Nowadays, it seems natural that the vast majority of individuals of our society knows little or even nothing about mathematics. Actually, mathematics is one of the first subjects that has to be covered by college students, however, it’s common that these students just take this subject in a light way. The reason of this situation relies on the lack of knowledge about the huge importance this subject has on their lives and careers.
Between the careers in which mathematics is fundamental and necessary, we can mainly find business-related careers.
From this point, the objective of this research essay is to demonstrate the importance of this subject on these careers, this will be achieved through the presentation of a set of applications of derivatives in business and economics.
Before starting, it’s necessary to have a certain knowledge of the concepts involved, so they are going to be presented below:
Economics is a social science which study object is the production, distribution, and consumption of goods and services.
This area mainly focuses on economic agents’ interactions and the way in which they work. CITATION Pau12 l 1033 (Krugman, 2012)Derivatives of functions of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. CITATION Jam08 l 1033 (Stewart, 2008)
Marginal rates
The derivative has several applications in economics and administration in relation to “marginal rates.” For being more specific, each marginal rate is going to be presented.
Marginal cost. The marginal cost is an important concept in the microeconomics, since it is used to determine the amount of production of a company and the prices of products.
It is clear that the marginal cost is nothing other than that derived from the cost function with respect to the quantity produced. It measures the rate with which the cost increases with respect to the increase in the quantity produced.
Average marginal cost. Average cost is the cost of production per unit of product. It is determined by dividing the total fixed costs and variable costs for the total number of units produced. While the marginal average cost is equal to that derived from the average cost.
Marginal income. The marginal revenue represents the additional entries of a company per additional item sold when there is a very small increase in the number of items sold. That is, the rate at which income grows with respect to sales volume.
Marginal utility. Under the economical point of view, marginal utility provides the consumption of an additional good; in mathematical terms it can be said that marginal Utility is the partial derivative of the function of the utility with respect to the amount consumed of a good. The marginal utility, represents the additional utility by article, if the production has a small increase. CITATION Mar16 l 1033 (Riofrio, 2016)
Maximum and minimum represent the highest and the lowest values ??that a function can compute respectively. An investor will always seek to maximize the performance of his portfolio. All companies use the maximum functions for their production, sales, profit or profit levels and the minimum functions to shorten production costs and losses. Some of the functions through which companies seek to apply the maximum and minimum theories for the optimization of their processes are presented below:
Maximum in a Production Function Companies operate in the market to sell their products and derive benefits from them. Only when the company produces optimal quantities is the business lucrative. The company needs to produce the most according to its limitations.
For that, use the production function. Companies face different scenarios with a production of different quantities of merchandise. If few items are produced, you risk losing market. If they produce in excess, the product may perish, lose its properties or become obsolete. In this sense, the company must analyze its balance point and its sales and profitability goals.
Maximum Sales Function Once the company produces its final product, it must be sold in the market. The product can attract customers only if they are aware of their convenience or need. To achieve this goal, the company needs to market and advertise the product. Sales also use the maximums function.
The company tries to sell to the maximum extent. There are many ways and methods in which sales can be maximized. The company has to take the best option between different markets and segments. You may be interested: Types of sales promotion
Maximum for a price function Optimization theory is also ideally used to assign the price to a company’s products. Customers are attracted to make a purchase only when they can adjust the price of the product with the appreciation of value derived from it.
If a company assigns premiums to its products and services, its customers will choose to buy them from their competitors. Therefore, you will lose market share. In any case, you should look for an equilibrium price. If the company assigns lower prices to the goods, it will not be able to cover its costs and will incur losses. The company needs to assign the optimal price based on its costs and market prices.
Minimum for a cost function Whatever the costs, all companies try to minimize their costs and losses. To do this, they use the minimum function.
The company analyzes all associated costs and tries to find ways and means to limit them. A numerical tool used to find the optimum of production levels that minimizes costs, based on existing limitations, is the linear programming of operational research.
Through algebraic operations, linear programming allows manufacturing companies to seek the mix of products that, based on the provision of raw materials and labor, ensure a minimum cost. Or propose an optimum of a shipping plan that covers the demand based on the supply or capacity available. CITATION Rau19 l 1033 (Moran, 2019)Elasticity of demand
Because of the importance of the relationship between the demand for a product and its price, it is necessary to study the elasticity of the demand, which is not difficult for the calculation of the variation that occurs in the demand when the prices of the product change. In fact, if the price goes up, sales decrease; If the price falls, the sales increase.
This situation occurs very frequently in real life in supermarkets, for employment, they are regularly registered in the products to attract a greater number of buyers. The elasticity of the demand is an approximation to the percentage change in demand caused by a 1% increase in the price. CITATION Ins l 1033 (Instituto Internacional de Tecnolog?a Educativa, n.d.)Application problems
1. Suppose the total cost C(x) (in millions of euros) for manufacturing x air-planes per year is given by the function Cx=6+ 4x+4 0?x?30
a) Find the marginal cost at a production level of x air-planes per year.
The marginal cost at a production level of x air-planes is C’x=6+ 4x+4’= 24x+4 b) Find the marginal cost at a production level of 15 and 24 air-planes per year, and interpret the results.
The marginal cost at a production level of 15 air-planes is C’15= 24(15)+4=0.25At a production level of 150 air-planes per year, the total cost is increasing at the rate of 250,000 per one air-plane.
The marginal cost at a production level of 24 air-planes is C’24= 24(24)+4=0.2At a production level of 150 air-planes per year, the total cost is increasing at the rate of 200,000 per one air-plane.
2. The total cost function (in thousands of euros) for manufacturing x manipulators per year is given Cx = 375 + 25 x – 0.25×2 0 ? x ? 50
a) Use the marginal cost function to approximate the cost of producing the 31st manipulator.
The marginal cost is C’x= 25-0.5x . Thus, C'(30)= 25-0.5(30)=10.The cost of producing of the 31st manipulator is approximately 10,000.
b) Use the total cost function to find the exact cost of producing the 31st manipulator.
The exact cost of producing the 31st manipulator is C31- C30= 909.75 – 900 = 9.75 , i.e. 9,750.The marginal cost of 10,000 per manipulator is a close approximation to his exact cost.
CITATION Jar l 1033 (Zahra?dka, n.d.)3. An apartment complex has 250 apartments to rent. If they rent x apartments then their monthly profit, in dollars, is given by
Px=-8×2+3200x-80,000How many apartments should they rent in order to maximize their profit?
All that we’re really being asked to do here is to maximize the profit subject to the constraint that x must be in the range of 0?x?2500?x?250First, we’ll need the derivative and the critical point(s) that fall in the range
0?x?2500?x?250P’x=-16x+32003200-16x=0x=200Since the profit function is continuous and we have an interval with finite bounds we can find the maximum value by simply plugging in the only critical point that we have (which nicely enough in the range of acceptable answers) and the end points of the range.
P0=-80,000P200=240,000P250=220,000So, it looks like they will generate the most profit if they only rent out 200 of the apartments instead of all 250 of them.
Note that with these problems you shouldn’t just assume that renting all the apartments will generate the most profit. Do not forget that there are all sorts of maintenance costs and that the more tenants renting apartments the more the maintenance costs will be.
With this analysis we can see that, for this complex at least, something probably needs to be done to get the maximum profit more towards full capacity. This kind of analysis can help them determine just what they need to do to move towards that goal whether it be raising rent or finding a way to reduce maintenance costs.
Note as well that because most apartment complexes have at least a few units empty after a tenant moves out and the like that it’s possible that they would actually like the maximum profit to fall slightly under full capacity to take this into account. Again, another reason to not just assume that maximum profit will always be at the upper limit of the range. CITATION Daw18 l 1033 (Dawkins, 2018)
Throughout the discussion of this document, it was proved that there exists a large number of applications of derivatives in economics. In this area, the derivative is a very useful tool since its nature allow to obtain the exchange rate when an additional unit is added to the total, whatever the economic amount being considered: cost, income, profit or production. In addition, it allows people to look for optimization values.
Finally, it could be established that, in latest years, decision-making in the area of economics has turned more and more mathematically oriented, as some topics of the subject allow to understand and analyze what is happening, anticipate the consequences of numerous policy options, and find or elect convenient courses of action from the multitude of possibilities.
By this way, business-related career’s students should pay more attention to these topics because, as it was demonstrated, they have a lot do at the moment of having the company’s finances clear and managing resources and, consequently, decisions in the most efficient way.
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