In the Vertical Y Axis measure Price in $
In the Horizontal X Axis measure Quantity to be sold in Units
| Description of the software | Theory where this model is based on |
| Screen Shot | Glossary of terms |
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The idea of this model is to understand the behavior of the variables related with a micro-economy, a company trading products and or services. For doing so, these variables are turned into mathematic equations that show us an instant moment (momentum) displaying a graphic based on those equations. The idea is to see a big picture of how price, income, cost and benefit, are affected by the quantity of units of product that such market will be able to consume, related with the capacity of such company to produce that amount of product units.
What we are looking to know is: How many units and at what price?, a company must sell / produce a good or service in order to achieve their goals based on a marketing strategy. Generally a company wants to get the "maximum income" or "maximum benefit", or "minimum marginal cost" in order to improve their productivity. All these different points are shown in the graphic, and they are located into an specific position determined by the circumstances that are represented by the functions. This model can be useful to teach and learn how a micro-economy works. Once we understand what each equation represents then we only need to change a few numbers to see what happen if. Bellow is the theory where this model is based on. I hope you to enjoy this program, any comments and suggestions are always appreciated.
When the price of any good or service change, it's demand will change too, generally when the Price go down the Demand go up, that means that if the price of a product go down then the quantity of units to be sold will go up proportionally, this relation is called elasticity. We understand that if we want to sell more units of a product (not changing anything else, but the price), then it's price must be lower. This phenomenon is represented in the model by the Demand Equation with the following shape: P = Da*Q + Db.
Where:
P is Price of the product
Q is Quantity to be sold
Da is the Elasticity or slope
Db is the price for selling zero units
Graphic of Demand will look like this:
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Demand Function: P = Da*Q + Db.
In the Vertical Y Axis measure Price in $ In the Horizontal X Axis measure Quantity to be sold in Units |
Now having the Demand Equation ready is possible to determine the Income for each different quantity of units sold. This Income will be equal to the number of units sold multiplied by it's price, in mathematical terms: Income = P*Q. As seen before P is a function so deploying the income function then Income = (Da*Q + Db) * Q , so the final Income Function will be: I = Da*Q^2 + Db*Q
Graphic of Income and Demand together, sharing the X axis:
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Income Function: I = Da*Q^2 + Db*Q
In the Vertical Y Axis measure Total Income in $ In the Horizontal X Axis measure Quantity to be sold in Units |
| Demand Function: P = Da*Q + Db.
In the Vertical Y Axis measure Price in $ In the Horizontal X Axis measure Quantity to be sold in Units |
Okay, the market volume is defined now by all possible Incomes selling a product that is tied to the Demand equation. As seen in the Income Graphic (Upper Green) there is a maximum point, that maximum is the maximum income that a product may produce in a defined market, to exactly know where this point is we need to know where this maximum occurs, the question to be made is: How much will I get if I sell one more unit ? this value will be a differential equation derived from Income function. This concept is called Marginal Income that is the pendent of the Income equation for each Quantity value, at the top of the function the pendent is zero, at it's left it positive and at it's right it is negative. So the maximum income will occur when the marginal income becomes zero. In mathematical Terms the derived Income Function (I = Da*Q^2 + Db*Q) will be dI/dQ = 2*Da*Q + Db, so the Marginal Income Function now is: MI = 2*Da*Q + Db
Graphic of Income, and Marginal Income, and Demand combined, will look like this:
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Income Function: I = Da*Q^2 + Db*Q
In the Vertical Y Axis measure Total Income in $ In the Horizontal X Axis measure Quantity to be sold in Units |
| Demand Function: P = Da*Q + Db.
(Yellow) Marginal Income Function: MI = 2*Da*Q + Db (Green) In the Vertical Y Axis measure Price in $ for Demand, and Marginal Income in $ for MI In the Horizontal X Axis measure Quantity to be sold in Units |
When Marginal Income (Lower Green) is zero then we are at the top of Income function (Upper Green), this means that if we want to have maximum income we can lower the prices until the marginal income becomes zero. After that point reducing prices and selling more units will generate a decrease in the total income for sales.
At this time we evaluate the market behavior for a product, based on it's Demand equation. Well, lets forget all that for a little while to concentrate now in our company behavior.
There is a concept called Scale Economy, that show us how the cost of producing a product will vary depending on it's production scale. A generic shape of a function that represent this concept will look like this
The best mathematical fit for this shape is a third degree equation, we will
call this equation Cost, so Cost function will be a third degree function of
Q.
C = Ca*Q^3 + Cb*Q^2 + Cc*Q + Cd where Cd is the fixed cost, cost for producing zero
units, when Q=0 C=Cd. Then Ca, Cb, and Cc, numbers are the coefficients that
represent our company behavior in the equation for variation of units
produced.
After processing the values of Total Cost against Produced Units taken from the reality and extracting Ca Cb Cc Cd coefficients, the Graphic of Cost function will be
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Cost Function: C = Ca*Q^3 + Cb*Q^2 + Cc*Q + Cd
In the Vertical Y Axis measure Total Cost in $ In the Horizontal X Axis measure Quantity to be Produced in Units |
From now on, this is our company's Cost behavior for a produced Qty variation.
The concavity of the function change in a particular point where our productivity is the best, now appears a concept called Marginal Cost that show how much will cost to produce one more unit of product ?, this question is answered by the function called Marginal Cost that is the pendent of the Cost equation for each Quantity value, the different values of that pendent will be expressed by the function Marginal Cost (MC), derived from Cost equation (C = Ca*Q^3 + Cb*Q^2 + Cc*Q + Cd) so dC/dQ = MC = 3*Ca*Q^2 + 2*Cb*Q + Cc
Both functions, Cost and Marginal Cost, combined sharing the X axis in a graphic
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Cost Function: C = Ca*Q^3 + Cb*Q^2 + Cc*Q + Cd
In the Vertical Y Axis measure Total Cost in $ In the Horizontal X Axis measure Quantity to be Produced in Units |
| Marginal Cost: MC = 3*Ca*Q^2 + 2*Cb*Q + Cc
In the Vertical Y Axis measure Marginal Cost in $ In the Horizontal X Axis measure Quantity to be produced in Units |
One last concept to take care about is the Medium Cost that is nothing more than the Unitary Cost, this Medium Cost function is calculated by dividing the Total Cost by the Qty of units produced C / Q. so Medium Cost MeC = Ca*Q^2 + Cb*Q + Cc +Cd/Q
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Cost Function: C = Ca*Q^3 + Cb*Q^2 + Cc*Q + Cd
In the Vertical Y Axis measure Total Cost in $ In the Horizontal X Axis measure Quantity to be Produced in Units |
| Marginal Cost: MC = 3*Ca*Q^2 + 2*Cb*Q + Cc
(Red) Medium Cost MeC = Ca*Q^2 + Cb*Q + Cc +Cd/Q (Orange) In the Vertical Y Axis measure Marginal Cost and Medium Cost in $ In the Horizontal X Axis measure Quantity to be produced in Units |
As seen in this graphic our productive factors are best used when the Medium Cost is minimum, this point can be easily located when Marginal Cost is equal to Medium Cost. MC = MeC then MeC is minimum.
The only thing less now is Benefit function, this equation is the difference between the Income and Cost.
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Income Function: I = Da*Q^2 + Db*Q Cost Function: C = Ca*Q^3 + Cb*Q^2 + Cc*Q + Cd Benefit = Income - Cost In the Vertical Y Axis measure Income, Benefit, Cost in $ In the Horizontal X Axis measure Quantity to be sold/produced in Units |
If we see what is really happening is that a maximum income is not in the same place as a maximum profit, so for each different case we will need to determine a price in order to sell a quantity of products that allows the company to step in the exact point.
Now if all graphics are combined we can see how all variables are related to each other.
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Income Function: I = Da*Q^2 + Db*Q Cost Function: C = Ca*Q^3 + Cb*Q^2 + Cc*Q + Cd Benefit = Income - Cost In the Vertical Y Axis measure Total Income in $ In the Horizontal X Axis measure Quantity to be sold in Units |
| Demand Function: P = Da*Q + Db.
(Yellow) Marginal Income Function: MI = 2*Da*Q + Db (Green) Marginal Cost: MC = 3*Ca*Q^2 + 2*Cb*Q + Cc (Red) Medium Cost MeC = Ca*Q^2 + Cb*Q + Cc +Cd/Q (Orange) In the Vertical Y Axis measure Price, MI, MeC, MC |
Looking into this simple screen is ratable where the equations cross each
other, for example in the lower graphic, if medium cost is equal to price then
benefit will be 0, this same point in the upper graphic appears when
income equals to cost.
When Marginal Income equals to Marginal Cost we are on maximum benefit.
When Marginal Cost is greater than Medium Cost we are entering in the 3rd stage
of production, this point starts when Medium Cost is minimum and equals to
Marginal Cost.
Moving left to right through the graphic looking into each variable, we can understand the big picture of the market and how it is related to a company, and how a modification in one can affect the other, and in what proportions, and how.
Many What happen if...? questions can be made once we see how these variables work together. Bellow is a list of possible questions, all these questions can be better answered using a mathematic model to define the scenario, in order to take more intelligent decisions every day.
What happen if a new competitor enter to the market ?
The elasticity of my products will be affected, how my demand will change?
What happen if a technology development make my cost structure to be
obsolete?
How will affect this to my benefits?
How do I set up my new cost structure based on marketing measures?
How many units my market allow me to sell / produce?
Can the structure support a market grow?
How do I have to watch into the cost behavior, do I need to change my installed
capacity ?
Is the price well defined?
Are we selling less than we have to sell ? or more ?
How big is the market, in terms of money?
Can I supply this market with my current structure? Why?
Is the company working at a considerable scale level?
Do I need to produce more or less ?
Are we loosing business?
Does the price needs to be higher or lower?
How we will support a price falling?
Is our cost structure capable of supporting a price falling?
In what situation my company's assets are best used?
What is the exact market share that my current structure may be able to support?
Do we have market enough to justify investments?
How is the scale of the market compared with a company?
It is a graphical interface that represents mathematical equations based on an economy theory. Economic variables are shown as equations in order to represent what the theory means. Using complex mathematical processes such as Gauss polynomial estimation for taking values from the reality, convert those values to mathematical equations and display them as a graphic. Making them fit in a relation where the factors are exposed in order to better understand the relationship between each other.
The software allow us to see the forest without loosing time counting individual trees, let the program do the numbers in order to focus in the concepts related to this represented situation.
This software tool can be used to teach and learn how a micro-economy works. Right now this tool is being used in the university with a big screen, asking pupils to solve situations based on imaginary cases. The interface has been designed for working in 800x600 pixels for dynamically display using a projector. Students are also using the software for studying in their homes solving problems, discussed in the class room.
This particular screen shot shows the price for selling 10.58 units (or thousands of units) must be $ 15.3 for having the maximum benefit in this conditions which is equal to $ 113.14, having a gross income of $ 161.98, spending $ 48.84. The medium cost or unitary cost is $ 4.61, marginal cost equal to marginal income in $ 6.94, that point is the modeled condition for maximum benefit.
Looking into the model we can see that this company may be able to invest in installed capacity making their cost structure more efficient at a higher production level. Why ? As shown in the graphic the product that this company is trading still have market to be developed until reach a maximum income condition, one practical decision could be investing in production assets, modifying the Cost behavior making the cost structure being more reliable at higher production levels.
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One time fee of only
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First we need to extract data from the reality, in order to estimate Demand and Total Costs functions. this data may become from marketing measures and studies, based on historical data, or supposed by ourselves, based on any method. The data format will be (Qty ; Cost) for Cost equation, and (Qty ; Price) for Demand equation. The value R^2 (correlation factor) will also be calculated and displayed when equations are estimated.
Click on "Demand Data" button, the following window will appear
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Enter the values for each (Price; Qty) in the
table and press calculate button. Then Da and Db variables will acquire a value, calculated by a linear regression. If "Display this values in the graphic" checkbox is activated then the points will be shown as X1 X2 X3 X4 and X5 How Demand Function is determined
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Demand coefficients will be shown in this editable text boxes in the up-left screen. Da will be the elasticity, and Db will be the price for selling zero units
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Click on "Cost Data" button, the following window will appear
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Enter the values for each (Price; Cost) in the
table and press calculate button. Then Ca Cb Cc and Cd variables will acquire a value, calculated by a polynomial regression. The Cost function will be drawn in the upper graphic in red. If "Display this values in the graphic" checkbox is activated then the points will be shown as X1 X2 X3 X4 and X5 How Cost Function is determined Cost coefficients will be shown in this editable text boxes in the up-left screen
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Cost coefficients will be shown in this editable text boxes in the up-left screen
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After the functions are drawn, then set the appropriate range for the graphic
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By using the arrows, or entering manually set the graphics range for the best equation fit |
The model have two graphics sharing the horizontal X axis that represents the Quantity of units to be sold or to be produced "q". And in the vertical Y axis measure money in both graphics. The upper graphic display TOTALS such as Total Cost and Total Income, and the lower graphic display UNITARY values such as price and Unitary Cost. These graphics are tied together by the X axis to see how they are related each other.
In order to see how a variable is affected by any other variable in the model, use the option buttons located at the left of the graphics. Using this option you can select what functions to track with a cross when moving the mouse over the graphic. Besides that all values will be calculated and displayed at side of each analyzed variable.
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Use option buttons to select what function top
follow with a cross.
Click in the colored labels to set On / Off each individual function to be displayed in the graphic. In order to see one or more at time. The number at side of each colored label is the value for that variable |
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Mouse position, X and Y values will be shown
here. Also set how many decimal digits will be used for the calculations |
For save and open models just click on "Save" or "Open" buttons and follow the standard Windows procedure. You can also reset the model to default values by clicking "Reset" button.
Suppose that we have five point of sales where we know that the market behavior is similar. In this five points we offer the same product at a different price. After a period of time we collect the data to know how many units were sold at what price, this measure will give us a table that will look like this:
| Point of Sale | Price | Quantity to be Sold |
| X1 | 26 | 2 |
| X2 | 17 | 8 |
| X3 | 10 | 14 |
| X4 | 8 | 15 |
| X5 | 2 | 19 |
Having this data the program will estimate a Demand Function by a first degree linear Gauss approximation. Extracting the values Da and Db for the Demand equation P = Da*Q + Db.
A total cost for producing such quantity of products, this information may come from a real balance sheet, or this points (Cost ; Qty) can also be estimated if a project is being analyzed.
To determine the Cost Function that will show the particular behavior of a company or a business unit, such a sales or production department. For generating this function the model will estimate a Cost function by taking values from the reality and making a third degree Gauss polynomial estimation. Giving as result an equation C = Ca*Q^3 + Cb*Q^2 + Cc*Q + Cd where Cd is the fixed cost, cost for producing zero units, when Q=0. Ca, Cb, and Cc, numbers are the coefficients that represent our company behavior, at a different quantity of produced units.
In both cases Cost and Demand estimation there is one more value that is calculated by the model, it is R^2 or R square, this is the correlation factor, means how good the calculated equation fits the given numbers. When R^2 = 1 means that all points belongs to the equation so the alignment is perfect, this is of course impossible in real situations so the value of R^2 will vary from 0 to 1, a number greater than 0.6 is ok for an accurate estimation.
| Example of two different correlation factors R^2 = 0.99 and 0.84 | ||
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R^2 = 0.99
X1 X2 X3 X4 X5 points are aligned with a correlation factor of 0.99 |
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R^2 = 0.84
X1 X2 X3 X4 X5 points are aligned with a correlation factor of 0.84 |
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How many products will I sell at what price ? that is the question to be made, understanding that the quantity of product to be sold will change with a price variation, where the (sold units / price) relation is determined by the product's elasticity.
How the quantity of units sold will be affected by a price variation, or in other words, How the price of a product will must change if the quantity to be sold of that product changes, this number is the pendent of the Demand equation. Practically in all cases it is negative.
The price of the product to be sold / produced, this value will be given by the Demand equation, a new price will exist for each new quantity of product to be sold.
Is the total income generated by sales of a product. knowing that price and quantity variables are subject to an equation represented by demand function, the income will also be an equation as result of multiplying the quantity of products sold by it's price. So Income will be equal to P*Q.
Marginal Income is: How much will income for selling one more unit of product? in mathematical terms this can be explained by following the pendent of total income function through each Q value. The pendent of a function in one point is a derivate of that function in that point, so the marginal income will be dI/dQ derivate of I resptect to Q.
What the total cost will be for producing such quantity of products?. The shape of this function is based in the Scale Economy concept for producing products at some scale level. The equation is being generated by doing a statistical estimation from point determined by values pairs (Qty ; TotCost). The resultant function is a 3rd degree function C = Ca*X^3 + C b*X^2 + Cc*X + Cd. The coefficients (Ca Cb Cc C d) are representing the micro-economy's cost behavior under study.
Marginal Cost is: How much will cost to produce one more unit of product? in mathematical terms this can be explained by following the pendent of total cost function through each Q value. The pendent of a function in one point is a derivate of that function in that point, so the marginal cost will be dC/dQ derivate of C resptect to Q.
The Medium Cost is the average cost for one unit of product, or in other words the unitary cost. To know how much cost each unit of product we only need to divide the total cost by the quantity of product produced that genertate that total cost. So medium cost function will be C/Q.
The Benefit is the difference between the total sales income against total costs. So benefit is calculated as follows: B=I-C where B is benefit, I is income and C is cost.
Suppose that I need to construct a
tall wall, say 2 meters Width by 4 meters Height, and I know that one
worker will finish the job in 8 hours, I also know that the cost of 1 hour is $
1 , to construct the wall my cost with one worker will be $ 8 for finishing the
wall in 8 hours.
If I use a second worker then the time for constructing the wall will NOT be the
half, (4 hours). Why ? because the second worker will throw the bricks to the
first worker upstairs, so this first worker will not have to go up and down
trough the stairway saving time by dividing functions. Using this method my wall
is finished now in 3 hours not 4. So the cost for constructing the same wall
using 2 workers will be 3 hs by $1 by 2 workers that is a total of $6 for
finishing the wall in 3 hours.
Let place a third worker in scene this third man will take care of cleaning
doing the cement mix and helping the other two men in general tasks, having this
third worker the wall has been finished in 2 hours by $1 by 3 workers, is a
total cost of $6 for finishing the wall in 2 hours.
Now I try a fourth worker with no much to do really, just to see what happens,
the wall is finished in 2 hours at a total cost of $8
If I insist placing a fifth worker who actually is distracting the first 3
working men, while chat with the fourth I can observe that the wall is taking
now 3 hours at a total cost of $15.
This simple example show us how the productivity is increased until one optimum
point, until this point the cost increases proportionally less than the last
value. Then productivity decreases making our costs go up proportionally more
than the last value. Almost all companies
have the same behavior if we look into the numbers.
Copyright © 2003 By Ricardo David Lerch rlerch@usa.net all Rights Reserved
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