SHORT-RUN COST FUNCTION

In previous post cost concept we distinguished between the short run and the long run. We also distinguished between fixed costs and variable costs. The distinction between fixed and variable costs is of great significance to the business manager. Variable costs are those costs, which the business manager can control or alter in the short run by changing levels of production. On the other hand, fixed costs are clearly beyond business manager’s control, such costs are incurred in the short run and must be paid regardless of output.

Total Costs

Three concepts of total cost in the short run must be considered: total fixed cost (TFC), total variable cost (TVC), and total cost (TC). Total fixed costs are the total costs per period of time incurred by the firm for fixed inputs. Since the amount of the fixed inputs is fixed, the total fixed cost will be the same regardless of the firm’s output rate. Table-1 shows the costs of a firm in the short run. According to this table, the firm’s total fixed costs are Rs. 100. The firm’s total fixed cost function is shown graphically in Figure-1.

Total variable costs are the total costs incurred by the firm for variable inputs. To obtain total variable cost we must know the price of the variable inputs. Suppose if we have two variable inputs viz. labour (V1) and raw material (V2) and the corresponding prices of these inputs are P1 and P2, then the total variable cost (TVC) = P1 * V1 + P2 * V2. They go up as the firm’s output rises, since higher output rates require higher variable input rates, which mean bigger variable costs. The firm’s total variable cost function corresponding to the data given in Table 1 is shown graphically in Figure 1.

Finally, total costs are the sum of total fixed costs and total variable costs. To derive the total cost column in Table 1, add total fixed cost and total variable cost at each output. The firm’s total cost function corresponding to the data given in Table 0.1 is shown graphically in Figure 1. Since total fixed costs are constant, the total fixed cost curve is simply a horizontal line at Rs.100. And because total cost is the sum of total variable costs and total fixed costs, the total cost curve has the same shape as the total variable cost curve but lies above it by a vertical distance of Rs. 100.

Corresponding to our discussion above we can define the following for the short run:

TC = TFC + TVC

Where,

TC = total cost

TFC = total fixed costs

TVC = total variable costs

Average Fixed Costs

While the total cost functions are of great importance, managers must be interested as well in the average cost functions and the marginal cost function as well. There are three average cost concepts corresponding to the three total cost concepts. These are average fixed cost (AFC), average variable cost (AVC), and average total cost (ATC). Figure-2 show typical average fixed cost function graphically. Average fixed cost is the total fixed cost divided by output. Average fixed cost declines as output (Q) increases. Thus we can write average fixed cost as:

AFC = TFC/Q

Average Variable Cost

Average variable cost is the total variable cost divided by output. Figure-2 shows the average variable cost function graphically. At first, output increases resulting in decrease in average variable cost, but beyond a point, they result in higher average variable cost.

AVC = TVC/Q

Where,

Q = output

TVC = total variable costs

AVC = average variable costs

Average Total Cost

Average total cost (ATC) is the sum of the average fixed cost and average variable cost. In other words, ATC is total cost divided by output. Thus,

ATC = AFC + AVC = TC/Q

Figure-2 shows the average total cost function graphically. Since ATC is sum of the AFC and AVC, ATC curve always exceeds AVC curve. Also, since AFC falls as output increases, AVC and ATC get closer as output rises. Note that ATC curve is nearer the AFC curve at initial levels of output, but is nearer the AVC curve at later levels of output. This indicates that at lower levels of output fixed costs are more important part of the total cost, while at higher levels of output the variable element of cost becomes more important.

Marginal Cost

Marginal cost (MC) is the addition to either total cost or total variable cost resulting from the addition of one unit of output. Thus,

The two definitions are the same because, when output increases, total cost increases by the same amount as the increase in total variable cost (since fixed cost remains constant). Figure-2 shows the marginal cost function graphically. At low output levels, marginal cost may decrease with increase in output, but after reaching a minimum, it goes up with further increase in output. The reason for this behaviour is found in diminishing marginal returns.

The marginal cost concept is very crucial from the manager’s point of view Marginal cost is a strategic concept because it designates those costs over which the firm has the most direct control. More specifically, MC indicates those costs which are incurred in the production of the last unit of output and therefore, also the cost which can be “saved” by reducing total output by the last unit. Average cost figures do not provide this information. A firm’s decisions as to what output level to produce is largely influenced by its marginal cost. When coupled with marginal revenue, which indicates the change in revenue from one more or one less unit of output, marginal cost allows a firm to determine whether it is profitable to expand or contract its level o production.

Relationship between Marginal Cost and Average Costs

The relationships between the various average and marginal cost curves are illustrated in Figure-2. The figure shows typical AFC, AVC, ATC, and MC curves but is not drawn to scale for the data given in Table-1. The MC cuts both AVC and ATC at their minimum. When both the MC and AVC are falling, AVC will fall at a slower rate. When both the MC and AVC are rising, MC will rise at a faster rate. As a result, MC will attain its minimum before the AVC. In other words, when MC is less than AVC, the AVC will fall, and when MC exceeds AVC, AVC will rise. This means that as long as MC lies below AVC, the latter will fall and where MC is above AVC, AVC will rise. Therefore, at the point of intersection where MC = AVC, AVC has just ceased to fall and attained its minimum, but has not yet begun to rise. Similarly, the MC curve cuts the ATC curve at the latter’s minimum point. This is because MC can be defined as the addition either to TC or TVC resulting from one more unit of output. However, no such relationship exists between MC and AFC, because the two are not related; MC by definition includes only those costs which change with output, and FC by definition is independent of output.

Relationship between Average Product and Marginal Product, and Average Variable Cost and Marginal Cost

There is a straightforward relationship between factor productivity and output costs. To see this, let us consider a single variable factor L say labour. All other inputs are fixed. AP and MP will denote the average and marginal products of labour, respectively. If W is the wage rate and L is the quantity of labour, then Consequently, since Q/W is the average product (AP), AVC = W AP Also, WTVC = W * WL (W does not change and is assumed to be given.). Dividing by WQ we get But, marginal product (MP) = WQ/ W L. Hence, MC = W/MP Figure-3 shows the relationship between average product and marginal product, and average variable cost and marginal cost. The relationship AVC = W/AP shows that AVC is at a minimum when AP is at maximum. Similarly, the relationship MC = W/MP shows that MC is at a minimum when MP is at a maximum. Also, when AP is at a maximum, AP = MP. Hence, when AVC is at a minimum, AVC = MC. It is clearly shown that when MP is rising, MC is falling. And when MP is falling, MC is rising.

The relevant costs to be considered for decision-making will differ from one situation to the other depending on the problem faced by the manager. In general, the TC concept is quite useful in finding out the breakeven quantity of output. The TC concept is also used to find out whether firm is making profits or not. The AC concept is important for calculating the per unit profit of a business firm. The MC concept is essential to decide whether a firm should expand its production or not.