LTC / Q
.Recall that in the long run, more productive inputs are variable. Thus plant sizes may be altered and entrepreneurs may enter or leave an industry.
Suppose a firm has decided to produce output level Q1 in the long run. How will it decide on a quantity of land to use in the production of Q1 so as to minimize total cost? The answer comes from considering the STC curves associated with every possible level of land input and selecting the lowest one at Q1.
The long-run total cost (LTC) curve is constructed by choosing the lowest STC curve for producing EACH possible output level. Consequently, it is the envelope of the family of STC curves: tangent to each but intersecting none.
The shape of the LTC curve depend on the assumed properties of the underlying production function. In general, there is some fixed factor in the production process that cannot be changed even in the long run -- the services of the entrepreneur, for example. This fixed factor causes the firm's LTC curve to have a shape resembling that of an STC curve. For low levels of Q, the fixed factor is underutilized and costs rise as a decreasing rate as Q expands. Beyond some point, however, the presence of the fixed factor causes other inputs to exhibit diminishing marginal productivities (analogous to the case of labour in short-run analysis) and costs will begin to rise at an increasing rate.
Long-run average total cost is the long-run total cost per unit of output produced:
Long-run marginal cost is the change in long-run total cost resulting from the production of an additional unit of output:
LTC / Q
.Since these cost curves are related to the LTC curve in the same way that the short-run average and marginal cost curves for a given level of land input are related to the corresponding STC curve, both are U-shaped and the latter cuts the former from below at its minimum point.
Since the land input level which minimizes the STC of producing a given level of output also minimizes the SATC of producing that output, the LATC curve is the envelope of the family of SATC curves. Note that the LMC is NOT the envelope of the family of SMC curves.