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学習ガイド > Business Calculus

Reading: Consumer and Producer Surplus

Here are a demand and a supply curve for a product. Which is which?
Figure 1 Figure 1
  • The demand curve is decreasing—lower prices are associated with higher quantities demanded, higher prices are associated with lower quantities demanded. Demand curves are often shown as if they were linear, but there’s no reason they have to be.
  • The supply curve is increasing—lower prices are associated with lower supply, and higher prices are associated with higher quantities supplied.
The point where the demand and supply curve cross is called the equilibrium point (q*, p*).
Figure 2 Figure 2
Suppose that the price is set at the equilibrium price, so that the quantity demanded equals the quantity supplied. Now think about the folks who are represented on the left of the equilibrium point. The consumers on the left would have been willing to pay a higher price than they ended up having to pay, so the equilibrium price saved them money. On the other hand, the producers represented on the left would have been willing to supply these goods for a lower price—they made more money than they expected to. Both of these groups ended up with extra cash in their pockets! Graphically, the amount of extra money that ended up in consumers’ pockets is the area between the demand curve and the horizontal line at p*. This is the difference in price, summed up over all the consumers who spent less than they expected to—a definite integral. The amount of extra money that ended up in producers’ pockets is the area between the supply curve and the horizontal line at p*. This is the difference in price, summed up over all the producers who received more than they expected to.

Consumer and Producer Surplus

Given a demand function p = f(q) and a supply function p = g(q), and the equilibrium point (q*, p*) The consumer surplus = [latex] \int_{0}^{q*} f(q)dq - p*q* [/latex] The producer surplus = [latex] p*q* - \int_{0}^{q*} g(q)dq [/latex] The sum of the consumer surplus and producer surplus is the total gains from trade.
What are the units of consumer and producer surplus? The units are (price units)(quantity units) = money!

Example

Suppose the demand for a product is given by p = −0.8q + 15o and the supply for the same product is given by p = 5.2q. For both functions, q is the quantity and p is the price, in dollars.
  1. Find the equilibrium point.
  2. Find the consumer surplus at the equilibrium price.
  3. Find the producer surplus at the equilibrium price.

Solution

  1. The equilibrium point is where the supply and demand functions are equal. Solving −0.8q + 15o = 5.2q gives q = 25. The price when q = 25 is p = 130; the equilibrium point is (25, 130).
  2. The consumer surplus is [latex] \int_{0}^{25}(-0.8q + 150)dq - (130)(25) = $250 [/latex]
  3. The producer surplus is [latex] (130)(25) - \int_{0}^{25} 5.2qdq = $1625 [/latex]

Example

The tables below show information about the demand and supply functions for a product. For both functions, q is the quantity and p is the price, in dollars.
q 0 100 200 300 400 500 600 700
p 70 61 53 46 40 35 31 28
q 0 100 200 300 400 500 600 700
p 14 21 28 33 40 47 54 61
  1. Which is which? That is, which table represents demand and which represents supply?
  2. What is the equilibrium price and quantity?
  3. Find the consumer and producer surplus at the equilibrium price. 

Solution

  1. The first table shows decreasing price associated with increasing quantity; that is the demand function.
  2. For both functions, q = 400 is associated with p = 40; the equilibrium price is $40 and the equilibrium quantity is 400 units. Notice that we were lucky here, because the equilibrium point is actually one of the points shown. In many cases with a table, we would have to estimate.
  3. The consumer surplus uses the demand function, which comes from the first table. We’ll have to approximate the value of the integral using rectangles. There are 4 rectangles, and I choose to use left endpoints.
The consumer surplus [latex] = \int_{0}^{400} \text{demand}dq-(40)(400)[/latex] [latex] \cong (70)(100)+(61)(100)+(53)(100)+(46)(100)-(40)(400)=7000 [/latex]. The consumer surplus is about $7,000. The producer surplus uses the supply function, which comes from the second table. I choose to use left endpoints for this integral also. The producer surplus [latex] = (40)(400) - \int_{0}^{400} \text{supply}dq [/latex] [latex] \cong (40)(400)- [(14)(100) +(21)(100) + (28)(100) + (33)(100)] =6400 [/latex]. The producer surplus is about $6400.

Licenses & Attributions

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  • Business Calculus. Provided by: Washington State Colleges Authored by: Dale Hoffman and Shana Calaway. Located at: https://docs.google.com/file/d/0B1lkHWwO61QEM0gwOFhES2N5Tlk/edit. License: CC BY: Attribution.