Exercises: 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?

Explain what the division of labor 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 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 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 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 $\$300\,000$ per month in total, and spends another $\$100\,000$ per month on fruit, packaging and energy. It produces $2,000$ liters of juice which it sells for $\$240$ 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 pesos of value does it generate for every pesos spent?

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 $\$24\,000$ per month and each machine costs $\$36\,000$ 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.

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.

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 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.

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 reorganising 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) Analyse two operational or organisational risks introduced by this specialisation (such as lack of flexibility or job monotony) and design a work organisation 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 specialising and trading, rather than attempting to manufacture complete bicycles under autarky.