The Production Function
Production is the process by which a firm transforms inputs (factors of production) into outputs that can be consumed or used for further production.
A production function specifically describes the relationship between the quantities of inputs used and the maximum quantity of output that can be produced from those inputs given a specific technology.
For a firm using two factors, Labour (L) and Capital(K), the function is expressed as:
q=f(L,K)
Where q is the maximum output. If technology improves, a new production function is created because the same level of inputs can produce more output.
Short Run vs. Long Run Production Function
Short Run: A time period where at least one factor of production (usually capital) remains fixed and cannot be varied. To change output, the firm can only vary the variable factor (usually labour).
Long Run: A time period in which all factors of production can be varied. There are no fixed factors in the long run.
Returns to Scale (Long Run Only)
In the long run, when all inputs are increased by the same proportion (t), the resulting output change determines the Returns to Scale:
Constant Returns to Scale (CRS): Output increases by the same proportion as inputs (f(tL,tK)=t⋅f(L,K)).
Increasing Returns to Scale (IRS): Output increases by a larger proportion than inputs (f(tL,tK)>t⋅f(L,K)).
Decreasing Returns to Scale (DRS): Output increases by a smaller proportion than inputs (f(tL,tK)<t⋅f(L,K))
2. Total, Marginal, and Average Product
When studying the short-run production function (where capital is fixed), three primary concepts are used to measure the contribution of the variable input:
Total Product (TP): The relationship between the variable input and total output, keeping all other inputs constant. TP is the sum of all Marginal Products.
Average Product (AP): Output per unit of the variable input (APL=TP/L).
Marginal Product (MP): The change in output resulting from a unit change in the variable input when all other inputs are held constant (MPL=ΔTP/ΔL).
3. Shapes of TP, MP, and AP Curves
The shapes of these curves are governed by the Law of Variable Proportions (also known as the Law of Diminishing Marginal Product).
The Law of Variable Proportions
This law states that as we increase the employment level of a variable factor (holding others fixed), the MP initially rises but eventually starts falling. This occurs because:
Initially, factor proportions become more suitable for production.
Eventually, the production process becomes "too crowded" with the variable input, leading to inefficiency.
Curve Characteristics
TP Curve: A positively sloped curve that increases as the variable input increases, though the rate of increase varies based on the MP.
MP Curve: It is inverse ‘U’-shaped. It rises initially, reaches a maximum, and then falls as diminishing returns set in.
AP Curve: Also inverse ‘U’-shaped. It represents the average of all marginal products up to a certain level.
Relationships between the Curves
MP and AP start at the same point for the first unit of input.
As long as MP > AP, the AP rises.
When MP < AP, the AP falls.
Crucial Point: The MP curve cuts the AP curve from above at the maximum point of the AP curve.
4. Concepts of Cost
To produce output, firms must pay for inputs, which constitutes the cost of production. A cost function describes the least cost of producing each level of output given input prices and technology.
Short-Run Cost Concepts
Total Fixed Cost (TFC): Costs incurred to employ fixed inputs. These costs do not change with the level of output and remain constant even at zero output. Graphically, TFC is a horizontal straight line.
Total Variable Cost (TVC): Costs incurred to employ variable inputs. TVC increases as output increases.
Total Cost (TC): The sum of fixed and variable costs (TC=TFC+TVC). TC increases as output increases.
Average Fixed Cost (AFC): TFC per unit of output (AFC=TFC/q). Since TFC is constant, AFC decreases as output increases, forming a rectangular hyperbola.
Average Variable Cost (AVC): TVC per unit of output (AVC=TVC/q).
Short-Run Average Cost (SAC): Total cost per unit of output (SAC=TC/q or SAC=AFC+AVC).
Short-Run Marginal Cost (SMC): The change in total cost per unit change in output (SMC=ΔTC/Δq). In the short run, SMC is entirely due to changes in TVC.
5. Shapes of Short-Run Cost Curves
Short-run cost curves (except AFC) are generally 'U'-shaped due to the Law of Variable Proportions.
SMC Curve: Initially falls and then rises as diminishing marginal product sets in.
AVC Curve: Initially falls, reaches a minimum, and then rises as SMC becomes greater than AVC.
SAC Curve: It is the vertical sum of AVC and AFC. It also falls initially, reaches a minimum point (to the right of AVC's minimum), and then rises.
Relationship between Marginal and Average Costs:
SMC cuts the AVC curve from below at the minimum point of AVC.
SMC cuts the SAC curve from below at the minimum point of SAC.
When SMC is less than Average Cost (SAC/AVC), the Average Cost is falling; when SMC is greater, the Average Cost is rising.
6. Concepts of Revenue
Revenue is the money earned by a firm through selling its output in the market.
Total Revenue (TR): The market price (p) multiplied by the quantity sold (q) (TR=p×q). Under perfect competition, price is constant, so TR is an upward-rising straight line starting from the origin.
Average Revenue (AR): Total revenue per unit of output (AR=TR/q). For a price-taking firm, AR = Market Price (p). The AR curve is a horizontal line at the market price, often called the price line.
Marginal Revenue (MR): The increase in total revenue for a unit increase in output (MR=ΔTR/Δq). For a perfectly competitive firm, MR = p, and thus AR = MR = p.
7. Producer’s Equilibrium: Profit Maximisation
A producer is in equilibrium when they produce the quantity that maximizes Profit (π), which is defined as TR−TC.
The MC and MR Approach
For a firm to maximize profit, three conditions must be satisfied simultaneously at the chosen output level (q0):
Condition 1 (MR=MC): Profit is increasing as long as MR>MC and falling if MR<MC. Therefore, at equilibrium, MR must equal MC. Since MR=P under perfect competition, this implies P=MC.
Condition 2 (MC must be non-decreasing): The MC curve must be upward-sloping at the equilibrium point. A firm will not produce in the range where MC is falling because increasing output would further reduce costs and increase profits.
Condition 3 (Continuity of Production):
Short Run: The market price must be greater than or equal to the minimum Average Variable Cost (P≥AVC). If P<AVC, the firm's total revenue cannot even cover its variable costs, and it will minimize losses by shutting down (producing zero output).
Long Run: The market price must be greater than or equal to the minimum Average Cost (P≥AC). If P<AC, the firm incurs a loss and will exit the market in the long run.
Key Equilibrium Points
Shut-down Point: In the short run, this is the point of minimum AVC.
Break-even Point: The point where the firm earns only normal profit (where P=minimum SAC or LAC). Normal profit is the minimum level needed to keep the firm in business and is considered part of total costs. Profits earned above this are called super-normal profits.