Chapter 11: Behind the Supply Curve: Inputs, Production Functions, and Cost Structures

The Production Function and Input Classifications

• Production Function Definition:

  • A production function is the mathematical and technical relationship between the quantity of inputs a firm uses and the quantity of output it produces.

• Input Classifications:

  • Variable Input: An input whose quantity the firm can vary and alter at any given point in time (for example, labor hours or raw materials).

  • Fixed Input: An input whose quantity is fixed and predetermined for a given period of time and cannot be varied immediately (for example, physical plant size, factory building, or farm land acreage).

• Operational Time Horizons:

  • The Short Run: The operational time horizon during which at least one input is fixed.

  • The Long Run: The operational time horizon in which all inputs can be varied, meaning no inputs are fixed.

• The Short-Run Total Product Curve ($TP$):

  • Shows how the quantity of output ($Q$ or $q$) depends on the quantity of the variable input employed, holding the quantity of the fixed input strictly constant.

Marginal and Average Productivity of Variable Inputs

• Marginal Product of Labour ($MPL$):

  • The marginal product of an input is the additional output produced by using one additional unit of that input, holding all other input levels fixed:     $MPL = \frac{\Delta q}{\Delta L} = \frac{\Delta Q}{\Delta L}$

• Diminishing Returns to an Input:

  • Diminishing returns occur when an increase in the quantity of a variable input, holding the levels of all other inputs fixed, leads to a progressive decline in that input's marginal product.

  • Although the total product curve slopes upward across its operational range, the slope diminishes and flattens from left to right as more units of the variable input are applied.

• Case Study Analysis: Alec and Janet’s Farm:

  • Production operates on a fixed land parcel of $10$ acres with variable farm labour ($L$) producing wheat ($Q$):

  • Employing $0$ workers yields $0$ bushels of wheat.

  • Employing $1$ worker yields $19$ bushels of wheat ($MPL = 19$ bushels per worker).

  • Employing $2$ workers yields $36$ bushels of wheat ($MPL = 36 - 19 = 17$ bushels per worker).

  • Employing $3$ workers yields $51$ bushels of wheat ($MPL = 51 - 36 = 15$ bushels per worker).

  • Employing $4$ workers yields $64$ bushels of wheat ($MPL = 64 - 51 = 13$ bushels per worker).

  • Employing $5$ workers yields $75$ bushels of wheat ($MPL = 75 - 64 = 11$ bushels per worker).

  • Employing $6$ workers yields $84$ bushels of wheat ($MPL = 84 - 75 = 9$ bushels per worker).

  • Employing $7$ workers yields $91$ bushels of wheat ($MPL = 91 - 84 = 7$ bushels per worker).

  • Employing $8$ workers yields $96$ bushels of wheat ($MPL = 96 - 91 = 5$ bushels per worker).

  • Adding a second worker yields an increase of $17$ bushels, whereas adding a seventh worker yields an increase of only $7$ bushels, illustrating diminishing returns to labor.

• Effects of Changing the Quantity of Fixed Inputs:

  • Increasing the fixed input (for example, expanding Alec and Janet's farm from $10$ acres to $20$ acres) allows each variable worker to become more productive because workers have more physical capital and space to work with.

  • An increase in acreage shifts the total product curve upward from $TP_{10}$ to $TP_{20}$.

  • Simultaneously, this increase shifts the marginal product of labour curve upward from $MPL_{10}$ to $MPL_{20}$.

• Economic Measurement Consistency ("What’s a Unit?"):

  • A "unit" of an input can be defined in hours, weeks, or worker-years, provided the measurement unit remains consistent throughout the entire economic analysis.

  • Failure to maintain consistency across inputs and outputs (such as contrasting output per hour with labor cost per month) produces calculation errors.

• Average Product of Labour ($APL$):

  • The average product of an input is the total output produced per unit of that input employed:     $APL = \frac{q}{L} = \frac{\text{Total quantity of output}}{\text{Total quantity of labour employed}}$

  • Alec and Janet's Farm $APL$ schedule:

  • At $L = 1$ worker: $Q = 19$, $APL = \frac{19}{1} = 19\,\text{bushels/worker}$

  • At $L = 2$ workers: $Q = 36$, $APL = \frac{36}{2} = 18\,\text{bushels/worker}$

  • At $L = 3$ workers: $Q = 51$, $APL = \frac{51}{3} = 17\,\text{bushels/worker}$

  • At $L = 4$ workers: $Q = 64$, $APL = \frac{64}{4} = 16\,\text{bushels/worker}$

  • At $L = 5$ workers: $Q = 75$, $APL = \frac{75}{5} = 15\,\text{bushels/worker}$

  • At $L = 6$ workers: $Q = 84$, $APL = \frac{84}{6} = 14\,\text{bushels/worker}$

  • At $L = 7$ workers: $Q = 91$, $APL = \frac{91}{7} = 13\,\text{bushels/worker}$

  • At $L = 8$ workers: $Q = 96$, $APL = \frac{96}{8} = 12\,\text{bushels/worker}$

• Relationship Between Average Product and Marginal Product:

  • When marginal product exceeds average product ($MPL > APL$), the average product curve is pulled upward and rises.

  • When marginal product is less than average product ($MPL < APL$), the average product curve is pulled downward and falls.

  • The marginal product curve intersects the average product curve precisely at the maximum point of average product (Point $A$).

Short-Run Total Cost Curves

• Foundational Cost Components:

  • Fixed Cost ($FC$): A cost that does not depend on the quantity of output produced. It represents the cost of acquiring and maintaining fixed inputs.

  • Variable Cost ($VC$): A cost that depends explicitly on the quantity of output produced. It represents the payments made to variable inputs.

  • Total Cost ($TC$): The total financial expense required to produce a specific quantity of output, computed as the direct sum of fixed and variable costs:     $TC = FC + VC$

• Cost Structure of Alec and Janet’s Farm:

  • Assumes fixed equipment/land costs $FC = \$400$ and variable labor wage rate $w = \$200$ per worker:

  • Point $A$: $L = 0$, $Q = 0$, $FC = \$400$, $VC = \$0$, $TC = \$400$

  • Point $B$: $L = 1$, $Q = 19$, $FC = \$400$, $VC = \$200$, $TC = \$600$

  • Point $C$: $L = 2$, $Q = 36$, $FC = \$400$, $VC = \$400$, $TC = \$800$

  • Point $D$: $L = 3$, $Q = 51$, $FC = \$400$, $VC = \$600$, $TC = \$1000$

  • Point $E$: $L = 4$, $Q = 64$, $FC = \$400$, $VC = \$800$, $TC = \$1200$

  • Point $F$: $L = 5$, $Q = 75$, $FC = \$400$, $VC = \$1000$, $TC = \$1400$

  • Point $G$: $L = 6$, $Q = 84$, $FC = \$400$, $VC = \$1200$, $TC = \$1600$

  • Point $H$: $L = 7$, $Q = 91$, $FC = \$400$, $VC = \$1400$, $TC = \$1800$

  • Point $I$: $L = 8$, $Q = 96$, $FC = \$400$, $VC = \$1600$, $TC = \$2000$

• Curvature of the Total Cost Curve:

  • The total cost curve slopes upward across all levels of output because producing additional output requires additional variable inputs, which incur additional costs.

  • The total cost curve becomes steeper as output increases due to diminishing returns to the variable input: because each additional worker produces less additional output, progressively larger quantities of variable input (and higher costs) are needed to produce equal successive increments of output.

Marginal Cost and Average Cost Fundamentals

• Marginal Cost ($MC$):

  • The additional financial expense generated by producing one additional unit of output:     $MC = \frac{\Delta TC}{\Delta q} = \frac{\text{Change in total cost}}{\text{Change in quantity of output}}$

  • Graphically, marginal cost represents the slope of the total cost curve ($\frac{\text{rise}}{\text{run}}$).

• Average Cost Metrics:

  • Average Total Cost ($ATC$): Total cost divided by total quantity of output produced:     $ATC = \frac{TC}{q}$

  • Average Fixed Cost ($AFC$): Fixed cost divided by total quantity of output produced:     $AFC = \frac{FC}{q}$

  • Average Variable Cost ($AVC$): Variable cost divided by total quantity of output produced:     $AVC = \frac{VC}{q}$

  • Additive Relationship:     $ATC = AFC + AVC$

• Case Study: Selena’s Gourmet Salsas (Fixed Cost $FC = \$108$):

  • Complete production and cost schedule:

  • Output $q = 0$: $FC = \$108$, $VC = \$0$, $TC = \$108$

  • Output $q = 1$: $FC = \$108$, $VC = \$12$, $TC = \$120$, $MC = \$12$, $ATC = \$120.00$, $AFC = \$108.00$, $AVC = \$12.00$

  • Output $q = 2$: $FC = \$108$, $VC = \$48$, $TC = \$156$, $MC = \$36$, $ATC = \$78.00$, $AFC = \$54.00$, $AVC = \$24.00$

  • Output $q = 3$: $FC = \$108$, $VC = \$108$, $TC = \$216$, $MC = \$60$, $ATC = \$72.00$, $AFC = \$36.00$, $AVC = \$36.00$

  • Output $q = 4$: $FC = \$108$, $VC = \$192$, $TC = \$300$, $MC = \$84$, $ATC = \$75.00$, $AFC = \$27.00$, $AVC = \$48.00$

  • Output $q = 5$: $FC = \$108$, $VC = \$300$, $TC = \$408$, $MC = \$108$, $ATC = \$81.60$, $AFC = \$21.60$, $AVC = \$60.00$

  • Output $q = 6$: $FC = \$108$, $VC = \$432$, $TC = \$540$, $MC = \$132$, $ATC = \$90.00$, $AFC = \$18.00$, $AVC = \$72.00$

  • Output $q = 7$: $FC = \$108$, $VC = \$588$, $TC = \$696$, $MC = \$156$, $ATC = \$99.43$, $AFC = \$15.43$, $AVC = \$84.00$

  • Output $q = 8$: $FC = \$108$, $VC = \$768$, $TC = \$876$, $MC = \$180$, $ATC = \$109.50$, $AFC = \$13.50$, $AVC = \$96.00$

  • Output $q = 9$: $FC = \$108$, $VC = \$972$, $TC = \$1080$, $MC = \$204$, $ATC = \$120.00$, $AFC = \$12.00$, $AVC = \$108.00$

  • Output $q = 10$: $FC = \$108$, $VC = \$1200$, $TC = \$1308$, $MC = \$228$, $ATC = \$130.80$, $AFC = \$10.80$, $AVC = \$120.00$

Dynamics of the U-Shaped Average Total Cost Curve

• Opposing Economic Forces Determining the Shape of Average Total Cost:

  • The Spreading Effect: As production volume increases, the fixed cost is divided across a larger quantity of output, lowering average fixed cost ($AFC$). This effect dominates at low levels of production, causing the initial downward slope of $ATC$.

  • The Diminishing Returns Effect: As production increases, diminishing returns to variable inputs require progressively larger quantities of variable inputs per additional unit of output, increasing average variable cost ($AVC$). This effect dominates at high levels of production, causing the subsequent upward slope of $ATC$.

• Minimum-Cost Output:

  • The quantity of output at which average total cost achieves its absolute minimum value, represented by the trough (Point $M$) of the U-shaped curve.

  • For Selena's Gourmet Salsas, minimum-cost output occurs at $q = 3$ cases, where $ATC$ reaches its lowest value of $\$72.00$.

• Universal Principles Governing Marginal Cost and Average Total Cost:

  • When $MC < ATC$: Average total cost is falling. Each incremental unit costs less to produce than the preceding average, dragging down the overall average.

  • When $MC > ATC$: Average total cost is rising. Each incremental unit costs more to produce than the preceding average, pulling up the overall average.

  • When $MC = ATC$: Average total cost is at its minimum point ($M$). Average total cost is neither falling nor rising.

  • Consequently, the marginal cost curve always intersects the average total cost curve from below at exactly the minimum of $ATC$.

Production Specialization and Realistic Cost Curves

• Specialization and Initial Increasing Returns:

  • In actual production environments, marginal cost curves frequently exhibit an initial downward slope before sloping upward.

  • When a firm employs only a tiny workforce, workers must divide their efforts across multiple disparate operations, failing to reap the productivity gains of specialization.

  • Adding workers initially permits division of labour and specialization, which raises marginal product and lowers marginal cost.

• The Transition to Diminishing Returns:

  • Once workforce expansion fully exhausts the technical and organizational gains from specialization, diminishing returns set in due to fixed capital limitations.

  • Marginal product declines, and marginal cost begins to ascend, giving the realistic $MC$ curve a distinct "checkmark" or U-shape.

  • The realistic average variable cost ($AVC$) curve and average total cost ($ATC$) curve remain U-shaped.

  • The marginal cost curve ($MC$) intersects both the $AVC$ curve and the $ATC$ curve at their respective minimum points.

Mathematical Duality Between Productivity and Cost

• Mathematical Derivation of Marginal Cost from Labor Productivity:

  • Total cost is defined by:     $TC = FC + VC = FC + w \times L$

  • Differentiating with respect to output ($q$):     $MC = \frac{\Delta TC}{\Delta q} = \frac{\Delta(FC + w \times L)}{\Delta q} = \frac{\Delta FC}{\Delta q} + \frac{\Delta(w \times L)}{\Delta q}$

  • Because fixed costs do not vary with output, $\Delta FC = 0$. Assuming a constant market wage rate ($w$):     $MC = w \times \frac{\Delta L}{\Delta q}$

  • By definition, the marginal product of labor is:     $MPL = \frac{\Delta q}{\Delta L}$

  • Substituting the reciprocal gives:     $\frac{\Delta L}{\Delta q} = \frac{1}{MPL}$

  • Yielding the fundamental inverse duality:     $MC = \frac{w}{MPL}$

  • When $MPL$ increases, $MC$ decreases. When $MPL$ reaches its maximum (Point $A$), $MC$ hits its minimum (Point $A'$). When diminishing returns cause $MPL$ to fall, $MC$ rises.

• Mathematical Derivation of Average Variable Cost from Labor Productivity:

  • Average variable cost is defined by:     $AVC = \frac{VC}{q} = \frac{w \times L}{q} = w \times \frac{L}{q}$

  • By definition, the average product of labor is:     $APL = \frac{q}{L}$

  • Substituting the reciprocal yields:     $AVC = \frac{w}{APL}$

  • When $APL$ rises, $AVC$ declines. When $APL$ reaches its maximum (Point $B$), $AVC$ reaches its minimum (Point $B'$). When $APL$ falls, $AVC$ increases.

Short-Run Versus Long-Run Cost Curves

• Flexibility of Capital in the Long Run:

  • In the short run, fixed cost is completely exogenous and unchangeable.

  • In the long run, all inputs are variable; therefore, a firm's fixed cost is an endogenous choice variable.

  • A firm chooses its level of fixed cost in advance based on the planned or anticipated volume of output it intends to sustain.

• Trade-off Between Fixed Cost and Variable Cost:

  • Investing in a higher level of fixed capital increases overhead expenses but reduces variable operating costs per unit, and vice versa.

  • Comparative analysis of Selena's Gourmet Salsas with Low Fixed Cost ($FC = \$108$, denoted as curve $ATC_1$) versus High Fixed Cost ($FC = \$216$, denoted as curve $ATC_2$):

  • At $q = 1$: $ATC_1 = \$120.00$, $ATC_2 = \$222.00$

  • At $q = 2$: $ATC_1 = \$78.00$, $ATC_2 = \$120.00$

  • At $q = 3$: $ATC_1 = \$72.00$, $ATC_2 = \$90.00$

  • At $q = 4$: $ATC_1 = \$75.00$, $ATC_2 = \$78.00$

  • At $q = 5$: $ATC_1 = \$81.60$, $ATC_2 = \$73.20$

  • At $q = 6$: $ATC_1 = \$90.00$, $ATC_2 = \$72.00$

  • At $q = 7$: $ATC_1 = \$99.43$, $ATC_2 = \$72.86$

  • At $q = 8$: $ATC_1 = \$109.50$, $ATC_2 = \$75.00$

  • At $q = 9$: $ATC_1 = \$120.00$, $ATC_2 = \$78.00$

  • At $q = 10$: $ATC_1 = \$130.80$, $ATC_2 = \$81.60$

  • Conclusion: At low output levels ($q \le 3$), lower fixed cost yields a lower average total cost. At high output levels ($q \ge 5$), higher fixed cost yields a lower average total cost because the variable cost savings outweigh the higher fixed charges.

• The Long-Run Average Total Cost Curve ($LRATC$):

  • Shows the relationship between output and average total cost when the level of fixed cost has been optimized to minimize average total cost at each specific output level.

  • Graphically, the $LRATC$ curve serves as the lower boundary envelope beneath the family of short-run average total cost curves ($ATC_3$, $ATC_6$, $ATC_9$).

Returns to Scale Classifications

• Increasing Returns to Scale (Economies of Scale):

  • Occurs when long-run average total cost declines as output increases.

  • Typically generated by technological indivisibilities, high initial fixed capital investments, and expanded opportunities for specialized labor and machinery.

• Decreasing Returns to Scale (Diseconomies of Scale):

  • Occurs when long-run average total cost increases as output increases.

  • Primarily arises from managerial bottlenecks, communication breakdowns, coordination difficulties, and bureaucratic inefficiencies inherent in exceptionally large enterprise hierarchies.

• Constant Returns to Scale:

  • Occurs when long-run average total cost remains constant as output increases.

  • Arises when an expansion in output can be achieved by replicating existing production facilities without changing input proportions or encountering organizational friction.

Comprehensive Cost Taxonomy

• Short-Run Measures:

  • Fixed Cost ($FC$): Cost that does not depend on the quantity of output produced.

  • Average Fixed Cost ($AFC$): Fixed cost per unit of output:     $AFC = \frac{FC}{Q}$

• Measures Relevant in Both Short Run and Long Run:

  • Variable Cost ($VC$): Cost that directly depends on the quantity of output produced.

  • Average Variable Cost ($AVC$): Variable cost per unit of output:     $AVC = \frac{VC}{Q}$

  • Total Cost ($TC$): Total economic cost incurred:     $TC = FC + VC$

  • Average Total Cost ($ATC$): Total cost per unit of output:     $ATC = \frac{TC}{Q}$

  • Marginal Cost ($MC$): Additional cost incurred from generating one more unit of output:     $MC = \frac{\Delta TC}{\Delta Q}$

• Long-Run Measure:

  • Long-Run Average Total Cost ($LRATC$): Average total cost achieved when fixed capital has been chosen specifically to minimize average total cost for each level of output.

Real-World Application: The Economics of Municipal Snow Removal

• Fixed Cost Decisions in Municipal Infrastructure (Victoria vs. Montreal):

  • Cities face production choices where the "output" is the removal of accumulated snowfall from roadways and sidewalks.

  • In Victoria, significant snowfall is an infrequent event. Victoria rationally selects a low fixed-cost strategy, maintaining minimal snowplows and specialized clearing vehicles. Consequently, an unexpected snowfall of an inch or two overnight produces severe disruption and high travel delay costs, but purchasing an extensive fleet would be economically inefficient under normal weather patterns.

  • In Montreal, heavy snowfall occurs regularly throughout the winter season. Montreal anticipates high output requirements and chooses a high fixed-cost strategy by maintaining a massive capital fleet of snow-clearing machinery and infrastructure. This large fixed investment minimizes the variable and average total cost of snow clearance per major storm, allowing ordinary municipal activity to proceed without systemic interruption.