Modeling of commodity packaging issues

First, the background of the problem is proposed

When we are shopping in our daily lives, we often find that large-packaged goods are cheaper than small-packaged goods. For example, when buying the same brand of biscuits, Master Kong's “3+2” biscuits with a 150g net content (W) sell for 3.5 bucks, and a pack of contents is exactly the same, but the net content (W) is increased to 600g. "3+2" only costs 10.2 yuan, which means that the average of "2.5+2" per 150g is only 2.55 yuan.

Here, the 150g "3+2" mentioned earlier belongs to the small package, and the 600g "3+2" mentioned later belongs to the large package. The same is a 150g “3+2” cookie, which costs 3.5 yuan for a small, individually packaged product. Why does it cost as little as 2.55 yuan from a large package? What are the mysteries in this? The purpose of this article is to explain this issue to everyone through the establishment of relevant mathematical models.

Second, the hypothesis of model related elements

1. Let the sales unit price of the product be p, and the sales volume (w) be the dynamic change amount x (g), then the sales revenue I = px. (1)
2. If the unit price of the production cost of the product is c, then the production cost C=cx. (2)
3. If the unit price of the transportation cost of a commodity is t, then the transportation cost T=tx. (3)
4. Assuming that the price per unit area of ​​the packaging material of the product is s, and the surface area of ​​the packaging material required for the product with the unit price p is m (cm2), the packaging cost is S=sm. (4)

Third, the establishment and analysis of the model

Under market economy conditions, the price (p) of a commodity is mainly determined by the production cost C of the commodity, the transportation cost T, and the packaging cost S. Some of these costs are directly proportional to the surface area (M) of the packaging material of the product, some are proportional to the weight (w) of the product, and some are not related to it. The following specific analysis.

First, the relationship between the price (p) of a commodity and the transportation cost T of a commodity is analyzed.

For the transportation cost of goods T, because the load (W) of a truck is fixed, if the goods are transported each time, the total weight of the goods W will always reach the maximum load of the truck, as long as the transportation cost of the goods. If the unit price (t) does not change, then what kind of packaging will be carried out on these transported goods, whether large or small, is independent of the sales price (p) of the goods, that is, the transportation of the goods. The cost T is only proportional to the total amount of the goods in the case that the unit price of t does not change, and it is not related to the size of the goods, namely its sales unit price (p).

Second, analyze the relationship between the price of goods (p) and the packaging cost of goods.

It is well known that the greater the surface area (M) of the packaging material, the higher the cost S required for the packaging and the proportional relationship between them. However, in the specific process of product packaging, because there is a certain "marginal packaging cost" (S side), there is no real cost of the part of the material used to package the goods. For example, if a product has a surface area of ​​M=100cm2, then the surface area of ​​the material used to wrap it M ́ must be greater than 100cm 2 , ie, M ́>M, where (M ́-M) is the marginal packaging material. Surface area value. It should be noted that the packaging cost S of a product is not completely proportional to the surface area (M) of the packaging material because of the existence of “marginal packaging material”, which we can set as a fixed value for the surface area of ​​the marginal packaging material. (M side) has nothing to do with the weight of the product. The packaging-related parts are illustrated below.

Assuming commodity A is a cube, the surface area of ​​each side (m) = 10cm2 (A = 100g), the surface area of ​​the marginal packaging material M side = 10cm2, the total area of ​​packaging costs M ́ = 70cm2; now, the use of two fake A goods are packaged together and sold as a product A ́ (A ́=200g). It is easy to know that the surface area M of the packaging material is the same as that of the A packaging material, but the two sides A are the same. For goods, at least two packaging materials with surface area (m=10 cm2) can be saved. That is, the total packaging material surface area of ​​A ́ M ́ = 110 cm2. Compared to the two small-package goods A, the cost of the goods A ́ saves 20cm2 of packaging material cost.

So, it is clear that large-packaged goods are cheaper than small-packaged goods. However, to what extent should packaging be large? In other words, to what extent is the packaging so large that the price cannot be smaller? This has a lot to do with the production cost of goods C. Let's talk about the relationship between them.

The reason why businesses will set the price of packaged goods is lower than the price of small packages, not that businesses do not want to make more money, on the contrary, this is precisely because they are pursuing the "best economic benefits." The price it is drafting is precisely the critical price (p*) that will allow it to obtain the best economic benefit. However, why is the price of commodities going down from the small package to the large package? Moreover, when will the gradually changing price (p*) change to the critical price and cannot be smaller? In theory, there will be a functional relationship among these, and it is a function of the drop in the sales price p. That is, the weight (w) of the merchandise being sold is a descending function with respect to the merchandise price (p).

It is written as: x=f(p), (change in weight of x product). (5)
Thus, in the case where the production cost price c is a constant, a functional relationship concerning profit can be obtained, that is, the profit is equal to the sales income I minus the production cost C, the transportation cost T, and the corresponding packaging cost S. therefore,
Profit U(P)==I(P)-C(P)-T(P)-S(P) (6)
Here, because the transportation cost T and the packaging cost S have been considered previously, for the sake of convenient discussion,
Let U*(P) = I(P)-T(P)-S(P). (7)
Then, we can approximately convert (6) to (7). That is, the main analysis of the relationship between profit U*(P) and production cost C(P): U*(P)=I(P)-C(P) (8)
U*(P)=I(P)-C(P) is known from (5)
U*(P)=(PC)x
=(PC)f(p) (9)
Obviously, the optimal price p* that maximizes the profit U*(P) can be solved by the equation du/dc=0.
That is, when p=p*, there are: di/dp|P=P*=dc/dp|P=P*. (10)
Note: The left side of the equation shows the amount of change in income when the price changes by one unit.
The right side of the equation shows the amount of cost change when the price changes by one unit.
In the circulation process, we have set the cost price C as a constant, x=f(p) as the drop function, and f as the demand function. Let f(p)=a-bp (a,b>0) (11)
At this point, U*(P)=(pc)(a-bp)
=(a+bc)p-bp2-ac (12)
By du/dp=0, solvable: p*=a+bc/2b
=c/2+a/2b (13)
Note: a can be interpreted as "absolute" demand.
b=|dx/dp| indicates the increase in sales when the price falls by one unit.
Why shouldn't the package of goods be any bigger when the price p reaches the critical price p*? In fact, it is not difficult to understand
From (11), we can see that f(P)=a-bp (a,b>0) shows:
p=a/b-y/b
Then, p ́=-1/b. That is to say, the derivative p ́ obtained after deriving the product's sales price P is negative. If so, the profit will not be the most profitable, so say that when the price When p is reduced to the critical price p*, the packaging of the commodity should not be large.

Fourth, the evaluation of the model

This mathematical model is generally not perfect. Due to the academic level, many things cannot be explained clearly. However, if this model is used to roughly analyze related economic issues, it is still possible, especially if By grasping the total "absolute" demand for a certain commodity in society, we can obtain a more accurate critical price p*. This is the advantage of this model.

Reprinted from: Marketing Forum