Having established that, for Marshallian firms, adjusting to market dynamics in short run periods where capital is assumed to be fixed, the productivity of labor services at the margin of employment is apt to diverge from the average, we may assume that, in terms of costs per unit of output, the firm's production expenses at the margin may diverge from their average total costs. Emphatically, the point of the next two sections will be to outline production by Marshallian firms under conditions where the marginal costs incurred by the firm for increasing production constrained by fixed factors rise above the firm's average costs. This section, in particular, seeks to develop the conceptual apparatus through which we will analyze Marshallian firms engaged in (roughly) competitive markets in the short run.
On the most general level, firms incur a set of expenses in producing a given short run scale of output that can be aggregated to constitute the firm's total costs, incorporating both the variable costs of factors that can be readily varied (labor and, perhaps, circulating raw and intermediate goods) and the fixed/sunk costs for factors that cannot be varied in the short run (capital equipment and infrastructure/plant). As such we can define the firm's total costs as:
Total Costs (TC) = Total Variable Costs (TVC) + Total Fixed/Sunk Costs (TFC)
To the extent that we approach the problem of short run costs from a simple two-factor model, where labor services are variable and capital is fixed, our aggregation of costs involves a set of fixed expenses for capital, unresponsive to output quantities, and a set of labor expenses that vary as we increase total output. Our total variable cost calculation, thus, measures changes in labor costs as we incrementally increase output quantities. In pricing terms, we express this sum as:
Which simply reiterates our total cost calculation for the Walrasian/Paretian firm with the capital quantity fixed.
In Marshallian theory, short run variable and total cost curves tend to assume a particular form, again, driven by the diminishing marginal productivity of labor. As the productivity of each additional labor unit rented by the firm at the margin of employment diminishes, labor costs incorporated in the production of successive units of output increase. We encounter increasing marginal costs per unit of output, which drive increases in average costs per unit of output. Graphically, at the level of total cost and total variable cost, this pattern of diminishing marginal productivity has a particular effect. If the short run total product curve approaches a horizontal asymptote at some maximum attainable output level, then the total cost and total variable cost curves, drawn in two-dimensional output-price/cost space, will approach a vertical asymptote at the same maximum attainable output level, at which total cost and total variable cost will approach infinity. This is demonstrated in figure 6.
Elaborating on figure 6, fixed costs are, by their nature, fixed for all quantities of output produced by the firm or for zero output - if the firm is predisposed to maintain a certain quantity of capital, then the expense for this capital must be borne by the firm whether or not it produces any outputs. The total fixed cost curve is, thus, drawn as a horizontal line, defined by the cost of capital. The total variable cost curve expresses the cost of labor for all quantities of output, which increases at changing rates as we increase outputs, reflecting the initially increasing marginal productivity of labor and its subsequent diminution. To the extent that labor costs are zero for the firm at zero output, the total variable cost curve emanates from the output-cost origin. Finally, the total cost curve, as the combination of fixed and variable costs, emanates from the cost-intercept for fixed costs, because, in the short run, the firm is entirely constrained to operate with the costs of its fixed factor. Graphically, it represents a shifting of the total variable cost curve to the fixed cost intercept and, thus, its curvature is identical to that of the total variable cost curve.
The horizontal asymptote xiMax on the total product curve in the upper frame of figure 6 is transposed as a vertical asymptote on the total cost graph. In the same way that the total product curve will increase at a diminishing rate asymptotically approaching xiMax, the total cost and total variable cost curves are asymptotically bounded by xiMax. As the total cost and total variable cost curves approach this quantity, they will take on infinitely large values. Reading between graphs to approach the broader logic embodied in this asymptotic form, we are presuming that there is some absolute limit on the productivity of labor in a particular production process, under constraints on capital and infrastructure. It may always be possible for the firm to introduce more units of labor, but, beyond a certain point, they will not add anything to total output. They represent pure additions to total cost and total variable cost. At this point, the slopes of these curves will approach infinity.
The point A on the total product curve in the upper panel, depicting the combination of labor and capital at which the average productivity of labor is maximized coincides with the total variable cost curve, TVCi1, where the total variable cost curve intersects with a ray from the origin at a single point of tangency. This output level, xi1, constitutes the minimum average variable cost for the firm. Acknowledging that the firm's variable labor cost is minimized here, we further need to consider that the firm's fixed costs are distributed across the span of its total output. As such, its average fixed cost decline continuously. When we account for average fixed costs, as a component of average total costs per unit of output, the level of output at which we realize the minimum of average total costs is different and higher than the level at which we realize the average variable/labor cost minimum. We can designate this level graphically by drawing a ray from the origin that intersects with the average total cost curve at a single point of tangency. This cost level is depicted as TCi2.
The total cost curves above generate a set of three average cost curves and the marginal cost curve for the firm. Again, the average total cost, average variable cost, and marginal cost curves strongly reflect a relationship to changes in the marginal productivity of labor derived from the firm's short run production function as the firm increases the margin of employment under a fixed capital base. These curves are depicted in figure 7.
Elaborating, the average fixed cost curve, as the simplest of the above relationships, declines continuously as the cost of fixed capital is spread over an increasing quantity of outputs. As outputs approach zero, each of the average cost curves approach positive infinity. As outputs approach positive infinity, average fixed cost approaches zero. Conversely, the average variable cost and average total cost curves are U-shaped curves. They decline at a diminishing rate from positive infinity for an initial range of output quantities, reaching a minimum value where the slope of each curve equals zero. From this minimum, the curves increase at an increasing rate until they again approach positive infinity. Relating back to the marginal productivity of labor, the point here is that, against a fixed base of capital, increasing marginal labor productivity reduces average costs per unit of output for an initial range but, beyond a point at which the marginal productivity of labor attains its maximum, average total and variable costs approach their minimum values and begin to increase because marginal labor productivity is diminishing. To the extent that diminishing marginal productivity of labor is a universal principle for all Marshallian firms in the short run, all Marshallian short run average total and variable cost curves assume a U-shape.
Our explanation for the shape of the marginal cost curve follows the same principle even more directly. The additional/marginal cost as we increase output by discrete units must, likewise, decline as the marginal productivity of labor increases. At the very point at which the marginal productivity of labor is maximized, marginal cost per unit is minimized. Beyond this point, as the marginal productivity of labor begins to decline, the cost of each additional unit of output begins to rise. The marginal cost curve then crosses the average variable and total cost curves at their minimum points. These points are each denoted in figure 7. The mathematical logic underlying the equality of marginal cost with average variable and average total costs at their minima is intuitive. If we are calculating average costs by adding the cost of each additional unit to a series of costs for successive discrete units, and if marginal costs increase monotonically from their minimum, then the average variable and total cost series must reach their minima where the cost of an additional unit of output equals the average. For all preceding units of output beyond the marginal cost minimum, marginal costs are below the average variable and total cost series, and, thus, their addition to the average variable and average total cost series must cause the averages to decline. Beyond the average variable and total cost minima, the monotonically increasing values of marginal cost must cause each of these series to rise because the cost of each additional unit beyond these minima must exceed each of the averages.
The principle that Marshallian firms universally operate with increasing short run marginal costs is theoretically definitive. To the extent that certain production factors must be held fixed in the short run and the marginal productivity of variable factors diminish as we increase employment, it is inevitable that we encounter increasing short run marginal costs per unit of output. Returning to the short run profit maximizing condition and the short run labor demand schedule introduced in the previous section, we can further relate the condition of increasing short run marginal costs to profit maximization by the firm. Specifically, short run profit maximization by a Marshallian firm operating within a relatively competitive output market in which the firm is incapable by itself of affecting output prices demands that the firm produce more units of output up to the quantity at which the marginal cost of the last unit of output equals the equilibrium output market price paid by consumers, expressed as:
Where, in turn, the firm's marginal revenues from the sale of the last additional unit produced and sold are equal to the output market price per unit, because the firm is incapable of affecting, by itself, the output market price in relation to its output decisions. Intuitively, the logic here is that, facing increasing marginal costs with each additional unit of output, the firm will suffer foregone revenues if it produces a quantity of output at which the marginal cost of the last unit is less than the price paid for the unit. Such foregone revenues are a cost to the firm that could be eliminated by increasing production. Alternatively, if the firm produces a quantity of output at which the marginal cost of the last unit exceeds the output price, then this last unit will be sold at a price less than its cost of production. The difference here constitutes a cost to the firm that could be minimized by decreasing production. The firm only minimizes these costs and, hence, maximizes profits if it produces up to the quantity at which the marginal cost of the last unit produced exactly equals the output market price that the firm receives for selling the unit. We can label this conclusion our short run marginal cost pricing rule.
Proceeding further, we can graphically limit the range of the marginal cost curve at which the firm will produce positive quantities of output. This is shown in figure 8.
Our reasoning for accenting this particular region of the firm's marginal cost curve, from the minimum of the average variable cost curve to positive infinity, requires some elaboration. Emphatically, at the minimum point of average variable cost, the marginal cost/output price per unit received by the firm just equals its average labor costs per unit. It can satisfactorily pay its workers for participating in production even if it loses the cost of employing its fixed factor capital. At this point the firm can cover zero sunk costs for capital. If the output market price declines from this point, then the firm will be unable to fully cover its average labor costs per unit of output, in addition to failing to cover any of its sunk costs. It will, therefore, choose not to produce at all, incurring the loss of its sunk costs but incurring zero labor costs. For this reason, the minimum point of the average variable cost curve is defined as the shut down boundary of the firm. Beyond it, the firm will produce zero output in the short run.
As we move up from the shut down boundary, the firm begins to offset some of its sunk costs. Between the minimum points of the average variable cost curve and the average total cost curve, the firm continues to incur some losses in sunk costs, but it can more than cover its average labor costs per unit. At the point where the marginal cost curve crosses the minimum of the average total cost curve, the firm fully covers both its average variable/labor costs and its sunk average fixed costs. In effect, the minimum point of the average total cost curve represents the Marshallian equivalent of a point of product exhaustion, where the revenues of the firm are fully expended paying the factors of production at their factor market compensation rates. In contrast to the circumstances of product exhaustion under linearly homogeneous production technologies for Walrasian/Paretian firms, we will argue that there is, emphatically, no reason to expect that Marshallian firms will operate at a minimum average total cost in the short run. Rather, short run market conditions may consign firms to earn revenues per unit of output above, below, or equal to minimum short run average total costs.
As the output market price continues up the marginal cost curve above our product exhaustion point, the firm begins earning revenues per unit for each of its outputs in excess of its average total costs per unit. It, therefore, begins to earn short term profits. This conclusion is important, if only because it represents the first moment in our consideration of the firm in Neoclassical economic theory when a firm actually earns positive economic profits, differentiated from economic rents, which accrue to the owners of absolutely scarce goods, services, and/or factors of production not otherwise subject to competitive pressures, and normal profits, constituted as a regular rate of return to entrepreneurship. Economic profits, by contrast, exist as short run excess returns over average total costs per unit, driven by fluctuations in market conditions, that we would expect to see competed away in the long run. Our analysis of the Walrasian/Paretian firm recognized only the possible existence of economic rents, insofar as it is always possible that the household owners of absolutely scarce production factors could command an excess return that might never be competed away. Such firms never encounter, however, normal profits because they are functionally absent an entrepreneur. Nor is it possible for Walrasian/Paretian firms to enjoy economic profits, because they operate in a timeless environment of seamless and continuous tâtonnement - no short run periods exist in which firms and industries may undertake discrete readjustments to changes in market conditions. Marshallian firms, on the other hand, are always capable of earning economic profits under the appropriate market conditions, which are always mitigated by the potential for competition to eliminate short run profits and, perhaps, generate short run losses.
Coming to the end of our analysis of short run costs, we can draw some basic analogies between output cost analysis and our prior analysis of labor market employment decisions. Emphatically, in our analysis of labor markets, the firms demands positive quantities of labor services only under conditions where the marginal revenue product it receives from the last unit of labor services employed is less than or equal to the average revenue product that it receives from all the labor services it employs. Translating this conclusion in reference to our cost analysis, this condition is equivalent to the conclusion that the firm will only supply positive quantities of output to the extent that the output market price that it receives is greater than or equal to the minimum of its average variable/labor cost. This minimum resides at the output quantity generated by labor where the marginal revenue product generated by the last unit of labor services hired is exactly equal to the average product of all labor services. Read in these terms, the firm's short run demand schedule for labor services mirrors, in certain respects, its short run output supply schedule. Both schedules describe different aspects of the short run production and market conditions faced by firms, indicating the conditions under which firms will hire positive quantities of labor services to produce positive quantities of output.
Short Run Production Analysis for a Marshallian Firm
Let us now develop an example of short run production analysis utilizing the analytic tools so far introduced in this document. In this regard, I want to build a fairly simple example around our hypothetical remanufacturer of master brake cylinders. Let us say that the entrepreneur of this firm faces a short run production function defined, for his purposes, in reference to an existing quantity of fixed capital in machinery and infrastructure and employment of one to seven employees for daily production schedules. As a matter of everyday expectations, regarding average exertion by individual employees with roughly equivalent skills and experience in the remanufacturing of brake cylinders, the entrepreneur estimates the production schedule in table 1 to obtain over an eight hour day:
| Labor | Total Product |
| 1 | 5 |
| 2 | 11 |
| 3 | 16 |
| 4 | 20 |
| 5 | 23 |
| 6 | 25 |
| 7 | 26 |
Graphing as closely to scale as possible, I get figure 9 below.
Perusing this schedule, there is a definite conformity to our theoretically formed expectations about the marginal productivity of labor as we increase employment. The first employee produces five units of output in a day. If we add a second employee, under the assumption that we are adding an individual with roughly equal skill, experience, and dexterity in the manufacturing process, the additional output that we get from this second employee over a day is six units. As such, cooperation between the two employees raises the average output from both. When we add a third employee, however, the additional output yielded from the increased labor services is back down to five units, resulting in a slight decrease in the average productivity of the three employees. I summarize the average and marginal productivity of labor in table 2:
| Labor | Ave. Product | Marg. Product |
| 1 | 5 | 5 |
| 2 | 5.5 | 6 |
| 3 | 5.33 | 5 |
| 4 | 5 | 4 |
| 5 | 4.6 | 3 |
| 6 | 4.17 | 2 |
| 7 | 3.71 | 1 |
Table 2: The Average and Marginal Products of Labor in Daily Remanufacturing of Master Brake Cylinders
Figure 10 graphically displays these schedules.
Figure 10: Average and Marginal Productivity Schedules for Remanufactured Master Brake Cylinders
Figure 10 graphically displays these schedules.
The maximum average productivity of labor, represented as a point of equality between the marginal and average productivity schedules, thus, occurs at an employment level between two and three full daily labor schedules. If we were to hold that the entrepreneur is not capable of hiring on an additional worker for less than an eight hour work day, then, by our conclusion that the firm's labor demand schedule under competitive market conditions extends outward from the maximum point on the average product of labor schedule, we would have to argue that the firm will always maintain a minimum of three employees to the extent that it produces positive outputs (i.e. if the market price for its outputs does not fall so low that it cannot justify producing any output under existing market wages). We will, further, see in our short run cost analysis that the firm's shut down boundary and output supply schedule mirrors this conclusion.
To enter into a short run cost analysis, we have to define and enumerate a number of cost categories. First, we know that the firm maintains and operates facilities for which it faces rental expenses and fixed capital costs, including maintenance and depreciation on existing machinery. For purposes of our analysis, let us say that these costs add up to $3,696 per month or, over twenty-two monthly production days, $168 per day, regardless of whether the firm produces any positive outputs on a given day. These amount to the firm's short run fixed/sunk costs. By contrast, if the firm produces positive outputs, we know that it will begin to accumulate costs for its variable factor of production, labor services. Again, for the sake of our analysis, let us say that hourly wages for labor services amount to $14 per hour or, for an eight hour day, $112 per worker. Combining these cost categories with our knowledge of production schedules, we should be able to construct some portrait of production costs per unit of output. Here, however, I am going to complicate our analysis slightly insofar as we are extrapolating from a production schedule defined per worker over an entire daily production schedule. In effect, undertaking a cost analysis forces us to ask how the cost of labor services per employee are distributed over an entire day's outputs. In this regard, I am advancing two separate variations constructed on different assumptions about the distribution of labor costs.
Variation number one, extrapolated in table 3, is developed according to the assumption that each additional employee added by the firm experiences increasing then diminishing marginal productivity as a function of the length of their working day. As such, for the first employee, who produces a total of five master cylinders, the first master cylinder produced takes a greater investment of time than the second and the second a greater investment of time than the third. At some point, however, this diminution of production time will turn around so that the fifth master cylinder takes more time to produce than the fourth. As we add additional workers, our broader pattern of increasing then diminishing marginal productivity is, similarly, reflected, even as we continue to account for increasing then diminishing marginal productivity over the course of an eight-hour day for each individual added employee. Measuring the total and average cost categories and marginal cost, in this respect, we can posit the following:
| Output | Total Variable | Total Fixed | Total Cost | Marginal Cost | Ave. Variable | Ave. Fixed | Ave. Total |
| 1 | 40 | 168 | 208 | 40 | 40 | 168 | 208 |
| 2 | 60 | 168 | 228 | 20 | 30 | 84 | 114 |
| 3 | 78 | 168 | 246 | 18 | 26 | 56 | 82 |
| 4 | 94 | 168 | 262 | 16 | 23.5 | 42 | 65.5 |
| 5 | 112 | 168 | 280 | 18 | 22.4 | 33.6 | 56 |
| 6 | 146 | 168 | 314 | 34 | 24.33 | 28 | 52.33 |
| 7 | 166 | 168 | 334 | 20 | 23.71 | 24 | 47.71 |
| 8 | 185 | 168 | 353 | 19 | 23.13 | 21 | 44.13 |
| 9 | 198 | 168 | 366 | 13 | 22 | 18.67 | 40.67 |
| 10 | 210 | 168 | 378 | 12 | 21 | 16.8 | 37.8 |
| 11 | 224 | 168 | 392 | 14 | 20.36 | 15.27 | 35.63 |
| 12 | 252 | 168 | 420 | 28 | 21 | 14 | 35 |
| 13 | 278 | 168 | 446 | 26 | 21.38 | 12.92 | 34.3 |
| 14 | 296 | 168 | 464 | 18 | 21.14 | 12 | 33.14 |
| 15 | 315 | 168 | 483 | 19 | 21 | 11.2 | 32.2 |
| 16 | 336 | 168 | 504 | 21 | 21 | 10.5 | 31.5 |
| 17 | 380 | 168 | 538 | 34 | 22.35 | 9.88 | 32.23 |
| 18 | 402 | 168 | 560 | 22 | 22.33 | 9.33 | 31.66 |
| 19 | 422 | 168 | 580 | 20 | 22.21 | 8.84 | 31.05 |
| 20 | 448 | 168 | 606 | 26 | 22.4 | 8.4 | 30.8 |
| 21 | 500 | 168 | 658 | 52 | 23.81 | 8 | 31.81 |
| 22 | 528 | 168 | 686 | 28 | 24 | 7.64 | 31.64 |
| 23 | 560 | 168 | 718 | 32 | 24.35 | 7.3 | 31.65 |
| 24 | 630 | 168 | 788 | 70 | 26.25 | 7 | 33.25 |
| 25 | 672 | 168 | 830 | 42 | 26.88 | 6.72 | 33.6 |
| 26 | 784 | 168 | 942 | 112 | 30.15 | 6.46 | 36.61 |
Table 3: Total, Average, and Marginal Costs under Labor Productivity Variation 1
Figure 11 conveys the total fixed cost, total variable cost, and total cost schedules for the firm under this variation of labor productivity.
Figure 11: Total Fixed, Total Variable, and Total Cost of Master Brake Cylinders under Labor Productivity Variation 1
The significance of this particular variation on labor productivity, in relation to the second variation that we will subsequently posit, resides in the fact that we encounter certain "hiccups" in average variable and marginal costs as we add additional employees, reflected in the fact that this variation precludes a "smooth" addition to total labor productivity when a new employee, with its own daily marginal productivity profile, is added. For example, the marginal cost per unit of output declines for the first four units of output with a single employee, reflecting the marginal productivity contour for this employee over the workday. As we move to the sixth unit of output, the marginal cost jumps from $18 to $34, because, having added a second employee, with a unique marginal productivity schedule, we have to account for the impact of an initially lower marginal productivity with the second employee. This pattern is similarly reflected in average variable cost, which jumps by almost two dollars from the fifth to the sixth unit of output. As such, we lack smooth, continuous marginal and average variable cost schedules. Figure 12 attempts to graph these schedules as closely to scale as possible(!).
Figure 12: Marginal, Average Variable, Average Fixed, and Average Total Cost Schedules for Master Brake Cylinder, Variation 1
The average total cost schedule, to a certain degree, lacks significant hiccups, at least initially. The main reason why average total cost tends to enjoy a more continuously smooth contour is because discontinuities in the marginal productivity of labor are counterbalanced by the continuous decline of average fixed costs. We only see an abrupt discontinuity in the average total cost schedule as the decline in average fixed costs begins to level off, as we transition from sixteen to seventeen units of output (integrating a fourth employee into production, a transition inadequately conveyed by figure 12!).
With regard to the key production cost thresholds defining the firm's short run output supply schedule, the discontinuities in marginal productivity built into our model generate certain complex incongruities. At eleven units of output, the average variable cost curve reaches its absolute minimum value at $20.36 in labor costs per unit of output. This is not, however, the shutdown boundary of the firm, because the marginal cost incurred by the firm at this level is only $14 for producing the eleventh unit. If, as a criterion for profit maximization, the firm should produce up to the point at which its marginal cost is equal to the output market price, then, at eleven units, such a condition would not allow the firm to fully recover its total variable costs, let alone any of its fixed costs. Instead, the actual shut down boundary for the firm is at sixteen units of output, where the average variable cost per unit of output is equal to the marginal cost at $21 per unit, an outcome that is highlighted in table 3. If the firm were to operate at this level in the short run, it would fully cover its total variable costs without covering any of its fixed/sunk costs of $10.50 per unit of output.
Returning briefly to connect our production/employment level analysis to our cost analysis, we previously stated that, if the firm produces any positive output, then it will always maintain a minimum of three employees. Again, to the extent that the firm can only hire employees for a full work day, this outcome is verified with respect to its cost analysis. In the short run, the firm will only produce positive quantities of output to the extent that the output market price meets or exceeds $21 per unit of output, where the marginal cost equals the average variable/labor cost of production. This outcome occurs at sixteen units of output, where the firm is operating with three employees on a full work day schedule. To the extent that the firm must operate with at least three employees if it produces any output, it will only produce output if its returns from production at least recover its full labor expenses. The construction of the firm's short run labor demand schedule, thus, mirrors the construction of its short run output supply schedule.
Finally, in table 3, I have highlighted the marginal cost ($32) and average total cost ($31.65) obtaining at twenty- three units of output. This level of output comes closest to realizing a minimum of short run average total cost at which the firm receives its total costs for employing both fixed and variable factors if it follows the rule that it should produce up to the point at which the output market price equals its marginal cost for the last unit of output. In fact, it the firm receives its marginal cost at twenty-three units of output, it will receive an additional revenue of $.35 per unit over and above its total costs. For now, I will ignore this as a mathematical peculiarity of the numbers I included, but I intend to return to it briefly as a subject for further inquiry. The larger point here is that, with the firm employing each of its employees for a full production day, this threshold implies that it will come closest to product exhaustion if it employees five workers, producing a total of twenty-three units of output. Again, the absolute minimum of the average total cost curve ($30.80) actually occurs at twenty units of output, where the marginal cost is $26 for the twentieth unit. If the firm followed its profit maximization rule here, then it would fully cover its labor costs but incur a loss of $4.80 per unit of fixed/sunk costs.
Summarizing, thus, the short run cost analysis evident under our first analytic variation, the firm will produce positive quantities of master brake cylinders only to the extent that it can receive at least $21 per cylinder. If it receives this price, it will produce sixteen cylinders per day with three employees in the short run, although, over time, if the market does not improve, the firm may cease operations altogether and liquidate its fixed capital, for which, at an output market price of $21, it cannot cover its costs. On the other hand, if the output price is at least $31.65, the firm will be able to fully cover its costs for both labor and capital by producing twenty-three cylinders per day with five employees. If the price goes above $32 to, say, $40 per cylinder, then the entrepreneur may decide to hire a sixth employee, and, following our condition that such an employee would be hired for a full work day, the firm might produce twenty-five cylinders at an average total cost of $33.60 per cylinder, earning $6.40 in economic profit per cylinder even though the marginal cost of the last cylinder exceeds the output market price by $2.
Our second analytic variation on cost analysis attempts to smooth out the variation in variable factor expenses per unit of output in order to create smooth average variable, average total, and marginal cost schedules. Rather than assume that each individual worker has its own daily marginal productivity profile, we assume that the marginal productivity schedule follows a particular profile across all workers such that the first two workers enjoy increasing marginal productivity and, thus, decreasing per unit marginal costs, and all subsequently workers enjoy diminishing marginal productivity and rising marginal costs. Table 4 encapsulates this cost analysis:
Table 4: Total, Average, and Marginal Costs under Labor Productivity Variation 2
Figure 13 displays the total variable, total fixed, and total costs for this variation graphically.
Figure 13: Total Fixed, Total Variable, and Total Costs for Master Brake Cylinders, Variation 2
Figure 14 displays the marginal, average total, average variable, and average fixed cost schedules for this variation.
Figure 14: Marginal, Average Total, Average Variable, and Average Fixed Costs for Master Brake Cylinders, Variation 2
My purposes in introducing a second variation on labor productivity was explicitly to achieve a smoother set of cost curves without discontinuities, however much these discontinuities appeared to more accurately represent, at least in my view, an actual, experiential Marshallian vision of the operation of real firms. As it stands, the second variation introduces some interesting problems, centered on the production cost thresholds. Most critically, we argued in the course of analyzing the first variation that the entrepreneur would necessarily hire at least three employees if he, in fact, produced any positive quantities of brake cylinders for the market. In the previous scenario, the rationale for this decision was clear, and the grounds for hiring workers for a full working day was, likewise, presented in the productivity data. In this case, spreading marginal productivity across all workers, by contrast, we face certain conundrums! In particular, the shut down boundary for the firm becomes unclear if we allow the entrepreneur to hire labor services on a part time basis. Under the current variation, the firm minimizes its average variable costs between thirteen and fourteen units of output, levels at which the firm requires more than two but less than three full work days of labor services. If we allowed the entrepreneur to hire less than a full work day of labor, then he might hire enough labor services to obtain fourteen units of output. Alternatively, forcing the entrepreneur to hire labor services on a full work day rate, the firm would produce sixteen units of output at its shut down boundary, generating revenues per unit of output in excess of minimum average variable costs but not achieving product exhaustion.
Similarly, under the current variation, average total cost is minimized at twenty units of output ($30.70 per unit), produced by four workers. At this level of output, the marginal cost of the twentieth unit of output remains $1.70 less than the average total cost, resulting in excess fixed costs per unit. If we allow the entrepreneur to hire another worker for part time, then the firm might produce twenty-one units of output, at which the firm's marginal cost of $33 for the twenty-first unit would exceed its average total cost ($30.81 per unit) by $2.19 per unit. Alternatively, if we force the entrepreneur to hire labor services only on a full work day basis, then he might hire a fifth worker to produce twenty-three units of output at a marginal cost of $46 for the twenty-third unit and an average total cost of $31.73, realizing an excess revenue of $14.27 per unit of output over average total cost. In this manner, if the firm strictly adheres to its marginal cost pricing rule for profit maximization and the entrepreneur is forced to hire only full work day units of labor services, then the firm will come closest to achieving product exhaustion with four workers and twenty units of output. However, insofar as the firm can achieve excess revenues per unit of output by producing twenty-three units of output with five workers at output market prices in excess of its average total costs but below its marginal cost for the twenty-third unit, such a production level remains a strategic option for the entrepreneur. Having acknowledged this result, our cost analysis in the second variation does not necessarily provide us with a clear answer to the question of how much to produce, under the constraint that the entrepreneur can only hire labor services at a full work day rate.
Concluding this section, I want to return briefly to the short run marginal cost pricing rule in our examples, especially in reference to the existence of excess revenues over average total costs. Again, it intuitively makes sense that the firm should always produce up to the point at which the additional revenues that it receives from the last unit of output sold are exactly equal to the cost of producing this last unit, provided the revenues per unit of output can at least cover the firm's variable/labor costs. As the cases above indicate, however, placing constraints on hiring can impact an entrepreneur's decision-making in relation to the marginal cost pricing rule. In particular, if the output market price faced by the firm in our second variation above stands at $46 per unit, then it makes sense that the firm should hire five workers to produce twenty-three master brake cylinders. What, however, should the firm do if the price falls to $40 per unit? The technical constraints of the model just do not allow the entrepreneur to adjust production quantities to shift to an output level at which the marginal cost of the last unit will equal the price. Consequently, the firm will likely continue to produce twenty-three units and endure a $6.00 loss in excess revenues relative to its marginal costs. It will still, however, continue to earn excess revenues in relation to its average total costs per unit. The point that I am seeking to make here is that, intuition notwithstanding, we do not necessarily need to treat the short run marginal cost pricing rule as a theoretic straitjacket provided the price is above the firm's shut-down boundary. At all prices above this level, entrepreneurs enjoy at least some margin of strategic decision space.
The significance of this particular variation on labor productivity, in relation to the second variation that we will subsequently posit, resides in the fact that we encounter certain "hiccups" in average variable and marginal costs as we add additional employees, reflected in the fact that this variation precludes a "smooth" addition to total labor productivity when a new employee, with its own daily marginal productivity profile, is added. For example, the marginal cost per unit of output declines for the first four units of output with a single employee, reflecting the marginal productivity contour for this employee over the workday. As we move to the sixth unit of output, the marginal cost jumps from $18 to $34, because, having added a second employee, with a unique marginal productivity schedule, we have to account for the impact of an initially lower marginal productivity with the second employee. This pattern is similarly reflected in average variable cost, which jumps by almost two dollars from the fifth to the sixth unit of output. As such, we lack smooth, continuous marginal and average variable cost schedules. Figure 12 attempts to graph these schedules as closely to scale as possible(!).
The average total cost schedule, to a certain degree, lacks significant hiccups, at least initially. The main reason why average total cost tends to enjoy a more continuously smooth contour is because discontinuities in the marginal productivity of labor are counterbalanced by the continuous decline of average fixed costs. We only see an abrupt discontinuity in the average total cost schedule as the decline in average fixed costs begins to level off, as we transition from sixteen to seventeen units of output (integrating a fourth employee into production, a transition inadequately conveyed by figure 12!).
With regard to the key production cost thresholds defining the firm's short run output supply schedule, the discontinuities in marginal productivity built into our model generate certain complex incongruities. At eleven units of output, the average variable cost curve reaches its absolute minimum value at $20.36 in labor costs per unit of output. This is not, however, the shutdown boundary of the firm, because the marginal cost incurred by the firm at this level is only $14 for producing the eleventh unit. If, as a criterion for profit maximization, the firm should produce up to the point at which its marginal cost is equal to the output market price, then, at eleven units, such a condition would not allow the firm to fully recover its total variable costs, let alone any of its fixed costs. Instead, the actual shut down boundary for the firm is at sixteen units of output, where the average variable cost per unit of output is equal to the marginal cost at $21 per unit, an outcome that is highlighted in table 3. If the firm were to operate at this level in the short run, it would fully cover its total variable costs without covering any of its fixed/sunk costs of $10.50 per unit of output.
Returning briefly to connect our production/employment level analysis to our cost analysis, we previously stated that, if the firm produces any positive output, then it will always maintain a minimum of three employees. Again, to the extent that the firm can only hire employees for a full work day, this outcome is verified with respect to its cost analysis. In the short run, the firm will only produce positive quantities of output to the extent that the output market price meets or exceeds $21 per unit of output, where the marginal cost equals the average variable/labor cost of production. This outcome occurs at sixteen units of output, where the firm is operating with three employees on a full work day schedule. To the extent that the firm must operate with at least three employees if it produces any output, it will only produce output if its returns from production at least recover its full labor expenses. The construction of the firm's short run labor demand schedule, thus, mirrors the construction of its short run output supply schedule.
Finally, in table 3, I have highlighted the marginal cost ($32) and average total cost ($31.65) obtaining at twenty- three units of output. This level of output comes closest to realizing a minimum of short run average total cost at which the firm receives its total costs for employing both fixed and variable factors if it follows the rule that it should produce up to the point at which the output market price equals its marginal cost for the last unit of output. In fact, it the firm receives its marginal cost at twenty-three units of output, it will receive an additional revenue of $.35 per unit over and above its total costs. For now, I will ignore this as a mathematical peculiarity of the numbers I included, but I intend to return to it briefly as a subject for further inquiry. The larger point here is that, with the firm employing each of its employees for a full production day, this threshold implies that it will come closest to product exhaustion if it employees five workers, producing a total of twenty-three units of output. Again, the absolute minimum of the average total cost curve ($30.80) actually occurs at twenty units of output, where the marginal cost is $26 for the twentieth unit. If the firm followed its profit maximization rule here, then it would fully cover its labor costs but incur a loss of $4.80 per unit of fixed/sunk costs.
Summarizing, thus, the short run cost analysis evident under our first analytic variation, the firm will produce positive quantities of master brake cylinders only to the extent that it can receive at least $21 per cylinder. If it receives this price, it will produce sixteen cylinders per day with three employees in the short run, although, over time, if the market does not improve, the firm may cease operations altogether and liquidate its fixed capital, for which, at an output market price of $21, it cannot cover its costs. On the other hand, if the output price is at least $31.65, the firm will be able to fully cover its costs for both labor and capital by producing twenty-three cylinders per day with five employees. If the price goes above $32 to, say, $40 per cylinder, then the entrepreneur may decide to hire a sixth employee, and, following our condition that such an employee would be hired for a full work day, the firm might produce twenty-five cylinders at an average total cost of $33.60 per cylinder, earning $6.40 in economic profit per cylinder even though the marginal cost of the last cylinder exceeds the output market price by $2.
Our second analytic variation on cost analysis attempts to smooth out the variation in variable factor expenses per unit of output in order to create smooth average variable, average total, and marginal cost schedules. Rather than assume that each individual worker has its own daily marginal productivity profile, we assume that the marginal productivity schedule follows a particular profile across all workers such that the first two workers enjoy increasing marginal productivity and, thus, decreasing per unit marginal costs, and all subsequently workers enjoy diminishing marginal productivity and rising marginal costs. Table 4 encapsulates this cost analysis:
| Output | Total Variable | Total Fixed | Total Costs | Marginal Cost | Ave. Variable | Ave. Fixed | Ave. Total |
| 1 | 24 | 168 | 192 | 24 | 24 | 168 | 192 |
| 2 | 47 | 168 | 215 | 23 | 23.5 | 84 | 107.5 |
| 3 | 69 | 168 | 237 | 22 | 23 | 56 | 79 |
| 4 | 91 | 168 | 259 | 22 | 22.75 | 42 | 66.75 |
| 5 | 112 | 168 | 280 | 21 | 22.4 | 33.6 | 56 |
| 6 | 132 | 168 | 300 | 20 | 22 | 28 | 50 |
| 7 | 152 | 168 | 320 | 20 | 21.71 | 24 | 45.71 |
| 8 | 171 | 168 | 339 | 19 | 21.38 | 21 | 42.38 |
| 9 | 189 | 168 | 357 | 18 | 21 | 18.67 | 39.67 |
| 10 | 207 | 168 | 375 | 18 | 20.7 | 16.8 | 37.5 |
| 11 | 224 | 168 | 392 | 17 | 20.36 | 15.27 | 35.63 |
| 12 | 243 | 168 | 411 | 19 | 20.25 | 14 | 34.25 |
| 13 | 263 | 168 | 431 | 20 | 20.23 | 12.92 | 33.15 |
| 14 | 284 | 168 | 452 | 21 | 20.29 | 12 | 32.29 |
| 15 | 307 | 168 | 475 | 23 | 20.47 | 11.2 | 31.67 |
| 16 | 334 | 168 | 502 | 27 | 20.88 | 10.5 | 31.38 |
| 17 | 361 | 168 | 529 | 27 | 21.24 | 9.88 | 31.12 |
| 18 | 389 | 168 | 557 | 28 | 21.61 | 9.33 | 30.94 |
| 19 | 417 | 168 | 585 | 28 | 21.95 | 8.84 | 30.79 |
| 20 | 446 | 168 | 614 | 29 | 22.3 | 8.4 | 30.7 |
| 21 | 479 | 168 | 647 | 33 | 22.81 | 8 | 30.81 |
| 22 | 516 | 168 | 684 | 37 | 23.45 | 7.64 | 31.09 |
| 23 | 562 | 168 | 730 | 46 | 24.43 | 7.3 | 31.73 |
| 24 | 617 | 168 | 785 | 55 | 25.71 | 7 | 32.71 |
| 25 | 674 | 168 | 830 | 57 | 26.96 | 6.72 | 33.68 |
| 26 | 784 | 168 | 942 | 112 | 30.15 | 6.46 | 36.61 |
Figure 13 displays the total variable, total fixed, and total costs for this variation graphically.
Figure 14 displays the marginal, average total, average variable, and average fixed cost schedules for this variation.
My purposes in introducing a second variation on labor productivity was explicitly to achieve a smoother set of cost curves without discontinuities, however much these discontinuities appeared to more accurately represent, at least in my view, an actual, experiential Marshallian vision of the operation of real firms. As it stands, the second variation introduces some interesting problems, centered on the production cost thresholds. Most critically, we argued in the course of analyzing the first variation that the entrepreneur would necessarily hire at least three employees if he, in fact, produced any positive quantities of brake cylinders for the market. In the previous scenario, the rationale for this decision was clear, and the grounds for hiring workers for a full working day was, likewise, presented in the productivity data. In this case, spreading marginal productivity across all workers, by contrast, we face certain conundrums! In particular, the shut down boundary for the firm becomes unclear if we allow the entrepreneur to hire labor services on a part time basis. Under the current variation, the firm minimizes its average variable costs between thirteen and fourteen units of output, levels at which the firm requires more than two but less than three full work days of labor services. If we allowed the entrepreneur to hire less than a full work day of labor, then he might hire enough labor services to obtain fourteen units of output. Alternatively, forcing the entrepreneur to hire labor services on a full work day rate, the firm would produce sixteen units of output at its shut down boundary, generating revenues per unit of output in excess of minimum average variable costs but not achieving product exhaustion.
Similarly, under the current variation, average total cost is minimized at twenty units of output ($30.70 per unit), produced by four workers. At this level of output, the marginal cost of the twentieth unit of output remains $1.70 less than the average total cost, resulting in excess fixed costs per unit. If we allow the entrepreneur to hire another worker for part time, then the firm might produce twenty-one units of output, at which the firm's marginal cost of $33 for the twenty-first unit would exceed its average total cost ($30.81 per unit) by $2.19 per unit. Alternatively, if we force the entrepreneur to hire labor services only on a full work day basis, then he might hire a fifth worker to produce twenty-three units of output at a marginal cost of $46 for the twenty-third unit and an average total cost of $31.73, realizing an excess revenue of $14.27 per unit of output over average total cost. In this manner, if the firm strictly adheres to its marginal cost pricing rule for profit maximization and the entrepreneur is forced to hire only full work day units of labor services, then the firm will come closest to achieving product exhaustion with four workers and twenty units of output. However, insofar as the firm can achieve excess revenues per unit of output by producing twenty-three units of output with five workers at output market prices in excess of its average total costs but below its marginal cost for the twenty-third unit, such a production level remains a strategic option for the entrepreneur. Having acknowledged this result, our cost analysis in the second variation does not necessarily provide us with a clear answer to the question of how much to produce, under the constraint that the entrepreneur can only hire labor services at a full work day rate.
Concluding this section, I want to return briefly to the short run marginal cost pricing rule in our examples, especially in reference to the existence of excess revenues over average total costs. Again, it intuitively makes sense that the firm should always produce up to the point at which the additional revenues that it receives from the last unit of output sold are exactly equal to the cost of producing this last unit, provided the revenues per unit of output can at least cover the firm's variable/labor costs. As the cases above indicate, however, placing constraints on hiring can impact an entrepreneur's decision-making in relation to the marginal cost pricing rule. In particular, if the output market price faced by the firm in our second variation above stands at $46 per unit, then it makes sense that the firm should hire five workers to produce twenty-three master brake cylinders. What, however, should the firm do if the price falls to $40 per unit? The technical constraints of the model just do not allow the entrepreneur to adjust production quantities to shift to an output level at which the marginal cost of the last unit will equal the price. Consequently, the firm will likely continue to produce twenty-three units and endure a $6.00 loss in excess revenues relative to its marginal costs. It will still, however, continue to earn excess revenues in relation to its average total costs per unit. The point that I am seeking to make here is that, intuition notwithstanding, we do not necessarily need to treat the short run marginal cost pricing rule as a theoretic straitjacket provided the price is above the firm's shut-down boundary. At all prices above this level, entrepreneurs enjoy at least some margin of strategic decision space.
