Production Function: Total, Marginal and Average Product, Diminishing Returns

A food truck hires a third cook, and output jumps by 30 meals an hour. Later it hires a fifth cook, and output rises by just 6. Same truck, same grill, same kind of cook. So why would one more person add almost nothing? This is Module 3.1, the production function.

A firm turns inputs, like workers and equipment, into output. The production function is the link between the inputs a firm uses and the output it makes. The truck and its grill are a fixed input: for now, the owner can't change how much of them she has. Cooks are a variable input: she can add or remove them any time. That's the short run: at least one input is fixed. In the long run, every input can change, even the truck.

Here's the truck's short-run production function: cooks in one column, meals per hour in the next. Total output is called total product. Zero cooks make zero meals. Then 10, 30, 60, 84, 90, and with a sixth cook, 78.

Quick refresher from Unit 1: marginal means the change that comes from one more unit. Here, the unit is one more cook.

Marginal product is the extra output from one more worker. The formula: the change in total product, divided by the change in workers. First cook: 10 minus 0 is 10 meals, divided by 1 cook: 10 meals an hour. Second cook: 30 minus 10 is 20. Third cook: 60 minus 30 is 30. Each new cook adds more than the last, because with a few cooks, one grills, one wraps and one serves. They specialize. Fourth cook: 84 minus 60 is 24, divided by 1 cook: 24 meals an hour. That means the fourth cook adds 24 meals, fewer than the third cook's 30.

Fifth cook: 90 minus 84 is 6. Sixth cook: 78 minus 90 is negative 12. Output actually falls.

So what just happened at the truck? One: each cook's marginal product is the change in total product. Two: it rose, 10, 20, 30. Three: then it fell, 24, 6, and negative 12.

That fall has a name: diminishing marginal returns. Add more of one input while the other inputs stay fixed, and at some point each new unit adds less output than the one before. Diminishing returns begin with the first cook whose marginal product falls. Here, that's the fourth: marginal product rose to 30 with the third cook, then fell to 24. Why? Four cooks now share one grill and one counter. Each new cook has less of that fixed equipment to work with, so each adds less.

That answers the hook. The fifth cook added only 6 meals because five people were sharing one grill. And it's a short-run idea: it needs a fixed input.

So what just happened at the truck? One: marginal product rose to 30 with the third cook. Two: it fell to 24 with the fourth, so diminishing returns began there. Three: the cause is more cooks sharing the same grill.

Exam trap one: the sixth cook made output fall, and its marginal product is negative. A common mistake is to say diminishing returns start there. They began two cooks earlier, at the fourth: the first fall in marginal product. Not where marginal product turns negative, and not where total product falls. Exam trap two: explain with marginal product, not total product. Total product still rose with the fourth cook, from 60 to 84, so it can't show the turn. And write marginal product, not marginal revenue or marginal benefit. Show it rising, then falling: 10, 20, 30, then 24 with the fourth cook.

Now, how the measures connect. Total product grows by each cook's marginal product: 60 meals plus the fourth cook's 24 is 84.

Quick refresher: an average is a total divided by the number of units. So average product is total product divided by the number of workers: output per worker.

With 3 cooks: 60 meals over 3 cooks is 20 meals per cook. With 4 cooks: 84 over 4 is 21. With 5: 90 over 5 is 18. Now look at the fourth cook. Marginal product fell, from 30 to 24. But average product still rose, from 20 to 21, and total product still rose, from 60 to 84. In this table, marginal product turns down first, at the fourth cook. Average product turns down next, at the fifth. Total product turns down last, at the sixth. On the graph, marginal product is the slope of the total product curve. The curve climbs more and more steeply up to the third cook, flattens from the fourth, and turns down at the sixth.

So what just happened with the three measures? One: average product is total over workers: 20, 21, then 18. Two: at the fourth cook, only marginal product fell, so the total product curve flattened. Three: diminishing returns are about marginal product, not average or total.

Now the long run. The owner trades up to a bigger truck with two grills. With more equipment, one cook or six, each crew makes more meals, so the whole total product curve shifts up. And each cook has more grill to work with, so the marginal product curve shifts up too.

So what just happened in the long run? One: even the truck and grill could change. Two: with more equipment, every crew made more meals, so total product shifted up. Three: each cook had more grill to use, so marginal product shifted up too.

One more link, for the next module. Once diminishing returns begin, each new cook adds less, so each extra meal takes more labor to make. That's why marginal cost rises, and Module 3.2 shows it.

Exam moment: two questions the way the AP exam asks them, then one free-response part.

A print shop with a fixed number of printers adds workers. When diminishing marginal returns begin, which of the following must be true? A) Total product falls. B) Average product falls. C) Marginal product is negative. D) Marginal product falls. E) Marginal product rises.

The answer is D. Diminishing marginal returns means each added worker adds less than the one before: marginal product falls. A and C need not happen at the turn: our truck hit both only at the sixth cook. B, the average, still rose at our fourth. E comes before the turn.

A hand car wash has a fixed number of bays. With 0 to 5 workers, it washes 0, 8, 17, 33, 40 and 44 cars a day. What is the average product of labor with 4 workers? A) 0.1 B) 7 C) 10 D) 11 E) 40

The answer is C. Average product is total product divided by workers: 40 cars over 4 workers is 10 cars per worker. A flips the ratio, 4 over 40. B is the fourth worker's marginal product, 40 minus 33. D uses the wrong row, 33 over 3. E is the total, never divided.

A flower shop has two workbenches. With 0 to 6 workers, it makes 0, 6, 14, 24, 36, 45 and 43 bouquets a day. With which worker do diminishing marginal returns begin? Explain using numbers.

Quick refresher: marginal product is the change in output from one more worker. Second worker: 14 minus 6 is 8 bouquets.

What earns the point: diminishing marginal returns begin with the fifth worker. Marginal product rises for workers 1 to 4: 6, 8, 10, 12. Then it falls to 9 with the fifth. The costly mistake: explaining with total output, which still rose, 36 to 45, or quoting only the fifth worker's 9. On a recent AP exam, only about half earned this point.

So what just happened in that answer? One: find each worker's marginal product. Two: spot the first fall, 12 to 9, at the fifth worker. Three: name that worker, and quote the marginal products rising, then falling.

Recap: the production function links inputs to output. Marginal product is the extra output from one more worker. And diminishing returns begin at the first fall in marginal product, in the short run. Next: Module 3.2, Short-Run Production Costs.