Derivation of Short-Run Average and Marginal Cost Curves From Total Cost Curves

Whether it is production or cost, there are three categories of concepts- total cost, average cost and marginal cost. In the short run, there are three types of total cost curves: total fixed cost curve, total variable cost curve and total cost curve, whereas, in the long run, there is no fixed cost, only variable and total cost. The average and marginal cost curves can be geometrically derived from a total cost curve.

Derivation of Cost Functions

1. Derivation of Average Fixed Cost Curve From Total Fixed Cost Curve

The Total fixed cost (TFC) curve is parallel to the X-axis since the costs remain the same irrespective of the level of output. In the table below, we find TFC is Rs 240 at all levels of output. The Average Fixed cost (AFC) equals TFC divided by output.

Geometrically, at the output level of one unit, AFC is equal to the slope of ray OA and refers to point ’a’ on the AFC curve. At the output level of two units, AFC is equal to the slope of ray OB and refers to point ‘b’ on the AFC curve. Similarly, we can plot for other levels of output and get the corresponding points on the AFC curve, as shown in Figure 1.

The AFC curve is a rectangular hyperbola – it is asymptotic to the axes, which means that as the AFC curve moves further away from the origin along the axis, it gets closer to the axis but never touches it.

Output	TFC	TVC	TC	AFC	AVC	ATC	MC
0	240	0	240	240	–	240	–
1	240	120	360	240	120	360	120
2	240	160	400	120	80	200	40
3	240	180	420	80	60	140	20
4	240	212	452	60	53	113	32
5	240	280	520	48	56	104	68
6	240	420	660	40	70	110	140
7	240	640	880	30.43	90.14	120.57	220

2. Derivation of Average Variable Cost Curve From Total Variable Cost Curve

The Average Variable cost (AVC) is equal to the TVC divided by output. The AVC is equal to Rs 120 at the output level of one unit (TVC is Rs 120). It is represented by the slope of the ray OA and is plotted as point ‘a’ on the AVC curve. At the output level of three units, AVC is Rs 60 and is plotted as point ‘c’ on the AVC curve.

Similarly, we can plot other points on the AVC curve. The AVC curve is downward sloping till point ‘d’ and then slopes upward. The slope of the ray OC from the origin to the TVC curve is the lowest at point ‘C’ on the TVC curve, which implies that AVC at the output level of 4 units is the lowest at d (figure no. 2).

3. Derivation of Average Total Cost Curve From Total Cost Curve

When the level of output is 0, there is no variable cost. Therefore, the TFC = TC=AFC. When the output level is one, TC is Rs 360, and ATC is also Rs 360. When the output level is 2, TC is Rs 400, and ATC is 400 ÷ 2 = Rs 200. The ATC at different levels of output is indicated by the slope of the ray from the origin to the TC curve. Thus, OA, OB, OC and OD are rays whose slope indicates the ATC at the corresponding level of output.

These points are plotted to obtain points ‘a’, ’b’, ‘c’, ’d’, and so on to get the AVC curve. The slope of a ray from the origin falls up to point ‘C’ on the TC curve and rises afterwards. The ray from the origin (OC) is tangent to the TC curve at point ‘C’. Thus, the ATC curve falls up to point ’C’ (point ‘c’ is the lowest point on the ATC curve) and then rises, giving it a ‘U’ shape (figure no. 3).

4. Derivation of Marginal Cost Curve From Total Cost and Total Variable Cost Curve

We can derive the marginal cost curve from the total cost curve and the total variable cost curve. Let the TFC and TVC be given as below in table no: 2; we can find the MC from it. We can draw the TC, TVC and derive the MC.

OUTPUT	TC	TFC	TVC	MC
0	120	120	0	–
1	180	120	60	60
2	200	120	80	20
3	210	120	90	10
4	230	120	110	20
5	270	120	150	40
6	360	120	240	90

From figure no. 4, we find that the slopes of the TVC curve and the TC curve are the same at every level of output. Points A and A¹ are the points of inflexion, respectively, on TC and TVC.

The MC falls up to 2.5 units of output and refers to point ‘a ‘on the MC curve and then rises. The MC is given by the slope of the TVC curve at point B. At 5 units of output, MC is equal to the slope of the TC curve at point C. Thus, point ‘b’ refers to the lowest AVC and point ’c’ refers to the lowest AC. Marginal cost is sometimes known as ‘incremental cost’ – as it is the increase in TC consequent upon a small increase in output. MC = ΔTC/ΔQ or ΔTVC/ΔQ.

ΔTC is the change in total cost due to a small change in output

ΔTVC is the change in total variable cost due to a small change in output

ΔQ is the small change in output

For example, the total cost of producing 4 units of output is Rs 1000, and the total cost of producing 5 units of output is Rs 1200; therefore the marginal cost of the fifth unit is Rs 200 (Rs 1200 – Rs 1000).

Because fixed cost remains unchanged in the short run, therefore, marginal cost is also the increase in total variable cost due to a small increase in output.

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