Derivation of long run cost curves

Derivation of long-run cost curves

        In the long run, all the factors of production are variable. There are no variable cost and fixed cost to be separated in the long run. There are only three types of costs in the long run which are explained as follows:

Long run total cost (LTC)

    It is the cost incurred by all the factors of production in the long run. Long run Total Cost (LTC) alludes to the minimum cost at which given level of yield can be delivered. LTC refers to the minimal cost of various amounts of yield. LTC is constantly, not exactly or equivalent to the short-run total cost, yet it is never more than short-run cost. According to Leibhafasky, “the long run total cost of production is the least possible cost of producing any given level of output when all inputs are variable.” It can be shown by the help of the following diagram:

In the above figure, we can see that the Long Run Total Cost Curve (LTC) is derived by joining the minimum points of short-run total cost curves, i.e; STC1, STC2, and STC3 respectively.

Long run average cost (LAC)

     LAC is obtained by dividing the long-run total cost (LTC) by the level of output.

Symbolically;
LAC = LTC / Q

As all the factors of production are variable in the long run, the firm can expand its output dividing any size of factors of production. Here, we take into consideration the division of plant to explain and derive the LAC.

 Suppose there are 3 plants which operate in three different short run average cost curves. The following diagram can make  more clear about it:

In the above figure there are three curves of short run average cost of production SAC1, SAC2,and SAC3 . The long run average cost curve (LAC) is drawn from each of the short run average cost curves, so that the LAC when declining becomes tangent to the left of the minimum point of SAC1 ; to the minimum point of SAC2 and when inclining, to the right of the minimum point of SAC3 . In other words, LAC envelops short run average cost curves from below. Therefore, it is also called envelop curve. LAC curve is also ‘U’ shaped because of the combination of short run cost curves but it is more flatter than SAC curve.

Long run marginal cost curves (LMC)

     It is defined as the addition in the long run total cost because of the increase in the output by 1 unit. It is expressed as;

LMC = ∆LTC / ∆Q

Long run marginal cost curves are derived also from the short-run marginal cost curves. It can be derived as follows:

In the above figure, to derive LMC curve, let us consider the points of tangency between all the SACs and LAC I.e. point A, B, and C. Perpendicular is drawn from the point A, B, and C. The corresponding levels of output are OQ1, OQ2, and OQ3 respectively. The three different perpendiculars intersect SMC1, SMC2, and SMC3 at point M, B, and N respectively. When these points are connected by a curve, they form LMC.