the labour 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 labour factor.
2. b) Evaluate whether the investment is economically efficient by calculating and comparing the direct cost per can (labour cost plus machinery cost per day) in both scenarios.
3. c) A trade 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.