Production Decision: Returns to a Factor and Scale

Introduction to Production Decisions

Definition of Production: Production is defined as the process of transforming physical inputs into tangible outputs (goods and services). The study of production provides insights into the efficient allocation of factors of production.
Producer Objectives: Producers employ the least-cost combination of inputs—such as land, labour, and capital—to maximize profit.
Profit Maximization Rule: A firm achieves maximum profit by producing a quantity of output where marginal revenue ( $MR$ ) equals marginal cost ( $MC$ ). Producing at the level that yields maximum output from the best combination of factor inputs is considered a significant challenge for producers.
Scope of Study: This chapter covers production functions in both the short-run and long-run contexts.

The Production Function

Functional Relationship: The production function explains the relationship between inputs used and the resulting output. It is mathematically expressed as: $Q_x = f(i_1, i_2, i_3, \dots, i_n)$ Where: * $Q_x$ is the quantity of a commodity produced. * $(i_1, i_2, i_3, \dots, i_n)$ represents the quantities of various inputs utilized.
Simplified Two-Input Model: Assuming only labour ( $L$ ) and capital ( $K$ ) are used, the function is simplified to: $Q = f(L, K)$
Fixed Technology Assumption: It is assumed that technology is given; therefore, output can only be increased by increasing input quantities.
Time-Based Classification: The relationship between input and output is divided based on the duration of production: * Short-run Production: Characterized by inadequate time for a firm to change its fixed inputs. At least one factor of production is fixed. Changes in output result from a change in input proportions. The production function is expressed as $Q_x = f(L)$ , where $K$ is fixed. * Long-run Production: Defined by sufficient time for a firm to adjust all inputs. All factors of production are variable. Changes in output result from a change in the scale of production. The production function is expressed as $Q_x = f(L, K)$ .

Basic Concepts of Production

Total Product (TP): Also referred to as Total Physical Product, Total Return, or Total Output. It is the total quantity of a commodity produced with a given unit of factor inputs and technology during a specific period. $TP = TP_1 + TP_2 + TP_3 + \dots + TP_n$ $TP = \sum MP$
Average Product (AP): This indicates the output produced per unit of the variable input employed. It is calculated by dividing the total product by units of the variable factor: $AP_L = \frac{TP}{L}$
Marginal Product (MP): This refers to the additional (stray) output produced by employing one more unit of the variable input. It is calculated as: $MP_L = \frac{\Delta TP}{\Delta L}$ Alternatively, for discrete units: $MP = TP_n - TP_{n-1}$

Law of Variable Proportions (Returns to a Factor)

Definition: This law explains the change in output as a result of applying different units of a variable factor to a fixed unit of a fixed factor in the short run.
Core Principle: As additional units of a variable factor are employed with a fixed factor, the total product initially increases at an increasing rate, then increases at a diminishing rate, and eventually falls perpetually.
Assumptions of the Law: 1. The firm operates in the short run. 2. The state of technology remains constant and unchanged. 3. At least one input is fixed while others vary. 4. It is possible to change the factor proportion. 5. All units of the variable factor are homogeneous and equally efficient.
The Three Stages of Production: * Stage I: Increasing Returns: Total product increases at an increasing rate initially. Marginal product increases, reaches a maximum, and then begins to decrease. Average product increases throughout this stage and reaches its maximum at the boundary of Stage I. * Stage II: Diminishing Returns: Total product continues to increase but at a diminishing rate until it reaches its maximum. Marginal product continues to decrease and eventually becomes zero. Average product begins to decrease consistently but remains positive. * Stage III: Negative Returns: Total product begins to decrease. Marginal product becomes negative. Average product continues to decrease but stays positive.
The Point of Inflexion: This is the point on the TP curve where the slope changes, indicating that total output will now increase at a lesser rate. This occurs exactly where the MP reaches its maximum. It serves as a turning point for the producer.

Causes for Different Types of Returns to a Factor

Causes of Increasing Returns: 1. Fuller Utilization of Fixed Factors: Initially, fixed factors (like land) are underutilized relative to variable factors. As more variable factors are added, the fixed factor is used more efficiently. 2. Division of Labour: Increasing variable factors allows for the division of labour, which leads to specialization and higher efficiency.
Causes of Diminishing Returns: 1. Disturbing the Optimum Proportion: There is an ideal combination of fixed and variable factors. Moving beyond this optimum point reduces the marginal and average products. 2. Imperfect Substitutability of Factor Inputs: Labour and capital cannot be substituted for each other indefinitely. Once the optimal combination is passed, adding more of one cannot compensate for the lack of the other.
Causes of Negative Returns: 1. Overcrowding: Too many variable units applied to a fixed factor lead to overcrowding, which hampers productivity. 2. Management Problems: An excessive number of workers creates difficulty in effective management, leading to responsibility-shifting and decreased efficiency.

Numerical Example: Learning Activity 2.1

Table 2.1: TP, AP, and MP Calculation

Capital (Fixed)	Labour (Variable)	Total Product (TP)	Average Product (AP)	Marginal Product (MP)
$1$	$0$	$0$	$0$	$-$
$1$	$1$	$2$	$2$	$2$
$1$	$2$	$6$	$3$	$4$
$1$	$3$	$12$	$4$	$6$
$1$	$4$	$16$	$4$	$4$
$1$	$5$	$18$	$3.6$	$2$
$1$	$6$	$18$	$3$	$0$
$1$	$7$	$16$	$2.3$	$-2$
$1$	$8$	$12$	$1.5$	$-4$

Law of Returns to Scale (Long Run)

Definition: Returns to scale explains the change in total output resulting from a proportionate change in all factor inputs. In the long run, all factors are variable ( $Q_x = f(L, K)$ ).
Three Possibilities of Returns to Scale: 1. Increasing Returns to Scale (IRS): Output increases by a greater proportion than the increase in inputs (e.g., inputs increase by $100\%$ , output increases by $150\%$ ). 2. Constant Returns to Scale (CRS): Output increases by the exact same proportion as the increase in inputs (e.g., doubling inputs doubles the output). 3. Decreasing Returns to Scale (DRS): Output increases by a smaller proportion than the increase in inputs (e.g., doubling inputs increases output by only $70\%$ or $80\%$ ).

Causes for Returns to Scale

Causes of Increasing Returns to Scale: 1. Indivisibility of Inputs: Large, bulky machinery cannot be divided into small units. At higher scales of production, these machines are utilized fully and efficiently. 2. Greater Division and Specialization: Expanding the scale allows for advanced division of labour, increasing productivity and management efficiency.
Causes of Constant Returns to Scale: 1. Limits of Economies of Scale: Economies of scale (cost reduction per unit) are not infinite. CRS represents a brief phase when economies are exhausted before diseconomies begin. 2. Perfect Divisibility of Inputs: In certain activities where inputs are perfectly divisible, replicate plants can be set up to double output by doubling inputs.
Causes of Decreasing Returns to Scale: 1. Entrepreneur as a Fixed Factor: Beyond a certain scale, the skills and management abilities of the entrepreneur become overstretched and less efficient. 2. Exhaustibility of Natural Resources: Limited natural resources (e.g., timber or minerals) mean that doubling inputs like labour and tools does not necessarily double the biological or physical resource extraction rate.

Questions & Discussion

Learning Activity 2.1 Questions:

Q: What happens to TP when MP decreases?
Q: Explain the relationship between TP and MP.
Q: Explain the relationship between MP and AP.

Learning Activity 2.2 Questions:

Q: If you are a producer, in which stage would you decide to operate? Why?
Q: Why the producers would not operate in other stages?
Q: Which one do you think is important for a producer; TP, AP or MP? Justify.
Q: Do you think a firm can increase the output after reaching the negative return in the short run? Justify.