Questions: Production Functions and Technological Relationships

Question 1

A factory's production function shows that 4 workers and 3 machines can produce a maximum of 200 units per day. The factory currently uses exactly 4 workers and 3 machines but produces only 150 units per day. What is the best explanation?

• A. The production function is wrong — it overstates what is technologically possible
• B. The factory is technically inefficient — it is operating below the technological frontier
• C. The production function describes average output, so some days will naturally be below 200
• D. Technology must have changed recently, shifting the production function downward

Answer: (B)

The production function describes the maximum output achievable with given inputs — it is the technological frontier. A firm operating below it is technically inefficient: the same inputs could produce more output. The function does not represent average or typical performance. Options C and D reflect a misunderstanding: the production function is not a statistical average, and a firm producing below potential does not imply the function has shifted — it implies the firm is not reaching it.

Question 2

A software update allows a manufacturing firm to produce 20% more output from the same capital and labor. In the Cobb-Douglas production function Q = AK^αL^β, what has changed?

• A. The exponents α and β increased, raising the output responsiveness of each input
• B. The total factor productivity parameter A increased, shifting the entire production function upward
• C. Capital K effectively increased because the machines work faster
• D. Labor L increased because workers can now accomplish more per hour

Answer: (B)

The parameter A in the Cobb-Douglas function captures total factor productivity (TFP) — the state of technology. When a process improvement, software update, or organizational change lets the same inputs produce more output, A rises. This is precisely what it means for technology to 'change' in production theory: the production function shifts upward, not because K or L changed, but because the same K and L now yield more Q. Treating this as a change in K or L would be a category error.

Question 3

The production function tells a firm which combination of inputs to actually use in production.

• T. True
• F. False

Answer: False

The production function describes what is technologically feasible — the constraint imposed by physics and engineering — not what a firm should do. It tells you the maximum output for every possible input combination, but many different combinations can yield the same output. The decision of which combination to actually use requires additional information: input prices. A firm minimizes cost by choosing the input mix given those prices, a separate optimization problem. The production function provides the feasible set; cost minimization picks from within it.

Question 4

Many different input combinations can yield the same level of output on a production function.

• T. True
• F. False

Answer: True

This is one of the most important properties of production functions, and it is the basis for isoquant analysis. A bakery could produce 1,000 loaves per day with 3 ovens and 5 bakers, or with 2 ovens and 8 bakers, depending on substitution possibilities. The production function maps all these feasible combinations; the set of combinations that all produce the same output level traces an isoquant. The firm uses input prices to determine which of these equivalent combinations is cheapest.

Question 5

Why is the production function described as a constraint rather than a decision? What is it constraining, and what additional information is needed to determine what a firm actually does?

Model answer: The production function constrains the firm's feasible set — it specifies the maximum output achievable from each input combination, encoding the current state of technology. The firm cannot produce more than the function allows. But the production function says nothing about which combination to choose among the many that could produce a given output level. That decision requires input prices (wages, rental cost of capital). Given prices and the production function, the firm solves a cost-minimization problem to find the cheapest way to produce its target output.

This separation — technology (the production function) versus economics (input prices and cost minimization) — is foundational to producer theory. The production function is like a map of all possible routes; input prices determine which route is cheapest. Confusing the two — for example, thinking the production function tells you to use more labor because labor is 'efficient' — ignores that efficiency without price is not a complete economic concept.