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How to Produce

A firm can produce its output by using a variety of inputs; which inputs should the firm use?

How should the business produce (Factor-Factor)

 

What if a manager could use different inputs to produce the same product? Restated, if a manger had to decide which of two variable inputs? Can economic theory help explain how much of each input should be used to maximize profit?

Example 1 -- a business is producing a crop (output) on a fixed quantity of land (fixed input) and a limited amount of cash to be used for fertilizer and irrigation (variable inputs). How much of the cash should be used to buy and apply fertilizer and how much of the cash should be used to operate the irrigation system?

Example 2 -- a business is manufacturing a consumer product (output) in a manufacturing facility (fixed input) with a limited amount of cash (again) that can be used to buy steel or to hire workers (variable inputs). How much of the cash should be spent on steel and how much of the cash should be spent on labor?

Using the second example, if the business spends most of the cash on steel and only a little on labor, there may not be enough workers so the steel remains unused at the end of the production period. Alternatively, if the cash is spent on workers and not enough steel, the workers may find themselves standing around because there is no steel with which to manufacture the products. Another alternative is to buy plenty of steel and hire plenty of workers, but two matters limit this strategy -- the size of the facility and the amount of cash. These two matters are the fixed inputs and cannot be altered during the production period.

Economic theory states that the answer to the question of how much steel and how much labor can be stated as

MPPx1/Px1 = MPPx2/Px2, or

MPPx1/MPPx2 = Px2/Px1

The ratios in the second formula are referred to in economic theory as the marginal rate of substitution and the inverse price ratio.

These formulas focuses the manager's attention on the relationship between the quantity of output produced (MPP) and the cost of the variable input (Px). Due to the reality of diminishing marginal productivity, the MPP for both inputs changes as more variable input is used. The manager does not want to use so much of one input and so little of the other input that neither of the levels of use is close to being the profit maximizing combination of inputs. However, the manager also needs to consider the relative cost of each variable input. Accordingly, the manager considers how adjusting the relative quantity of the variable input impacts total production of the output and the relative cost of the variable inputs.

This mathematical statement is based on the premise that there is a production function for the steel and for labor; that is, how much output is there for the steel and how much output is there for the labor.

In these two disjointed relationship (that is, the relationship between steel and output is presumably not impacted by the amount of labor, and the production function for labor is presumably not impacted by the availability of steel), there is an individual production function for each variable input.

The primary fixed input for the individual production functions for steel and labor is presumed to be the manufacturing facility, not the cash.

The solution, however, is limited by the amount of cash available to the business, therefore, (Px1 * X1) + (Px2 * X2) must not exceed the amount of available cash.

Because of the cash limitation, the firm may not produce where MVP = MIC, but instead has to settle for producing at some point where MVP > MIC. This is based on the assumption that even though the business is profitable and there is room to expand, there is no opportunity during this production period to acquire more cash with which to purchase more variable inputs.

 

Restating the Concept

Restated, the manager considers the MVP for each of the alternative inputs that can be used to produce the output.  A portion of the firm’s resource (usually capital) is then directed to the input with the greatest MVP.  The next portion of the firm’s capital is directed to the input with the next greatest MVP.  This process is continued until all of the firm’s capital is invested or the MVPs for each of the inputs being considered is less than the MIC.

 

An Example of Deciding “How to Produce”

The question that needs to be answered (the decision that needs to be made) is what combination of inputs should a business use to produce its product.  The following example uses a simple example from production agriculture.

A farmer has $200 to invest in raising a crop.  This $200 can be used for seed, fertilizer or chemical.  The farmer mentally divides the $200 of capital into $10 units (thus the farmer has 20 units of capital to allocate among the three inputs).  The question is how much of each input should the farmer use?

To answer this question, the farmer will apply the concepts of Marginal Input Cost (MIC) and Marginal Value Product MVP for each input.  In this case, the MIC is $10 for each unit of capital.  The MVPs for the inputs are summarized in the following table.

Each column reflects the production function for that particular input.

Unit of Capital  

MIC per unit of Capital

MVP for Seed

MVP for fertilizer

MVP for chemical

1

$10

$20 (1)

$17 (3)

$16 (5)

2

$10

$18 (2)

$16 (4)

$15 (8)

3

$10

$15 (7)

$15 (6)

$13 (11)

4

$10

$11 (13)

$14 (9)

$11 (14)

5

$10

$7

$13 (10)

$8

6

$10

$3

$12 (12)

$5

7

$10

$0

$10 (15)

$2

8

$10

$0

$8

$0

9

$10

$-5

$6

$-4

10

$10

 

$3

 

11

$10

 

 

 

12

$10

 

 

 

The numbers in parenthesis indicate the order in which the inputs will be acquired with the units of capital.  These decisions, though, are based on sequence of considerations for each unit of capital, as described in the following points.

Decision rule:  buy the input with the greatest MVP as long as the MVP exceeds the MIC.

 

Unit of Capital:  1

Options

Buy first unit of seed with an MVP of $20

Buy first unit of fertilizer with an MVP of $17

Buy first unit of chemical with an MVP of $16

Decision:  Buy first unit of seed with an MVP of $20

 

Unit of Capital:  2

Options

Buy second unit of seed with an MVP of $18

Buy first unit of fertilizer with an MVP of $17

Buy first unit of chemical with an MVP of $16

Decision:  Buy second unit of seed with an MVP of $18

 

Unit of Capital:  3

Options

Buy third unit of seed with an MVP of $15

Buy first unit of fertilizer with an MVP of $17

Buy first unit of chemical with an MVP of $16

Decision:  Buy first unit of fertilizer with an MVP of $17

 

Unit of Capital:  4

Options

Buy third unit of seed with an MVP of $15

Buy second unit of fertilizer with an MVP of $16

Buy first unit of chemical with an MVP of $16

Decision:  Buy EITHER the second unit of fertilizer or the first unit of chemical with MVPs of $16 (for this example, the second unit of fertilizer will be purchased with the fourth unit of capital)

 

Unit of Capital:  5

Options

Buy third unit of seed with an MVP of $15

Buy third unit of fertilizer with an MVP of $15

Buy first unit of chemical with an MVP of $16

Decision:  Buy first unit of chemical with an MVP of $16

 

Unit of Capital:  6, 7 & 8

Options

Buy third unit of seed with an MVP of $15

Buy third unit of fertilizer with an MVP of $15

Buy second unit of chemical with an MVP of $15

Decision:  Buy third unit of seed, third unit of fertilizer, or second unit of chemical in no particular order because each has an MVP of $15

 

Unit of Capital:  9

Options

Buy fourth unit of seed with an MVP of $11

Buy fourth unit of fertilizer with an MVP of $14

Buy third unit of chemical with an MVP of $13

Decision:  Buy fourth unit of fertilizer with MVP of $14

 

Unit of Capital:  10 & 11

Options

Buy fourth unit of seed with an MVP of $11

Buy fifth unit of fertilizer with an MVP of $13

Buy third unit of chemical with an MVP of $13

Decision:  Buy fifth unit of fertilizer and third unit of chemical in no particular order because each has an MVP of $13

 

Unit of Capital:  12

Options

Buy fourth unit of seed with an MVP of $11

Buy sixth unit of fertilizer with an MVP of $12

Buy fourth unit of chemical with an MVP of $11

Decision:  Buy sixth unit of fertilizer with an MVP of $12

 

Unit of Capital:  13 & 14

Options

Buy fourth unit of seed with an MVP of $11

Buy seventh unit of fertilizer with an MVP of $10

Buy fourth unit of chemical with an MVP of $11

Decision:  Buy fourth unit of seed and fourth unit of chemical in no particular order because each has an MVP of $11

 

Unit of Capital:  15

Options

Buy fifth unit of seed with an MVP of $7

Buy seventh unit of fertilizer with an MVP of $10

Buy fifth unit of chemical with an MVP of $8

Decision:  Buy seventh unit of fertilizer with an MVP of $10

 

Unit of Capital:  16

Options

Buy fifth unit of seed with an MVP of $7

Buy eighth unit of fertilizer with an MVP of $8

Buy fifth unit of chemical with an MVP of $8

Decision:  Do not invest any more units of capital in these inputs for this crop because each MVP is less than the $10 MIC.  Move the remaining capital to other inputs or other enterprises.

 

 

What to do if there is not enough capital to maximize profit for all three inputs?

If the farmer has only $120 (12 units of capital), for example, the farmer would stop with 3 units of seed, 6 units of fertilizer and 3 units of chemical.  That is, the farmer would have to stop short of profit maximization because the business ran out of capital for these inputs for this crop.

The farmer would use the same analysis or thought process as described above, but stop when the capital is fully spent (or invested).

 

What to do if profit is maximized before all the capital is expended?

As illustrated with the example above, the farmer would stop investing capital at profit maximization (that is, where MVP for each crop is equal to MIC).  The farmer would invest any remaining capital in something other than these three inputs for this crop.

Again, the farmer would use the same analysis or thought process as described above, but stop when the MIC is greater than any of the available MVPs.

 

 

Managers also may want to consider the economic theory that explains deciding how much to produce and what to produce.

 

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