Production with Diminishing Marginal Products

Introduction to Marginal Cost and Productivity

In microeconomics, understanding how production costs change with output is essential. Marginal cost can increase for two main reasons: (1) extensive changes, where different individuals with varying costs begin producing, and (2) intensive changes, where an individual’s productivity decreases as more output is produced due to diminishing returns.

• Extensive changes: Costs rise as higher-cost producers enter the market.

• Intensive changes: Costs rise for an individual producer as input productivity falls with increased output.

The Production Function

Definition and Inputs

The production function describes the maximum output achievable from given quantities of inputs. Common inputs include land, labor, capital, and raw materials. For simplicity, we often focus on two inputs:

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

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

The general form of the production function is:

where is the quantity of output.

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.

Additional info: Returns to scale help firms decide on the optimal size of operation.

Short Run vs. Long Run

• Short Run: At least one input (usually capital) is fixed.

• Long Run: All inputs can be varied.

Short-Run Production Relationships: TP, MP, and AP

Key Concepts

• Total Product (TP): Total output produced with given inputs.

• 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 (like capital), the additional output from each new unit eventually decreases. This is known as the Law of Diminishing Marginal Returns.

• Stage I: Output increases at an increasing rate.

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

• Stage III: Output decreases.

Additional info: Stage II is where most firms operate, as it is the range of efficient production.

Relationship Between MP and AP

• When MP > AP, AP is rising.

• When MP < AP, AP is falling.

• When MP = AP, AP is at its maximum.

Marginal Product and Demand for Labor

Value of the Marginal Product (VMP)

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

where is the price of output.

• The firm hires labor up to the point where the wage () equals the value of the marginal product:

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

Example: Hiring Decision

• Given a table of marginal products and wage/output price, the firm should hire workers up to the point where .

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:

where is the wage rate and is the rental rate of capital.

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

Additional info: This is analogous to consumer utility maximization, where marginal utility per dollar is equalized across goods.

Marginal Costs 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 decreases (due to diminishing returns), marginal cost increases.

Relationship Between MC and AVC (or 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 labor productivity: Shifts MC and AVC downward.

• Increased wages: Shifts MC and AVC upward.