Derivation of long-run average cost curve

In the long run, all inputs (factors of production) are variable and firms can enter or exit any industry or market. Consequently, a firm's output and costs are unconstrained in the sense that the firm can produce any output level it chooses by employing the needed quantities of inputs (such as labor and capital) and incurring the total costs of producing that output level.

The Long Run Average Cost, LRAC, curve of a firm shows the minimum or lowest average total cost at which a firm can produce any given level of output in the long run (when all inputs are variable).

Explanation

In the long run, all inputs (factors of production) are variable and firms can enter or exit any industry or market.

Assumption - A firm is uncertain about the demand in the long run and is considering four alternate plants. The short run curves are given by SAC1, SAC2, SAC3 and SAC4.

Look at the following figure. In this figure, we have 4 short run curves SAC1, SAC2, SAC3 and SAC4.

- To produce Q0, The firm will use SAC1 curve. At this output cost is P0

- To produce Q1, The firm will again use SAC1 curve. At this output cost is P0

- To produce Q2, The firm will use SAC2 curve. If it will continue to use SAC1 curve, then the cost will increase to P2. So, it would be better for the firm to bring second plant into the production.

- At SAC2 curve, the cost of producing Q2 would be P1, much less than P2

- Lets see Q3: to Produce Q3, firm can either use SAC3 or SAC4 curve.

- For Q3, the firm will use SAC4 curve, as it has low cost.

- For Q3, the cost of producing at SAC3 is much higher then SAC4.

In the long run, a firm will use the level of inputs that can produce a given level of output at the lowest possible average cost. Consequently, the LRAC curve is the envelope of the short run average cost (SAC) curves, where each SRAC curve is defined by a specific quantity of inputs

The Long Run Cost Function describes the least-cost method of producing a given amount of output. The "Long Run" part of the cost function means that all inputs are variable. In the simple case, you'd consider capital and labor. In the long run, both capital and labor may be adjusted. In the short-run, however, capital may not be adjusted. (You can't buy and install new machinery by next week, or sell a factory and be moved out.) You can, however, hire new employees to start work tomorrow..

Summary

· In the long run all inputs are flexible, while in the short run some inputs are not flexible, long-run cost will always be less than or equal to short-run cost.

· In the short run the firm faces an additional constraint: all expansion must proceed using only the variable input. And additional constraints increase cost.

· The envelope relationship is the relationship explaining that, at the planned output level, short-run average total cost equals long-run average total cost, but at all other levels of output, short-run average total cost is higher than long-run average total cost.