Cost Curve - Relationship Between Short Run and Long Run Cost Curves

Relationship Between Short Run and Long Run Cost Curves

Basic: For each quantity of output there is one cost minimizing level of capital and a unique short run average cost curve associated with producing the given quantity.

• Each STC curve can be tangent to the LRTC curve at only one point. The STC curve cannot cross (intersect) the LRTC curve. The STC curve can lie wholly “above” the LRTC curve with no tangency point.
• One STC curve is tangent to LRTC at the long-run cost minimizing level of production. At the point of tangency LRTC = STC. At all other levels of production STC will exceed LRTC.
• Average cost functions are the total cost function divided by the level of output. Therefore the SATC curveis also tangent to the LRATC curve at the cost-minimizing level of output. At the point of tangency LRATC = SATC. At all other levels of production SATC > LRATC To the left of the point of tangency the firm is using too much capital and fixed costs are too high. To the right of the point of tangency the firm is using too little capital and diminishing returns to labor are causing costs to increase.
• The slope of the total cost curves equals marginal cost. Therefore when STC is tangent to LTC, SMC = LRMC.
• At the long run cost minimizing level of output LRTC = STC; LRATC = SATC and LRMC = SMC, .
• The long run cost minimizing level of output may be different from minimum SATC, .
• With fixed unit costs of inputs, if the production function has constant returns to scale, then at the minimal level of the SATC curve we have SATC = LRATC = SMC = LRMC.
• With fixed unit costs of inputs, if the production function has increasing returns to scale, the minimum of the SATC curve is to the right of the point of tangency between the LRAC and the SATC curves. Where LRTC = STC, LRATC = SATC and LRMC = SMC.
• With fixed unit costs of inputs and decreasing returns the minimum of the SATC curve is to the left of the point of tangency between LRAC and SATC. Where LRTC = STC, LRATC = SATC and LRMC = SMC.
• With fixed unit input costs, a firm that is experiencing increasing (decreasing) returns to scale and is producing at its minimum SAC can always reduce average cost in the long run by expanding (reducing) the use of the fixed input.
• LRATC will always equal to or be less than SATC.
• If production process is exhibiting constant returns to scale then minimum SRAC equals minimum long run average cost. The LRAC and SRAC intersect at their common minimum values. Thus under constant returns to scale SRMC = LRMC = LRAC = SRAC .
• If the production process is experiencing decreasing or increasing, minimum short run average cost does not equal minimum long run average cost. If increasing returns to scale exist long run minimum will occur at a lower level of output than SRAC. This is because there are economies of scale that have not been exploited so in the long run a firm could always produce a quantity at a price lower than minimum short run aveage cost simply by using a larger plant.
• With decreasing returns, minimum SRAC occurs at a lower production level than minimum LRAC because a firm could reduce average costs by simply decreasing the size or its operations.
• The minimum of a SRAC occurs when the slope is zero. Thus the points of tangency between the U-shaped LRAC curve and the minimum of the SRAC curve would coincide only with that portion of the LRAC curve exhibiting constant economies of scale. For increasing returns to scale the point of tangency between the LRAC and the SRAc would have to occur at a level of output below level associated with the minimum of the SRAC curve.

These statements assume that the firm is using the optimal level of capital for the quantity produced. If not, then the SRAC curve would lie "wholly above" the LRAC and would not be tangent at any point.