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Numeric Example

This page offers hypothetical data to help illustrate the production theory concepts described in this folder.
 

Numeric Example to Illustrate Profit Maximizing Level of Variable Input & Level of Output and Related Concepts

 

Managers produce in Stage 2; that is, they will use enough variable input so the firm reaches Stage 2 of its production function.  In Stage 2, adding variable input during the production period increases output (TPP) but at a declining rate; that is, MPP is declining.  APP also declines as more variable input is used during the production period.

The following tables repeat the data that corresponds to Stage 2 of the hypothetical production function introduced on a previous page (Production Function).  In the first table, a column has been added (Py) for the price of the output (Py).  Another column has been added for the marginal value product (MVP).  Three additional columns have been added to illustrate alternative cost scenarios for the variable input (Px1, Px2 & Px3). 

The purpose of the table is to offer data to illustrate how converting Stage 2 production data into "dollars"  allows managers to

  1. identify the profit maximizing level of variable input for a firm that uses the variable input to produce its output, and
  2. understand the demand for the input, that is, how different prices for the input affect the quantity of input that will be purchased (demanded) by firms that use the input.

Price of the output (Py) is assumed to be $4 per unit.  MVP is NOT the total revenue; it is the additional revenue that will result from using an additional unit of the variable input.

The three price scenarios for the variable input are $18 per unit of variable input (Px1), $13 per unit (Px2) and $8 per unit (Px3).

 

Profit Maximizing

Recall, the level of variable input that maximizes profit is where MVP is greater than or equal to Px.

For Px1, the profit maximizing level of variable input is 12 units (why?  the $18 cost of variable input is less than the $20 MVP from using a 12th unit of variable input but greater than the $15 MVP from using a 13th unit of variable input). 

At Px2, the level of variable input would be 13 ($13<$15 but >$10); and at Px3, the profit maximizing level of variable input would be 14 units ($8<$10 but >$5).

Note that profit is maximized at 58, 61 or 63 units of output (Y), depending on the cost of the variable input (Px).

 

Qty. of Var. Input Qty. of Output              
. (X) . (Y or TPP) APP MPP Py MVP Px1 Px2 Px3
11
54
4.91
 

         
   
 
4
$5 $20 $18 $13 $8
12
58
4.83
 

         
   
 
3
 $5  $15  $18  $13  $8
13
61
4.69
 

         
   
 
2
 $5 $10  $18  $13 $8
14
63
4.5
 

         
   
 
1
 $5  $5  $18  $13  $8
15
64
4.26
 

         
   
 
0
 $5  $0  $18  $13  $8
16
64
4
 

         

 

 

 

 

 

 

 

 

 

 

 

 

Demand for Variable Input

The manager was willing to pay $18 for the 12th unit of variable input because the 12th unit increased revenue by $20 while the 12th unit only cost $18. Profit increased by $2 by using the 12th unit of variable input.  The manager would pay $19 for the 12th unit; even in that case, the firm's profit would increase by $1.  If the cost of the variable input would rise to $21 and the manager would NOT use the 12th unit because that would DECREASE profit by $1.  Bottom line:  the manager would pay as much as $20 for the 12th unit of variable input (the MVP for the 12th unit).

The manager would pay as much as $15 for the 13th unit (its MVP); as much as $10 for the 14th unit (its MVP); etc.

Thus the MVP curve reveals the manager's demand for the variable input.

 

Setting a Price for the Variable Input

If the MVP is the purchaser's demand for the variable input, can the seller of the variable input use the same information to set a market price for the variable input?  Yes, especially if the market for the variable input is "less than perfectly competitive"; that is, the seller (supplier) can extract an economic profit in setting the price.

If the MVP for buyer/user of the product is $15 per unit, the seller of the product (who knows the buyer's MVP) will charge as much as $15 for the product.  This strategy allows the supplier to extract as much revenue as possible from the buyer/user but leave just the minimum amount of potential additional profit to motivate the buyer/user to use the product/input.

 

Calculating Profit

Profit is revenue minus total cost.  Can the hypothetical data be further analyzed to calculate profit?  Yes, but...

Recall that total cost is total variable cost plus total fixed cost.  The table above provides enough to calculate revenue and total variable cost.  Additional information is needed about total fixed cost.

Price of the output (Py) will be held at $5; price of the variable input will be set at $18 (i.e., Px1 in the example).  The cost of fixed inputs (that is, the fixed cost) is set at $50 for the production period in this example.  From this data, total revenue (Y x Py), total variable cost (X x Px), total cost (TVC + TFC) and profit (TR - TC) can be calculated.

 

Qty. of Var. Input Qty. of Output              
. (X) . (Y or TPP) APP MPP Py Total Revenue Px TVC TFC TC Profit
11
54
4.91
 

$5
 $270 $18
$198
$50
$248 $22
   
 
4

12
58
4.83
 

$5  $290  $18 $216 $50 $266 $24
   
 
3
     
13
61
4.69
 

$5
 $305  $18 $234 $50 $284 $21
   
 
2
   
14
63
4.5
 

 $5  $315  $18 $252 $50 $302 $13
   
 
1
 
15
64
4.26
 

 $5  $320  $18 $270 $50 $320 $0
   
 
0
   
16
64
4
 

$5
$320
 $18 $288
 $50 $338 $-18

 

 

 

 

 

 

 

 

 

 

 

 

Twelve (12) units of variable input and 58 units for output maximizes profit when price of output is $5 per unit and cost of variable input is $18 per unit.

 

Stretch These Concepts Further

Can this hypothetical data be used to illustrate the concepts of Marginal Revenue, Marginal Cost and Profit Maximizing Level of Output?  The answer is "yes" but we need to "stretch" the data by making a simple assumption of a linear relationship between input and output as the firm adds each unit of variable input.  The following table provides detailed data based on that assumption and continuing to use $5 and $18 as the prices for the output and variable input, respectively.  Fixed costs also continue to be held at $50 for the production period.

 

Qty. of Var. Input Qty. of Output          
. (X) . (Y or TPP) Py or MR
Total Revenue Px TVC TFC TC Profit MC
11
54
   $270 $18 $198 $50 $248 $22
$5 $4.50
11.25 55 $275 $18 $202.50 $50 $252.50 $22.50
$5 $4.50
11.5 56 $280 $18 $207 $50 $257 $23
$5 $4.50
11.75 57 $285 $18 $211.50 $50 $261.50 $23.50
$5 $4.50
12 58 $290 $18 $216 $50 $266 $24
$5 $6
12.33 59    $295 $18 $222 $50 $272 $23
$5 $6
12.66 60 $300 $18 $228 $50 $278 $22
$5 $6
13 61    $305  $18 $234 $50 $284 $21
$5 $9
13.5 62   $310 $18 $243 $50 $293 $17
$5 $9
14 63 $315 $18 $252 $50 $302 $13
    $5 $18
15 64 $320 $18 $270 $50 $320 $0
$5 ????
16 64 $320 $18 $288 $50 $338 $-18

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

As long as MR exceeds MC, the firm will continue to increase profit by increasing output through the use of additional variable input.  Note the profit maximizing level of output is 58 units (12 units of variable input).  The same answer as reached in a previous subsection.

 

 

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