Question

Let a firm’s production function be given by K^0.7 L^0.3.
(i) Sketch (without specific numbers) the shape of the long run average and long-run marginal cost curves of the firm; explain the key features of the sketch.
(ii) in the same graph, please also sketch the firm’s short run average and marginal cost curves (when the amount of capital is fixed). Comment on the relationship between the long- and the short-run curves depicted in your graph.

Answer 1

(i) Long run Marginal Cost is the change in the Long run total cost due to one unit change in output whereas Long run Average Cost is the per unit cost of producing a good or a service in the long run when all the inputs are variable.
The shape of the long run average and long-run marginal cost depends on whether there are increasing returns to scale, constant returns to scale or decreasing returns to scale. The key features of the sketch are:

(a) When there is increasing returns to scale i.e. when the Long run Average Cost (LAC) is downward sloping then the Long run Marginal Cost (LMC) is less than the Long run Average Cost.

(b) When there is constant returns to scale i.e. when then Average Cost is minimum, then Long run Marginal Cost = Long run Average Cost.

(c) When there is decreasing returns to scale i.e. when the Average Cost is upward rising, then the Long run Marginal Cost is greater than the Long run Average Cost.

(ii) When Short run Average Cost (SAC) = Long run Average Cost (LAC), then Short run Marginal Cost (SMC) = Long run Marginal Cost (LMC). Hence, finding out the SMC where SAC = LAC we can derive the LMC. The derivation of LMC curve is shown in th figure below.

In the production process, we have different SAC curves and their corresponding SMC curves. First, we will consider the points where SAC = LAC i.e. the point of tangency of SAC and LAC curves. From these points of tangency we will draw the perpendicular on the horizontal axis. Now, we will take the points of intersection of these vertical lines with the corresponding SMC curves. By joining these points of intersection we will get the LMC curve of the firm.

In the figure, SAC₁ is tangent to LAC at point A. From A, we drop the perpendicular AQ_1. This perpendicular cuts the SMC₁ at point C. Then point C is a point on the LMC curve. Repeating this procedure, we get other points on the LMC curve. That the LMC curve is not the envelope of the SMC curve can be shown as following. Let us take any point A which is to the left of the SAC₁ curve. At point A^', SAC is greater than LAC while at point A, SAC = LAC. Thus Short run Total Cost is greater than Long run Total Cost at point A^' and Short run Total Cost = Long run Total Cost at point A. Now when we move from A^' to A, we move from a point where Short run Total Cost is higher than the Long run Total Cost to a point where Short run Total Cost = Long run Total Cost. The addition to Short run Total Cost must therefore be smaller than addition to Long run Total Cost. In other words, Short run Marginal Cost will be lower than Long run Marginal Cost to the left of A. Again considering the point A^'' on SAC₁, SAC > LAC i.e. SRTC > LRTC. Thus, when we move from A to A^''we actually move from a situation where SRTC = LRTC. The addition to SRTC must be higher than the addition to LRTC. This means that to the right of A, SRMC is higher than LRMC. Hence, LMC cannot be tangent to SMC for if it had been tangent to SMC, then SMC would have been greater than LMC both to the left of A or to the right of A.