Another in our WWWTF (What in the World Will I ever use This For?) series:
Math Concept: Integration
Having spent multiple hours over the past week helping official oldest daughter of Env-Econ (OODEE) work through calculus problem after calculus problem in preparation for the pending AP exam, I thought it might be helpful to see how she might eventually move past solving math problems to applying the concepts. Right now I'm going to focus on integration, and hopefully later I will come back and talk about differentiation (seems backwards, but integration is fresh in my mind right now). In what follows, I assume a working knowledge of basic integral calculus--otherwise why would you care what you use this for?
As a reminder, a definite integral is defined as:
where f'(x) is the derivative of the function f(x) with respect to the variable x.
Graphically, the integral represents the area under the function y=f'(x) over the closed domain [a,b].
An example might help. Consider the simple linear function f'(x)=6-2X. What is the definite integral of f'(x) over the closed domain [1,3]?
What does this look like graphically?
Notice that for this case of a linear function (a line), the area under the function makes a triangle. A simple way to check your integration is to just find the area of the triangle on the graph.
The area of the triangle is .5(Base x Height) = .5(2 x 4) = 4. This is the same result we got for the intergral. For more complicated functions (non-linear), the area under the function isn't a nice neat geometric shape.
I know what you're thinking: Nice pictures and it's great fun to think about math, but WWWTF? (What in the World Will I ever use This For?). Keep reading for an application to environmental economics.
Application: Measuring total willingness to pay for changes in environmental quality
Economists use demand functions to represent the relationship between how much someone is willing to pay for something and the quantity of the good consumed. It is generally believed (and accepted) that there is an inverse relationship between what someone is willing to pay for something and the quantity demanded. This is known as the law of demand.
For example, suppose the demand for trips to a local beach can be represented by the demand function:
where Q is the total number of trips demanded for by travelers (measured in thousands of trips) at a trip cost (price) of P. If the average trip cost (including gas and time costs) is $3.00 per trip, then the total number of trips demanded will be 1,289.
In addition to knowing how many trips will be taken at a particular price, economists are sometimes interested in knowing the total value of those trips beyond the price paid for the trip. The total value of the trips could be found by adding up the willingness to pay for each individual trip and then subtracting the price. We call this 'consumer surplus.'
For example, from the demand function, the willingness to pay for the first trip (the first trip is Q=0.001 because Q is measured in thousands) is $18.19. Subtracting the $3.00 price of a trip, the consumer surplus for the first trip is $15.19. The willingness to pay for the second trip (Q=0.002) is $16.71 with a consumer surplus of $13.71.
While this method would work, it would be tedious to calculate calculate consumer surplues for 1,289 trips. Instead, we can find the total consumer surplus by taking the integral of the demand function between the current price ($3.00) and the maximum price that will cause the number of trips taken to go to zero, called the choke price. This is the equivalent of adding up all the willingnesses to pay for each trip.
In the current example, the choke price is infinity since the exponential function is asymptotic to Q=0 as the price goes to infinity.
Graphically, this looks like:
(Notice that this graph is drawn with Q on the horizontal axis and P on the vertical axis. This is a flaw of economists as we draw demand functions on the wrong axes. I apologize for those who preceded me who established this convention and just note that the range of integration is now on the vertical axis.)
Mathemaically, consumer surplus is measured as:
Multiplying by 1,000, the total consumer surplus for the $1,298 trips is $2,678.
What does this have to do with valuation of environmental quality?
Consumer surplus gives us a measure of the value that trip takers place on trips above and beyond the cost of the trips. Suppose that the government decides to invest in a clean up project at the local beach. Once the project is complete, we would expect the demand for trips to increase. If the demand increases, that means that the total amount consumers are willing to pay for the beach trips has now increased. The difference between the change in consumer surplus from before to after the clean up is a measure of the value of the clean up project itself.
To see how this works, suppose after the clean up project, the demand for trips increase to:
Using the integral technique, the consumer surplus after the clean up project is $7,279 (assuming the price of trips stays the same at $3.00). So consumer surplus has increased by $4,601 due to the clean up project. This is a value we can now compare to the cost of the clean up project to begin to assess whether the project passes a benefit cost test.
And just to complete the circle, here is one last graph that shows you what the valuation of the environmental quality improvement looks like:
