Basics of Benefits and Costs

Editor's Notes

• #2: This presentation is going to go over some of those more difficult concepts and the math behind benefits and costs.
• #3: Let’s get started with a few economic definitions.When economists thinks about efficiency they want to maximize the net benefits associated with endeavor being considered. In the case of Environmental Economics, economists want to see policies that maximizes the net benefits to society through pollution reduction. The abatement level that achieves this purpose is called the efficient abatement level. Net benefits is simply Total Benefits minus Total costs. Total benefits and costs can be represented as a function of some quantity x, where the quantity x represents the level of abatement. Abatement is the term we use in economics to refer to the level of pollution reduction.
• #4: “Recall from you text that a normal total cost curve is shaped essentially as depicted here in the upper right hand corner. On our x-axis we have some quantity whether it be the # of cookies produced or the amount of pollution that is reduced. The y-axis represents our dollar values associated with the quantities under consideration. Think of going into best buy you are planning on purchasing a new TV. It may only cost you a few hundred dollars to obtain decent TV that meet nearly all of you needs. But as you start adding more bells and whistles that price starts jumping higher and higher and higher. It gets to the point that maybe there is only a difference in one feature and there is a huge price jump. Likewise, when companies or government consider reducing pollution, it may not be extremely expensive to abate a considerable amount, but as we try to get the last little bits, the technology required becomes more and more expensive to acquire and operate. This is why the cost function is shaped as it is. At first as quantity increases costs do not rise all that fast, but at a certain point, as quantities continue to increase the costs begin to rise quite fast. At the beginning of the curve, the slope of the line is shallow and as you increase in quantities, the slope of the line becomes more and more steep until it is nearly vertical! A nearly horizontal line represents fairly unchanging costs as quantities increase. A nearly vertical line represents ever increasing costs with a very small increase in quantity.”Let’s actually see how this line equates to a function!
• #5: Let’s consider a scenario where you are looking at the costs associated with the Clean Air Act. You are going to consider your abatement levels in terms of a 0-100 percent scale. Either no abatement or complete pollution elimination! The costs that your team find are in millions of dollars. Now your team wants to see how this looks on a graph. They open up a new sheet in excel and copy/paste these basic figures. (Try this yourself as we go!)Remember when working with excel, you enter the x-axis data in the first column and the y-axis data in the second column. The reverse of what you see here! Then select the data columns, go to “insert” and select a scatter chart (without lines) and you should get a chart that essentially looks like the one above. (I have done some additional formatting). Now any line such as the one above has a “function’,” some equation that correlates the abatement levels to the costs. This equation would be much more concise than always have to deal with a table of numbers! To find this function in excel right click on a data point and select “Add a Trendline.” Choose a polynomial of the order 2 and select the box so that the equation is displayed on the chart.
• #6: As you can see we have a line that is added and an equation that reads y = 0.85x2 + 25x + 1E-11. The last 1E -11 means 1 x 10 to the negative 11. or 0.000-000-000-01. This last bit is a rounding error, so you can remove it from the equation. You see for it to be absolutely accurate you would need to have all costs values from 1 to 100, not just in 10s. Since costs = C(X) and represented on the y axis, we can substitute “y” for C(X) to represent the cost “cost function.” Now we have an equation instead of tables of data to work with.
• #7: Why don’t you grab your calculator. Let’s do an example: Our table does not tell us how much it would cost for a 53% reduction in pollution. But we can find this out from our equation! If we plug in an abatement level of 53% for X into our equation, we can find out what the cost would be! 0.85 times 53 squared plus 25 times 53, we get approximately 3713 million dollars! From our table we know that 50% abatement will cost $3375 million and 60% will cost $4560 million. So, a cost of 3713 million for a 53% reduction in pollution is consistent! Being able to use a function allows us to be more precise. When we write C(53), as shown above, we are simply saying that we are evaluating the cost function at an abatement level of 53%.
• #8: Now that we have looked at how the cost function works, lets consider Total Benefits. While total benefits do indeed continue to increase as quantities increase, the benefits the benefits received actually “slow down” that is the additional benefits become smaller and smaller causing our total benefit curve to start looking more horizontal. Recall a horizontal line means little change in the $$ as quantities increase. A good way to start thinking about total benefits is to make you think of something personal that you receive benefit from. Lets take food, pizza specifically. Consider a situation where you’ve had a very long day. You’re tired and hungry and all you want to do is curl up on your couch for your favorite episode and some pizza. You get all situated, queue up your show, and pull out your first slice of your favorite pizza. I could imagine that that first bite is fabulous! Your tasetbuds and your tummy are extremely happy as you consume that first slice of culinary delight. You have received a lot of satisfaction from that first 10th of a pizza! Now your on to slice #2 and 20% of your pizza is gone. You still got a lot of satisfaction from that second slice of pie, but as you consume more and more pizza that satisfaction is going to start diminishing. The benefits you receive are going to start leveling off…and beware if you eat too much your benefits might just take a nose-dive on that chart above! It is the same with reducing pollution. If we reduce pollution by 50% and get that really nasty stuff out of the water, we are going to see a real improvement! But as we start removing the pollution that is less harmful or less visible, the benefits to society of this additional reduction are going to start leveling off as well. Like cleaning up your room. At first it looks like a tornado hit the place. But, once you’ve picked up all the clothes, made your bed, etc organizing your closet and vacuuming under the bed are not going to keep your benefits increasing at the rate those first, very visible tasks did.
• #9: Now your same team that evaluated the costs of the Clean Air Act also came up with some values for the benefits received for every 10% of pollution reduction.You can also do this example in excel as well. Do it on the same sheet as your cost data. In fact you only need to add a new column with the benefit values next to the cost data column. Then select both the abatement level column and the benefit data column. Use CTRL button to select individual columns. Go to Insert and select a scatter plot without lines. The table above should be quite similar to the one that you get. Note that again the abatement levels are on the X-axis and the $ values are on the y-axis. In economics this is always the case. Quantities on the x-axis and dollars on the y-axis. Now let’s go about finding the equation of this line. In excel, right click on a data point and select “Add Trendline.” Then choose a polynomial of the order 2 and also check the box so that the equation is displayed in the chart.
• #10: As you can see we again have a small rounding error of 7 times 10 to the negative 12th. This is the equivalent of 0.000-000-000-007. For our purposes we can ignore this very small value. Our equation then, is B(X) = -0.4x2 + 75x, were x represent the abatement level in % just like for the cost function.As with having a cost function, by having a benefit function were are able to extrapolate actual costs of abatement levels that are not on our chart by solving our benefit function at the abatement level under consideration.
• #11: If we graph the data for both total benefits and costs together, we get the graph shown. If you would like to do this, first make sure you data is in 3 columns, where the first column is abatement, the second column is the cost data, and the third column is your benefit data. Select all three columns and Go to Insert and select a scatter plot with lines. This should give you a graph similar to the one displayed. To find that “Efficient” abatement level remember that we must find the level at which NET Benefits are maximized. Recall that net benefits are Total Benefits minus Total Costs. Looking at our graph we can see that there are two points in which Total Benefits equal total costs. One where abatement is zero And One where abatement is at 40%. We want to maximize net benefits and if total benefits = total costs then our NET BENEFITS are ZERO.So our maximized net benefits are going to be somewhere between these to points where the gap between the lines are greatest. By approximating on the graph we can say that this would be around a 20% abatement level. However to know the real value we would have to subtract total benefits from total costs from 0 to 40% and that can be a long and tedious process. There is a better method!
• #12: If we were to look carefully at this area we would see that where the gap is largest, the slope of the line for both total costs and benefits nearly the same!Recall the equation for the slope of a line is the change in y over the change in x.There are two central concepts in Economics and these are the concepts of marginal cost and marginal benefit. Marginal costs are equal to the change in total costs divided by the change in the level of abatementMarginal benefits are equal to the change in total benefits divided by the change in the level of abatement. Now consider what our x’s and y’s are. Our X’s are the abatement levels and our y’s are either the cost or the benefits depending on which curve you are looking at. That means that the marginal costs and benefits are simply the slopes of the total cost and benefit curves respectively!!! To maximize net benefits then we want to find where the two slopes are equal or where MC = MB!!! The abatement level where this occurs is our “efficient level of abatement.” The MB = MC rule is also known as the equimarginal rule. If were using calculus we would simply take the derivative of the total benefit and cost curve to find the equation for the marginal benefit and marginal cost curve. But as this course does not require calculus we will do it algebraically!
• #13: Let us figure out what our Marginal Costs are first. Lets use our equation the change in total costs divided by the change in abatement level. Since we are looking at “changes’ between levels we have no value when x = 0. we start with X = 10. The change in total costs is 335 minus 0 = 335 and the change in abatement level is 10-0 equaling 10. We get a marginal cost of 33.5 million. If we go down the table in the same manner we can fill in marginal cost. For example MC at 20% would be 840 minus 335 divided by 10 equaling 50.5 million. Please note that the change in abatement level will always be 10. Use this same method to solve for marginal benefits, except use the total benefit data. See if you get the same numbers that I have in the table. Note that the last entry in the table for MB is negative. This can happen, what this means is that your Total Benefits are now decreasing rather than continuing to increase. Now, we can actually plot the MC and MB curves!
• #14: We find where Marginal benefit and Marginal cost are equal is actually at about 25% on the graph. We can double check this using algebra.We found the MC equation to be equal to 1.7 times X plus 16.5We found the MB equation to be equal to negative 0.8 times X plus 79Recall the y-axis represents the marginal costs and the marginal benefits. If we want to find where MC = MB, then we can actually put these two equations together. When we do this we get 1.7 times X plus 16.5 equals negative 0.8 times x plus 79.Next we need to combine like terms. We want to get all the X terms on one side. So we add 0.8x to the left hand side of the equation and subtract 16.5 from the right hand side of the equation. We then get 2.5 x = 62.5Where x is equal to 62.5 divided by 2.5 equaling 25What we found is that what looked like the greatest gap was at a 20% abatement level , we found it to be at 25% instead! This is because we found the answer algebraically. For those math inclined individuals, if we had used calculus and therefore not divided by increments of 10 for our marginal costs and benefits, we would have indeed found 20% to be our maximized net benefits.
• #15: We have now found that based on our data, our optimal abatement level is 25%.Now we will want to know what our net benefits will be! Recall our equation from the beginning of this presentation.Net benefits equals total benefits minus total costs. Our total benefits equation was equal to negative 0.4 times x squared plus 75 times x and our total costs equation was equal to 0.85 times x squared plus 25 times x. If we plug 25 in for x for both of these equations then we receive $1625 million for the benefits received with a 25% abatement level and $1156.25 million In costs when the abatement level is 25%. In addition, if you want to find the Marginal costs and the Marginal benefits at a particular abatement level, then all you need to do is plug that abatement value into the appropriate equation and the resulting value will be your answer.
• #16: What if you are only given data for marginal benefits and marginal costs? Well you could sum up all the marginal costs and benefits at each increments in the table OR you could simply plot your MC and MB curves and find the area under those curves by using areas of rectangles and squares. The second method tends to be easier, especially when your marginal values are in increments other than “per 1 unit.” What you need to know are the equations for finding area of the rectangle and triangle. The area of a rectangle is base time the height and the area of triangle is one half the base times the height.
• #17: Let’s do an example:Let’s assume that we only have the equation for Marginal cost but wish to find total cost when the abatement level is set at 50%First thing we need to is find the marginal cost when abatement is at 0. This is not necessarily zero as it is for total costs. What we need to do is plug zero into our MC equation to find out the value. If MC = 16 + 0.5 times zero) our marginal cost is $16.The next thing we need to so is find what the marginal cost is at an abatement level of 50!We plug 50 into our marginal cost equation and we get $41. Now we need to find the area under the curveWhat we can do is break this shape into one rectangle and one triangleCalculating the rectangle first we use the formula for the area of a rectangle. The base is 50 minus 0 and the height is 16 minus zero. This gives us 50 times 16 equaling $800Calculating the triangle next we get one half times 50 minus zero times 41 minus 16. Note that the base of the triangle is the top of the rectangle which is at 16. We then get one half times 50 times 25 equalling $625. Obviously my graph is not to quite to scale as the triangle looks bigger than the rectangle.Finally, we sum the two areas together for a grand total of $1425 for our Total abatement Costs. This same method can be used to find the area underneath the marginal benefit curve in order to obtain total benefits.
• #18: That’s it for this presentation. Take a moment to go through Chapter 2 of Markets for the Environment again to help solidify concepts.