# Create a simple decision tree for a water authority associated with the tradeoff of issuing a Boil Water Advisory.

###### Create a simple decision tree for a water authority associated with the tradeoff of issuing a Boil Water Advisory.

This problem set is an individual assignment worth 70 points, In 2006, EPA issued its Long Term 2 Enhanced Surface Water Treatment Rule to the control of microbial pathogens at public water systems. This rule was intended to reduce illness linked with the contaminant Cryptosporidium and other pathogenic microorganisms in drinking water. Cryptosporidium is a microscopic parasite that causes a watery diarrheal illness (intestinal cryptosporidiosis) with or without a persistent cough. For people with weakened immune systems, symptoms can be severe and could lead to death. According to EPA “Cryptosporidium is a significant concern in drinking water because it contaminates most surface waters used as drinking water sources, it is resistant to chlorine and other disinfectants, and it has caused waterborne disease outbreaks. Consuming water with Cryptosporidium can cause gastrointestinal illness, which may be severe and sometimes fatal for people with weakened immune systems (which may include infants, the elderly, and people who have AIDS).” (EPA Fact Sheet – Long Term 2 Enhanced Surface Water Treatment Rule) Even with additional monitoring and protection, Cryptosporidium may still occasionally be detected in the finished water. If the local water treatment plant deems it necessary, it will issue a “boil water order” (BWO) (or a “boil water advisory”) to the surrounding community. Under a BWO, individuals are advised to boil all water used for drinking, preparing food, beverages, ice cubes, washing fruits and vegetables, or brushing teeth. The water should be brought to a rolling boil for a minimum of one minute, cooled and refrigerated in a clean container. This process is considered to be the best way to remove Cryptosporidium however it may not be foolproof if the water is not boiled long enough or if contaminated pots or storage containers are used. At present, there is no standard risk level or environmental concentration of Cryptosporidium (measured in oocysts per liter, or oocysts/L) that triggers a BWO. Public officials must weigh the costs associated with the BWO against the risk of disease or death. Implicitly, there is some threshold beyond which the costs of the BWO are outweighed by the benefits of the reduced risks. You will conduct a simplified economic analysis to determine the risk of illness and the level of Cryptosporidium (in oocysts/L) that would justify a Boil Water Advisory from an economic perspective. 1.Create a simple decision tree for a water authority associated with the tradeoff of issuing a Boil Water Advisory. After some level of Cryptosporidium has been detected in the water, the water authority must decide if that level warrants issuing a Boil Water Advisory (which you can label as A). From an economic perspective, that decision is a comparison of the expected value of issuing the advisory, (labeled EV[A]), against the expected value of not issuing the advisory, (labeled EV[no A]). For this part, we assume that this expected value is for a one-month period, but you do not need to include time discounting. Simplifying a bit, the decision is a comparison of the following two possible outcomes: •If the advisory is not issued, then there is a probability, M, for each individual that he or she will become ill from ingesting the parasite. If she becomes ill, then there is a probability, p, of dying from the illness and a complementary probability, (1-p), of not dying but developing intestinal cryptosporidiosis. The economic loss from dying is represented by the value of statistical life (VSL), V. The economic loss from developing cryptosporidiosis but not dying is the cost of illness from that disease, E. If the individual does not become ill, we assume that the economic loss is zero. •If the advisory is issued, then every individual suffers the cost of boil, hauling, or purchasing water, B. While boiling is generally effective, there is some probability, Q, that the boiling water will be ineffective against the parasite. Then, just like the outcome above, there is some probability, M, that an individual will become ill. If he or she becomes ill, then there is a probability of death, p, with an economic loss of the VSL, V; and a probability (1-p) of cryptosporidiosis with a loss of the cost of illness, E. Create a simple decision tree of this tradeoff. A decision tree is “a schematic, tree-shaped diagram used to determine a course of action or show a statistical probability. Each branch of the decision tree represents a possible decision, occurrence or reaction. The tree is structured to show how and why one choice may lead to the next, with the use of the branches indicating each option is mutually exclusive.” (Investopedia) If you don’t know what a decision tree is, you can refer to this Investopedia link ((https://www.investopedia.com/terms/d/decision-tree.asp) or the Wikipedia page (https://en.wikipedia.org/wiki/Decision_tree). Don’t worry too much about the exact convention of the branches and nodes (you won’t be graded on whether you use squares, circles, or triangles), just make sure you convey all the possible outcomes. At the end of each branch, list the economic cost associated with that outcome. 2.(27 points) Assume a boil water advisory lasts for one month and that it takes that amount of time for the water authority to improve the filtering system or to switch to a different water source. Estimate the threshold monthly probability of illness, M, from Cryptosporidium exposure which would justify a boil water advisory. That is, calculate the break-even probability that would make the expected value of issuing an advisory, EV[A], equal to the expected value of not issuing an advisory, EV[no A]. a.(3 points) Write out the formula for the expected value of issuing a boil water advisory, EV[A], using the variables that you use in your decision tree in part 1. b.(3 points) Write out the formula for the expected value of not issuing a boil water advisory, EV[no A], using the variables that you use in your decision tree in part 1. c.(3 points) Assume that you have all of the variables except the monthly probability of illness, M. Equate EV[A] and EV[no A] and solve for the break-even probability of illness, M, in terms of all the other variable, that would make the economic loss of the two outcomes equal. Report those calculations here. d.For this exercise, assume that the probability that boiling is ineffective, Q, is 1%. That is Q=0.01. e.(3 points) Calculate the monthly cost of a boil water advisory. •This cost can be obtained from the Kocagil et al. article posted on the class website. That article reports the “total averting behavior cost (AC) to one statistical individual” of boiling, hauling, or purchasing water for a 30 day period based on a study in Lancaster, PA. •The value from the Kocagil et al article is the monthly cost reported in 1996 dollars, so you will have to inflate it to 2020 dollars using the Consumer Price Index (CPI) found at https://www.bls.gov/data/. Use the Consumer Price Index (CPI) for All Urban Consumers and select the “U.S. City Average,” for “All items,” “Not Seasonally Adjusted.” You should inflate using the value for the annual average. •Report the value you obtained from the Kocagil, et al. article in 1996 dollars, the inflation factor you calculated from the CPI, and the cost of a boil water advisory in 2020 dollars here. f.(3 points) Calculate the average probability of dying from a cryptosporidium-related illness. •This probably can be calculated using the EPA’s Economic Analysis for the Final Long Term 2 Enhanced Surface Water Treatment Rule (LT2ESWTR). This is a large document and has been posted on the class website. •In section 5.2.3 of Chapter 5, the Benefits Analysis, there is a sub-section entitled “Mortality Rate.” This sub-section details EPA’s calculations. Near the end is a paragraph with estimates of the “overall mortality rate” (reported as deaths per 100,000 cryptosporidiosis illnesses). This is a combination of AIDS-related and non-AIDS-related mortality from cryptosporidiosis illnesses. Two overall mortality rates are reported: one rate for unfiltered systems and one rate for filtered systems. Calculate the probability of dying from a cryptosporidium-related illness as the simple average of these two individual probabilities. •Report the two individual probabilities (for unfiltered and filtered systems) and your calculated average here. g. (3 points) Calculate the cost of illness from a non-fatal Cryptosporidium-related illness. •This value can also be found in the Economic Analysis for the LT2ESWTR. •Section 5.3.1.1 details the calculation of the value of illnesses avoided. In the subsection on the “Total Morbidity Cost of Illness,” there is a table for the “Total Loss Per Case.” Obtain the total loss per case for the “Enhanced COI” (that is, the Enhanced Cost of Illness) in 2003 dollars. •This cost will have to be inflated from 2003 to 2020 dollars using the same CPI series that you used in part 2.b. •Report the value you obtained from the EPA economic analysis in 2003 dollars, the inflation factor you calculated from the CPI, and the cost of illness from a non-fatal Cryptosporidium-related illness in 2020 dollars here. h.(3 points) Calculate the Value of a Statistical Life (VSL). Obtain the EPA’s recommended central estimate for the VSL from https://www.epa.gov/environmental-economics/mortality-risk-valuation. This value is in $2006, so, again, it will have to be inflated to 2020 dollars using the CPI index. Report the recommend value for the VSL that you obtained from the EPA website, the inflation factor that you calculated from the CPI, and the VSL in 2015 dollars here. i.(3 points) Substitute all of the values that you calculated in parts 2.d through 2.h. and calculate the break-even probability of illness, M, using the formula that you created in part 2.c. Report that break-even probability here. j.(3 points) Calculate the expected value of issuing a boil water advisory, EV[A], using the formula that you developed in part 2.a. Confirm that the expected value of not issuing a boil water advisory, EV[no A], using the formula that you developed in part 2.b. is the same as EV[A]. Report your calculations and the expected value from EV[A] and EV[no A] here. 3. (18 points) Calculate the average daily probability of an illness, i, the daily probability of morbidity, y, the daily risk of an infection, n, and the average daily dose of Cryptosporidium that would justify a boil water advisory. a.(3 points) Calculate the average probability of an illness. The monthly probability of illness, M, that you calculated in part 2.i is a composite probability of the daily probability of illness, i. Specifically, the monthly probability is Monthly probability = 1 – [1 – Daily probability]30 ?M = 1 – [1 – i]30 Solve for the daily probability of illness, i, and report that value here. b.(3 points) Obtain the probability of morbidity, y. The probability of illness, i, that you calculated in 3.a is actually a product of two probabilities: the probability of an infection from ingesting Cryptosporidium parasites, n, and the probability of illness conditional upon an infection, y. That is prob[illness] = prob[infection] * prob[illness | infection] ?i = n * y The probability of illness conditional upon an infection, y, is also known as the “infectivity rate” or the “morbidity rate.” In section 5.2.3 of Chapter 5 of the Economic Analysis for the LT2ESWTR, there is a sub-section entitled “Morbidity Rate.” Report the “central tendency (mode) for the distribution” the morbidity rate from this sub-section. c.(3 points) Calculate the daily risk of an infection, n. Using the formula given in part 3.b, calculate the daily risk of infection (the prob[infection]), n. Report that value here.