Production Functions, Marginal and Average Product of Labor: Microeconomics Study Guide

Production Functions and Labor Productivity

Inputs, Outputs, and Time Horizons

Production in microeconomics refers to the process by which firms transform inputs into outputs. The main inputs are Labor (L) and Capital (K). Understanding how these inputs affect output is fundamental to analyzing firm behavior.

• Labor (L): The human effort used in production, such as workers or employees.

• Capital (K): All other productive resources, including machines, tools, buildings, and equipment.

The production function expresses output as a function of inputs:

$Q = f(L, K)$

• Q: Total quantity of output produced.

• L: Labor input.

• K: Capital input.

Time horizons in production:

• Short Run: At least one input (usually capital) is fixed. Firms can only adjust variable inputs (typically labor).

• Long Run: All inputs are variable. Firms can change the scale of operation, such as expanding facilities or acquiring more equipment.

Short-run production function:

$Q = f(L, \overline{K})$

where $\overline{K}$ is the fixed level of capital.

Total Product Curve: Output vs. Labor

The Total Product (TP) curve shows the relationship between labor input and total output, holding capital fixed. The curve typically has three distinct phases:

1. Phase 1: Increasing at an Increasing Rate

   • Output rises faster as more workers are added.

   • Caused by learning, specialization, and improved coordination.

2. Phase 2: Increasing at a Constant Rate

   • Output increases steadily with each additional worker.

3. Phase 3: Increasing at a Decreasing Rate (Diminishing Returns)

   • Output continues to rise, but at a slower pace.

   • Diminishing marginal returns occur as more workers are added to a fixed amount of capital.

Important: "Increasing at a decreasing rate" means output is still rising, just more slowly—not that output is falling.

Marginal Product of Labor (MPL)

The Marginal Product of Labor (MPL) measures the additional output generated by hiring one more worker.

• Definition: $MPL = \frac{\Delta Q}{\Delta L}$

• Interpretation: MPL is the contribution of the next worker to total output.

• Graphical intuition: The slope of the Total Product curve at any point is the MPL.

Example: If output increases from 10 to 15 when a third worker is hired, $MPL = 15 - 10 = 5$.

Average Product of Labor (APL)

The Average Product of Labor (APL) is the output per worker.

• Definition: $APL = \frac{Q}{L}$

• Example: If 2 workers produce 30 units, $APL = \frac{30}{2} = 15$ units per worker.

Numerical Example: Car Wash Production

Consider a car wash with fixed capital and varying labor. Output is measured in clean cars per hour.

Labor (L)	Q (Clean cars/hr)	MPL	APL
2	10	--	5
3	15	5	5
4	18	3	4.5
5	19	1	3.8
6	19	0	3.1

Observation: As more workers are hired, MPL falls, indicating diminishing marginal returns.

Optimal Hiring Decision: Marginal Benefit vs. Marginal Cost

Firms hire workers to maximize profit, not just output. The decision depends on the value of the marginal product and the wage rate.

• Profit: $\text{Profit} = \text{Revenue} - \text{Cost}$

• Value of MPL: $\text{Value of MPL} = MPL \times \text{Price per unit}$

• Hiring rule: Hire the next worker if $\text{Value of MPL} \geq \text{Wage}$

Example: If wage = $15/hour and price per car wash = $10:

• Worker 3: MPL = 5, Value = $50, Cost = $15, Net gain = $35 → Hire

• Worker 4: MPL = 3, Value = $30, Cost = $15, Net gain = $15 → Hire

• Worker 5: MPL = 1, Value = $10, Cost = $15, Net gain = -$5 → Do NOT hire

Optimal labor: Hire up to L = 4.

Marginal–Average Relationship (Marginal–Average Rule)

The relationship between marginal and average product is a fundamental concept:

• If Marginal > Average, then Average rises.

• If Marginal < Average, then Average falls.

• If Marginal = Average, then Average stays the same (this is the peak or trough).

MPL and APL Curves: Shape and Relationship

The MPL and APL curves are typically hump-shaped:

• MPL Curve: Rises initially, becomes constant, then falls due to diminishing returns.

• APL Curve: Rises while MPL is above it, falls when MPL drops below it.

Key Rule: The MPL curve intersects the APL curve at the maximum point of the APL curve:

• At this point, $MPL = APL$.

• Before this point: $MPL > APL$ → APL rising.

• After this point: $MPL < APL$ → APL falling.

This relationship is essential for understanding how firms evaluate productivity and make hiring decisions.

Additional info: The notes cover core microeconomic concepts from Chapter 11: Technology, Production, and Costs, including production functions, marginal and average product, and the logic of profit-maximizing input choices.