Model Research on Commodity Price and Packing Model

With the rapid development of the economy, human society is highly prosperous, and the relationship between people and people, people and things and things in society is becoming more and more complex, and everything is becoming dazzling. In order to make the contours of this relationship clearer, humans use various methods to analyze the model. Moreover, the analysis of the model becomes more and more important in today's society. Psychologically, it can give people a sense of security, and the things that are already confusing have changed in accordance with rules and regulations and are no longer chaotic. In terms of material, it can make people maximize their utility or income. Using models can grasp the laws of economic operation and predict its direction. Here, models are also used to analyze an economic phenomenon—the relationship between commodity prices and package models.

I believe that there will be no unfamiliarity with the issue of commodity prices and packaging models, and there must be a lot of people who find that large-packaged goods are cheaper than small-packaged goods. What is the reason? Here we use the product of toothpaste for model analysis.

Problems and their formation

For example, the blue sky toothpaste 60 grams each loaded 0.96 yuan, 150 grams loaded 2.15 yuan each, the price ratio between the two unit weight is 1.17:1, try to establish a mathematical model and explain this phenomenon. Requirements: 1 Analyze the relationship between price C and product weight W. Prices are determined by production costs, transportation costs, and packaging costs. Some of these costs are directly proportional to weight W, and some are proportional to surface area S, and others are related to W. Unrelated factors. 2 Write the relationship between unit weight price C and W, indicating that W is larger and C is smaller. 3 Explain how the rate of decline in unit price as W increases is negative and what is its practical significance.

Simplification of the problem and establishment of assumptions

Solve this problem for a specific toothpaste problem. First of all, the price of toothpaste is determined by production costs, transportation costs, and packaging costs. It is assumed that there are no restrictions on the rate of profit in the economy and non-natural price participation methods such as price manipulation. Secondly, the toothpaste that appears on the market is almost cylindrical in its packaging, so it is treated as a bottomless cylinder. Finally, as toothpaste packaging models expand, their length and radius are correspondingly expanded. In view of the proportion of aesthetics, there is a fixed ratio between length and radius, and they both increase or decrease in the same factor. The problem here is a lot simpler. Here are some assumptions about the variables used in the model. Commodity prices are expressed in P, and product weights in W. The production cost of each toothpaste is X, the transportation cost is Y, and the packaging cost is Z. The toothpaste package has a surface area of ​​S, a volume of V, a density of ρ, a length of L, and a radius of R. Unit weight price is C.

The establishment of a model

Since the production cost X is proportional to W, then X=K1W. The actual meaning of K1 is the unit cost of production cost. The transportation cost Y is proportional to W, ie Y=K2W. K2 is the transportation cost per unit weight. Whereas the packaging cost is proportional to its surface area, Z = K3S. Since the actual toothpaste shape does not have a bottom surface, S=2πRL refers to the side area of ​​the cylinder, and its volume V=W/ρ=πR2L. Since the length and radius are stretched from the same ratio, R=K4L, and K4 is a constant coefficient of proportionality. According to the above, S=2W/(ρR) is available, and since R=V1/3/(πk4)1/3, the expression of Z of the variable W is obtained.
Z=2K3(ρπK4)1/3/ρ*W2/3 (1.1)
According to the above analysis of the price of goods P = X + Y + Z, according to the above analysis shows that production costs, transportation costs, packaging costs are a function of the weight of goods, then the price of goods also has a function of weight W, the expression is as follows :
F(W)=P=K1W+K2W+2K3(ρπK4)1/3/ρ*W2/3 (1.2)
According to the above formula, the unit weight price C
C=F(W)/W=P/W=K1+K2+2K3(ρπK4)1/3/ρ*W-1/3 (1.3)
In order to study the problem and get the size of the speed of change of unit weight, formula 1.4 is derived for the formula of 1.3.
T=dc/dw=(-2/3)K3(ρπK4)1/3/ρ*W-4/3 (1.4)
These equations are all introduced with the formula 1.2. The model is now established. The remaining question is what does the model explain and what can it explain?

Model interpretation and application

The first look at formula 1.2 shows that K1, K2, 2K3(ρπK4)1/3/ρ are non-negative constants. The function is a monotonically increasing function. In reality, the price of 150 grams of toothpaste is less than 60 grams, otherwise it will become a joke.

Secondly, when we look at formula 1.3, the commodity weight W is also a function of the commodity unit weight price C. This function is a monotonically decreasing function. As W increases, the third term of the formula gradually decreases, and C decreases. .
Finally, look at the 1.4 formula to know that the unit price weight per unit weight change caused by the number of changes in the price is also a function of W, and as W increases, but its value is negative. This shows that as the size of the package becomes larger, the unit price of the toothpaste gradually decreases, but the decreasing range of the toothpaste decreases as the weight increases, that is, the decreasing speed of the toothpaste decreases until the product. The ratio of the unit weight price to 1:1, but a long process.

In short, the most important thing for model analysis is to analyze the relationship between variables. There are generally three models for the relationship between commodity prices and package types. 1) As the size of the package increases, the price per unit weight will decrease. 2) As the size of the package increases, the unit weight price will become smaller and larger. 3) As the size of the package increases, the price per unit weight will increase. For different objects and problems, as long as the important factors affecting them are found out and found out, the establishment of all models becomes much easier.