Understanding the production function

The production function describes how a firm transforms inputs into outputs, helping to explain the decisions firms make about resource use. This relationship applies in both the short run and the long run, shaping how firms optimize production to minimize costs and maximize efficiency.

Key concepts in production

Before exploring the production function, it's important to define the basic terms that form its foundation.

Inputs and outputs:

• Inputs - Resources used in production, such as labor (workers), capital (machinery and equipment), land, and entrepreneurship.
• Outputs - The goods or services produced from these inputs.

Time periods in production:

• Short run - A time period where at least one input is fixed (cannot be changed), while others can vary. For example, a factory might increase workers but cannot quickly expand its building size.
• Long run - A time period where all inputs can be varied, allowing the firm to adjust everything, such as building a larger factory or buying more equipment.

The production function shows the maximum output achievable from different combinations of inputs, assuming efficient use. In the short run, it focuses on varying one input while holding others constant. In the long run, it considers scaling all inputs together, which can lead to different cost behaviors as the firm grows.

Key measures of productivity

Productivity measures how effectively inputs are converted into outputs. These concepts build on each other, starting with the total amount produced and then breaking it down into averages and changes per additional input.

Total product

Total product (TP), also known as total physical product, is the overall quantity of output produced from a given amount of inputs. As inputs increase, TP typically rises, but the rate of increase can vary due to factors like efficiency.

Average product

Average product (AP) measures the output per unit of input, showing overall efficiency.

Formula for average product:

$\text{AP}=\frac{\text{total product}}{\text{units of input}}$

Where:

• Total product = Overall output quantity
• Units of input = Amount of the variable input used (e.g., number of workers)

AP helps firms assess how productive their inputs are on average. It often rises initially as inputs are added but may fall if overcrowding or inefficiencies occur.

Marginal product

Marginal product (MP) measures the additional output gained from using one more unit of input, while holding other inputs constant.

Formula for marginal product:

$\text{MP}=\frac{\text{change in total product}}{\text{change in units of input}}$

Where:

• Change in total product = Increase in output from adding the input
• Change in units of input = Increase in the variable input (usually 1 unit)

MP shows the incremental benefit of adding more inputs. Changes in MP directly affect how TP grows—when MP is positive and rising, TP increases at an accelerating rate; when MP falls but remains positive, TP still grows but more slowly.

Relationships between productivity measures

How the measures connect:

• TP changes based on MP: Each additional input adds the MP amount to TP.
• AP is influenced by MP: When MP is above AP, AP rises; when MP is below AP, AP falls.
• These measures are often graphed with inputs on the x-axis and output on the y-axis. The TP curve starts at the origin and rises, while AP and MP curves are U-shaped, peaking before declining.

Diminishing marginal returns

Diminishing marginal returns is a key principle in the short run that explains why adding more of one input doesn't always yield proportional increases in output.

This occurs when a firm increases one variable input (like labor) while keeping other inputs fixed (like capital). Initially, MP may rise due to better specialization, but eventually, it decreases as the fixed inputs become overcrowded— for example, too many workers sharing limited machinery leads to inefficiencies.

As a result, TP continues to increase but at a slowing rate, and AP may also start to fall after a point. This law highlights why firms can't endlessly expand production in the short run without adjusting fixed inputs.

Production and costs in the short run and long run

Production levels directly influence a firm's costs, as the efficiency of turning inputs into outputs affects expenses per unit. Costs are tied to productivity measures and differ between the short run and long run.

Short-run costs

In the short run, costs are divided into fixed and variable categories, reflecting the fixed inputs.

Types of short-run costs:

• Fixed costs (FC) - Costs that do not change with output level, such as rent or machinery depreciation.
• Variable costs (VC) - Costs that vary with output, such as wages for additional workers or raw materials.
• Total cost (TC) - The sum of fixed and variable costs (TC = FC + VC).
• Average total cost (ATC) - Total cost per unit of output (ATC = TC / output).
• Average fixed cost (AFC) - Fixed cost per unit (AFC = FC / output), which decreases as output rises.
• Average variable cost (AVC) - Variable cost per unit (AVC = VC / output).
• Marginal cost (MC) - The additional cost of producing one more unit (MC = change in TC / change in output).

Short-run costs relate to production through diminishing marginal returns: As MP falls, MC rises because more inputs are needed for each additional output unit. Graphs typically show MC, AVC, and ATC as U-shaped curves, with MC intersecting AVC and ATC at their minimum points.

Long-run costs

In the long run, all costs are variable since all inputs can change. Firms can achieve economies of scale, where increasing output lowers average costs due to efficiencies like bulk purchasing or specialization.

The long-run average cost (LRAC) curve is often U-shaped, showing decreasing costs initially (economies of scale), constant costs (constant returns to scale), and then increasing costs (diseconomies of scale) as the firm grows too large and faces inefficiencies.

Production in the long run allows firms to choose the optimal scale, minimizing LRAC for a given output level.

Worked example - Calculating productivity measures

A firm produces widgets using labor as the variable input, with capital fixed. The table below shows total product for different labor units.

Calculate the marginal product and average product for 3 labor units, then find MP when labor increases from 4 to 5 units.

Step 1: Identify the values for average product at 3 units

• Total product at 3 units = 18 widgets
• Labor units = 3

Step 2: Calculate average product at 3 units

$\text{AP}=\frac{18}{3}=6\text{ widgets per labor unit}$

Step 3: Calculate marginal product at 3 units

• Change in total product from 2 to 3 units = 18 - 12 = 6 widgets
• Change in labor = 1 unit

$\text{MP}=\frac{6}{1}=6\text{ widgets}$

Step 4: Calculate marginal product from 4 to 5 units

• Change in total product = 24 - 22 = 2 widgets
• Change in labor = 1 unit

$\text{MP}=\frac{2}{1}=2\text{ widgets}$

This shows diminishing marginal returns starting, as MP falls from 6 to 2.

Worked example - Calculating short-run costs

A firm has fixed costs of $100 and the following variable costs for different output levels.

Calculate total cost, average total cost, and marginal cost for 3 units of output.

Step 1: Identify the values

• Fixed cost = $100
• Variable cost at 3 units = $120

Step 2: Calculate total cost at 3 units

TC = FC + VC = $100 + $120 = $220

Step 3: Calculate average total cost at 3 units

$\text{ATC}=\frac{220}{3}\approx \$73.33\text{ per unit}$

Step 4: Calculate marginal cost for the 3rd unit

• Change in TC from 2 to 3 units: At 2 units, TC = $100 + $90 = $190; at 3 units, TC = $220
• Change in TC = $220 - $190 = $30
• Change in output = 1 unit

$\text{MC}=\frac{30}{1}=\$30$