Background to Supply: The Theory of Production and Cost Study Notes

Introduction to the Theory of Production and Cost

The background to supply is rooted in the Theory of Production and Cost. To understand how firms make supply decisions, one must examine the types of firms that exist and the primary goal of the firm, which is typically to maximize profit. This analysis involves evaluating production Returns to Scale, Economies of Scale, Diseconomies of Scale, and Economies of Scope. Furthermore, understanding the distinction between basic cost and profit concepts is essential, specifically the differences between explicit and implicit costs, accounting costs versus economic costs, and accounting profit versus economic profit.

Revenue, Profit, and Cost Relationships

To analyze a company's financial performance, economists use several fundamental equations relating to revenue and profit. Total Revenue ($TR$) is calculated by multiplying the price of the product ($P$) by the quantity sold ($Q$), expressed as $TR = P \times Q$. Average Revenue ($AR$) represents the total revenue divided by the quantity sold, shown as $AR = \frac{TR}{Q}$. Marginal Revenue ($MR$) is the additional revenue earned by selling an additional unit of product, calculated as $MR = \frac{\Delta TR}{\Delta Q}$. The relationship between profit, revenue, and cost is captured in the company equation where Total Profit is equal to Total Revenue minus Total Cost ($\text{Total Profit} = TR - TC$). This equation can be rearranged to state that $TR = \text{Total Profit} + TC$ or $TC = TR - \text{Total Profit}$.

Cost and Profit Concepts in Economic Theory

Economists distinguish between costs and profits differently than accountants. Explicit costs are those that involve a direct monetary payment, while implicit costs relate to the opportunity costs of using resources the firm already owns. In an accountant's view, profit is simply Total Revenue minus Explicit Expenses. However, in the economist's world, Economic Profit is calculated as $TR - (\text{Explicit Costs} + \text{Implicit Costs})$.

Accounting Profit (also referred to as Total Profit in some contexts) is defined as $TR - \text{Total Explicit Cost}$. Normal profit is a critical concept in economics, defined as being equal to the best return that a firm's resources could have earned elsewhere. Normal profit forms part of the cost of production. When Total Revenue is exactly equal to the sum of Total Explicit Costs and Implicit Costs ($TR = \text{Explicit Costs} + \text{Implicit Costs}$), the firm is said to be breaking even in economic terms, achieving a normal profit. Any revenue beyond this total represents an Economic Profit.

Production and Cost in the Short Run and Long Run

Production theory distinguishes between two critical time periods: the short run and the long run. The short run is defined as a period during which at least one of the inputs (usually capital or land) is fixed in quantity. The long run is defined as a period in which all inputs are variable, allowing the firm to change the scale of its operations.

In the short run, Total Cost ($TC$) is the total cost of producing a certain quantity of the firm’s product. Average Cost ($AC$) is the total cost divided by the number of units produced ($AC = \frac{TC}{Q}$). Marginal Cost ($MC$) is the additional or extra cost required to produce one additional unit of the product, expressed as $MC = \frac{\Delta TC}{\Delta Q}$.

Short Run Production Analysis and the Law of Diminishing Returns

To analyze supply decisions, we assume a typical firm, such as a farmer, who operates with a fixed quantity of land and utilizes labour as a variable input. Assumptions for this model include a single homogenous and divisible product, a given production function, fixed prices for inputs and outputs, and the presence of one variable input. As more labour is added to a fixed unit of land, the firm eventually experiences the Law of Diminishing Returns.

The Law of Diminishing Returns states that as more of a variable input (such as labour) is combined with one or more fixed inputs (such as land) in a production process, points will eventually be reached where first the marginal product ($MP$), then the average product ($AP$), and finally the total product ($TP$) start to decline. This phenomenon is often illustrated by the congestion on a fixed piece of land, where too many workers interfere with one another (metaphorically described as workers yelling "GET OUT OF MY WAY").

Quantitative Short Run Production Data

The following data illustrates the production function where land is fixed at $1$ unit and labour varies from $0$ to $10$ units. At $0$ units of labour, total product ($TP$) is $0$. With $1$ unit of labour, $TP$ is $16$ tons, the Marginal Product ($MP$) is $+16$, and the Average Product ($AP$) is $16.00$. With $2$ units of labour, $TP$ rises to $44$ tons ($MP = +28$, $AP = 22.00$). At $3$ units, $TP$ is $78$ tons ($MP = +34$, $AP = 26.00$). At $4$ units, $TP$ is $113$ tons ($MP = +35$, $AP = 28.25$). At $5$ units, $TP$ is $145$ tons ($MP = +32$, $AP = 29.00$).

Continuing the addition of labour, at $6$ units, $TP$ is $171$ tons ($MP = +26$, $AP = 28.50$). At $7$ units, $TP$ is $190$ tons ($MP = +19$, $AP = 27.14$). At $8$ units, $TP$ is $200$ tons ($MP = +10$, $AP = 25.00$). At $9$ units, $TP$ reaches its peak at $200$ tons ($MP = 0$, $AP = 22.22$). Finally, at $10$ units of labour, $TP$ declines to $187$ tons ($MP = -13$, $AP = 18.70$), demonstrating the stage where total product explicitly declines.

Short Run Cost Structure and Numerical Values

In the short run cost model, we assume the cost of labour is $R2400$ per unit and the Total Fixed Cost ($TFC$) for land is $R9000$. Total Variable Cost ($TVC$) is calculated by multiplying units of labour by the unit cost ($R2400$). Total Cost ($TC$) is the sum of $TFC$ and $TVC$.

Starting with $0$ labour units and $0$ TP, the costs are: $TFC = R9000$, $TVC = 0$, $TC = R9000$. At $1$ labour unit ($TP = 16$): $TVC = R2400$, $TC = R11400$. At $2$ labour units ($TP = 44$): $TVC = R4800$, $TC = R13800$. At $3$ labour units ($TP = 78$): $TVC = R7200$, $TC = R16200$. At $4$ labour units ($TP = 113$): $TVC = R9600$, $TC = R18600$. At $5$ labour units ($TP = 145$): $TVC = R12000$, $TC = R21000$. At $6$ labour units ($TP = 171$): $TVC = R14400$, $TC = R23400$. At $7$ labour units ($TP = 190$): $TVC = R16800$, $TC = R25800$. At $8$ labour units ($TP = 200$): $TVC = R19200$, $TC = R28200$. At $9$ labour units ($TP = 200$): $TVC = R21600$, $TC = R30600$. At $10$ labour units ($TP = 187$): $TVC = R24000$, $TC = R33000$.

Unit Cost Analysis: AFC, AVC, AC, and MC

Average Fixed Cost ($AFC$) is calculated as $\frac{TFC}{TP}$. Average Variable Cost ($AVC$) is $\frac{TVC}{TP}$. Average (Total) Cost ($AC$) is $\frac{TC}{TP}$. Marginal Cost ($MC$) is calculated as $\frac{\Delta TC}{\Delta TP}$.

For the data provided: At $TP = 16$, $AFC = 562.50$, $AVC = 150.00$, $AC = 712.50$, and $MC = 150.00$. At $TP = 44$, $AFC = 204.55$, $AVC = 109.09$, $AC = 313.64$, and $MC = 85.71$. At $TP = 78$, $AFC = 115.38$, $AVC = 92.31$, $AC = 207.69$, and $MC = 70.59$. At $TP = 113$, $AFC = 79.65$, $AVC = 84.96$, $AC = 164.60$, and $MC = 68.57$. At $TP = 145$, $AFC = 62.07$, $AVC = 82.76$, $AC = 144.83$, and $MC = 75.00$.

As production continues to increase: At $TP = 171$, $AFC = 52.63$, $AVC = 84.21$, $AC = 136.84$, and $MC = 92.31$. At $TP = 190$, $AFC = 47.37$, $AVC = 88.42$, $AC = 135.79$, and $MC = 126.32$. At $TP = 200$, $AFC = 45.00$, $AVC = 96.00$, $AC = 141.00$, and $MC = 240.00$. For labour unit $9$ where output does not increase ($TP = 200$), the marginal cost becomes undefined. Similarly, at labour unit $10$ where output drops to $187$, the marginal cost calculation is undefined.

The Geometry of Cost and Product Curves

The shapes of these cost curves are distinct and interrelated. $AFC$ is L-shaped; as output increases from zero, it starts at a very high value and declines continuously until maximum output is reached. Conversely, $AVC$, $AC$, and $MC$ curves are U-shaped. This means that as output increases from zero, these costs start high, decline at decreasing rates until they reach a minimum point, and then increase at increasing rates.

Notably, the $AC$ curve always lies above both the $AFC$ and $AVC$ curves because it is the summation of the two. A critical geometric property is that the $MC$ curve intersects both the $AVC$ and $AC$ curves at their respective minimum points. Before these intersections, $MC$ lies below $AVC$ and $AC$. Beyond these minimum points, as total product increases further, $MC$ lies above $AVC$ and $AC$.

There is also a reciprocal relationship between production and cost curves. When Marginal Product ($MP$) is at its maximum, Marginal Cost ($MC$) is at its minimum. Similarly, when Average Product ($AP$) is at its maximum, Average Variable Cost ($AVC$) is at its minimum. This shows that the efficiency of labour (product per unit of labour) is directly tied to the unit costs of output.

Production and Costs in the Long Run

In the long run, all inputs are variable, and the firm focuses on Returns to Scale. There are three types of Returns to Scale: Constant Returns to Scale occur when output increases in the same proportion as inputs; Increasing Returns to Scale occur when output increases by a larger proportion than inputs; and Decreasing Returns to Scale occur when output increases by a smaller proportion than inputs.

Economies of Scale refer to the situation where cost per unit of output falls as the scale of production increases. This is represented by a downward-sloping Long-Run Average Cost ($LRAC$) curve. Diseconomies of Scale occur when unit costs rise as output increases, represented by an upward-sloping $LRAC$ curve. Constant costs occur when the unit costs remain the same regardless of scale, resulting in a horizontal $LRAC$. A typical $LRAC$ curve is U-shaped, exhibiting economies of scale at low output levels, constant costs at intermediate levels, and diseconomies of scale at high output levels.

Finally, Economies of Scope represent cost savings achieved by producing related goods within a single firm rather than producing them in two separate firms. This efficiency arises from sharing resources or production processes between the different products.