# Production Function

The economic concepts of Total Physical Product (TPP), Average Physical Product (APP), Marginal Physical Product (MPP), and the Stages of the Production Function

#### Production Function and Stages of Production -- Applying the Concept of Diminishing Marginal Productivity

Based on the assumptions of a goal of profit maximization and making decisions in the short run, combined with our understanding of diminishing marginal productivity, the question is "what level of input should a manager use and what level of output should the manager produce to maximize profit."

• The answer for one business will be different than the answer for another business.  Each business uses a slightly different combination of inputs to produce similar outputs.  For example, you use a different recipe and a different combination of ingredients than I do, but we both can produce a delicious chocolate cake.  Seldom do businesses use identical “recipes” to produce similar (substitutable) products.

The relationship between the level of variable input and level of output can be illustrated with a production function.

• A table of data can be used to present this relationship. This table does not identify the fixed inputs, but it indicates how the level of output changes if the manager changes the quantity of variable input used during the production period. In this example, using 2 units of variable input will result in producing 3 units of output. However, using 7 units of variable input during the production period would allow the business to produce 28 units of output.  Qty. of Var. Input Qty. of Output . (X) . (Y or TPP) 0 0 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 43 10 49 11 54 12 58 13 61 14 63 15 64 16 64 17 63 18 61
• A graph may improve our understanding of the concept (graph 1). The axes represent the number of physical units used (variable input or X) and the number of physical units produced (output or Y).
• Total physical product (TPP) -- Quantity of output (Y) that is produced from a firm's fixed inputs and a specified level of variable inputs (X).
• Production function-- illustrates the relationship between the quantity of variable input and the level of output.

The production function could be described as a combination or series of enterprise analyses wherein each point on the production function represents a different enterprise; that is, a different recipe or combination of fixed inputs and variable input. The idea that the production function is a series of enterprises is expanded on in subsequent sections.

No business operates with one variable input and one fixed input. Instead, it may be easier to think about fixed and variable inputs as a collection of resources. Any resource or input that cannot be altered during the production period would be considered part of the fixed inputs and inputs that can be varied would be considered variable inputs. In a farm setting during a production season, there may not be enough time to acquire more land, buildings, equipment or labor. These would be fixed inputs. But there may be enough time to borrow more capital with which to buy more fertilizer, seed, pesticides, fuel. These would be the variable inputs. However to simplify illustrating the concept of diminishing marginal productivity, the examples often assume a collection or group of fixed inputs and one variable input.

What can we learn by looking at the data or graph?

First, as the level of variable input is increased, the level of output:

• Increases at an increasing rate, then
• Increases at a decreasing rate, and at some point,
• decreases.

Second, managers should not use so much variable input that the output actually declines. In this example, the manager would not use more than 15 units because the 16th unit does not increase production, and using more than 16 units actually decreases production. The economic concept of marginal physical product can help explain this point. (Graph 2)

Marginal physical product (MPP) is the change in the level of output due to a change in the level of variable input; restated, the MPP is the change in TPP for each unit of change in quantity of variable input.

• MPP = (TPP2 - TPP1)/(X2 - X1)
• A firm will not produce in stage III because using additional units of variable input decreases output; that is, TPP decreases as more variable input is used; MPP < 0.
Economic theory refers to stage III as the portion of the production function where additional variable input results in decreased output. Managers do not produce in Stage III. In this situation, the boundary between Stage II (not yet defined) and Stage III is at 15 units of variable input.

Graph 5

Third, there is a minimum level of variable input that the manager should use. If a manager decides to use some of the variable input; is there a minimum quantity of variable input the manager should use? The answer is yes, but why is the answer yes?

Consider the example illustrated in the table. Using 1 unit of variable input will result in the production of 1 unit of output. However, using 2 units of variable input will result in the production of 3 units of output. At the first level of production, the variable input, on the average produces just one unit of output. At the second level, each unit of variable input produces 1.5 units of output (Y/X). Thus increasing the level of input increases that quantity of output for each unit of variable input. Economic theory refers to quantity of output per unit of variable input as the average physical product (APP).

Continuing the example, using 3 units of variable input will result in an APP of 2 (6/3); this too is better than using only 2 units of variable input.

Graph 3

As long as the APP is increasing, the manager will use more units of the variable input. In this situation, APP increases until the manager is using 11 units of variable input. This is the minimum number of units of variable input the manager will use, if the variable input is used.

Economic theory refers to the portion of the production function where the APP is increasing as Stage I. The boundary between Stage I and Stage II, in this example, is 11 units of variable input. This is the level of variable input where the APP is maximized. Managers will not produce in Stage I because using more variable input will increase the output for each unit of variable input.

Graph 4

Average physical product (APP) -- quantity of output per unit of variable input.

How much output is each unit of variable input producing?

• APP = TPP/X
• A firm will not produce in stage I because using additional units of variable input improves the productivity of the variable input (the APP is increasing as more units of variable input are used).  It is not until the firm reaches stage II (declining APP) that the answer to the question of whether to use more variable input is unclear.

 Qty. of Var. Input Qty. of Output . (X) . (Y or TPP) APP MPP 0 0 ?? 1 1 1 1 2 2 3 1.5 3 3 6 2 4 4 10 2.5 5 5 15 3 6 6 21 3.5 7 7 28 4 8 8 36 4.5 7 9 43 4.78 6 10 49 4.9 5 11 54 4.91 4 12 58 4.83 3 13 61 4.69 2 14 63 4.5 1 15 64 4.26 0 16 64 4 -1 17 63 3.7 -2 18 61 3.39

Managers will produce only in Stage II: where APP declines if more variable input is used but MPP is still positive; that is, TPP still increases as a result of using more variable input.

Accordingly, the manager will produce somewhere in Stage II; where the APP decreases if more variable input is used, but MPP is still greater than 0.

This information still does not reveal what level of variable input or level of output within stage II maximizes profit – we need to convert the information about physical units into dollars in order to determine the profit maximizing level of input and output.

How does the business manager know the relationship between level of output and level of variable input for the business?

Each business is different. The relationship between productivity (output) and the quantity of input is different for each business. There is no information source about this relationship. Yes, for some industries there may be some published data on this relationship but even in those cases, each business in the industry has a different experience.

Bottom line -- the manager needs to track data for the business to develop the information needed to reveal the relationship between quantity of input and quantity of production or output. This is one small part of developing a business inventory.

Example to illustrate impact of technology

The quantity of output resulting from the use of the variable input is impacted by the production technology the business is employing. A change in the technology, for example, an improvement in production technology, is illustrated by an upward shift in the production function. New technology, for example, may allow a farmer to produce more wheat (output) from the same acre (fixed input) and fertilizer (variable input).

Graph 29

The cost of employing the new technology is discussed in the cost section of these web pages.

In summary --

During a production period, diminishing marginal returns "occurs when equal increases of variable resources are successively added to some fixed resource; marginal physical products eventually decline”.

The next page discusses the relationship between input and output in terms of dollar value rather than physical units (as it was discussed on this page). Discussing the relationship between variable input and output in terms of dollars allows a manager to consider the profit maximizing level of variable input.

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