Chapter 8
Cost Theory and Estimation
8.1 THE NATURE OF COSTS
Economic costs include explicit and implicit costs. Explicit costs are the actual expenditures of the firm to purchase or hire inputs. Implicit costs refer to the value of the inputs owned and used by the firm in its own production activity. The value of these inputs is imputed or estimated by what they could earn in their best alternative use (see Problem 8.1). The firm must pay each input (whether owned or purchased) a price equal to what the same inputs could earn in their best alternative use. This is the alternative or opportunity cost theory. Economic or opportunity costs must be distinguished from accounting costs, which refer only to the firm’s actual expenditures, or explicit costs, incurred for purchased or hired inputs. Accounting or historical costs are important for financial reporting by the firm and for tax purposes. For managerial decision-making purposes, however, economic or opportunity costs are the relevant cost concepts to use.
We must also distinguish between marginal and incremental costs. Marginal cost refers to the change in total cost for a one-unit change in output. Incremental cost is a broader concept and refers to the change in total costs that results from implementing a particular management decision, such as the introduction of a new product line. The costs that are not affected by the decision are irrelevant and are called sunk costs.
EXAMPLE 1. Suppose that John Smith, an accountant working for an accounting firm and earning $50,000 per year, is contemplating opening his own accounting office. To do so, he would have to give up his present job and use as his office a store that he owns and rents out for $15,000 per year. Smith estimates that he would earn an income of $100,000 for the year and incur $40,000 of out-of-pocket expenses to hire a secretary, rent office equipment, and pay utility costs and taxes Should Smith open his own accounting office? The answer is. no because his total economic costs of $105,000 (explicit costs of $40,000 plus the implicit costs of $65,000) exceed his total earnings of $100,000.
8.2 SHORT-RUN COST FUNCTIONS
In the short run, some of the firm’s inputs are fixed and some are variable, and this leads to fixed and variable costs. Total fixed costs (TFC) are the total obligations of the firm per time period for all fixed inputs. Total variable costs (TVC) are the total obligations of the firm for all variable inputs. Total costs (TC) equal total fixed costs plus total variable costs. That is,
TC = TFC + TVC (8-1)
Within the limits imposed by the given plant and equipment, the firm can vary its output in the short run by varying the quantity of the variable inputs used. This gives rise to the TFC, TVC, and TC schedules or functions. These show the minimum costs (explicit plus implicit) of producing various levels of output. From the total cost curves we can derive the per unit cost curves. Average fixed cost (AFC) equals total fixed costs divided by the level of output (Q). Average variable cost (AVC) equals total variable costs divided by output. Average total cost (ATC) equals total cost divided by output. ATC also equals AFC plus AVC. Marginal cost (MC) is the change in TC or TVC per unit change in output. That is,
(8-2)
(8-3)
(8-4)
(8-5)
EXAMPLE 2. Table 8.1 gives the hypothetical short-run total and per unit cost schedules of a firm. These are plotted in Figure 8-1. From column (2) of the table we see that TFC are $120 regardless of the level of output. TVC [column (3)] is zero when output is zero and rises as output rises. At point H' (the point of inflection in the top panel of Fig. 8-1), the law of diminishing returns begins to operate, and the TVC curve faces up or increases at a growing rate. The TC curve has the same shape as the TVC curve but is $120 (the TFC) above it at each output level. MC is plotted halfway between the various levels of output in the bottom panel of Fig. 8-1. The AVC, ATC, and MC curves are U-shaped. AFC is equal to the vertical distance between the ATC and AVC curves. Graphically, AVC is the slope of a ray from the origin to the TVC curve, ATC is the slope of a ray from the origin to the TC curve, and the MC is the slope of the TC or TVC curve. Note that the MC curve reaches its minimum at a lower level of output than, and intercepts from below, the AVC and ATC curves at their lowest point. The U-shape of the AVC and MC curves can be explained, respectively, from the inverted U-shape of the AP and MP curves (see Problem 8.4).
Table 8.1
Q
( 1 ) / TFC( 2 ) / TVC
( 3 ) / TC
( 4 ) / AFC
( 5 ) / AVC
( 6 ) / ATC
( 7 ) / MC
( 8 )
0 / $120 / $ 0 / $120 / –– / –– / –– / ––
1 / 120 / 60 / 180 / $120 / $60 / $180 / $60
2 / 120 / 80 / 200 / 60 / 40 / 100 / 20
3 / 120 / 90 / 210 / 40 / 30 / 70 / 10
4 / 120 / 104 / 224 / 30 / 26 / 56 / 14
5 / 120 / 140 / 260 / 24 / 28 / 52 / 36
6 / 120 / 210 / 330 / 20 / 35 / 55 / 70
Fig. 8-1
8.3 LONG-RUN COST FUNCTIONS
The long run is the time period during which all inputs and costs are variable (i.e., the firm faces no fixed inputs or costs). The firm’s long-run total cost (LTC) curve is derived from the firm’s expansion path and shows the minimum long-run total costs of producing various levels of output. From the LTC curve we can then derive the firm’s long-run average cost (LAC) and long-run marginal cost (LMC) curves. LAC is equal to LTC divided by output (Q) and is given by the slope of a ray from the origin to the LTC curve. That is,
(8-6)
whereas LMC measures the change in LTC per unit change in output and is given by the slope of the LTC curve. That is,
(8-7)
EXAMPLE 3. The top panel of Fig. 8-2 shows the expansion path of the firm. Point A shows that the optimal combination of inputs to produce one unit of output
(1Q) is three units of labor (3L) and three units of capital (3K). If the wage of labor (w) is $10 per unit and the rental price of capital (r) is also $10 per unit, the minimum total cost of producing lQ is $60. This is shown as point A' on the long-run total cost (LTC) curve in the middle panel. Other points on the LTC curve are sirnilarly obtained. [At point E' on the LTC curve the firm uses 4.8L and 4.8K to produce 3Q, the isoquant for 3Q (and, hence, E') are not shown in order not to clutter the figure.] Note that the LTC curve starts at the origin because there are no fixed costs in the long run. The LAC to produce 1Q is obtained by dividing the LTC of $60 (point A' on the LTC curve in ‘the middle panel) by 1. This is plotted as point A" in the bottom panel. Note that the LAC curve declines up to point G" (4Q) because of increasing returns to scale and rises thereafter because of decreasing returns to scale. For an increase in output from 0 to lQ, LTC increases from $0 to $60. Therefore, LMC is $60 and is plotted at 0.5Q (i.e., halfway between 0Q and 1Q) in the bottom panel. The LMC curve intersects the LAC curve from below at the lowest point (G") on the LAC curve.
Fig. 8-2
8.4 PLANT SIZE AND ECONOMIES OF SCALE
The long-run average cost (LAC) curve is the tangent or “envelope” to the short-run average cost (SAC) curves and shows the lowest average cost of producing each level of output when the firm can build any scale of plant. If the firm can build only a few scales of plants, its LAC curve will not be smooth as in the bottom panel of Fig. 8-2 but will have kinks at the points where the SAC curves cross (see Example 4). The firm operates in the short run and plans for the long run (the planning horizon). Declining LACs reflect increasing returns to scale. These arise because as the scale of operation increases, a greater division of labor and greater specialization can take place, and more specialized and productive machinery can be used (technological reasons). Furthermore, large firms can receive quantity discounts by purchasing raw material in bulk, can usually borrow at lower rates than small firms, and can achieve economies in their promotional efforts (financial reasons). However, rising LAC reflects decreasing returns to scale. These arise because, as the scale of operation increases, effective management of the firm becomes ever more difficult [see Problem 8.11(d)]. In the real world, the LAC curve is often found to have a nearly flat bottom and to be L-shaped rather than U-shaped.
Fig. 8-3
EXAMPLE 4. If the firm can build only the four scales of plant given by SAC1, SAC2, SAC3, and SAC4 in Fig. 8-3. the LAC curve of the firm is A" B*C"E*G"J*R". This shows that the minimum LAC of producing 1Q is $60 and arises when the firm operates plant 1 at point A". The firm can produce 1.5Q at LAC = $55 by utilizing either plant 1 or plant 2 at point B*. To produce 2Q, the firm will utilize plant 2 at point C" ($40) rather than plant 1 at point C* (the lowest point on SAC1, which refers to the average cost of $53). If the firm could build many more scales of plant, the kinks at points B*, E*, and J* would become less pronounced and at the limit would disappear, thus giving a smooth LAC curve (as in the bottom panel of Fig. 8-2).
8.5 COST-VOLUME-PROFIT ANALYSIS AND OPERATING LEVERAGE
Cost-volume-profit (or break-even) analysis examines the relationship among total revenue, total costs, and total profits of the firm at various levels of output. This technique is often used by business executives to determine the sales volume required for the firm in order to break even, as well as to determine the total profits and losses at other sales levels. The analysis utilizes a chart in which the total revenue (TR) and the total cost (TC) curves are represented by straight lines, and the break-even output (QB) is determined at their intersection (see Example 5). QB can also be determined algebraically by
(8-8)
where TFC refers to total fixed costs, P refers to price, and AVC to average variable costs. The denominator in equation (8-8), i.e., P – AVC, is called the contribution margin per unit because it represents the portion of the selling price that can be applied to cover the fixed costs of the firm and to provide for profits. The target output (QT) to earn a specific target profit (pT) is then given by
(8-9)
The ratio of the firm’s total fixed costs to total variable costs is called its operating leverage. When a firm becomes more highly leveraged, its total fixed costs increase, its average variable costs decline, its break-even output is larger, and its profitability becomes more variable. The degree of operating leverage (DOL), or sales elasticity of profits, measures the percentage change in the firm’s total profits resulting from a one-percentage-point change in the firm’s output or sales and can be obtained from
(8-10)
Linear cost-volume-profit analysis is applicable only if prices and average variable costs are constant.
EXAMPLE 5. In Fig. 8-4, the slope of the TR curve refers to P = $20, the price at which the firm can sell its output. TFC = $300 (the vertical intercept of the TC curve) and AVC = $10 (the slope of the TC curve). The firm breaks even (with TR = TC = $600) at Q = 30 (point B in the figure). The firm has losses at smaller outputs and profits at higher outputs. Algebraically,
Fig. 8-4
The contribution margin per unit (i.e., P – AVC) is $10.
The target output (QT) to earn the target profit of pT = $100 is
This is shown at the intersection of the TR and TC + pT curves in Fig. 8-4. At Q = 50 on the TC curve,
DOL increases as the firm becomes more highly leveraged (i.e., more capital intensive) and as it moves closer to the break-even output (see Problem 8.15). The effect of an increase in P can be shown by increasing the slope of the TR curve; an increase in TFC, by an increase in the vertical intercept of the TC curve; and an increase in AVC, by an increase in the slope of the TC curve. The chart will then show the change in QB and the profits or losses at other output or sales levels (see Problem 8.14).
8.6 EMPIRICAL ESTIMATION OF COST FUNCTIONS
The firm’s short-run cost functions are usually estimated by regression analysis, whereby total variable costs are regressed on output, input prices, and other operating conditions, for the time period within which the size of the plant is fixed. To do this, opportunity costs must be extracted from the available accounting cost data. Costs must be correctly apportioned to the various products produced and matched to output over time. The period of time for the estimation must be chosen, and costs must be deflated to correct for inflation. While economic theory postulates an S-shaped (cubic) TVC curve, a linear approximation often gives a better empirical fit for the observed range of outputs (see Problem 8.21).
Since the type of product that the firm produces and the technology it uses are likely to change in the long run, the long-run average cost curve is estimated by cross-sectional regression analysis (i.e., with cost-quantity data for a number of firms at a given point in time). Since firms in different geographic areas are likely to pay different prices for their inputs, input prices must be included, together with output, as independent explanatory variables in the regression. The different accounting practices of the firms in the sample must be reconciled, and we must ensure that the firms are operating the optimal scale of plant at the optimal level of output (i.e., at the point on the SAC curve of the firm that forms part of its LAC curve). Empirical studies indicate that the LAC curve is L-shaped (see Problem 8.13).
The LAC curve can also be estimated by the engineering technique and the survival technique. The first method estimates the LAC curve by determining the optimal input combinations necessary to produce various levels of output, given input prices and the available technology. The survival technique determines the existence of increasing, decreasing, or constant returns to scale depending on whether the share of industry output coming from large firms (as compared with the share of industry output coming from small firms) increases, decreases, or remains the same over time.