Question

The production function of a crop where the input is fixed is given by:

• A. \(Y = f(X_1 / X_2, X_3, \dots, X_n)\)
• B. \(Y = f(X_1, X_2 / X_3, X_4, \dots, X_n)\)
• C. \(Y = f(X_1, X_2, X_3, \dots, X_n)\)
• D. \(Y = f(Y_1, Y_2, Y_3 / Y_4, \dots, Y_n)\)

Hint:

Exam Tip: In production function analysis:
Short Run: At least one input is fixed.
Long Run: All inputs are variable.
The notation \(X_1 / X_2\) indicates that \(X_2\) is the fixed input.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question tests the understanding of production functions in agricultural economics, specifically how to represent a production function when one input is fixed.

Step 2: Analyzing Production Function Notation:
A general production function is represented as: \[ Y = f(X_1, X_2, X_3, \dots, X_n) \] Where:
• \(Y\) is the output (crop yield).
• \(X_1, X_2, X_3, \dots, X_n\) are the inputs (e.g., land, labor, capital, water, fertilizers). When an input is fixed, it means its quantity does not change.
The notation \(X_1 / X_2\) indicates that the ratio of two inputs is held constant or that one input is fixed relative to the other.
In production economics, when a particular input is fixed, it is often represented by placing it in the denominator or separating it from the variable inputs.
The notation \(X_1 / X_2\) implies that \(X_2\) is fixed or that the relationship between \(X_1\) and \(X_2\) is constant.
Option (A) shows \(Y = f(X_1 / X_2, X_3, \dots, X_n)\), indicating that \(X_1\) varies while \(X_2\) is fixed, and other inputs (\(X_3\) to \(X_n\)) vary.
This is the correct representation when one input is fixed.

Step 3: Evaluating Other Options:

• (B) \(Y = f(X_1, X_2 / X_3, X_4, \dots, X_n)\): This is less standard and would imply a different fixed relationship.
• (C) \(Y = f(X_1, X_2, X_3, \dots, X_n)\): This represents a general production function with all inputs variable.
• (D) \(Y = f(Y_1, Y_2, Y_3 / Y_4, \dots, Y_n)\): This is incorrect as it uses \(Y\) for inputs, which is typically used for output.

Step 4: Final Answer:
The production function where one input is fixed is represented as \(Y = f(X_1 / X_2, X_3, \dots, X_n)\). Therefore, option (A) is correct.