Production Function
What is Production Function?
A production function shows the maximum output a firm can produce from given quantities of inputs.
It links inputs such as labor and capital to output. In the short run at least one input is fixed, which leads to diminishing marginal returns. In the long run all inputs are variable.
Production Function: a worked example
Hold capital fixed at 2 machines and read the short-run function. One worker yields 20 units, two yield 44, three yield 60, and four yield 70. The extra output per worker runs 20, 24, 16, then 10, so this technology shows rising and then falling marginal returns while capital cannot move. Now switch to the long run and double both inputs, going to 4 machines and 8 workers. Constant returns to scale would deliver 2 × 70 = 140 units. If the expanded plant actually turns out 154 units, output more than doubled, so the function displays increasing returns to scale across that range. One function answers both questions, but only after deciding whether capital is frozen or scaled.
The mistake students make with production function
Students apply diminishing marginal returns to a change in plant size and conclude that a firm doubling every input must end up with less than double the output. The two ideas answer different questions. Diminishing marginal returns describe adding one input while others stay frozen, a short-run statement. Returns to scale describe multiplying every input by the same factor, a long-run statement, and output can rise by more than, less than, or exactly that proportion. Check first whether anything in the problem is being held constant.
Production Function questions
Why does a production function show maximum output rather than any output?
Production functions assume technical efficiency, meaning the firm wastes none of its inputs. Two workers and one machine could produce almost nothing if everyone stands idle, and that outcome carries no useful information. Defining the function as the largest output obtainable from each input bundle makes the relationship well defined, so inefficiency appears as a point below the curve instead of a second value on it.
How does a production function turn into a cost curve?
Read off the labor needed for each output level, then multiply by the wage. If 3 workers produce 60 units at a wage of $200 each, variable cost at 60 units is $600, so average variable cost is $10 per unit. Where extra workers add less output, each additional unit requires more labor, and cost per unit climbs. Cost curves are the production function viewed from the other side.
What does a short-run production function look like on a graph?
Total product goes on the vertical axis and labor on the horizontal axis, with capital held fixed. The curve climbs steeply while marginal product is increasing, then keeps climbing at a decreasing rate once diminishing marginal returns begin, which flattens its slope. Crowd enough workers onto the fixed equipment and the curve can peak and bend downward, the region of negative marginal product.
Formula / Example
This is the live Production Costs sandbox. Drag the curves, or open the full version.
Related terms
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