Profit Maximization

This is the third post in a series of articles focused on deconstructing the economic forces behind the operative performance of a business. 

 

Economic Articles

 

Supply & Demand

Profit Theory

Profit Maximization

Cost Theory

Unit Econometrics

Labor

(Links Coming Soon)

 

Profit Maximization

 

One of the basic objectives for any entrepreneur is to make sure their business not only generates revenue, but that their company makes a profit on the revenues that it generates. 

This fundamental objective is a reflection of basic economic theory, namely the economics of profit maximization. 

Profit Maximization represents the study of the short term and long term processes through which a company determines the price and output levels   that yield the greatest profit. 

Here, we deconstruct the elements of profit maximization in simple terms to illuminate the forces behind generating profit. 

Traditionally, entrepreneurs adopt two approaches to this task; the total revenue minus total cost perspective, and the marginal revenue minus marginal cost perspective. 

 

The Total Perspective:

The total revenue minus total cost perspective focuses on maximizing the difference between revenue and costs to enhance profits. 

A company whose revenues exceeds costs will benefit from greater profitability. 

 

The Marginal Perspective:

The marginal revenue minus marginal cost perspective focuses on achieving an equilibrium between marginal revenue and marginal cost to maximize potential profits. 

If marginal revenue is greater than marginal cost, then marginal profit is positive and a company should produce a greater quantity of it’s solution. 

Conversely, If marginal revenue is less than marginal cost, then marginal profit is negative and a company should produce a lesser quantity of it’s solution. 

At a production level where marginal revenue equals marginal costs, the marginal profit of the company is zero, which is the output level and equilibrium that maximizes profit. 

Because total profit increases when marginal profit is positive, and decreases when marginal profit is negative, the affect of changing levels of production on total profit must reach a maximum where marginal profit is zero, or where marginal costs equal marginal revenues. 

 

Both perspectives are equally valid and effective, however applying the marginal perspective to your profit maximization analysis requires a more sophisticated understanding of economic theory. 

 

Definitions

 

To facilitate our analysis, lets cover a few definitions of terms associated with Profit Maximization:

 

Revenue: 

Revenue represents the amount of money that a company receives during a specific period. It is sometimes referred as the "top line" or "gross income" figure, from which costs are subtracted to determine net income.

Revenue is calculated by multiplying the price of a good or service by the number of units or amount sold.

 

Profit:


Profit represents the financial benefit realized when the revenues gained from a business activity exceeds the expenses, costs and taxes required to sustain the activity. 

Profit is calculated by subtracted total expenses from total revenue. 

 

Fixed Costs:

Fixed Costs represent the accumulated economic expenses that occur only in the short term, and that are incurred by the company at any level of production, including zero output. These costs include; rent, wages, equipment maintenance, and other overheads.

 

Variable Costs:

Variable Costs represent that accumulated economic expenses that occur in both the short and long term, and that are incurred by the company when levels of output or production change. Typically variable costs increase when more of a product or solution is generated. 

Variable costs typically include; materials or resources for production, wages associated to production levels, and any equipment required to produce varying or increasing levels of production. 

 

Total Costs:

Total Costs represent the sum of all the costs a business incurs. It is calculated by adding up all of a business’ fixed and variable expenses. 

 

Cost Chart

 

 
Cost Chart (White).png
 

The cost chart illustrates the relationship between fixed, variable and total costs in respect to a changing level of unit production.

 

Profit Maximization Formulas

 

Now, beginning with revenue, the economic formula called the Profit Function, can be represented by the following equation:

Revenue (R) can be expressed as:         

 

R (Q) = P (Q) - C (Q)

        

Where:

R = Revenue

P= Price of your solution 

C = Cost to create and/or sell your solution

Q = Quantity of units of your solution sold. 

 

For Example:

Company Z sells 50 hats for $10 each at a cost of $2 per hat. Applying our revenue formula, Company Z generates a revenue of;

400 = $10 (50) - $2(50)

 

With revenue established, the challenge is determining how many units, or the quantity of units, must be sold in order to generate a profit. 

Calculating the how changes in quantity (Q) affect profitability can be done in 3 ways:

 

A. Marginal Profits

 

Marginal profits represents how a company’s profit changes if the level of sales increases or decreases by one unit. In other words, the question is; if one more unit is sold, how is profitability affected?

Examining the difference between the economic result from a change in activity requires using a derivative of the revenue function with respect to the quantity of sales (Q). 

 

dπ(Q1) / d (Q2)

 

Where:

d= Derivative

π = Change

Q1 = Quantity (increased or decreased)

Q2 =Quantity (constant)

 

For Example:

Company Z wants to discover the effect of selling 51 hats versus 50 hats. 

$10(51) /  $10(50)

 

In this case the ‘d’ represents sales. A derivative can represent any value that derives its false form an underlying term, which in our case includes; profit, sales, costs, and revenues.

Therefore, a unit increase in sales, compared to ‘normal’ sales is expressed in economic terms by the following formula: 

 

MPS (Q) = dP (Q1) / d (Q2)

 

Where:

MPS (Q) = Marginal Profit of a change in Sales of a certain Quantity

dP (Q1) = Affect on Profit from Change in Sales

d (Q2) = Profit from Regular Sales

 

For Example:

Continuing with Company Z, the Marginal Price of selling one more $10 hat compared to normal sales levels is:

$1.02 = $10(51) / $10(50)

 

So, If our Profit Formula is represented by the following equation:

 

P (Q) = R (Q) - C (Q) 

 

Where:

P = Profit

R = Revenue

C = Costs

Q = Quantity

 

For Example:

Continuing with Company Z, the Marginal Price of selling one more $10 hat compared to normal sales levels is:

$1.02 = $10(51) / $10(50)

 

Marginal Profit is calculated using the same formula, the trick is to alter the quantity (Q) by 1 unit.

 

MP (Q) = MR (Q) - MC (Q)

 

Where:

MP = Marginal Profit

MR = Marginal Revenue

MC = Marginal Costs

Q = Quantity

 

Then, subtracting the original sales volume with the adjusted volume in quantity of sales will reveal the $ amount increase or decrease in profitability. Likewise, dividing the original sales volume with the adjusted will yield the % difference in profitability.

 

For Example:

Adjusting the quantity sold of Company Z’s $10 hats, we find profitability or Marginal Profit per Sale affected in the following way;

$408 = $10(51) - $2(51)

Dividing the $400 by $408 we find that one more unit sold yields $8 additional dollars in profit, which represents an increase of .009% in profitability.

 

 

B. Marginal Revenues

Marginal Revenues represents how much revenues change if the level of sales increase or decreases by one unit sold. In other words, the question is; if one more unit is sold, how is revenue affected?

Again, examining the difference between the economic result from a change in activity requires using a derivative of the revenue function with respect to the quantity of sales (Q).

Applying the same logic used for the Marginal Profit, the formula for Marginal Revenues is represented by the following equation:

 

MRS (Q) = dR (Q1) / d(Q2)

 

Where:

MR (Q) = Marginal Revenue of a change in Sales of a certain Quantity

dR (Q1) = Affect on Revenue from Change in Sales

d (Q2) = Revenue from Regular Sales

 

For Example:

Using the same Company Z and corresponding data, the marginal revenue of a change in the quantity of sales is;

$1.02 = $10(51) / $10(50)

 

So, If our Revenue Formula is represented by the following equation:

 

R (Q) = P (Q) - C (Q)

 

Where:

R = Revenue

P = Price

C = Costs

Q = Quantity

 

Marginal Revenue is calculated using the same formula, the trick is to alter the quantity (Q) by 1 unit.

 

MR (Q) = MP (Q) - MC (Q)

 

Where:

MP = Marginal Revenue

MP = Marginal Price

MC = Marginal Costs

Q = Quantity

 

For Example:

Continuing with Company Z. The Marginal Revenue of a change in ales affects revenue in the following way;

$408 = $10(51) - $2(51)

Dividing the $400 by $408 we find that one more unit sold yields $8 additional dollars in profit, which represents an increase of .009% in profitability.

 

* It should be noted that the simplicity of our example generates the same results for profitability and revenue. More complex financial data will yield different results.

 

C. Marginal Costs

Marginal Costs represents how much costs change if the level of sales increase or decreases by one unit sold. Other words, the question is; if one more unit is sold, how are the costs affected?

Applying the same logic used for the Marginal Profit & Marginal Revenues, the formula for Marginal Costs is represented by the following equation:

 

MC (Q) = dC (Q) / d(Q)

 

Where:

MC (Q) = Marginal Costs of a change in Sales of a certain Quantity

dC (Q1) = Affect on Costs from Change in Sales

d (Q2) = Costs from Regular Sales

 

For Example:

Using our Company Z example, which sells 50 hats for $10 each at a cost of $2 per hat, we find that the marginal cost is affected in the following way;

$1.02 = $2(51) / $2(50)

 

So, if our Cost Formula is represented by the following equation:

 

C (Q) = R (Q) - P (Q)

 

Where:

C = Costs

R = Revenue

P = Profit

Q = Quantity

 

Marginal Cost is calculated using the same formula, the trick is to alter the quantity (Q) by 1 unit.

 

MC (Q) = MR (Q) = MP (Q)

 

Where: 

MC = Marginal Cost

MR = Marginal Revenue

MP = Marginal Profit

Q = Quantity

 

For Example:

With Company Z’s 50 $10 dollar hats, by adjusting the quantity we find that the Marginal Cost is;

$102 = $10(51) - $8(51)

Compared to Company Z’s normal sales volume cost of $100 dollars, we find that an increase of one unit of sales, increased overall costs by $2 or .009%

 

Key Insights

 

Examining the marginal profit, marginal revenues and marginal costs or your business will illuminate interesting insights on the specific forces that drive your companies profitability.  

First, that in order to break even, a companies Marginal Cost of a certain quantity (Q) must equal the Price of the same quantity (Q). 

Break Even Equation:

Marginal Cost (Q) = Price (Q)

 

This assumes, rather obviously that to generate revenues and profit, the amount of units sold (Q) must be greater (>) than 0. And, in order to generate a profit, the price of a solution must, at the very least, equal the marginal cost associated with selling one more unit of the solution. 

Second, if the Price of a certain quantity (Q) is greater than (>) the Marginal Cost of the same quantity (Q), then a company is able to increase it’s profits by increasing the sales of (Q). 

 

If P (Q) > MC (Q), then profit can be increased by increasing sales. 

 

Where: 

P = Price

MC = Marginal Cost

Q = Quantity

 

This is rather straight forward. If the price of your solution is greater than the cost associated with selling one more unit of the same solution, than your company can enhance its profitability by selling more units of the solution. 

Third, if the Price of a certain quantity (Q) is less than (<) the Marginal Cost of the same quantity (Q), than a company can increase its profitability by decreasing its sales of (Q). 

 

If P (Q) < MC (Q), then profit can be increased by decreasing sales. 

 

Where:

P = Price

MC = Marginal Cost

Q = Quantity

 

This last insight may appear counterintuitive, however think about it this way. If the price of your solution is less than the cost to sell one more of unit of the same solution, then a company would enhance it’s profitability by not selling additional units of the solution. 

 

Conclusion

 

In the effort to produce a profit, entrepreneurs should confidently explore how the forces behind profitability; revenue, costs and price affect the performance of their company. 

The interplay of these 3 elements will likely dictate the viability of your enterprise as well as it’s economic potential.

It is often the case that unforeseen factors affect the bottom line of a business, and it is only by examining each component individually that entrepreneurs can gain the right insights and make necessary adjustments to improve their company's profitability. 

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Send me a question: moebius@zenofwuwei.com