Kaya Ota Math178 2/27/2015

UMAP #2 Price Distribution and Consumer Surplus The method of calculus is used to study a behavior of economics and

contributes to the development of economic theories. In this article “Price Distribution and Consumer Surplus,” it leads readers to understand the analysis of price distribution and consumer surplus in a competitive market using the methods of calculus. Furthermore, it leads us to extend both ideas, price distribution and consumer surplus, to two-tier-price discrimination.

Before we actually apply calculus to the economic problems, we need to clarify assumptions. We can only apply differentiation and integration to continuous or differentiable functions, so the functions that describe economic behaviors have to be continuous or approximately continuous. In the real world, economic functions are typically not continuous due to following two reasons: money is not infinitely divisible and some products are sold with discrete quantities. Two functions we addressed, the supply functions and the demand functions, have to be either a step function or a discrete function to approximate they are continuous. Only in the case, we can apply the tools of the calculus to analyze the behavior of economics. One more additional assumption we have is that the supply and the demand functions have no exception.

As we mention it previously, we now introduce to two functions, the supply and the demand, deeper. They help us determine the price of the products sold in a competitive market. We define the supply function 𝑝 = 𝑆(𝑞) where a price of products that will be sufficient to attract exactly that quantity of the product in the market. We typically assume that the higher the price, the larger the quantity supplied. The demand function 𝑝 = 𝐷(𝑞) associates with each quantity, q, and returns the price at which the market will be cleared of that quantity. The demand function is usually assumed as a decreasing function, which tells us that the less people will buy (i.e. Will not need many quantities) as the price goes higher. Additionally, we have a remarkable point. When the price the suppliers offer and the price consumers ask are equal, which means 𝑝∗ = 𝐷(𝑞∗) = 𝑆(𝑞∗) holds, it is said to be an equilibrium price: denoted 𝑝∗and quantity, denoted 𝑞∗. Here we define two of important words from economic fields that we discussed in this article, “price discrimination” and “consumer supply.” The price discrimination is the practice of charging a different price for the same goods or services. It happens partially when some people buy a certain product under a cheep market, like outlets and some other buy the exact same product with expensive price. Ideally, a perfect price distribution may happen when every customer was charged a different price according to the price the customer willingly pay for. The total values of the product to customers are geometrically represented in the area under the demand function and algebraically

Figure 1 discrete demand function

∑ 𝐷(𝑞∗)𝑛𝑖=1 𝛥𝑞𝑖 in discrete. We approximate each step of 𝑖 is small enough. So, we

can integrate as: 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑟𝑒𝑣𝑒𝑛𝑢𝑒 = ∫ 𝐷(𝑞𝑖)𝑑𝑞 = lim𝑛→∞

∑ 𝐷(𝑞∗)𝑛𝑖=1 𝛥𝑞𝑖

𝑞 ∗

0

The next word, the customer supply is the difference between the maxima price that a consumer willingly pays for and the price he or she actually pays for. So, when a certain product is being sold at the equilibrium price 𝑝∗with corresponding 𝑞∗ then we have the total revenue paid by customers = p∗ 𝑞∗

and so 𝑐𝑜𝑛𝑠𝑢𝑚𝑒𝑟 𝑠𝑢𝑟𝑝𝑙𝑢𝑠 = (𝐷(𝑞𝑖) − 𝑝∗)𝑞∗ where 𝑖 = 𝑠𝑚𝑎𝑙𝑙 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑠𝑡𝑒𝑝.

In counties, consumer surplus is represented by 𝑐. 𝑠 = ∫ (𝐷(𝑞) − 𝑝∗)𝑑𝑞𝑞 ∗

0 . In the

figure 2, we can clearly see the area we are interested in is in interval of [0, 𝑞∗] as it is seen in the limitation of integration as well.

Up to here, we are interested in the specific price with the specific product. Now we are expand those ideas to ”Two-tire price discrimination.” Almost everybody has this experience; you regret when you see the product you bought few days ago is sold in the different store with cheaper price. As the experience, the price on a specific product is varied. In Two tire price discrimination is the simplest form of price discrimination where the sellers charges two different prices to the customers. The perfect two-tire price discrimination occurs when all consumers willingly pay for the higher price, 𝑝1. This happens when 𝑝 = 𝑝1 intersects with the demand curve 𝑝 = 𝐷 (𝑞). When we assume we have two prices that are 𝑝1 and 𝑝∗(i.e. the equilibrium price), the total revenue to the seller is 𝑝1 𝑞1 + 𝑝∗(𝑞∗ − 𝑞1).

This figure 3 tells: one group of the customer buys quantities 𝑞1at the higher price 𝑝1, and the other group buys the quantities 𝑞∗ − 𝑞1, with the equilibrium price 𝑝∗.

A theoretical remark in this article is; in this particular application of the calculus, our actual functions we are considering have actually discrete nature, which means their steps jump from value to

value and we can approximate them as continuous. Therefore, actual answer is the summation of areas of the all rectangles

rather than the integration. However, when we take a look of definition of the

integration, we see: ∫ 𝐷 (𝑞)𝑏

𝑎 𝑑𝑞 = lim

𝑛→∞𝛥𝑞𝑘→0

∑ 𝐷(𝑞𝑘∗ ) × 𝛥𝑞𝑘

𝑛𝑘=1 . This definition implies

that if each step of 𝛥𝑘 is small enough and the number of the rectangles, 𝑛, is very large then the method of integration gives us good approximation to the summation. This idea describes a characteristic of application using the method of calculus to economics.

Figure 2 consumer surplus

Figure 3