6 – Production Theory

Introduction

The firm’s production analysis focuses on the manufacturing process and associated production costs. We must consider the inputs used in the production process and the output. A commodity can be produced in a variety of ways. To avoid resource waste, the company must identify technically efficient production processes. These technically efficient production processes offer the option of selecting the least expensive process.

Firms are required to take different but interrelated production decisions like:
1. Whether or not to actually produce or shut down?
2. How much to produce?
3. What input combination to use?
4. What type of technology to use?

Firms produce the majority of the goods and services consumed in a modern economy. A firm is an organisation that combines and organises resources to produce goods and services for profit. The most important reason for the existence of a firm or business enterprise is that firms are specialised organisations dedicated to managing the production process.

6.1 Definition of Production

“A production function refers to the functional relationship, under the given technology, between physical rates of input and output of firm, per unit of time”.
Mathematically, production function can be express as: Q = f (N, L, K, E, T, etc.)

The transformation of inputs or resources into outputs or goods and services is referred to as production. Entrepreneurs combine economic resources or inputs (composed of natural resources such as labour, land, and capital equipment) to create economic goods and services (outputs or products).

In reality, production theory is nothing more than a constrained optimization technique in action. The firm attempts to either minimise production costs at a given level of output or maximise output at a given level of cost.

Inputs are the resources used in the production of goods and services, and they are broadly classified into three categories: labour, capital, and land or natural resources. They can be either fixed or variable.

Fixed inputs are those that cannot be changed quickly during the time period under consideration, except at great expense (e.g., a firm’s plant).

Variable inputs are those that can be changed quickly and easily (e.g., most raw materials and unskilled labour).

The time period in which at least one input is fixed is referred to as the short run, whereas the time period in which all inputs are variable is referred to as the long run. The length of the long run varies by industry; for example, the long run for a dry-cleaning business may be a few weeks or months. In general, a firm operates in the short run and plans long-term increases or decreases in its scale of operation. Over time, technology generally improves so that more output can be obtained from a given quantity of inputs or the same output can be obtained from fewer inputs.

6.2 Production Function with One Variable Input

A production function is a function that specifies a firm’s, an industry’s, or an entire economy’s output for all combinations of inputs. In other words, it depicts the functional relationship between the inputs and outputs.

The production function can be represented mathematically as follows:

Q = f (X1, X2………………. XK)

Where

Q = Output,

X1,….. XK = Inputs used.

For purposes of analysis, the equation can be reduced to two inputs X and Y.

Restating, Q = f (X, Y)

where

Q = Output

X = Labour

Y = Capital

“A production function defines the relationship between inputs and the maximum amount that can be produced within a given period of time with a given level of technology.”

A production function can be expressed as a table, schedule, or mathematical equation. But first, two unique characteristics of a production function are listed below:

1. Labour and capital are both unavoidable inputs in the production of any quantity of a good.

2. In the production process, labour and capital are substitutes for one another.

CES Function:

Represents a form of production function with the equation Q = B[gL–h + (1 – g)K–h]–1h.

Constants B, g, and h are involved, where h > –1.

If h is variable, it transforms into the Variable Elasticity of Substitution (VES) function.

Leontief Function (Fixed Proportion Production Function):

Expressed as Q = minimum(K/a, L/b), where a and b are constants.

‘Minimum’ denotes the smaller of the two ratios between K/a and L/b.

Linear Production Function:

Represents a simple form assuming perfect substitutes.

Given by Q = aL, where ‘a’ is a constant term, and L stands for labor.

Time Period Classification:

Economists analyze the relationship between factor inputs and outputs in short and long runs.

Time periods are categorized into short runs and long runs for this analysis.

Before proceeding, three terms must be defined: total product, average product, and marginal product.

1. Total product refers to the total amount produced by that many units of a variable factor (i.e., labour). For example, if ten men produce 2000 kg of wheat on a farm, the total product is 2000 kg.

2. The average product is calculated by dividing the total output by the number of units of the variable factor (or the number of men). As a result, AP = TP/L. The average product on the same farm would be 2000/10 = 200 kg.

3. The marginal product is the variation in total output resulting from changing the variable factor (using one more or one fewer unit). If an eleventh man is now added to this farm and output increases to 2,100 kg, the marginal product (of labour) is 100 kg. As a result, MP = d(TP)/dL.

6.3 Production Two variable inputs are used in this function

A firm’s output can be increased by using more of two variable inputs that are substitutes for each other, such as labour and capital. There may be several technical ways to produce a given output by combining different factor combinations. The prices of the production factors and the technical possibilities of factor substitution both influence which particular factor combination the firm chooses.

The technical possibilities of producing an output level by various combinations of the two factors can be represented graphically in terms of isoquants.

6.4 Types of Production Functions

6.5 Producer’s Equilibrium

Before delving into the producer’s equilibrium, it’s essential to grasp concepts like Isoquants, Marginal Rate of Technical Substitution, and Isocost Line. This foundational understanding will enhance comprehension of the producer’s equilibrium.

Isoquants (Iso-product Curves):

Isoquants serve as a graphical representation of the production function, illustrating diverse combinations of factor inputs yielding the same output level. By envisioning varying combinations of labour and capital, an isoquant curve is plotted to showcase all possible alternatives for a given output.

Any quantity of a good can be produced through multiple labour and capital combinations. An isoquant, representing a specific output level, connects various factor combinations capable of producing that amount. For instance, an isoquant for 100 Kg. of output encompasses numerous combinations, like 10 units of capital and 5 units of labour (A), equivalent to 3 units of capital and 20 units of labour (B).

The isoquant doesn’t dictate the specific factor input combination a firm employs; rather, it delineates technically feasible combinations necessary to achieve a designated output. Various points on an isoquant depict different methods of production—capital-intensive methods (e.g., point A) require more capital and less labour, while labour-intensive methods (e.g., point B) demand less capital and more labour. Understanding isoquants is crucial for comprehending the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

When moving along an isoquant, the level of output remains constant while the capital-labour ratio changes continuously. However, switching from one isoquant to another changes the level of output.

Types of Isoquants:

1. Linear Isoquants:

Description: Assumes perfect substitutability of factors of production, allowing for the production of a commodity using only capital, only labor, or any infinite combination of both (K and L).

Characteristics: Represents a straight line, indicating that factors can be substituted without limit.

2. Input-Output Isoquants:

Description: Assumes strict complementarity with zero substitutability between factors of production, resulting in only one method of production for any commodity.

Characteristics: Takes the shape of a right angle, known as a “Leontief isoquant.”

3. Kinked Isoquants:

Description: Assumes limited substitutability between capital (K) and labor (L), with only a few processes available for producing a commodity. Substitutability occurs only at specific points, known as kinks.

Other Names: Also referred to as “activity analysis isoquant” or “linear-programming isoquant” due to its use in linear programming.

4. Smooth, Convex Isoquants:

Description: Assumes continuous substitutability between capital and labor over a certain range, beyond which factors cannot substitute each other. The isoquant appears as a smooth curve convex to the origin.

Characteristics: Represents a curved and smooth shape, indicating variable substitution possibilities within a defined range.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Marginal Rate of Technical Substitution (MRTS):

The Marginal Rate of Technical Substitution (MRTS) quantifies the reduction in one input (– x₂) required when an additional unit of another input is utilized (x₁ = 1), ensuring constant output (y = y).

$<span\; style="text-transform:\; math-auto;">MRTS</span>{}_{}$

${\mathrm{(x}}_1,x_2)={\displaystyle \frac{\mathrm{\Delta }{x}_1}{\mathrm{\Delta }{x}_2}}=-{\displaystyle \frac{M{P}_1}{M{P}_2}}$​​

Here,

MP1​ and MP2​ represent the marginal products of input 1 and input 2, respectively. Along an isoquant, MRTS reflects the rate of substitution between inputs while maintaining consistent output and corresponds to the slope of the isoquant at that point.

Isocost Line:

For a firm employing only labour and capital, the total cost (C) can be expressed as the sum of expenditures on labour (wL) and capital (rK).

$C=wL+rK$

This equation defines the firm’s isocost line, illustrating diverse combinations of labour and capital available at a given total cost. The isocost line’s slope, derived from the equation in a more useful form, represents the rate at which the firm can trade one input for another while keeping costs constant.

$K={\displaystyle \frac{C}{r}}-{\displaystyle \frac{w}{r}}L$

Thus, different total costs or relative input prices yield distinct isocost lines, each parallel or with varying slopes, providing a comprehensive understanding of the firm’s cost structure.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Properties of Isoquants:

1. Negatively Sloped: Isoquants slope downward from left to right, indicating the negative relationship between inputs. The mathematical expression is

${\displaystyle \frac{\mathrm{\Delta }L}{\mathrm{\Delta }K}}=-{\displaystyle \frac{M{P}_L}{M{P}_K}}$

​​, illustrating that as one input increases, the other must decrease to maintain constant output.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2. Convexity: Isoquants exhibit convexity to the origin, implying diminishing marginal rate of technical substitution. The mathematical representation is

${\displaystyle \frac{M{P}_L}{M{P}_K}}$

decreases as the quantity of input K increases while the quantity of input L remains constant.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3. Cannot Intersect: Isoquants cannot intersect each other, ensuring consistency with the law of diminishing marginal returns. If isoquants intersected, it would imply contradictory relations between inputs, violating the principle of diminishing returns.

 

 

 

 

 

 

 

 

 

 

 

 

4. Higher Isoquant, Higher Output: Higher isoquants represent higher levels of output. Mathematically, if isoquant Q₁ corresponds to a higher output level than Q₂, then Q1​>Q2​.

 

 

 

 

 

 

 

 

 

 

 

 

5. Non-Intersecting Isocost Line: Isocost lines, representing various cost combinations, do not intersect with isoquants. This ensures that the cost-minimizing point occurs where the isocost line is tangent to the highest possible isoquant.

6. Isoquants Do Not Touch the Axes: Isoquants do not touch the axes because, in the absence of both inputs, output is zero. This property aligns with economic logic, where inputs are essential for production.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Producers Equilibrium:

Producer’s equilibrium, also termed the least-cost combination of inputs and optimal input combination, constitutes a pivotal objective for any firm or producer. The primary goal is to maximize profits, achieved by either enhancing output levels or sales or by producing output at reduced costs. By scrutinizing its production function, a firm can strategically select the combination of factor inputs that incurs the least cost while maintaining technical efficiency, thereby optimizing profit. Two approaches exist for determining the least-cost combination of factors to produce a given output:

a) Finding the Total Cost of Factor Combinations:

This approach empowers the producer to select a combination by assessing the total cost of production. The cost of each factor combination is computed by multiplying the price of each factor by its quantity and then summing these values for all inputs. The firm opts for the input combination where the total cost is minimal. Further elaboration on this method is provided through the following illustration:

Illustration: Finding the Total Cost of Factor Combinations

Consider a scenario where a firm is aiming to determine the least-cost combination of factor inputs to produce a given output. The firm has two factors, labour (L) and capital (K), with associated prices represented as w (wage rate) and r (rental price), respectively.

The cost (C) of a particular factor combination (L, K) can be expressed as follows:

C=wL+rK

To find the least-cost combination, the firm calculates the total cost for various combinations of labour and capital, considering different levels of output. For each combination, the cost is determined by multiplying the quantity of each factor by its respective price and then summing these values.

For instance, if the firm is producing 100 units of output and the prices are $w=10$ and

$$r=20, the cost for a specific combination (L, K) is computed as:

$C=10L+20K$

By systematically evaluating the total cost of different combinations, the firm identifies the combination that minimizes production costs while meeting the desired output level. This method allows the producer to make economically sound decisions regarding factor inputs, ultimately optimizing the firm’s profitability.

b) Geometrical Method:

Another method to determine the least-cost combination of factors is the geometrical method. This approach is explained using the iso-quant map and iso-cost line.

Iso-quant Map: An iso-quant map illustrates all possible combinations of factors that can produce different levels of output. Higher iso-quants represent higher output levels, with iso-quants closer to the origin indicating lower output levels.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Iso-cost Line: Similar to a budget line in consumer theory, the iso-cost line depicts various combinations of labour and capital that a producer can afford given the total expenditure on these factors and their respective prices. It represents combinations resulting in the same total cost and is defined by the ratio of factor prices (slope): $\frac{\mathrm{PK}}{\mathrm{PL<span\; class="math\; math-inline">\; </span>}}$

 

 

 

 

 

 

 

 

 

 

 

 

 

In the diagram, the iso-cost line (PL) shows combinations of capital (OP) and labour (OL) that the firm can afford, considering given factor prices.

Optimal Input Combination for Minimizing Cost: To minimize cost while producing a given output, the firm selects the optimal method. The iso-quant is tangent to the iso-cost line at point ‘E’, representing the least-cost combination. The ratio of the marginal product of labour to capital equals the ratio of their prices ($\frac{\mathrm{MPL}}{\mathrm{MPK}}=\frac{\mathrm{PL}}{\mathrm{PK}}$).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Optimal Input Combination for Maximizing Output: For maximizing output with a given cost, the equilibrium condition remains the same as the minimization method. The maximum output is achieved at point ‘E’, where iso-quant Q1 is tangent to the iso-cost line AB. Points above ‘E’ are unattainable, and those below are less productive.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

These analyses help firms optimize profits based on factor costs and prices.

6.6 Ridge Lines

The ridge lines delineate points on an isoquant where the marginal product of factors becomes zero. While the isoquant takes an oval shape, the region of practical operation is confined between these ridge lines. The firm opts for production solely within the convex segments of isoquants situated between the upper and lower ridge lines. These ridge lines signify points on the isoquants where the marginal products (MP) of factors drop to zero, with the upper ridge line indicating zero MP of capital and the lower ridge line indicating zero MP of labour. Efficient production techniques are confined to the region inside the ridge lines, where the marginal products of factors are negative. In contrast, outside the ridge lines, production methods become inefficient.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The ridge lines are illustrated in the diagram, where curves OA and OB serve as the ridge lines on the oval-shaped isoquants. Between these lines, at points G, J, L, N, and H, K, M, P, economically viable combinations of capital and labour can be employed to produce 100, 200, 300, and 400 units of the product. For instance, OT units of labour and ST units of capital can produce 100 units of the product, but the same output can be achieved with the same quantity of labour OT and a reduced quantity of capital VT. Therefore, production in the dotted region of isoquant 100 is considered uneconomical.

These dotted segments of isoquants represent uneconomical production regions as they necessitate an increase in the use of both factors without a corresponding increase in output. Connecting points G, J, L, N, H, K, M, and P with lines OA and OB forms the ridge lines. On both sides of these lines, production is deemed uneconomical, while it remains economically feasible within the ridge lines.

6.7 Expansion Path

6.8 Total, Marginal, and Average Revenue

Before any costs or expenses are deducted, revenue is the amount generated from the sale of goods or services, or any other use of capital or assets, associated with the main operations of the firm. In economics, there are three types of revenues: total revenue, average revenue, and marginal revenue, which are discussed further below.

6.8.1 Total Revenue (TR)

Total revenue is the total amount of money earned from the sale of any given amount of output. Total revenue is calculated by multiplying the sale price by the quantity sold, i.e.,

TR = Price x Quantity.

For example, if the price is ten dollars and the quantity sold is one hundred dollars, the total revenue is one thousand dollars. A total revenue curve is depicted in the figure.

 

 

 

 

 

 

 

 

 

 

 

 

6.8.2 Average Revenue (AR)

The average revenue received for selling a good is the revenue received per unit of output sold. It is calculated by dividing total revenue by output quantity, i.e., AR= TR/Quantity.

A simpler and more widely used term for average revenue is price. Using average revenue over time rather than price provides a link to other related terms, particularly total revenue and marginal revenue. Average revenue, when compared to average cost, shows the amount of profit generated per unit of output produced. An average revenue curve, as shown in Figure, is frequently used to depict average revenue.

The figure shows an AR curve in a perfect market.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.8.3 Marginal Revenue (MR)

The change in total revenue caused by a change in the quantity of output sold is referred to as marginal revenue. Marginal revenue is the sum of additional funds a business receives for selling an additional unit of output. It is calculated by dividing the change in total revenue by the change in output quantity. Marginal revenue is one of two revenue concepts derived from total revenue and is the slope of the total revenue curve. The other factor is revenue on a per-capita basis. A company’s profit is maximised by equating marginal revenue and marginal cost.

Change in TR/Change in Quantity = MR

The figure shows an MR curve in a perfect market.