Week 3 - Lecture on Forms and Marginal Rate of Technical Substitution (MRTS)

Recap from Last Week

• Discussed the firm representation as a technological black box using inputs to produce outputs.
• Key concepts from last week:
  • Technical and Economic Efficiency
  • Diminishing Marginal Returns: The phenomenon where the marginal productivity of an input decreases as the quantity of the input increases.

One Input and One Output Model

• Simple representation: Input (X) → Output (Y)
• Production function: Y = f(X)
• Marginal productivity (MP): How output changes as the input changes, denoted as f'(X).
• Diminishing marginal returns: MP decreases with an increase in X.

Two Inputs Model: Labor (L) and Capital (K)

• Production function expands to include two inputs: black box model takes labor and capital to produce output (Y).
• Marginal Productivity of Labor (MPL): Keeping capital constant and varying labor.
• Marginal Productivity of Capital (MPK): Keeping labor constant and varying capital.
• Diminishing returns apply to both inputs: Marginal productivity decreases beyond a certain point.

Isoquants

• Definition: Curve representing all combinations of L and K that produce the same level of output, showing technical efficiency.
• Example: Isoquant for producing 10 units could be represented by various combinations of labor and capital.
• Key Point: Isoquants already account for technical efficiency.

Economic Efficiency

• Objective: Minimize production cost while maintaining output level.
• Costs: Wages (W) and capital rental (R).
• Cost minimization formula: Minimize WL + RK subject to production constraints.

Marginal Rate of Technical Substitution (MRTS)

• Definition: Rate at which one input can be exchanged for another without changing the output level.
• Illustrated through changes in L and K along an isoquant.
• Slope of the isoquant gives the MRTS.
• Mathematically represented as: MRTS = MPL / MPK

Calculating MRTS

• Linear Technology: Y = aX1 + bX2
  • MPL = a, MPK = b
  • MRTS = a/b
• Cobb-Douglas Technology: Y = K^α * L^β
  • MPL = α * K^(α-1) * L^β
  • MPK = β * K^α * L^(β-1)
  • MRTS = MPL / MPK*

Important Notes on MRTS

• MRTS is not constant and changes along the isoquant.
• To maintain output, adjustments must be made in input quantities reflecting diminishing returns.
• MRTS can be calculated using both algebraic and calculus approaches, though calculus provides a more precise rate of change.

Examples and Practice

• Discussed isoquants for different levels (8, 10, 12 units of output).
• Demonstrated calculations for linear and Cobb-Douglas technologies.
• Mentioned Leontief technology and pointed out it's trickier due to its kinked isoquants.

Conclusion

• Emphasized the significance of MRTS in production optimization and cost minimization.
• Preview of future topics: Further optimization techniques using MRTS.

Next Steps

• Assignment: Derive MRTS for the Leontief technology as practice.