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M18.L01 — Technology and Production Sets

Module: Module 18 — Technology, Production, and Profit Maximisation Lesson: L01 of 05 Duration: ~30 minutes Level: Intermediate Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics

Learning Objective

Understand the concept of a production set and how it defines feasible input-output combinations.

Production Sets and Functions

A production set represents all feasible combinations of inputs (e.g., labor, capital) and outputs (e.g., goods). For two inputs \(x_1\) and \(x_2\), the production function \(y = f(x_1, x_2)\) maps inputs to output. Isoquants are curves showing all input combinations yielding the same output level.

Key Properties:

  • Marginal Rate of Technical Substitution (MRTS): The rate at which one input can be substituted for another while maintaining output: \(MRTS = -\frac{dx_2}{dx_1} = \frac{MP_1}{MP_2}\), where \(MP_i\) is the marginal product of input \(i\).
  • Returns to Scale: Describes how output scales with inputs:
  • Increasing (IRS): \(f(tx_1, tx_2) > tf(x_1, x_2)\)
  • Constant (CRS): \(f(tx_1, tx_2) = tf(x_1, x_2)\)
  • Decreasing (DRS): \(f(tx_1, tx_2) < tf(x_1, x_2)\)
  • Cobb-Douglas: A common functional form: \(f(x_1, x_2) = x_1^a x_2^b\).

Worked Example

[Australian context]

Suppose an Australian wheat farm uses labor (\(x_1\)) and machinery (\(x_2\)) with the production function \(y = 2x_1^{0.5} x_2^{0.5}\) (Cobb-Douglas). 1. Calculate MRTS: [ MP_1 = \frac{\partial y}{\partial x_1} = x_1^{-0.5} x_2^{0.5}, \quad MP_2 = \frac{\partial y}{\partial x_2} = x_1^{0.5} x_2^{-0.5} ] [ MRTS = \frac{MP_1}{MP_2} = \frac{x_2}{x_1} ] 2. Returns to Scale: [ f(tx_1, tx_2) = 2(tx_1)^{0.5} (tx_2)^{0.5} = t \cdot 2x_1^{0.5} x_2^{0.5} = t f(x_1, x_2) \quad \text{(CRS)} ]

Common Misconception

"Isoquants show output levels; higher isoquants mean higher output."

Isoquants represent equal output levels. Higher output is represented by moving to a new isoquant.

Key Takeaways

  • Production functions map inputs to outputs, and isoquants represent equal output levels.
  • MRTS measures the substitutability of inputs while holding output constant (slope of isoquant).
  • Cobb-Douglas functions exhibit CRS if exponents sum to 1.

Practice

  1. For \(y = x_1^{0.3} x_2^{0.7}\), find MRTS and returns to scale.
  2. Sketch the isoquant for \(y = 10\) with \(x_1\) on the x-axis and \(x_2\) on the y-axis.

Further Resources

Provenance: Intermediate Microeconomics with Excel (Barreto) | MIT OCW 14.01 Principles of Microeconomics