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Consumer and Producer Surplus

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Bingda Huang

Economics is a science that can help humans solve the problem of rational allocation and full utilization of resources. At a micro level, it can help us better understand our world and use economic thinking to view and analyze some people and things rationally. Even if we can’t know that we are making the right choices, it can help us avoid falling into some traps in life. This short article will help you understand the basic concepts of consumer surplus and producer surplus in economics and how they relate to calculus.

Before we get into consumer and producer surplus, let’s take a quick look at demand and supply curves. Basically, the quantity of a certain item produced and sold can be described by the supply and demand curves of the item. The supply curve shows what quantity, q, of the item the producers supply at different prices, p. The consumers’ behavior is reflected in the demand curve, which shows what quantity of goods are bought at various prices. See graph:

Supply and Demand Curves

The logic of the model of demand and supply is simple. By putting the two curves together, we should be able to find a price at which the quantity buyers are willing and able to purchase equals the quantity sellers will offer for sale (This price is called the equilibrium price). At equilibrium, a quantity q^* of an item is produced and sold for a price of p^* each. Notice that, under this condition, a number of consumers have bought the item at a lower price than they would have been willing to pay. (For example, there are some consumers who would have been willing to pay prices up to p_1.) Similarly, there are some suppliers who would have been willing to produce the item at a lower price (down to p_0, in fact). We define the following terms:

  • Consumer surplus measures the consumers’ gain from trade. It is the total amount gained by consumers by buying the item at the current price rather than at the price they would have been willing to pay.
  • Producer surplus measures the suppliers’ gain from trade. It is the total amount gained by producers by selling at the current price, rather than at the price they would have been willing to accept.

In brief, Both consumers and producers are richer for having traded. The consumer and producer surplus measures how much richer they are. See graph:

Consumer and Producer Surplus

In reality, sometimes, markets can produce changes. In a free market, the price of a product usually moves toward the equilibrium price, but the presence of external forces can make the price artificially high or artificially low. For example, rent control keeps prices below market value, while cartel pricing or minimum wage laws keep prices above market value. What happens to consumer and producer surplus at disequilibrium prices? Here is an example to help us find out:

The dairy industry has cartel pricing: the government has set milk prices artificially high. What effect does raising the price to p^+ from the equilibrium price have on.

Figure (1)

Figure (1) above gives a graph of possible supply and demand curves for the milk industry. Assume that the price is fixed at p^+, above the equilibrium price. Consumer surplus is the difference between the amount consumers pay (p^+) and the amount they would have been willing to pay (given the demand curve). This is the shaded area in Figure (2) below. This consumer surplus is less than the consumer surplus at the equilibrium price, as shown in Figure (2):

Figure (2)

For producer surplus: at price p^+, the quantity sold q^+ is less than at the equilibrium price. Producer surplus is represented by the area between p^+ and the supply curve when demand decreases. This area is shaded in the Figure (3) below. In this case, the producer surplus is larger at the artificial price than at the equilibrium price. (However, different supply and demand curves may lead to different answers).

Figure (3)

In summary, for the change in overall returns: the shaded area in Figure (4) below indicates the total gains from trade at a price p^+ (consumer surplus + producer surplus). The shaded area in Figure (5) below represents the total gains from trade at the equilibrium price p^*. At artificial prices, you can clearly see that the total gain from trade decreases. The total financial impact of artificially high prices is negative for all producers and consumers.

Figure (4)

Figure (5)

Since we know all about the concepts of consumer and producer surplus, it is time to introduce how they relate to the fields of calculus and mathematics. Under the field of calculus, we can use the knowledge of relevant integrals to help us quickly find the consumer or producer surplus.

Now assume that all consumers buy the item at the highest price they are willing to pay. And divide the interval from 0 to q^* into intervals of length \Delta q.

Figure (6)

As Figure (6) shows, a certain number of \Delta q items are sold at a price of approximately p_1, another \Delta q is sold at a slightly lower price of approximately p_2, the next \Delta q is sold at a price of approximately p_3, and so on. Thus, the total consumer expenditure is approximately.

Expressed in integral form:

If the equation of the demand curve is p = f (q), and if all consumers willing to pay more than p^* pay the amount they are willing to pay, then when \Delta q approaches 0, we will have consumer expenditure, which is:

We know that consumer surplus represents the extra money that ends up in the consumer’s pocket, and it is the area between the demand curve and the horizontal line p^*. Expressed in terms of the integral formula it is:

\int_ {0} ^{q^*} f(q) dq - p^*q^*

Similarly, the producer surplus represents the amount of extra money that ends up in the producer’s pocket, and it is the area between the supply curve and the horizontal line at p^* , and we assume that the supply curves have the equation: p = g (q). Then expressed in the integral equation is:

p^*q^*-\int_ {0} ^{q^*} g(q) dq

In summary, we can easily find the consumer and producer surplus using the two integral formulas above.

As stated at the beginning of this blog, economics can help you analyze things rationally, and I hope these consumer and producer surplus contents can help you solve economics or similar problems in the future.

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