Deriving the long-run average cost curve; why it is a planning curve

Short run and long run

In the short run a firm works with a given plant (fixed capital), and its costs per unit for different outputs are shown by that plant's short-run average cost curve (SAC). SAC is U-shaped: it falls at first as fixed costs are spread over more units and output gains from fuller use of the plant, then rises as diminishing returns set in. In the long run the firm can choose any plant size, since all inputs are variable. Each possible plant has its own SAC.

Deriving the LAC curve from the SAC curves

Step 1: a few plant sizes. Suppose only three plants can be built: small (SAC₁), medium (SAC₂) and large (SAC₃).

Diagram. Output on the horizontal axis, cost per unit on the vertical axis; three U-shaped curves SAC₁, SAC₂, SAC₃ overlapping, from left to right.

• For small outputs the firm would build the small plant, because SAC₁ lies lowest there.
• Where SAC₁ and SAC₂ cross, the firm is indifferent; beyond that output the medium plant is cheaper, and so on.
• The long-run average cost curve is then the scalloped lower boundary of the three curves (the solid parts of SAC₁, SAC₂ and SAC₃).

Step 2: many plant sizes. If plant size can be varied continuously, there is a SAC for every size, and the lower boundary becomes a smooth U-shaped curve that touches each SAC at one point. This is the LAC: an envelope curve.

Features of the envelope:

• Each point on LAC is the least cost per unit of producing that output when the plant can be chosen freely.
• To the left of LAC's minimum, LAC touches each SAC on SAC's falling part: for such an output, the least-cost plant is a slightly larger one run below its own minimum point, which is cheaper than the smaller plant whose own SAC minimum falls at that output. The plant LAC picks is still the lowest-cost one for that output.
• To the right of LAC's minimum, LAC touches each SAC on SAC's rising part.
• Only at the minimum point of LAC (output OQ) does LAC touch an SAC at that SAC's own minimum. That plant is the optimum plant, and OQ the optimum scale.
• LAC is U-shaped because of economies of scale (falling part) and diseconomies of scale (rising part), not because of diminishing returns to a fixed factor.

Why the LAC curve is the planning curve

1. It is used before the firm commits. At the planning stage every plant size is still open. For the output the firm expects to sell, the LAC curve marks the lowest unit cost any plant can achieve, and so points to the plant to build; once that plant is installed, the firm is tied to its SAC.
2. It sets a floor under unit cost. With today's methods and input prices no plant can make a given output for less than the LAC curve shows. A firm whose unit cost for its output sits higher than that floor is not producing that output at the least possible cost; one reason can be a plant of the wrong size, and the curve then shows how much choosing the least-cost plant would save.
3. It shows the choices, not current operations. At any time the firm actually operates on one SAC; LAC is the menu of all the SACs it could choose from.
4. It guides expansion. A firm expecting demand to grow can see whether a larger plant would lower unit costs (economies of scale) or raise them.
5. It is an ex ante concept: it describes planned, not realised, costs.

Example (our own). A dairy planning a processing unit in Haryana compares the unit cost of plants of 20,000, 50,000 and 1,00,000 litres a day against its expected milk procurement before choosing one.

Conclusion

The LAC curve is the envelope of the short-run average cost curves, touching each at one point and showing the least unit cost of every output when plant size can vary. Because it is used to choose the plant before investment, it is called the firm's planning curve.