Production with Diminishing Marginal Products

Introduction to Marginal Cost and Productivity

In microeconomics, understanding how production costs change with output is essential for analyzing firm behavior. Marginal cost can increase due to two main reasons: (1) extensive changes, where different individuals with varying costs begin producing, and (2) intensive changes, where the productivity of inputs diminishes as more output is produced from the same set of resources.

• Extensive Changes: Costs rise as higher-cost producers enter production.

• Intensive Changes: Costs rise due to diminishing productivity of inputs, even for a single producer.

The Production Function

Definition and Inputs

The production function describes the maximum output that can be produced from a given set of inputs. It is typically expressed as:

• Labour (L): Human effort used in production.

• Capital (K): Long-lived assets such as machinery and buildings.

Returns to Scale

• Constant Returns to Scale (CRS): Doubling all inputs doubles output.

• Increasing Returns to Scale (IRS): Doubling all inputs more than doubles output.

• Decreasing Returns to Scale (DRS): Doubling all inputs less than doubles output.

Short Run vs. Long Run

• Short Run: At least one input is fixed (e.g., capital).

• Long Run: All inputs can be varied.

Short-Run Production Relationships

Total, Marginal, and Average Product

When capital is fixed, changes in labor input affect output (total product). Key measures include:

• Marginal Product of Labor (MPL): The additional output from one more unit of labor.

• Average Product of Labor (APL): Output per unit of labor.

Diminishing Marginal Products

As more units of a variable input (like labor) are added to fixed inputs, the additional output from each new unit eventually decreases. This is known as the Law of Diminishing Marginal Returns.

Three-Stage Production Function

• Stage I: Output increases at an increasing rate.

• Stage II: Output increases at a decreasing rate (diminishing marginal product).

• Stage III: Output decreases.

Relationship between MP and AP:

• When MP > AP, AP is rising.

• When MP < AP, AP is falling.

• When MP = AP, AP is maximized.

Marginal Product and Demand for Labour

Value of the Marginal Product (VMP)

The value of the marginal product is the additional revenue generated by employing one more unit of input:

• Labour Demand Curve: The downward-sloping portion of the VMP curve represents the firm's demand for labor.

• Profit Maximization: The firm hires labor up to the point where .

Example Calculation

Given a table of marginal products and wage/output prices, the optimal number of workers is where but adding another worker would make .

Input Demand in the Long Run

Profit Maximization with Multiple Inputs

In the long run, firms can adjust all inputs. The profit-maximizing condition is to equate the marginal product per dollar spent across all inputs:

• Equilibrium:

This ensures that the last dollar spent on each input yields the same additional output.

Marginal and Average Costs in the Short Run

Cost Definitions

• Fixed Cost (FC): Costs that do not vary with output.

• Variable Cost (VC): Costs that change with output.

• Total Cost (TC):

• Average Fixed Cost (AFC):

• Average Variable Cost (AVC):

• Average Total Cost (AC):

• Marginal Cost (MC):

Relationship between Marginal Product and Marginal Cost

Marginal product and marginal cost are inversely related: as marginal product falls, marginal cost rises.

Relationship between MC, AVC, and AC

• When MC < AVC, AVC is decreasing.

• When MC > AVC, AVC is increasing.

• When MC = AVC, AVC is minimized.

Shifts in Cost Curves

• Per-unit tax: Shifts MC and AVC upward.

• Lump-sum tax: Shifts AFC and AC upward, but not MC or AVC.

• Increased productivity: Shifts MC and AVC downward.

• Higher wages: Shifts MC and AVC upward.

Relationship between AP and AVC: Average product and average variable cost are also inversely related.

Additional info: The included image visually demonstrates the relationship between marginal/average product and marginal/average cost curves, reinforcing the inverse relationship and the points where MC intersects AVC at its minimum.