Envelope Curve (Long-Run ATC)
What is Envelope Curve (Long-Run ATC)?
The envelope curve is the long-run average total cost curve, which 'wraps around' and is tangent to every short-run ATC curve, lying on or below all of them.
In the long run a firm can choose any plant size, so each plant size has its own short-run ATC curve. The LRATC is the lower boundary (the 'envelope') formed by the lowest achievable cost at each output, tangent to each short-run curve at the output where that plant is optimal. Importantly, the tangency point is usually NOT the minimum of the short-run ATC, except at the single output where LRATC is itself at its minimum. The downward part reflects economies of scale and the upward part diseconomies of scale.
Envelope Curve (Long-Run ATC): a worked example
A firm can build a small, medium, or large plant. Short run average total cost at three output levels runs like this. Small plant: 12 dollars at 100 units, 15 at 200, 24 at 300. Medium plant: 16 dollars at 100 units, 8 at 200, 11 at 300. Large plant: 25 dollars at 100 units, 13 at 200, 9 at 300. The long run curve takes the lowest cell in each column, so it reads 12 dollars at 100 units, 8 at 200, and 9 at 300. Now the subtle part. The small plant's own cheapest output is 11 dollars at 140 units, yet a firm planning to make 100 units still builds small and pays 12 dollars. The envelope touches that plant's curve on its downward slope, left of the plant's own minimum, because long run cost is still falling at 100 units.
The mistake students make with envelope curve (long-run atc)
The standard drawing error is running the long run curve through the minimum point of every short run curve, like connecting dots. When plant size can be varied continuously, exactly one short run curve is touched at its own minimum, the plant whose lowest cost equals the lowest point of the long run curve. Where the long run curve slopes down it touches short run curves to the left of their minimums, and where it slopes up it touches them to the right. A second slip is calling the long run curve an average of the short run curves. Take the lower boundary instead.
Envelope Curve (Long-Run ATC) questions
Why is the long-run average total cost curve called an envelope curve?
The long run curve envelops a whole family of short run cost curves, one per plant size, by wrapping underneath them. Each point on it answers a different question from a short run curve. Not what a given plant costs at that output, but what the best plant for that output costs once the firm is free to build any size. Reading it means choosing a quantity first, then asking which plant a firm would build to produce it.
Is long-run ATC ever above a short-run ATC curve?
Long run average total cost never rises above any short run curve, because building the plant that produced that short run cost is always an option once every input is variable. The most the long run curve can do is equal a short run curve, at the single output where that plant size is the right choice. At any other output some other plant is cheaper, so the long run curve dips below.
What makes the envelope curve slope down and then up?
Economies of scale drive the downward stretch, as bigger plants spread fixed costs over more units, allow specialized workers and machinery, and secure bulk terms on inputs. The upward stretch comes from diseconomies of scale, where communication delays, extra layers of management, and coordination failures push cost per unit back up. The lowest region between them marks minimum efficient scale, the smallest output at which a firm reaches the bottom of its long run costs.
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