{"text_format":"1","workbook":"xlsx","n_sheets":10,"sheets":["5CD","Intro","Conversation","1x","2x","2xBC","2xy","5xyS","50xyS","End"],"not_rendered":{"charts":7,"images":5,"macros":1,"external_links":0,"defined_names":256,"comments":4,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"5CD","index":0,"n_rows":14,"n_cols":8,"rows_kept":14,"cut":false,"hidden":true,"merged":[]}
["Solver cannot find the solution for a Cobb-Douglas functional form."]
[]
["a",0.5]
["b",0.5]
["c",0.5]
[null,null,null,null,null,null,"m growth rate",0.1]
[null,null,null,null,{"$f":"SUM(E10:E14)","v":1.7097072176369963},null,"yearly m",1]
[null,null,null,"discount rate",0.1,0.5,null,{"$f":"SUM(H10:H14)","v":-1.4574624174201745E-8}]
["Time","S","x","y","U","dS","DS","constraint"]
[0,0,1.4391357628878629,0,{"$f":"(C10^$B$3*D10^$B$4*B10^$B$5)/((1+dr)^A10)","v":0},{"$f":"$F$8*B10","v":0},{"$f":"C10-F10","v":1.4391357628878629},{"$f":"((1+$H$6)^A10)*$H$7/((1+dr)^A10)-C10-D10","v":-0.43913576288786293}]
[1,{"$f":"B10+G10","v":1.4391357628878629},1.5411854830901821,1.3616366697607682,{"$f":"(C11^$B$3*D11^$B$4*B11^$B$5)/((1+dr)^A11)","v":1.5798515181186918},{"$f":"$F$8*B11","v":0.71956788144393147},{"$f":"C11-F11","v":0.82161760164625064},{"$f":"((1+$H$6)^A11)*$H$7/((1+dr)^A11)-C11-D11","v":-1.9028221528509504}]
[2,{"$f":"B11+G11","v":2.2607533645341134},1.7462527061385042E-5,1.4028328841673469E-5,{"$f":"(C12^$B$3*D12^$B$4*B12^$B$5)/((1+dr)^A12)","v":1.9449020654986581E-5},{"$f":"$F$8*B12","v":1.1303766822670567},{"$f":"C12-F12","v":-1.1303592197399952},{"$f":"((1+$H$6)^A12)*$H$7/((1+dr)^A12)-C12-D12","v":0.99996850914409696}]
[3,{"$f":"B12+G12","v":1.1303941447941181},0,1.9126271369721557E-6,{"$f":"(C13^$B$3*D13^$B$4*B13^$B$5)/((1+dr)^A13)","v":0},{"$f":"$F$8*B13","v":0.56519707239705907},{"$f":"C13-F13","v":-0.56519707239705907},{"$f":"((1+$H$6)^A13)*$H$7/((1+dr)^A13)-C13-D13","v":0.999998087372863}]
[4,{"$f":"B13+G13","v":0.56519707239705907},0.11850584256517115,0.53950285278759957,{"$f":"(C14^$B$3*D14^$B$4*B14^$B$5)/((1+dr)^A14)","v":0.12983625049764944},{"$f":"$F$8*B14","v":0.28259853619852954},{"$f":"C14-F14","v":-0.16409269363335838},{"$f":"((1+$H$6)^A14)*$H$7/((1+dr)^A14)-C14-D14","v":0.34199130464722927}]

{"sheet":"Intro","index":1,"n_rows":26,"n_cols":3,"rows_kept":26,"cut":false,"hidden":false,"merged":[]}
["Rational Addiction"]
["Humberto Barreto"]
["DePauw University"]
[]
["This Excel workbook explains key ideas in the model of rational addiction."]
[null,"It uses Excel's Solver to find an optimal solution and makes clear how the model works."]
[]
["Software Requirements:"]
[null,"You need Microsoft Excel, Windows version 2007 or greater to get full functionality from this file."]
[null,null,"The file can be opened with Mac Excel, but it uses Visual Basic code that may not be fully supported on a Mac. "]
[]
[null,"You must be sure to Enable Content or the buttons and code in this macro-enabled workbook will not work. -->"]
[null,null,"Search \"enable macros\" in Excel's Help for more information on how to do this."]
[]
["As you read and scroll down a sheet, the workbook will unfold, displaying new sheets, and I will ask you to click buttons and change things. "]
[null,"You are free to take notes on the sheets or change cells as you desire."]
[null,"You can modify charts or create new ones -- there is nothing special about this workbook, nothing is password protected or otherwise blocked."]
[null,"You cannot really hurt anything -- if something does break, just close the workbook and re-open it. You can also Save As if you want to be extra careful and have an original copy available."]
[null,"If you know Visual Basic, go ahead and look at the code. It is freely available for you to inspect and alter as you wish."]
[null,"If you are reading this on your own, check out the Conversation sheet; otherwise your professor "]
[]
["I welcome all feedback, positive and negative. Drop me an email and let me know what you think or if you have any questions or suggestions."]
["I really hope this helps."]
["Enjoy!"]
["Humberto Barreto"]
["hbarreto@depauw.edu"]

{"sheet":"Conversation","index":2,"n_rows":107,"n_cols":14,"rows_kept":107,"cut":false,"hidden":true,"merged":[]}
["Talking Through Some Basic Ideas"]
[]
["Before diving into the deep end, let's take a few minutes to ponder what this is all about."]
["This will help you understand the model."]
[]
["--> In the beginning . . . "]
["Becker and Murphy (1988) \"A Theory of Rational Addiction\" Journal of Political Economy"]
["https://www.jstor.org/stable/1830469"]
["This article is the foundation of models that are optimization-based (that's the \"rational\" part) of goods whose consumption influences future consumption (i.e., that's \"addiction\")."]
[]
["--> Wait! Rational addiction sounds really silly -- no one chooses and plans to be an opioid or meth addict."]
["Right, no argument here. If a person knew for certain that a single puff as a teenager would unleash a cascade of events that immediately destroyed their lives, they would not do it."]
["The phrase \"rational addiction\" is catchy and attracts attention , but it's also confusing and can cause some people to stop listening because it seems so obviously wrong."]
["After all, it sounds like it's saying that people purposely choose harmful obsession. That makes no sense. "]
["Right again. "]
["The words are really getting in the way here."]
["You have to open your mind and be willing to see rationality and addiction in a different way than usual. Let's begin."]
[]
["--> This is not narrow -- there are a whole class of goods and services that can be analyzed by rational addiction models."]
["When you think of addiction, you immediately think of cigarettes, alcohol and drugs. Ingesting these things does something biological to make you want more of it, but what about gambling? "]
["You don't actually swallow, snort or inject yourself with anything when you buy a lottery ticket, play poker for money, or bet on sports, yet we know some people are addicted to gambling."]
["And speaking of sports, what about super-involved fans that know insane trivia about their team and spend outrageous sums of money to see games?"]
["We can conclude that sports fandom is also a form of addiction and we can study it as part of this category of goods and services that are addictive."]
["To broaden the idea even further, although we think of addiction as bad, we can be addicted to good things, like exercise, cleaning, and going to church. "]
["We might even say that marathon runners, germaphobes, and religious fundamentalists are essentially addicts."]
["You can become addicted to video games, the internet, and food."]
["You can even become addicted to other people -- non-economists call that love. Nearby is sex, to which we can also form an addiction."]
["There is no doubt that addiction is everywhere and remember that it's OK to be somewhat addicted to some things, but don't overdo it."]
["If you like work, we might then say you have a good work ethic, but too much of it and you're a workaholic -- that's bad."]
[]
["--> \"How use doth breed a habit in a man!\""]
["In their introduction, Becker and Murphy (1988) included the quotation above from William Shakespeare's The Two Gentlemen of Verona (let's agree that not many Econ articles quote Shakespeare)."]
["The line seems on point, but what exactly does doth mean? Well, actually, it means does, but even then, what exactly is Shakespeare saying here?"]
["Lucky for us, there are translations of his plays to modern English. \"How use doth breed a habit in a man!\" translates to:"]
["\"Repetition can make a man accustomed to anything!\"",null,null,null,null,null,"https://www.sparknotes.com/nofear/shakespeare/twogentlemen/page_174/"]
["So use actually means doing something repeatedly and Shakespeare is saying that this is how habits are formed.  "]
["In other words, consumption influences future consumption -- that's the hallmark of addictive things."]
["Addicts do not appear out of thin air as fully-formed maniacs. They became intensely involved over time. Addiction is repeated use, along with the idea that you can't stop."]
["But before you are addicted, you have to have repeated experience and knowledge. It's a cumulative process."]
["I once took a bunch of European professors to a baseball game -- wow, what a disaster! I had to explain the most ridiculous details to them: \"He didn't run because it's a foul ball.\""]
["On the other hand, thank goodness my first cricket game was T20 -- imagine five days of that! Even the score was baffling. Click to see -->",null,null,null,null,null,null,null,null,null,null,null,null,"https://www.google.com/search?q=cricket+t20"]
["So, some goods require learning and experience and we get more and more utility (satisfaction) the more we consume them."]
["You can think of it as some goods having a more durable effect on us than others. We build up a stock of familiarity that makes consumption more enjoyable."]
["And, yes, the addiction can have a biological foundation, like caffeine and coffee, but it's much bigger than that: anything where we get more out of it next time we do it is a candidate for addiction."]
[]
["--> \"Acuerdate de la rutina.\""]
["\"Remember the routine\" -- that's what my mom used to tell me. She spoke Spanish because, like me, she was born in Cuba. "]
["She believed we are creatures of habit. Familiarity reduces stress and increases confidence."]
["We do the same things the same ways, over and over (just like Shakespeare said) and we get comfortable in our rut. We are addicted."]
["We eat roughly the same foods day after day, follow pretty repetitive schedules, and have nightly procedures for bed (brush teeth, etc.). We get grouchy when our routines are disrupted."]
["Addiction is an extreme type of habit and habits are formed by repetition. Sounds perfectly reasonable, right?"]
["But -- and this is huge for understanding the word rational in rational addiction -- my mom also believed that you can have a meta-moment where you can literally choose your routine."]
["Like you, my mom never heard of rational addiction or Becker and Murphy, but she understood the core of the idea:"]
["She believed that we can step back and look at the whole timeline, past, present, and future, and plan the habits you will have."]
["That's what rational addiction is all about! "]
["There's an intentionality there that we can control. Like our friends -- we can randomly bump into people and they become our friends, but in another sense, we choose our friends."]
["My mom also anticipated another critical aspect of rational addiction: unexpected shocks."]
["She knew that a big life event might shake us out of the routine. These shocks give us a chance to reassess and reconsider what we are doing."]
[]
["--> Football concussions make some fans quit, but others hang on."]
["I know a lot about football. I mean, really, a lot. In fact, it's embarrassing. I grew up in Miami when the Dolphins were undefeated in 1972. They went 17 - 0."]
["Many people know they won the Super Bowl by a score of 14 - 7 and can name many of the stars on the team, but I bet only a few know this:"]
["Their coach, Don Shula, was ballistically upset at their kicker, Garo Yepremian, because up 14 - 0 and the game almost over, he tried a 42-year field goal."]
["Shula had a big smile on his face because he was about to win the game 17 - 0 and end the season 17 - 0 and this was some kind of cosmic numerological convergence."]
["Alas, the gods weren't having it and we got one of the wackiest plays in football history:"]
["https://www.youtube.com/watch?v=d-pxMm_UdY4"]
["But the last few years have been tough on football fans. I don't mean protesting players and charges of domestic violence. The game itself is facing an existential challenge at its core."]
["We now know that concussions cause CTE and football is a game in which giant, super-fast humans give themselves brain damage. Who wants to watch that?"]
["In the rational addiction model, that's a shock to the consumer's stock of invested capital."]
["Some addicts, I mean fans, will quit watching (and tell other fans they are terrible for continuing to watch -- former smokers act the same way), but others will hang on."]
["The hangers on (like me, so far) probably would not be fans except for the investment they have made: the built-up knowledge and experience is key."]
["When I watch a game, I know all kinds of things about what's going on: I have historical comparisons I can make; I know records and rare events; and I know poor and great plays."]
["Can you imagine selling this game from scratch, without any built-up fan base, but when everyone knows the game causes brain trauma? I mean, blown knees are one thing, but dementia?"]
["It seems pretty obvious that football is alive right now because of addicts and the existential threat is that its addicts might quit."]
["Figuring out how a shock can cause some people to quit while others hang on is pretty interesting, in general, and absolutely essential for football."]
[]
["--> Past ↔ Present ↔ Future consumption being connected makes for a brutal optimization problem."]
["You have come a long way already."]
["You are aware that rational addiction isn't as goofy as it first sounds."]
["You know that addictive goods are everywhere and that it relies on consumption today affecting consumption tomorrow."]
["Sure, we are creatures of habit, but we also control, to some extent, habit formation. There is an intentionality there that is not obvious when you look at a fully-formed addict."]
["And, finally, you know that quitting is another part of this issue -- some people quit and others don't. Why?"]
["Perhaps you think it would be a simple matter to extend the basic utility maximization model to include addiction. Just make consumption depend on past consumption."]
["You couldn't be more wrong."]
["The problem becomes much, much harder when past choices influence future ones."]
["If we weren't economists, then sure, the explanation would be easy. The consumer would accidently stumble into some kind of addiction vortex, then consume more and more until addicted. QED."]
["But, wait, how come some people do the repeated use thing until it becomes utterly destructive, spinning wildly out of control, while others are more benignly addicted? "]
["And what about quitting? How come some can escape the vortex?"]
["Falling into an addiction by mistake might work for some things, but it does not make sense for addictive goods where the consumer knows today's decision will influence future consumption."]
["Remember my mom, she knew we could step back and see the big picture. That's what she meant when she said, \"Acuerdate de la rutina.\""]
["The rational addiction model has a tough task: it needs to explain why someone would, on purpose, choose to consume an addictive good and become addicted."]
["We can't exactly pinpoint when \"everyone\" knew cigarettes were bad for you, but accidently becoming addicted to cigarettes is a hard sell today."]
["In economics, the rational consumer has full information -- they know the good is addictive. They can see the future."]
["The rational, optimizing consumer knows that the good is addictive, yet they choose a path that leads to addiction."]
["Why?"]
["Because they solve a lifetime maximization of utility problem and pick a path of consumption that yields addiction as the optimal solution."]
["That lifetime optimization problem is difficult to set up, solve, and understand."]
["It also has a stability aspect where a shock can knock you off the path and then you might be attracted back to it or leave it altogether."]
[]
["--> Numerical methods to the rescue!"]
["Instead of the usual approach involving intense math (solving involves a Hamiltonian, differential equations, and phase diagrams), we will use Excel's Solver."]
["We will implement the problem in Excel, which will require replacing abstract notation with numbers. So, instead of U=f(x), for example, we might use the formula =18x - x^2."]
["Excel will also force us to make plain how consumption in one period affects next period's consumption and what it means to maximize lifetime utility subject to a budget constraint."]
["Once we have the problem on the spreadsheet, we can use Solver, a numerical optimization add-in to give us the answer. We can avoid the intense math and focus on the model's results."]
["Make no mistake, the problem is still really difficult and you have to pay close attention, but you now have a realistic shot at understanding an advanced economic theory."]
[]
["Thanks for your investment so far. Let's hope you get addicted!"]

{"sheet":"1x","index":3,"n_rows":79,"n_cols":13,"rows_kept":79,"cut":false,"hidden":true,"merged":[]}
["This is an example of a quadratic utility function:"]
[]
[]
["We want to choose x to maximize utility. This can be done with calculus by taking the derivative of U with respect to x and setting it equal to zero, but Excel's Solver can also do it."]
[]
["To use Solver, we have to \"implement the problem in Excel,\" which means we have to enter the function in a way that Excel can deal with it:"]
[null,"x","U",null,"x is blue to represent that it is a choice variable, determined by the consumer"]
[null,1,{"$f":"18*B8-3*B8^2","v":15}]
["Change x in cell B8 to 2 and you will see that cell C8, which shows the value of utility, becomes 24 since 18*2 - 3*(2^2) = 24."]
[]
["Now that we have the problem on the sheet, we can use Solver to find the value of x that maximizes U."]
[]
["In the Data tab, click Solver to bring up the Solver Parameters dialog box."]
[null,"If you do not see Solver, you have to install it. Bring up the Add-Ins Manager (use Excel's Help if needed) and make sure Solver is checked."]
["In the Solver Parameters dialog box, enter cell C8 as the Objective and B8 in the Changing Variable Cells as shown to the right and click Solve (near the bottom)."]
[]
["Excel displays a Solver Results dialog box announcing success. Click OK to keep Solver's solution."]
[]
["Cell B8 now displays 3, the optimal value of x, and U* is 27 in cell C8. This is the answer."]
[]
["If you do this problem analytically, using calculus and algebra, you will get the same answer."]
[]
["What is important here is that Solver gets us the answer. It is a numerical optimization tool that gives us an alternative to math."]
["There are some additional complications and issues with this technique, the most important of which is that sometimes it does not work."]
["We will stay with our extremely simple quadratic utility function to ensure that Solver is giving us reliable answers."]
[]
["To display the solution on a graph, we plot U as a function of x -- it's pretty obvious that x* = 3 is the right answer:"]
[null,null,null,null,null,null,null,null,null,null,null,"x","U"]
["Now that we know how to use",null,null,null,null,null,null,null,null,null,null,0,{"$f":"18*L29-3*L29^2","v":0}]
["Solver, we can move on to a",null,null,null,null,null,null,null,null,null,null,0.1,{"$f":"18*L30-3*L30^2","v":1.77}]
["more complicated problem.",null,null,null,null,null,null,null,null,null,null,0.2,3.48]
[null,null,null,null,null,null,null,null,null,null,null,0.3,5.129999999999999]
[null,null,null,null,null,null,null,null,null,null,null,0.4,6.72]
[null,null,null,null,null,null,null,null,null,null,null,0.5,8.25]
[null,null,null,null,null,null,null,null,null,null,null,0.6,9.7199999999999989]
[null,null,null,null,null,null,null,null,null,null,null,0.7,11.129999999999999]
[null,null,null,null,null,null,null,null,null,null,null,0.8,12.48]
[null,null,null,null,null,null,null,null,null,null,null,0.9,13.77]
[null,null,null,null,null,null,null,null,null,null,null,1,15]
[null,null,null,null,null,null,null,null,null,null,null,1.1000000000000001,16.170000000000002]
[null,null,null,null,null,null,null,null,null,null,null,1.2,17.279999999999998]
[null,null,null,null,null,null,null,null,null,null,null,1.3,18.330000000000002]
[null,null,null,null,null,null,null,null,null,null,null,1.4,19.32]
[null,null,null,null,null,null,null,null,null,null,null,1.5,20.25]
[null,null,null,null,null,null,null,null,null,null,null,1.6,21.119999999999997]
[null,null,null,null,null,null,null,null,null,null,null,1.7,21.93]
[null,null,null,null,null,null,null,null,null,null,null,1.8,22.68]
[null,null,null,null,null,null,null,null,null,null,null,1.9,23.369999999999997]
[null,null,null,null,null,null,null,null,null,null,null,2,24]
[null,null,null,null,null,null,null,null,null,null,null,2.1,24.570000000000004]
[null,null,null,null,null,null,null,null,null,null,null,2.2000000000000002,25.08]
[null,null,null,null,null,null,null,null,null,null,null,2.2999999999999998,25.53]
[null,null,null,null,null,null,null,null,null,null,null,2.4,25.919999999999995]
[null,null,null,null,null,null,null,null,null,null,null,2.5,26.25]
[null,null,null,null,null,null,null,null,null,null,null,2.6,26.520000000000003]
[null,null,null,null,null,null,null,null,null,null,null,2.7,26.729999999999997]
[null,null,null,null,null,null,null,null,null,null,null,2.8,26.880000000000003]
[null,null,null,null,null,null,null,null,null,null,null,2.9,26.969999999999995]
[null,null,null,null,null,null,null,null,null,null,null,3,27]
[null,null,null,null,null,null,null,null,null,null,null,3.1,26.97]
[null,null,null,null,null,null,null,null,null,null,null,3.2,26.879999999999995]
[null,null,null,null,null,null,null,null,null,null,null,3.3,26.730000000000004]
[null,null,null,null,null,null,null,null,null,null,null,3.4,26.520000000000003]
[null,null,null,null,null,null,null,null,null,null,null,3.5,26.25]
[null,null,null,null,null,null,null,null,null,null,null,3.6,25.919999999999995]
[null,null,null,null,null,null,null,null,null,null,null,3.7,25.53]
[null,null,null,null,null,null,null,null,null,null,null,3.8,25.079999999999991]
[null,null,null,null,null,null,null,null,null,null,null,3.9,24.570000000000007]
[null,null,null,null,null,null,null,null,null,null,null,4,24]
[null,null,null,null,null,null,null,null,null,null,null,4.0999999999999996,23.370000000000005]
[null,null,null,null,null,null,null,null,null,null,null,4.2,22.680000000000007]
[null,null,null,null,null,null,null,null,null,null,null,4.3,21.929999999999993]
[null,null,null,null,null,null,null,null,null,null,null,4.4000000000000004,21.11999999999999]
[null,null,null,null,null,null,null,null,null,null,null,4.5,20.25]
[null,null,null,null,null,null,null,null,null,null,null,4.5999999999999996,19.320000000000007]
[null,null,null,null,null,null,null,null,null,null,null,4.7,18.329999999999998]
[null,null,null,null,null,null,null,null,null,null,null,4.8,17.279999999999987]
[null,null,null,null,null,null,null,null,null,null,null,4.9000000000000004,16.169999999999987]
[null,null,null,null,null,null,null,null,null,null,null,5,15]

{"sheet":"2x","index":4,"n_rows":114,"n_cols":62,"rows_kept":114,"cut":false,"hidden":true,"merged":[]}
["Now we will extend the model: we use the same utility function, but we have TWO time periods:"]
[]
[]
["So, we have to choose x for now (period 0, the present) and tomorrow (period 1, the future)."]
[]
["We can use the spreadsheet to make the implementation a little more flexible by using cells with parameter values, like this:"]
[null,null,null,null,"a0",18,null,"a0 and a1 are green because they are exogenous variables (they cannot be changed by the consumer) in this model."]
[null,null,null,null,"a1",3,null,"Below (D23:D25), x is blue because it is endogenous (a choice variable for the consumer)."]
["This way, we can change these coefficients easily if needed.",null,null,null,null,null,null,"Remember: green for exogenous and blue for endogenous."]
[]
["Introducing time means we have to account for the fact that people prefer present to future consumption. "]
["So, satisfaction today for future consumption has to be marked down."]
["The easiest way to do this is to adjust downward future utility by a factor called the discount rate."]
["Say utility from consuming x is 100 today, then utility now from consuming x one year from now might be 100/1.1, or 90.9. That's a 10% discount rate."]
["Different people can have different discount rates. If you are super impatient, then your discount rate is high, say 50%."]
["Then consuming x one year from now would give you utility right now of 100/1.5 = 67 or so."]
["Notice that we are computing utility now, at the present, of consumption in the future and that's why it is not worth as much--we have to wait for it."]
[]
["So now the optimization problem looks like this:"]
[]
[null,null,null,"discount rate (dr)",0.1]
[]
[null,null,"Time","x","U"]
[null,null,0,1,{"$f":"(a0*D24-_a1*D24^2)/((1+dr)^C24)","v":15},"<-- Click on cells E24 and E25 to see their formulas. For E24, dividing by (1+dr) doesn't do anything because it's raised to the zero power, making the denominator 1."]
[null,null,1,1,{"$f":"(a0*D25-_a1*D25^2)/((1+dr)^C25)","v":13.636363636363635},"But for E25, present discounted utility is smaller because we are dividing by (1+dr)."]
[null,null,null,"sum U",{"$f":"E24+E25","v":28.636363636363633},"Change E21 to a bigger number, say 0.5, to see that present discounted utility gets smaller. Put it back to 0.1 before continuing."]
[]
["Change cells D24 and D25 to 2 to see that the sum of present discounted utility rises so 2,2 provides more utility than 1,1."]
["As usual, we want the values of x in time 0 and 1 that maximize present discounted utility."]
[]
["We could solve this problem with math, but let's use Solver again."]
["All you have to do is call Solver (from the Data tab) and click Solve."]
["Notice that it has been set up for you."]
[]
["A graph of the solution looks like this:",null,null,null,null,null,null,null,"Economists like the 2D graph (it's a contour plot) better because it is easier to see the solution."]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[]
[null,null,null,null,null,null,null,null,"Those bands of color should remind you of indifference curves."]
["Next, we extend the model again, as we climb toward the model of rational addiction."]
[]
[]
["Data for graphs above:"]
[null,0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1,1.1000000000000001,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2,2.1,2.2000000000000002,2.2999999999999998,2.4,2.5,2.6,2.7,2.8,2.9,3,3.1,3.2,3.3,3.4,3.5,3.6,3.7,3.8,3.9,4,4.0999999999999996,4.2,4.3,4.4000000000000004,4.5,4.5999999999999996,4.7,4.8,4.9000000000000004,5,5.0999999999999996,5.2,5.3,5.4,5.5,5.6,5.7,5.8,5.9,6]
[0,{"$f":"a0*$A54-_a1*$A54^2+(a0*B$53-_a1*B$53^2)/(1+dr)","v":0},1.6090909090909089,3.1636363636363636,4.6636363636363622,6.1090909090909085,7.4999999999999991,8.8363636363636342,10.118181818181817,11.345454545454546,12.518181818181818,{"$f":"a0*$A54-_a1*$A54^2+(a0*L$53-_a1*L$53^2)/(1+dr)","v":13.636363636363635},14.700000000000001,15.709090909090905,16.663636363636364,17.563636363636363,18.409090909090907,19.199999999999996,19.936363636363634,20.618181818181817,21.245454545454542,{"$f":"a0*$A54-_a1*$A54^2+(a0*V$53-_a1*V$53^2)/(1+dr)","v":21.818181818181817},22.33636363636364,22.799999999999997,23.209090909090907,23.563636363636355,23.863636363636363,24.109090909090909,24.299999999999994,24.436363636363637,24.518181818181812,{"$f":"a0*$A54-_a1*$A54^2+(a0*AF$53-_a1*AF$53^2)/(1+dr)","v":24.545454545454543},24.518181818181816,24.43636363636363,24.3,24.109090909090909,23.863636363636363,23.563636363636355,23.209090909090907,22.79999999999999,22.33636363636364,{"$f":"a0*$A54-_a1*$A54^2+(a0*AP$53-_a1*AP$53^2)/(1+dr)","v":21.818181818181817},21.24545454545455,20.618181818181824,19.936363636363627,19.199999999999989,18.409090909090907,17.56363636363637,16.66363636363636,15.709090909090897,14.699999999999987,{"$f":"a0*$A54-_a1*$A54^2+(a0*AZ$53-_a1*AZ$53^2)/(1+dr)","v":13.636363636363635},12.518181818181814,11.345454545454547,10.118181818181814,8.8363636363636218,7.4999999999999991,6.1090909090909209,4.663636363636372,3.1636363636363543,1.6090909090909054,0]
[0.1,1.77,3.3790909090909089,4.9336363636363636,6.4336363636363618,7.8790909090909089,9.27,10.606363636363634,11.888181818181817,13.115454545454545,14.288181818181817,15.406363636363634,16.470000000000002,17.479090909090907,18.433636363636364,19.333636363636362,20.179090909090906,20.969999999999995,21.706363636363633,22.388181818181817,23.015454545454542,23.588181818181816,24.106363636363639,24.569999999999997,24.979090909090907,25.333636363636355,25.633636363636363,25.879090909090909,26.069999999999993,26.206363636363637,26.288181818181812,26.315454545454543,26.288181818181815,26.20636363636363,26.07,25.879090909090909,25.633636363636363,25.333636363636355,24.979090909090907,24.56999999999999,24.106363636363639,23.588181818181816,23.015454545454549,22.388181818181824,21.706363636363626,20.969999999999988,20.179090909090906,19.333636363636369,18.43363636363636,17.479090909090896,16.469999999999988,15.406363636363634,14.288181818181814,13.115454545454547,11.888181818181813,10.606363636363621,9.27,7.8790909090909214,6.4336363636363725,4.9336363636363547,3.3790909090909054,1.77]
[0.2,3.48,5.0890909090909089,6.6436363636363636,8.1436363636363627,9.589090909090908,10.979999999999999,12.316363636363635,13.598181818181818,14.825454545454546,15.998181818181818,17.116363636363634,18.18,19.189090909090904,20.143636363636364,21.043636363636363,21.889090909090907,22.679999999999996,23.416363636363634,24.098181818181818,24.725454545454543,25.298181818181817,25.81636363636364,26.279999999999998,26.689090909090908,27.043636363636356,27.343636363636364,27.58909090909091,27.779999999999994,27.916363636363638,27.998181818181813,28.025454545454544,27.998181818181816,27.916363636363631,27.78,27.58909090909091,27.343636363636364,27.043636363636356,26.689090909090908,26.27999999999999,25.81636363636364,25.298181818181817,24.72545454545455,24.098181818181825,23.416363636363627,22.679999999999989,21.889090909090907,21.04363636363637,20.143636363636361,19.189090909090897,18.179999999999986,17.116363636363634,15.998181818181815,14.825454545454548,13.598181818181814,12.316363636363622,10.979999999999999,9.5890909090909204,8.1436363636363716,6.6436363636363538,5.0890909090909053,3.48]
[0.3,5.129999999999999,6.7390909090909084,8.293636363636363,9.7936363636363613,11.239090909090908,12.629999999999999,13.966363636363633,15.248181818181816,16.475454545454546,17.648181818181818,18.766363636363636,19.829999999999998,20.839090909090906,21.793636363636363,22.693636363636362,23.539090909090906,24.329999999999995,25.066363636363633,25.748181818181816,26.375454545454541,26.948181818181816,27.466363636363639,27.929999999999996,28.339090909090906,28.693636363636355,28.993636363636362,29.239090909090908,29.429999999999993,29.566363636363636,29.648181818181811,29.675454545454542,29.648181818181815,29.566363636363629,29.43,29.239090909090908,28.993636363636362,28.693636363636355,28.339090909090906,27.929999999999989,27.466363636363639,26.948181818181816,26.375454545454549,25.748181818181823,25.066363636363626,24.329999999999988,23.539090909090906,22.693636363636369,21.793636363636359,20.839090909090896,19.829999999999984,18.766363636363636,17.648181818181811,16.475454545454546,15.248181818181813,13.966363636363621,12.629999999999999,11.239090909090919,9.7936363636363701,8.2936363636363524,6.7390909090909048,5.129999999999999]
[0.4,6.72,8.3290909090909082,9.8836363636363629,11.383636363636363,12.829090909090908,14.219999999999999,15.556363636363635,16.838181818181816,18.065454545454546,19.238181818181818,20.356363636363636,21.42,22.429090909090906,23.383636363636363,24.283636363636361,25.129090909090905,25.919999999999995,26.656363636363633,27.338181818181816,27.965454545454541,28.538181818181815,29.056363636363638,29.519999999999996,29.929090909090906,30.283636363636354,30.583636363636362,30.829090909090908,31.019999999999992,31.156363636363636,31.238181818181811,31.265454545454542,31.238181818181815,31.156363636363629,31.02,30.829090909090908,30.583636363636362,30.283636363636354,29.929090909090906,29.519999999999989,29.056363636363638,28.538181818181815,27.965454545454548,27.338181818181823,26.656363636363626,25.919999999999987,25.129090909090905,24.283636363636369,23.383636363636359,22.429090909090895,21.419999999999987,20.356363636363636,19.238181818181815,18.065454545454546,16.838181818181813,15.556363636363621,14.219999999999999,12.829090909090921,11.383636363636372,9.883636363636354,8.3290909090909047,6.72]
[0.5,8.25,9.8590909090909093,11.413636363636364,12.913636363636362,14.359090909090909,15.75,17.086363636363636,18.368181818181817,19.595454545454544,20.768181818181816,21.886363636363633,22.950000000000003,23.959090909090904,24.913636363636364,25.813636363636363,26.659090909090907,27.449999999999996,28.186363636363634,28.868181818181817,29.495454545454542,30.068181818181817,30.58636363636364,31.049999999999997,31.459090909090907,31.813636363636355,32.11363636363636,32.359090909090909,32.549999999999997,32.686363636363637,32.768181818181816,32.795454545454547,32.768181818181816,32.68636363636363,32.549999999999997,32.359090909090909,32.11363636363636,31.813636363636355,31.459090909090907,31.04999999999999,30.58636363636364,30.068181818181817,29.49545454545455,28.868181818181824,28.186363636363627,27.449999999999989,26.659090909090907,25.81363636363637,24.91363636363636,23.959090909090897,22.949999999999989,21.886363636363633,20.768181818181816,19.595454545454547,18.368181818181814,17.086363636363622,15.75,14.35909090909092,12.913636363636371,11.413636363636353,9.8590909090909058,8.25]
[0.6,9.7199999999999989,11.329090909090908,12.883636363636363,14.383636363636361,15.829090909090908,17.22,18.556363636363635,19.838181818181816,21.065454545454543,22.238181818181815,23.356363636363632,24.42,25.429090909090903,26.383636363636363,27.283636363636361,28.129090909090905,28.919999999999995,29.656363636363633,30.338181818181816,30.965454545454541,31.538181818181815,32.056363636363642,32.519999999999996,32.929090909090903,33.283636363636354,33.583636363636359,33.829090909090908,34.019999999999996,34.156363636363636,34.238181818181815,34.265454545454546,34.238181818181815,34.156363636363629,34.019999999999996,33.829090909090908,33.583636363636359,33.283636363636354,32.929090909090903,32.519999999999989,32.056363636363642,31.538181818181815,30.965454545454548,30.338181818181823,29.656363636363626,28.919999999999987,28.129090909090905,27.283636363636369,26.383636363636359,25.429090909090895,24.419999999999987,23.356363636363632,22.238181818181815,21.065454545454546,19.838181818181813,18.556363636363621,17.22,15.829090909090919,14.38363636363637,12.883636363636352,11.329090909090905,9.7199999999999989]
[0.7,11.129999999999999,12.739090909090908,14.293636363636363,15.793636363636361,17.239090909090908,18.63,19.966363636363631,21.248181818181816,22.475454545454546,23.648181818181818,24.766363636363636,25.83,26.839090909090906,27.793636363636363,28.693636363636362,29.539090909090906,30.329999999999995,31.066363636363633,31.748181818181816,32.375454545454545,32.948181818181816,33.466363636363639,33.929999999999993,34.339090909090906,34.693636363636358,34.993636363636362,35.239090909090905,35.429999999999993,35.566363636363633,35.648181818181811,35.675454545454542,35.648181818181811,35.566363636363633,35.43,35.239090909090905,34.993636363636362,34.693636363636358,34.339090909090906,33.929999999999993,33.466363636363639,32.948181818181816,32.375454545454545,31.748181818181823,31.066363636363626,30.329999999999988,29.539090909090906,28.693636363636369,27.793636363636359,26.839090909090896,25.829999999999984,24.766363636363636,23.648181818181811,22.475454545454546,21.248181818181813,19.966363636363621,18.63,17.239090909090919,15.79363636363637,14.293636363636352,12.739090909090905,11.129999999999999]
[0.8,12.48,14.08909090909091,15.643636363636364,17.143636363636361,18.58909090909091,19.98,21.316363636363633,22.598181818181818,23.825454545454548,24.99818181818182,26.116363636363637,27.18,28.189090909090908,29.143636363636364,30.043636363636363,30.889090909090907,31.679999999999996,32.416363636363634,33.098181818181814,33.725454545454539,34.298181818181817,34.81636363636364,35.28,35.689090909090908,36.043636363636352,36.343636363636364,36.589090909090913,36.779999999999994,36.916363636363641,36.998181818181813,37.025454545454544,36.99818181818182,36.916363636363627,36.78,36.589090909090913,36.343636363636364,36.043636363636352,35.689090909090908,35.279999999999987,34.81636363636364,34.298181818181817,33.725454545454554,33.098181818181828,32.416363636363627,31.679999999999989,30.889090909090907,30.04363636363637,29.143636363636361,28.189090909090897,27.179999999999986,26.116363636363637,24.998181818181813,23.825454545454548,22.598181818181814,21.316363636363622,19.98,18.58909090909092,17.143636363636372,15.643636363636354,14.089090909090906,12.48]
[0.9,13.77,15.379090909090909,16.933636363636364,18.43363636363636,19.879090909090909,21.27,22.606363636363632,23.888181818181817,25.115454545454547,26.288181818181819,27.406363636363636,28.47,29.479090909090907,30.433636363636364,31.333636363636362,32.179090909090903,32.97,33.706363636363633,34.38818181818182,35.015454545454546,35.588181818181816,36.106363636363639,36.569999999999993,36.979090909090907,37.333636363636359,37.633636363636363,37.879090909090905,38.069999999999993,38.206363636363633,38.288181818181812,38.315454545454543,38.288181818181812,38.206363636363633,38.07,37.879090909090905,37.633636363636363,37.333636363636359,36.979090909090907,36.569999999999993,36.106363636363639,35.588181818181816,35.015454545454546,34.38818181818182,33.706363636363626,32.969999999999985,32.179090909090903,31.333636363636369,30.43363636363636,29.479090909090896,28.469999999999985,27.406363636363636,26.288181818181812,25.115454545454547,23.888181818181813,22.606363636363621,21.27,19.87909090909092,18.433636363636371,16.933636363636353,15.379090909090905,13.77]
[1,{"$f":"a0*$A64-_a1*$A64^2+(a0*B$53-_a1*B$53^2)/(1+dr)","v":15},16.609090909090909,18.163636363636364,19.663636363636364,21.109090909090909,22.5,23.836363636363636,25.118181818181817,26.345454545454544,27.518181818181816,{"$f":"a0*$A64-_a1*$A64^2+(a0*L$53-_a1*L$53^2)/(1+dr)","v":28.636363636363633},29.700000000000003,30.709090909090904,31.663636363636364,32.563636363636363,33.409090909090907,34.199999999999996,34.936363636363637,35.618181818181817,36.245454545454542,{"$f":"a0*$A64-_a1*$A64^2+(a0*V$53-_a1*V$53^2)/(1+dr)","v":36.818181818181813},37.336363636363643,37.799999999999997,38.209090909090904,38.563636363636355,38.86363636363636,39.109090909090909,39.299999999999997,39.436363636363637,39.518181818181816,{"$f":"a0*$A64-_a1*$A64^2+(a0*AF$53-_a1*AF$53^2)/(1+dr)","v":39.545454545454547},39.518181818181816,39.43636363636363,39.299999999999997,39.109090909090909,38.86363636363636,38.563636363636355,38.209090909090904,37.79999999999999,37.336363636363643,{"$f":"a0*$A64-_a1*$A64^2+(a0*AP$53-_a1*AP$53^2)/(1+dr)","v":36.818181818181813},36.24545454545455,35.618181818181824,34.936363636363623,34.199999999999989,33.409090909090907,32.56363636363637,31.66363636363636,30.709090909090897,29.699999999999989,{"$f":"a0*$A64-_a1*$A64^2+(a0*AZ$53-_a1*AZ$53^2)/(1+dr)","v":28.636363636363633},27.518181818181816,26.345454545454547,25.118181818181814,23.836363636363622,22.5,21.10909090909092,19.663636363636371,18.163636363636353,16.609090909090906,15]
[1.1000000000000001,16.170000000000002,17.779090909090911,19.333636363636366,20.833636363636366,22.279090909090911,23.67,25.006363636363638,26.288181818181819,27.515454545454546,28.688181818181818,29.806363636363635,30.870000000000005,31.879090909090905,32.833636363636366,33.733636363636364,34.579090909090908,35.369999999999997,36.106363636363639,36.788181818181819,37.415454545454544,37.988181818181815,38.506363636363645,38.97,39.379090909090905,39.733636363636357,40.033636363636361,40.279090909090911,40.47,40.606363636363639,40.688181818181818,40.715454545454548,40.688181818181818,40.606363636363632,40.47,40.279090909090911,40.033636363636361,39.733636363636357,39.379090909090905,38.969999999999992,38.506363636363645,37.988181818181815,37.415454545454551,36.788181818181826,36.106363636363625,35.36999999999999,34.579090909090908,33.733636363636371,32.833636363636359,31.879090909090898,30.86999999999999,29.806363636363635,28.688181818181818,27.515454545454549,26.288181818181815,25.006363636363623,23.67,22.279090909090922,20.833636363636373,19.333636363636355,17.779090909090908,16.170000000000002]
[1.2,17.279999999999998,18.889090909090907,20.443636363636362,21.943636363636358,23.389090909090907,24.779999999999998,26.11636363636363,27.398181818181815,28.625454545454545,29.798181818181817,30.916363636363634,31.979999999999997,32.989090909090905,33.943636363636358,34.843636363636364,35.689090909090908,36.47999999999999,37.216363636363631,37.898181818181811,38.525454545454537,39.098181818181814,39.616363636363637,40.08,40.489090909090905,40.84363636363635,41.143636363636361,41.38909090909091,41.579999999999991,41.716363636363639,41.79818181818181,41.825454545454541,41.798181818181817,41.716363636363624,41.58,41.38909090909091,41.143636363636361,40.84363636363635,40.489090909090905,40.079999999999984,39.616363636363637,39.098181818181814,38.525454545454551,37.898181818181826,37.216363636363624,36.47999999999999,35.689090909090908,34.843636363636364,33.943636363636358,32.989090909090891,31.979999999999983,30.916363636363634,29.79818181818181,28.625454545454545,27.398181818181811,26.116363636363619,24.779999999999998,23.389090909090918,21.943636363636369,20.443636363636351,18.889090909090903,17.279999999999998]
[1.3,18.330000000000002,19.939090909090911,21.493636363636366,22.993636363636362,24.439090909090911,25.830000000000002,27.166363636363634,28.448181818181819,29.675454545454549,30.848181818181821,31.966363636363639,33.03,34.039090909090909,34.993636363636369,35.893636363636361,36.739090909090905,37.53,38.266363636363636,38.948181818181823,39.575454545454548,40.148181818181818,40.666363636363641,41.129999999999995,41.539090909090909,41.893636363636361,42.193636363636365,42.439090909090908,42.629999999999995,42.766363636363636,42.848181818181814,42.875454545454545,42.848181818181814,42.766363636363636,42.63,42.439090909090908,42.193636363636365,41.893636363636361,41.539090909090909,41.129999999999995,40.666363636363641,40.148181818181818,39.575454545454548,38.948181818181823,38.266363636363629,37.529999999999987,36.739090909090905,35.893636363636375,34.993636363636362,34.039090909090902,33.029999999999987,31.966363636363639,30.848181818181814,29.675454545454549,28.448181818181816,27.166363636363624,25.830000000000002,24.439090909090922,22.993636363636373,21.493636363636355,19.939090909090908,18.330000000000002]
[1.4,19.32,20.92909090909091,22.483636363636364,23.983636363636364,25.42909090909091,26.82,28.156363636363636,29.438181818181818,30.665454545454544,31.838181818181816,32.956363636363633,34.020000000000003,35.029090909090904,35.983636363636364,36.883636363636363,37.729090909090907,38.519999999999996,39.256363636363631,39.938181818181818,40.565454545454543,41.13818181818182,41.656363636363636,42.12,42.529090909090911,42.883636363636356,43.183636363636367,43.42909090909091,43.61999999999999,43.756363636363638,43.838181818181809,43.86545454545454,43.838181818181816,43.756363636363631,43.620000000000005,43.42909090909091,43.183636363636367,42.883636363636356,42.529090909090911,42.11999999999999,41.656363636363636,41.13818181818182,40.56545454545455,39.938181818181825,39.256363636363631,38.519999999999989,37.729090909090907,36.88363636363637,35.983636363636364,35.029090909090897,34.019999999999989,32.956363636363633,31.838181818181816,30.665454545454548,29.438181818181814,28.156363636363622,26.82,25.42909090909092,23.983636363636371,22.483636363636354,20.929090909090906,19.32]
[1.5,20.25,21.859090909090909,23.413636363636364,24.913636363636364,26.359090909090909,27.75,29.086363636363636,30.368181818181817,31.595454545454544,32.768181818181816,33.886363636363633,34.950000000000003,35.959090909090904,36.913636363636364,37.813636363636363,38.659090909090907,39.449999999999996,40.186363636363637,40.868181818181817,41.495454545454542,42.068181818181813,42.586363636363643,43.05,43.459090909090904,43.813636363636355,44.11363636363636,44.359090909090909,44.55,44.686363636363637,44.768181818181816,44.795454545454547,44.768181818181816,44.68636363636363,44.55,44.359090909090909,44.11363636363636,43.813636363636355,43.459090909090904,43.04999999999999,42.586363636363643,42.068181818181813,41.49545454545455,40.868181818181824,40.186363636363623,39.449999999999989,38.659090909090907,37.81363636363637,36.913636363636357,35.959090909090897,34.949999999999989,33.886363636363633,32.768181818181816,31.595454545454547,30.368181818181814,29.086363636363622,27.75,26.35909090909092,24.913636363636371,23.413636363636353,21.859090909090906,20.25]
[1.6,21.119999999999997,22.729090909090907,24.283636363636361,25.783636363636361,27.229090909090907,28.619999999999997,29.956363636363633,31.238181818181815,32.465454545454541,33.638181818181813,34.756363636363631,35.82,36.829090909090901,37.783636363636361,38.68363636363636,39.529090909090904,40.319999999999993,41.056363636363628,41.738181818181815,42.36545454545454,42.938181818181818,43.456363636363633,43.919999999999995,44.329090909090908,44.683636363636353,44.983636363636364,45.229090909090907,45.419999999999987,45.556363636363635,45.638181818181806,45.665454545454537,45.638181818181813,45.556363636363628,45.42,45.229090909090907,44.983636363636364,44.683636363636353,44.329090909090908,43.919999999999987,43.456363636363633,42.938181818181818,42.365454545454547,41.738181818181822,41.056363636363628,40.319999999999986,39.529090909090904,38.683636363636367,37.783636363636361,36.829090909090894,35.819999999999986,34.756363636363631,33.638181818181813,32.465454545454548,31.238181818181811,29.956363636363619,28.619999999999997,27.229090909090917,25.783636363636369,24.283636363636351,22.729090909090903,21.119999999999997]
[1.7,21.93,23.539090909090909,25.093636363636364,26.593636363636364,28.039090909090909,29.43,30.766363636363636,32.048181818181817,33.275454545454544,34.448181818181816,35.566363636363633,36.630000000000003,37.639090909090903,38.593636363636364,39.493636363636362,40.339090909090906,41.129999999999995,41.86636363636363,42.548181818181817,43.175454545454542,43.74818181818182,44.266363636363636,44.73,45.13909090909091,45.493636363636355,45.793636363636367,46.039090909090909,46.22999999999999,46.366363636363637,46.448181818181808,46.475454545454539,46.448181818181816,46.36636363636363,46.230000000000004,46.039090909090909,45.793636363636367,45.493636363636355,45.13909090909091,44.72999999999999,44.266363636363636,43.74818181818182,43.175454545454549,42.548181818181824,41.86636363636363,41.129999999999988,40.339090909090906,39.493636363636369,38.593636363636364,37.639090909090896,36.629999999999988,35.566363636363633,34.448181818181816,33.275454545454551,32.048181818181817,30.766363636363621,29.43,28.03909090909092,26.593636363636371,25.093636363636353,23.539090909090906,21.93]
[1.8,22.68,24.289090909090909,25.843636363636364,27.343636363636364,28.789090909090909,30.18,31.516363636363636,32.798181818181817,34.025454545454544,35.198181818181816,36.316363636363633,37.380000000000003,38.389090909090903,39.343636363636364,40.243636363636362,41.089090909090906,41.879999999999995,42.61636363636363,43.298181818181817,43.925454545454542,44.49818181818182,45.016363636363636,45.48,45.88909090909091,46.243636363636355,46.543636363636367,46.789090909090909,46.97999999999999,47.116363636363637,47.198181818181808,47.225454545454539,47.198181818181816,47.11636363636363,46.980000000000004,46.789090909090909,46.543636363636367,46.243636363636355,45.88909090909091,45.47999999999999,45.016363636363636,44.49818181818182,43.925454545454549,43.298181818181824,42.61636363636363,41.879999999999988,41.089090909090906,40.243636363636369,39.343636363636364,38.389090909090896,37.379999999999988,36.316363636363633,35.198181818181816,34.025454545454551,32.798181818181817,31.516363636363621,30.18,28.78909090909092,27.343636363636371,25.843636363636353,24.289090909090906,22.68]
[1.9,23.369999999999997,24.979090909090907,26.533636363636361,28.033636363636361,29.479090909090907,30.869999999999997,32.206363636363633,33.488181818181815,34.715454545454541,35.888181818181813,37.006363636363631,38.07,39.079090909090901,40.033636363636361,40.93363636363636,41.779090909090904,42.569999999999993,43.306363636363628,43.988181818181815,44.61545454545454,45.188181818181818,45.706363636363633,46.169999999999995,46.579090909090908,46.933636363636353,47.233636363636364,47.479090909090907,47.669999999999987,47.806363636363635,47.888181818181806,47.915454545454537,47.888181818181813,47.806363636363628,47.67,47.479090909090907,47.233636363636364,46.933636363636353,46.579090909090908,46.169999999999987,45.706363636363633,45.188181818181818,44.615454545454547,43.988181818181822,43.306363636363628,42.569999999999986,41.779090909090904,40.933636363636367,40.033636363636361,39.079090909090894,38.069999999999986,37.006363636363631,35.888181818181813,34.715454545454548,33.488181818181815,32.206363636363619,30.869999999999997,29.479090909090917,28.033636363636369,26.533636363636351,24.979090909090903,23.369999999999997]
[2,{"$f":"a0*$A74-_a1*$A74^2+(a0*B$53-_a1*B$53^2)/(1+dr)","v":24},25.609090909090909,27.163636363636364,28.663636363636364,30.109090909090909,31.5,32.836363636363636,34.118181818181817,35.345454545454544,36.518181818181816,{"$f":"a0*$A74-_a1*$A74^2+(a0*L$53-_a1*L$53^2)/(1+dr)","v":37.636363636363633},38.700000000000003,39.709090909090904,40.663636363636364,41.563636363636363,42.409090909090907,43.199999999999996,43.936363636363637,44.618181818181817,45.245454545454542,{"$f":"a0*$A74-_a1*$A74^2+(a0*V$53-_a1*V$53^2)/(1+dr)","v":45.818181818181813},46.336363636363643,46.8,47.209090909090904,47.563636363636355,47.86363636363636,48.109090909090909,48.3,48.436363636363637,48.518181818181816,{"$f":"a0*$A74-_a1*$A74^2+(a0*AF$53-_a1*AF$53^2)/(1+dr)","v":48.545454545454547},48.518181818181816,48.43636363636363,48.3,48.109090909090909,47.86363636363636,47.563636363636355,47.209090909090904,46.79999999999999,46.336363636363643,{"$f":"a0*$A74-_a1*$A74^2+(a0*AP$53-_a1*AP$53^2)/(1+dr)","v":45.818181818181813},45.24545454545455,44.618181818181824,43.936363636363623,43.199999999999989,42.409090909090907,41.56363636363637,40.663636363636357,39.709090909090897,38.699999999999989,{"$f":"a0*$A74-_a1*$A74^2+(a0*AZ$53-_a1*AZ$53^2)/(1+dr)","v":37.636363636363633},36.518181818181816,35.345454545454544,34.11818181818181,32.836363636363622,31.5,30.10909090909092,28.663636363636371,27.163636363636353,25.609090909090906,24]
[2.1,24.570000000000004,26.179090909090913,27.733636363636368,29.233636363636364,30.679090909090913,32.07,33.406363636363636,34.688181818181818,35.915454545454551,37.088181818181823,38.206363636363641,39.270000000000003,40.279090909090911,41.233636363636364,42.13363636363637,42.979090909090914,43.769999999999996,44.506363636363638,45.188181818181818,45.815454545454543,46.38818181818182,46.906363636363643,47.370000000000005,47.779090909090911,48.133636363636356,48.433636363636367,48.679090909090917,48.87,49.006363636363645,49.088181818181816,49.115454545454547,49.088181818181823,49.006363636363631,48.870000000000005,48.679090909090917,48.433636363636367,48.133636363636356,47.779090909090911,47.36999999999999,46.906363636363643,46.38818181818182,45.815454545454557,45.188181818181832,44.506363636363631,43.769999999999996,42.979090909090914,42.13363636363637,41.233636363636364,40.279090909090897,39.269999999999989,38.206363636363641,37.088181818181816,35.915454545454551,34.688181818181818,33.406363636363622,32.07,30.679090909090924,29.233636363636375,27.733636363636357,26.17909090909091,24.570000000000004]
[2.2000000000000002,25.08,26.689090909090908,28.243636363636362,29.743636363636362,31.189090909090908,32.58,33.916363636363634,35.198181818181816,36.425454545454542,37.598181818181814,38.716363636363631,39.78,40.789090909090902,41.743636363636362,42.643636363636361,43.489090909090905,44.279999999999994,45.016363636363636,45.698181818181816,46.325454545454541,46.898181818181811,47.416363636363641,47.879999999999995,48.289090909090902,48.643636363636354,48.943636363636358,49.189090909090908,49.379999999999995,49.516363636363636,49.598181818181814,49.625454545454545,49.598181818181814,49.516363636363629,49.379999999999995,49.189090909090908,48.943636363636358,48.643636363636354,48.289090909090902,47.879999999999988,47.416363636363641,46.898181818181811,46.325454545454548,45.698181818181823,45.016363636363621,44.279999999999987,43.489090909090905,42.643636363636368,41.743636363636355,40.789090909090895,39.779999999999987,38.716363636363631,37.598181818181814,36.425454545454542,35.198181818181808,33.91636363636362,32.58,31.189090909090918,29.743636363636369,28.243636363636352,26.689090909090904,25.08]
[2.2999999999999998,25.53,27.13909090909091,28.693636363636365,30.193636363636365,31.63909090909091,33.03,34.366363636363637,35.648181818181818,36.875454545454545,38.048181818181817,39.166363636363634,40.230000000000004,41.239090909090905,42.193636363636365,43.093636363636364,43.939090909090908,44.73,45.466363636363639,46.148181818181818,46.775454545454544,47.348181818181814,47.866363636363644,48.33,48.739090909090905,49.093636363636357,49.393636363636361,49.63909090909091,49.83,49.966363636363639,50.048181818181817,50.075454545454548,50.048181818181817,49.966363636363631,49.83,49.63909090909091,49.393636363636361,49.093636363636357,48.739090909090905,48.329999999999991,47.866363636363644,47.348181818181814,46.775454545454551,46.148181818181826,45.466363636363624,44.72999999999999,43.939090909090908,43.093636363636371,42.193636363636358,41.239090909090898,40.22999999999999,39.166363636363634,38.048181818181817,36.875454545454545,35.648181818181811,34.366363636363623,33.03,31.639090909090921,30.193636363636372,28.693636363636355,27.139090909090907,25.53]
[2.4,25.919999999999995,27.529090909090904,29.083636363636359,30.583636363636359,32.029090909090904,33.419999999999995,34.756363636363631,36.038181818181812,37.265454545454539,38.43818181818181,39.556363636363628,40.619999999999997,41.629090909090898,42.583636363636359,43.483636363636357,44.329090909090901,45.11999999999999,45.856363636363625,46.538181818181812,47.165454545454537,47.738181818181815,48.256363636363631,48.719999999999992,49.129090909090905,49.48363636363635,49.783636363636361,50.029090909090904,50.219999999999985,50.356363636363632,50.438181818181803,50.465454545454534,50.43818181818181,50.356363636363625,50.22,50.029090909090904,49.783636363636361,49.48363636363635,49.129090909090905,48.719999999999985,48.256363636363631,47.738181818181815,47.165454545454544,46.538181818181819,45.856363636363625,45.119999999999983,44.329090909090901,43.483636363636364,42.583636363636359,41.629090909090891,40.619999999999983,39.556363636363628,38.43818181818181,37.265454545454546,36.038181818181812,34.756363636363616,33.419999999999995,32.029090909090918,30.583636363636366,29.083636363636348,27.5290909090909,25.919999999999995]
[2.5,26.25,27.859090909090909,29.413636363636364,30.913636363636364,32.359090909090909,33.75,35.086363636363636,36.368181818181817,37.595454545454544,38.768181818181816,39.886363636363633,40.950000000000003,41.959090909090904,42.913636363636364,43.813636363636363,44.659090909090907,45.449999999999996,46.186363636363637,46.868181818181817,47.495454545454542,48.068181818181813,48.586363636363643,49.05,49.459090909090904,49.813636363636355,50.11363636363636,50.359090909090909,50.55,50.686363636363637,50.768181818181816,50.795454545454547,50.768181818181816,50.68636363636363,50.55,50.359090909090909,50.11363636363636,49.813636363636355,49.459090909090904,49.04999999999999,48.586363636363643,48.068181818181813,47.49545454545455,46.868181818181824,46.186363636363623,45.449999999999989,44.659090909090907,43.81363636363637,42.913636363636357,41.959090909090897,40.949999999999989,39.886363636363633,38.768181818181816,37.595454545454544,36.36818181818181,35.086363636363622,33.75,32.359090909090924,30.913636363636371,29.413636363636353,27.859090909090906,26.25]
[2.6,26.520000000000003,28.129090909090912,29.683636363636367,31.183636363636367,32.629090909090912,34.020000000000003,35.356363636363639,36.63818181818182,37.865454545454547,39.038181818181819,40.156363636363636,41.220000000000006,42.229090909090907,43.183636363636367,44.083636363636366,44.92909090909091,45.72,46.456363636363633,47.13818181818182,47.765454545454546,48.338181818181823,48.856363636363639,49.32,49.729090909090914,50.083636363636359,50.38363636363637,50.629090909090912,50.819999999999993,50.956363636363641,51.038181818181812,51.065454545454543,51.038181818181819,50.956363636363633,50.820000000000007,50.629090909090912,50.38363636363637,50.083636363636359,49.729090909090914,49.319999999999993,48.856363636363639,48.338181818181823,47.765454545454553,47.138181818181828,46.456363636363633,45.719999999999992,44.92909090909091,44.083636363636373,43.183636363636367,42.2290909090909,41.219999999999992,40.156363636363636,39.038181818181819,37.865454545454554,36.63818181818182,35.356363636363625,34.020000000000003,32.629090909090927,31.183636363636374,29.683636363636356,28.129090909090909,26.520000000000003]
[2.7,26.729999999999997,28.339090909090906,29.893636363636361,31.393636363636361,32.839090909090906,34.229999999999997,35.566363636363633,36.848181818181814,38.075454545454541,39.248181818181813,40.36636363636363,41.43,42.439090909090901,43.393636363636361,44.293636363636359,45.139090909090903,45.929999999999993,46.666363636363627,47.348181818181814,47.975454545454539,48.548181818181817,49.066363636363633,49.529999999999994,49.939090909090908,50.293636363636352,50.593636363636364,50.839090909090906,51.029999999999987,51.166363636363634,51.248181818181806,51.275454545454537,51.248181818181813,51.166363636363627,51.03,50.839090909090906,50.593636363636364,50.293636363636352,49.939090909090908,49.529999999999987,49.066363636363633,48.548181818181817,47.975454545454546,47.348181818181821,46.666363636363627,45.929999999999986,45.139090909090903,44.293636363636367,43.393636363636361,42.439090909090893,41.429999999999986,40.36636363636363,39.248181818181813,38.075454545454548,36.848181818181814,35.566363636363619,34.229999999999997,32.83909090909092,31.393636363636368,29.89363636363635,28.339090909090903,26.729999999999997]
[2.8,26.880000000000003,28.489090909090912,30.043636363636367,31.543636363636367,32.989090909090912,34.380000000000003,35.716363636363639,36.99818181818182,38.225454545454546,39.398181818181818,40.516363636363636,41.580000000000005,42.589090909090906,43.543636363636367,44.443636363636365,45.289090909090909,46.08,46.816363636363633,47.49818181818182,48.125454545454545,48.698181818181823,49.216363636363639,49.68,50.089090909090913,50.443636363636358,50.743636363636369,50.989090909090912,51.179999999999993,51.31636363636364,51.398181818181811,51.425454545454542,51.398181818181818,51.316363636363633,51.180000000000007,50.989090909090912,50.743636363636369,50.443636363636358,50.089090909090913,49.679999999999993,49.216363636363639,48.698181818181823,48.125454545454552,47.498181818181827,46.816363636363633,46.079999999999991,45.289090909090909,44.443636363636372,43.543636363636367,42.589090909090899,41.579999999999991,40.516363636363636,39.398181818181818,38.225454545454554,36.99818181818182,35.716363636363624,34.380000000000003,32.989090909090926,31.543636363636374,30.043636363636356,28.489090909090908,26.880000000000003]
[2.9,26.969999999999995,28.579090909090905,30.133636363636359,31.633636363636356,33.079090909090901,34.469999999999992,35.806363636363628,37.088181818181809,38.315454545454543,39.488181818181815,40.606363636363632,41.669999999999995,42.679090909090903,43.633636363636356,44.533636363636361,45.379090909090905,46.169999999999987,46.906363636363629,47.588181818181809,48.215454545454534,48.788181818181812,49.306363636363635,49.769999999999996,50.179090909090903,50.533636363636347,50.833636363636359,51.079090909090908,51.269999999999989,51.406363636363636,51.488181818181808,51.515454545454539,51.488181818181815,51.406363636363622,51.269999999999996,51.079090909090908,50.833636363636359,50.533636363636347,50.179090909090903,49.769999999999982,49.306363636363635,48.788181818181812,48.215454545454548,47.588181818181823,46.906363636363622,46.169999999999987,45.379090909090905,44.533636363636361,43.633636363636356,42.679090909090888,41.66999999999998,40.606363636363632,39.488181818181808,38.315454545454543,37.088181818181809,35.806363636363614,34.469999999999992,33.079090909090915,31.633636363636366,30.133636363636349,28.579090909090901,26.969999999999995]
[3,{"$f":"a0*$A84-_a1*$A84^2+(a0*B$53-_a1*B$53^2)/(1+dr)","v":27},28.609090909090909,30.163636363636364,31.663636363636364,33.109090909090909,34.5,35.836363636363636,37.118181818181817,38.345454545454544,39.518181818181816,{"$f":"a0*$A84-_a1*$A84^2+(a0*L$53-_a1*L$53^2)/(1+dr)","v":40.636363636363633},41.7,42.709090909090904,43.663636363636364,44.563636363636363,45.409090909090907,46.199999999999996,46.936363636363637,47.618181818181817,48.245454545454542,{"$f":"a0*$A84-_a1*$A84^2+(a0*V$53-_a1*V$53^2)/(1+dr)","v":48.818181818181813},49.336363636363643,49.8,50.209090909090904,50.563636363636355,50.86363636363636,51.109090909090909,51.3,51.436363636363637,51.518181818181816,{"$f":"a0*$A84-_a1*$A84^2+(a0*AF$53-_a1*AF$53^2)/(1+dr)","v":51.545454545454547},51.518181818181816,51.43636363636363,51.3,51.109090909090909,50.86363636363636,50.563636363636355,50.209090909090904,49.79999999999999,49.336363636363643,{"$f":"a0*$A84-_a1*$A84^2+(a0*AP$53-_a1*AP$53^2)/(1+dr)","v":48.818181818181813},48.24545454545455,47.618181818181824,46.936363636363623,46.199999999999989,45.409090909090907,44.56363636363637,43.663636363636357,42.709090909090897,41.699999999999989,{"$f":"a0*$A84-_a1*$A84^2+(a0*AZ$53-_a1*AZ$53^2)/(1+dr)","v":40.636363636363633},39.518181818181816,38.345454545454544,37.11818181818181,35.836363636363622,34.5,33.109090909090924,31.663636363636371,30.163636363636353,28.609090909090906,27]
[3.1,26.97,28.579090909090908,30.133636363636363,31.633636363636363,33.079090909090908,34.47,35.806363636363635,37.088181818181816,38.315454545454543,39.488181818181815,40.606363636363632,41.67,42.679090909090903,43.633636363636363,44.533636363636361,45.379090909090905,46.169999999999995,46.906363636363636,47.588181818181816,48.215454545454541,48.788181818181812,49.306363636363642,49.769999999999996,50.179090909090903,50.533636363636354,50.833636363636359,51.079090909090908,51.269999999999996,51.406363636363636,51.488181818181815,51.515454545454546,51.488181818181815,51.406363636363629,51.269999999999996,51.079090909090908,50.833636363636359,50.533636363636354,50.179090909090903,49.769999999999989,49.306363636363642,48.788181818181812,48.215454545454548,47.588181818181823,46.906363636363622,46.169999999999987,45.379090909090905,44.533636363636369,43.633636363636356,42.679090909090895,41.669999999999987,40.606363636363632,39.488181818181815,38.315454545454543,37.088181818181809,35.806363636363621,34.47,33.079090909090922,31.63363636363637,30.133636363636352,28.579090909090905,26.97]
[3.2,26.879999999999995,28.489090909090905,30.043636363636359,31.543636363636359,32.989090909090905,34.379999999999995,35.716363636363631,36.998181818181813,38.225454545454539,39.398181818181811,40.516363636363629,41.58,42.589090909090899,43.543636363636359,44.443636363636358,45.289090909090902,46.079999999999991,46.816363636363633,47.498181818181813,48.125454545454538,48.698181818181808,49.216363636363639,49.679999999999993,50.089090909090899,50.443636363636351,50.743636363636355,50.989090909090905,51.179999999999993,51.316363636363633,51.398181818181811,51.425454545454542,51.398181818181811,51.316363636363626,51.179999999999993,50.989090909090905,50.743636363636355,50.443636363636351,50.089090909090899,49.679999999999986,49.216363636363639,48.698181818181808,48.125454545454545,47.49818181818182,46.816363636363619,46.079999999999984,45.289090909090902,44.443636363636365,43.543636363636352,42.589090909090892,41.579999999999984,40.516363636363629,39.398181818181811,38.225454545454539,36.998181818181806,35.716363636363617,34.379999999999995,32.989090909090919,31.543636363636367,30.043636363636349,28.489090909090901,26.879999999999995]
[3.3,26.730000000000004,28.339090909090913,29.893636363636368,31.393636363636368,32.839090909090913,34.230000000000004,35.56636363636364,36.848181818181821,38.075454545454548,39.24818181818182,40.366363636363637,41.430000000000007,42.439090909090908,43.393636363636368,44.293636363636367,45.13909090909091,45.93,46.666363636363641,47.348181818181821,47.975454545454546,48.548181818181817,49.066363636363647,49.53,49.939090909090908,50.293636363636359,50.593636363636364,50.839090909090913,51.03,51.166363636363641,51.24818181818182,51.275454545454551,51.24818181818182,51.166363636363634,51.03,50.839090909090913,50.593636363636364,50.293636363636359,49.939090909090908,49.529999999999994,49.066363636363647,48.548181818181817,47.975454545454554,47.348181818181828,46.666363636363627,45.929999999999993,45.13909090909091,44.293636363636374,43.393636363636361,42.439090909090901,41.429999999999993,40.366363636363637,39.24818181818182,38.075454545454548,36.848181818181814,35.566363636363626,34.230000000000004,32.839090909090928,31.393636363636375,29.893636363636357,28.33909090909091,26.730000000000004]
[3.4,26.520000000000003,28.129090909090912,29.683636363636367,31.183636363636367,32.629090909090912,34.020000000000003,35.356363636363639,36.63818181818182,37.865454545454547,39.038181818181819,40.156363636363636,41.220000000000006,42.229090909090907,43.183636363636367,44.083636363636366,44.92909090909091,45.72,46.456363636363633,47.13818181818182,47.765454545454546,48.338181818181823,48.856363636363639,49.32,49.729090909090914,50.083636363636359,50.38363636363637,50.629090909090912,50.819999999999993,50.956363636363641,51.038181818181812,51.065454545454543,51.038181818181819,50.956363636363633,50.820000000000007,50.629090909090912,50.38363636363637,50.083636363636359,49.729090909090914,49.319999999999993,48.856363636363639,48.338181818181823,47.765454545454553,47.138181818181828,46.456363636363633,45.719999999999992,44.92909090909091,44.083636363636373,43.183636363636367,42.2290909090909,41.219999999999992,40.156363636363636,39.038181818181819,37.865454545454554,36.63818181818182,35.356363636363625,34.020000000000003,32.629090909090927,31.183636363636374,29.683636363636356,28.129090909090909,26.520000000000003]
[3.5,26.25,27.859090909090909,29.413636363636364,30.913636363636364,32.359090909090909,33.75,35.086363636363636,36.368181818181817,37.595454545454544,38.768181818181816,39.886363636363633,40.950000000000003,41.959090909090904,42.913636363636364,43.813636363636363,44.659090909090907,45.449999999999996,46.186363636363637,46.868181818181817,47.495454545454542,48.068181818181813,48.586363636363643,49.05,49.459090909090904,49.813636363636355,50.11363636363636,50.359090909090909,50.55,50.686363636363637,50.768181818181816,50.795454545454547,50.768181818181816,50.68636363636363,50.55,50.359090909090909,50.11363636363636,49.813636363636355,49.459090909090904,49.04999999999999,48.586363636363643,48.068181818181813,47.49545454545455,46.868181818181824,46.186363636363623,45.449999999999989,44.659090909090907,43.81363636363637,42.913636363636357,41.959090909090897,40.949999999999989,39.886363636363633,38.768181818181816,37.595454545454544,36.36818181818181,35.086363636363622,33.75,32.359090909090924,30.913636363636371,29.413636363636353,27.859090909090906,26.25]
[3.6,25.919999999999995,27.529090909090904,29.083636363636359,30.583636363636359,32.029090909090904,33.419999999999995,34.756363636363631,36.038181818181812,37.265454545454539,38.43818181818181,39.556363636363628,40.619999999999997,41.629090909090898,42.583636363636359,43.483636363636357,44.329090909090901,45.11999999999999,45.856363636363625,46.538181818181812,47.165454545454537,47.738181818181815,48.256363636363631,48.719999999999992,49.129090909090905,49.48363636363635,49.783636363636361,50.029090909090904,50.219999999999985,50.356363636363632,50.438181818181803,50.465454545454534,50.43818181818181,50.356363636363625,50.22,50.029090909090904,49.783636363636361,49.48363636363635,49.129090909090905,48.719999999999985,48.256363636363631,47.738181818181815,47.165454545454544,46.538181818181819,45.856363636363625,45.119999999999983,44.329090909090901,43.483636363636364,42.583636363636359,41.629090909090891,40.619999999999983,39.556363636363628,38.43818181818181,37.265454545454546,36.038181818181812,34.756363636363616,33.419999999999995,32.029090909090918,30.583636363636366,29.083636363636348,27.5290909090909,25.919999999999995]
[3.7,25.53,27.13909090909091,28.693636363636365,30.193636363636365,31.63909090909091,33.03,34.366363636363637,35.648181818181818,36.875454545454545,38.048181818181817,39.166363636363634,40.230000000000004,41.239090909090905,42.193636363636365,43.093636363636364,43.939090909090908,44.73,45.466363636363639,46.148181818181818,46.775454545454544,47.348181818181814,47.866363636363644,48.33,48.739090909090905,49.093636363636357,49.393636363636361,49.63909090909091,49.83,49.966363636363639,50.048181818181817,50.075454545454548,50.048181818181817,49.966363636363631,49.83,49.63909090909091,49.393636363636361,49.093636363636357,48.739090909090905,48.329999999999991,47.866363636363644,47.348181818181814,46.775454545454551,46.148181818181826,45.466363636363624,44.72999999999999,43.939090909090908,43.093636363636371,42.193636363636358,41.239090909090898,40.22999999999999,39.166363636363634,38.048181818181817,36.875454545454545,35.648181818181811,34.366363636363623,33.03,31.639090909090921,30.193636363636372,28.693636363636355,27.139090909090907,25.53]
[3.8,25.079999999999991,26.689090909090901,28.243636363636355,29.743636363636355,31.189090909090901,32.579999999999991,33.916363636363627,35.198181818181808,36.425454545454535,37.598181818181807,38.716363636363624,39.779999999999994,40.789090909090895,41.743636363636355,42.643636363636354,43.489090909090898,44.279999999999987,45.016363636363621,45.698181818181808,46.325454545454534,46.898181818181811,47.416363636363627,47.879999999999988,48.289090909090902,48.643636363636347,48.943636363636358,49.189090909090901,49.379999999999981,49.516363636363629,49.5981818181818,49.625454545454531,49.598181818181807,49.516363636363621,49.379999999999995,49.189090909090901,48.943636363636358,48.643636363636347,48.289090909090902,47.879999999999981,47.416363636363627,46.898181818181811,46.325454545454541,45.698181818181816,45.016363636363621,44.27999999999998,43.489090909090898,42.643636363636361,41.743636363636355,40.789090909090888,39.77999999999998,38.716363636363624,37.598181818181807,36.425454545454542,35.198181818181808,33.916363636363613,32.579999999999991,31.189090909090911,29.743636363636362,28.243636363636345,26.689090909090897,25.079999999999991]
[3.9,24.570000000000007,26.179090909090917,27.733636363636371,29.233636363636371,30.679090909090917,32.070000000000007,33.406363636363643,34.688181818181825,35.915454545454551,37.088181818181823,38.206363636363641,39.27000000000001,40.279090909090911,41.233636363636371,42.13363636363637,42.979090909090914,43.77,44.506363636363645,45.188181818181825,45.81545454545455,46.38818181818182,46.90636363636365,47.370000000000005,47.779090909090911,48.133636363636363,48.433636363636367,48.679090909090917,48.870000000000005,49.006363636363645,49.088181818181823,49.115454545454554,49.088181818181823,49.006363636363638,48.870000000000005,48.679090909090917,48.433636363636367,48.133636363636363,47.779090909090911,47.37,46.90636363636365,46.38818181818182,45.815454545454557,45.188181818181832,44.506363636363631,43.769999999999996,42.979090909090914,42.133636363636377,41.233636363636364,40.279090909090904,39.269999999999996,38.206363636363641,37.088181818181823,35.915454545454551,34.688181818181818,33.406363636363629,32.070000000000007,30.679090909090927,29.233636363636379,27.733636363636361,26.179090909090913,24.570000000000007]
[4,{"$f":"a0*$A94-_a1*$A94^2+(a0*B$53-_a1*B$53^2)/(1+dr)","v":24},25.609090909090909,27.163636363636364,28.663636363636364,30.109090909090909,31.5,32.836363636363636,34.118181818181817,35.345454545454544,36.518181818181816,{"$f":"a0*$A94-_a1*$A94^2+(a0*L$53-_a1*L$53^2)/(1+dr)","v":37.636363636363633},38.700000000000003,39.709090909090904,40.663636363636364,41.563636363636363,42.409090909090907,43.199999999999996,43.936363636363637,44.618181818181817,45.245454545454542,{"$f":"a0*$A94-_a1*$A94^2+(a0*V$53-_a1*V$53^2)/(1+dr)","v":45.818181818181813},46.336363636363643,46.8,47.209090909090904,47.563636363636355,47.86363636363636,48.109090909090909,48.3,48.436363636363637,48.518181818181816,{"$f":"a0*$A94-_a1*$A94^2+(a0*AF$53-_a1*AF$53^2)/(1+dr)","v":48.545454545454547},48.518181818181816,48.43636363636363,48.3,48.109090909090909,47.86363636363636,47.563636363636355,47.209090909090904,46.79999999999999,46.336363636363643,{"$f":"a0*$A94-_a1*$A94^2+(a0*AP$53-_a1*AP$53^2)/(1+dr)","v":45.818181818181813},45.24545454545455,44.618181818181824,43.936363636363623,43.199999999999989,42.409090909090907,41.56363636363637,40.663636363636357,39.709090909090897,38.699999999999989,{"$f":"a0*$A94-_a1*$A94^2+(a0*AZ$53-_a1*AZ$53^2)/(1+dr)","v":37.636363636363633},36.518181818181816,35.345454545454544,34.11818181818181,32.836363636363622,31.5,30.10909090909092,28.663636363636371,27.163636363636353,25.609090909090906,24]
[4.0999999999999996,23.370000000000005,24.979090909090914,26.533636363636369,28.033636363636369,29.479090909090914,30.870000000000005,32.206363636363641,33.488181818181822,34.715454545454548,35.88818181818182,37.006363636363638,38.070000000000007,39.079090909090908,40.033636363636369,40.933636363636367,41.779090909090911,42.57,43.306363636363642,43.988181818181822,44.615454545454547,45.188181818181818,45.706363636363648,46.17,46.579090909090908,46.93363636363636,47.233636363636364,47.479090909090914,47.67,47.806363636363642,47.88818181818182,47.915454545454551,47.88818181818182,47.806363636363635,47.67,47.479090909090914,47.233636363636364,46.93363636363636,46.579090909090908,46.169999999999995,45.706363636363648,45.188181818181818,44.615454545454554,43.988181818181829,43.306363636363628,42.569999999999993,41.779090909090911,40.933636363636374,40.033636363636361,39.079090909090901,38.069999999999993,37.006363636363638,35.88818181818182,34.715454545454548,33.488181818181815,32.206363636363626,30.870000000000005,29.479090909090925,28.033636363636376,26.533636363636358,24.97909090909091,23.370000000000005]
[4.2,22.680000000000007,24.289090909090916,25.843636363636371,27.343636363636371,28.789090909090916,30.180000000000007,31.516363636363643,32.798181818181824,34.025454545454551,35.198181818181823,36.31636363636364,37.38000000000001,38.38909090909091,39.343636363636371,40.243636363636369,41.089090909090913,41.88,42.616363636363644,43.298181818181824,43.925454545454549,44.49818181818182,45.01636363636365,45.480000000000004,45.88909090909091,46.243636363636362,46.543636363636367,46.789090909090916,46.980000000000004,47.116363636363644,47.198181818181823,47.225454545454554,47.198181818181823,47.116363636363637,46.980000000000004,46.789090909090916,46.543636363636367,46.243636363636362,45.88909090909091,45.48,45.01636363636365,44.49818181818182,43.925454545454556,43.298181818181831,42.61636363636363,41.879999999999995,41.089090909090913,40.243636363636377,39.343636363636364,38.389090909090903,37.379999999999995,36.31636363636364,35.198181818181823,34.025454545454551,32.798181818181817,31.516363636363629,30.180000000000007,28.789090909090927,27.343636363636378,25.84363636363636,24.289090909090913,22.680000000000007]
[4.3,21.929999999999993,23.539090909090902,25.093636363636357,26.593636363636357,28.039090909090902,29.429999999999993,30.766363636363629,32.04818181818181,33.275454545454537,34.448181818181808,35.566363636363626,36.629999999999995,37.639090909090896,38.593636363636357,39.493636363636355,40.339090909090899,41.129999999999988,41.86636363636363,42.54818181818181,43.175454545454535,43.748181818181806,44.266363636363636,44.72999999999999,45.139090909090896,45.493636363636348,45.793636363636352,46.039090909090902,46.22999999999999,46.36636363636363,46.448181818181808,46.475454545454539,46.448181818181808,46.366363636363623,46.22999999999999,46.039090909090902,45.793636363636352,45.493636363636348,45.139090909090896,44.729999999999983,44.266363636363636,43.748181818181806,43.175454545454542,42.548181818181817,41.866363636363616,41.129999999999981,40.339090909090899,39.493636363636362,38.59363636363635,37.639090909090889,36.629999999999981,35.566363636363626,34.448181818181808,33.275454545454537,32.048181818181803,30.766363636363614,29.429999999999993,28.039090909090913,26.593636363636364,25.093636363636346,23.539090909090898,21.929999999999993]
[4.4000000000000004,21.11999999999999,22.7290909090909,24.283636363636354,25.783636363636354,27.2290909090909,28.61999999999999,29.956363636363626,31.238181818181808,32.465454545454534,33.638181818181806,34.756363636363623,35.819999999999993,36.829090909090894,37.783636363636354,38.683636363636353,39.529090909090897,40.319999999999986,41.056363636363628,41.738181818181808,42.365454545454533,42.938181818181803,43.456363636363633,43.919999999999987,44.329090909090894,44.683636363636346,44.98363636363635,45.2290909090909,45.419999999999987,45.556363636363628,45.638181818181806,45.665454545454537,45.638181818181806,45.556363636363621,45.419999999999987,45.2290909090909,44.98363636363635,44.683636363636346,44.329090909090894,43.91999999999998,43.456363636363633,42.938181818181803,42.36545454545454,41.738181818181815,41.056363636363614,40.319999999999979,39.529090909090897,38.68363636363636,37.783636363636347,36.829090909090887,35.819999999999979,34.756363636363623,33.638181818181806,32.465454545454534,31.238181818181804,29.956363636363612,28.61999999999999,27.22909090909091,25.783636363636361,24.283636363636344,22.729090909090896,21.11999999999999]
[4.5,20.25,21.859090909090909,23.413636363636364,24.913636363636364,26.359090909090909,27.75,29.086363636363636,30.368181818181817,31.595454545454544,32.768181818181816,33.886363636363633,34.950000000000003,35.959090909090904,36.913636363636364,37.813636363636363,38.659090909090907,39.449999999999996,40.186363636363637,40.868181818181817,41.495454545454542,42.068181818181813,42.586363636363643,43.05,43.459090909090904,43.813636363636355,44.11363636363636,44.359090909090909,44.55,44.686363636363637,44.768181818181816,44.795454545454547,44.768181818181816,44.68636363636363,44.55,44.359090909090909,44.11363636363636,43.813636363636355,43.459090909090904,43.04999999999999,42.586363636363643,42.068181818181813,41.49545454545455,40.868181818181824,40.186363636363623,39.449999999999989,38.659090909090907,37.81363636363637,36.913636363636357,35.959090909090897,34.949999999999989,33.886363636363633,32.768181818181816,31.595454545454547,30.368181818181814,29.086363636363622,27.75,26.35909090909092,24.913636363636371,23.413636363636353,21.859090909090906,20.25]
[4.5999999999999996,19.320000000000007,20.929090909090917,22.483636363636371,23.983636363636371,25.429090909090917,26.820000000000007,28.156363636363643,29.438181818181825,30.665454545454551,31.838181818181823,32.956363636363641,34.02000000000001,35.029090909090911,35.983636363636371,36.88363636363637,37.729090909090914,38.520000000000003,39.256363636363645,39.938181818181825,40.56545454545455,41.13818181818182,41.65636363636365,42.120000000000005,42.529090909090911,42.883636363636363,43.183636363636367,43.429090909090917,43.620000000000005,43.756363636363645,43.838181818181823,43.865454545454554,43.838181818181823,43.756363636363638,43.620000000000005,43.429090909090917,43.183636363636367,42.883636363636363,42.529090909090911,42.12,41.65636363636365,41.13818181818182,40.565454545454557,39.938181818181832,39.256363636363631,38.519999999999996,37.729090909090914,36.883636363636377,35.983636363636364,35.029090909090904,34.019999999999996,32.956363636363641,31.838181818181823,30.665454545454555,29.438181818181821,28.156363636363629,26.820000000000007,25.429090909090927,23.983636363636379,22.483636363636361,20.929090909090913,19.320000000000007]
[4.7,18.329999999999998,19.939090909090908,21.493636363636362,22.993636363636362,24.439090909090908,25.83,27.166363636363634,28.448181818181816,29.675454545454542,30.848181818181814,31.966363636363631,33.03,34.039090909090902,34.993636363636362,35.893636363636361,36.739090909090905,37.529999999999994,38.266363636363636,38.948181818181816,39.575454545454541,40.148181818181811,40.666363636363641,41.129999999999995,41.539090909090902,41.893636363636354,42.193636363636358,42.439090909090908,42.629999999999995,42.766363636363636,42.848181818181814,42.875454545454545,42.848181818181814,42.766363636363629,42.629999999999995,42.439090909090908,42.193636363636358,41.893636363636354,41.539090909090902,41.129999999999988,40.666363636363641,40.148181818181811,39.575454545454548,38.948181818181823,38.266363636363621,37.529999999999987,36.739090909090905,35.893636363636368,34.993636363636355,34.039090909090895,33.029999999999987,31.966363636363631,30.848181818181814,29.675454545454546,28.448181818181812,27.16636363636362,25.83,24.439090909090918,22.993636363636369,21.493636363636352,19.939090909090904,18.329999999999998]
[4.8,17.279999999999987,18.889090909090896,20.443636363636351,21.943636363636351,23.389090909090896,24.779999999999987,26.116363636363623,27.398181818181804,28.625454545454531,29.798181818181803,30.91636363636362,31.97999999999999,32.989090909090891,33.943636363636351,34.84363636363635,35.689090909090893,36.479999999999983,37.216363636363624,37.898181818181804,38.525454545454529,39.0981818181818,39.61636363636363,40.079999999999984,40.489090909090891,40.843636363636342,41.143636363636347,41.389090909090896,41.579999999999984,41.716363636363624,41.798181818181803,41.825454545454534,41.798181818181803,41.716363636363617,41.579999999999984,41.389090909090896,41.143636363636347,40.843636363636342,40.489090909090891,40.079999999999977,39.61636363636363,39.0981818181818,38.525454545454537,37.898181818181811,37.21636363636361,36.479999999999976,35.689090909090893,34.843636363636357,33.943636363636344,32.989090909090883,31.979999999999976,30.91636363636362,29.798181818181803,28.625454545454534,27.398181818181801,26.116363636363609,24.779999999999987,23.389090909090907,21.943636363636358,20.44363636363634,18.889090909090893,17.279999999999987]
[4.9000000000000004,16.169999999999987,17.779090909090897,19.333636363636352,20.833636363636352,22.279090909090897,23.669999999999987,25.006363636363623,26.288181818181805,27.515454545454531,28.688181818181803,29.806363636363621,30.86999999999999,31.879090909090891,32.833636363636352,33.73363636363635,34.579090909090894,35.369999999999983,36.106363636363625,36.788181818181805,37.41545454545453,37.988181818181801,38.506363636363631,38.969999999999985,39.379090909090891,39.733636363636343,40.033636363636347,40.279090909090897,40.469999999999985,40.606363636363625,40.688181818181803,40.715454545454534,40.688181818181803,40.606363636363618,40.469999999999985,40.279090909090897,40.033636363636347,39.733636363636343,39.379090909090891,38.969999999999978,38.506363636363631,37.988181818181801,37.415454545454537,36.788181818181812,36.106363636363611,35.369999999999976,34.579090909090894,33.733636363636357,32.833636363636344,31.879090909090884,30.869999999999976,29.806363636363621,28.688181818181803,27.515454545454535,26.288181818181801,25.006363636363609,23.669999999999987,22.279090909090908,20.833636363636359,19.333636363636341,17.779090909090893,16.169999999999987]
[5,{"$f":"a0*$A104-_a1*$A104^2+(a0*B$53-_a1*B$53^2)/(1+dr)","v":15},16.609090909090909,18.163636363636364,19.663636363636364,21.109090909090909,22.5,23.836363636363636,25.118181818181817,26.345454545454544,27.518181818181816,{"$f":"a0*$A104-_a1*$A104^2+(a0*L$53-_a1*L$53^2)/(1+dr)","v":28.636363636363633},29.700000000000003,30.709090909090904,31.663636363636364,32.563636363636363,33.409090909090907,34.199999999999996,34.936363636363637,35.618181818181817,36.245454545454542,{"$f":"a0*$A104-_a1*$A104^2+(a0*V$53-_a1*V$53^2)/(1+dr)","v":36.818181818181813},37.336363636363643,37.799999999999997,38.209090909090904,38.563636363636355,38.86363636363636,39.109090909090909,39.299999999999997,39.436363636363637,39.518181818181816,{"$f":"a0*$A104-_a1*$A104^2+(a0*AF$53-_a1*AF$53^2)/(1+dr)","v":39.545454545454547},39.518181818181816,39.43636363636363,39.299999999999997,39.109090909090909,38.86363636363636,38.563636363636355,38.209090909090904,37.79999999999999,37.336363636363643,{"$f":"a0*$A104-_a1*$A104^2+(a0*AP$53-_a1*AP$53^2)/(1+dr)","v":36.818181818181813},36.24545454545455,35.618181818181824,34.936363636363623,34.199999999999989,33.409090909090907,32.56363636363637,31.66363636363636,30.709090909090897,29.699999999999989,{"$f":"a0*$A104-_a1*$A104^2+(a0*AZ$53-_a1*AZ$53^2)/(1+dr)","v":28.636363636363633},27.518181818181816,26.345454545454547,25.118181818181814,23.836363636363622,22.5,21.10909090909092,19.663636363636371,18.163636363636353,16.609090909090906,15]
[5.0999999999999996,13.769999999999996,15.379090909090905,16.93363636363636,18.43363636363636,19.879090909090905,21.269999999999996,22.606363636363632,23.888181818181813,25.11545454545454,26.288181818181812,27.406363636363629,28.47,29.4790909090909,30.43363636363636,31.333636363636359,32.179090909090903,32.969999999999992,33.706363636363633,34.388181818181813,35.015454545454539,35.588181818181809,36.106363636363639,36.569999999999993,36.9790909090909,37.333636363636352,37.633636363636356,37.879090909090905,38.069999999999993,38.206363636363633,38.288181818181812,38.315454545454543,38.288181818181812,38.206363636363626,38.069999999999993,37.879090909090905,37.633636363636356,37.333636363636352,36.9790909090909,36.569999999999986,36.106363636363639,35.588181818181809,35.015454545454546,34.38818181818182,33.706363636363619,32.969999999999985,32.179090909090903,31.333636363636366,30.433636363636356,29.479090909090893,28.469999999999985,27.406363636363629,26.288181818181812,25.115454545454543,23.88818181818181,22.606363636363618,21.269999999999996,19.879090909090916,18.433636363636367,16.933636363636349,15.379090909090902,13.769999999999996]
[5.2,12.480000000000004,14.089090909090913,15.643636363636368,17.143636363636368,18.589090909090913,19.980000000000004,21.31636363636364,22.598181818181821,23.825454545454548,24.99818181818182,26.116363636363637,27.180000000000007,28.189090909090908,29.143636363636368,30.043636363636367,30.88909090909091,31.68,32.416363636363641,33.098181818181821,33.725454545454546,34.298181818181817,34.816363636363647,35.28,35.689090909090908,36.043636363636359,36.343636363636364,36.589090909090913,36.78,36.916363636363641,36.99818181818182,37.025454545454551,36.99818181818182,36.916363636363634,36.78,36.589090909090913,36.343636363636364,36.043636363636359,35.689090909090908,35.279999999999994,34.816363636363647,34.298181818181817,33.725454545454554,33.098181818181828,32.416363636363627,31.679999999999993,30.88909090909091,30.043636363636374,29.143636363636364,28.189090909090901,27.179999999999993,26.116363636363637,24.99818181818182,23.825454545454551,22.598181818181818,21.316363636363626,19.980000000000004,18.589090909090924,17.143636363636375,15.643636363636357,14.08909090909091,12.480000000000004]
[5.3,11.129999999999995,12.739090909090905,14.293636363636359,15.793636363636358,17.239090909090905,18.629999999999995,19.966363636363631,21.248181818181813,22.475454545454539,23.648181818181811,24.766363636363629,25.83,26.839090909090899,27.793636363636359,28.693636363636358,29.539090909090902,30.329999999999991,31.066363636363629,31.748181818181813,32.375454545454538,32.948181818181808,33.466363636363639,33.929999999999993,34.339090909090899,34.693636363636351,34.993636363636355,35.239090909090905,35.429999999999993,35.566363636363633,35.648181818181811,35.675454545454542,35.648181818181811,35.566363636363626,35.429999999999993,35.239090909090905,34.993636363636355,34.693636363636351,34.339090909090899,33.929999999999986,33.466363636363639,32.948181818181808,32.375454545454545,31.74818181818182,31.066363636363622,30.329999999999984,29.539090909090902,28.693636363636365,27.793636363636356,26.839090909090892,25.829999999999984,24.766363636363629,23.648181818181811,22.475454545454543,21.248181818181809,19.966363636363617,18.629999999999995,17.239090909090915,15.793636363636367,14.293636363636349,12.739090909090901,11.129999999999995]
[5.4,9.7199999999999847,11.329090909090894,12.883636363636349,14.383636363636347,15.829090909090894,17.219999999999985,18.556363636363621,19.838181818181802,21.065454545454529,22.238181818181801,23.356363636363618,24.419999999999987,25.429090909090888,26.383636363636349,27.283636363636347,28.129090909090891,28.91999999999998,29.656363636363618,30.338181818181802,30.965454545454527,31.538181818181801,32.056363636363628,32.519999999999982,32.929090909090888,33.28363636363634,33.583636363636344,33.829090909090894,34.019999999999982,34.156363636363622,34.238181818181801,34.265454545454531,34.238181818181801,34.156363636363615,34.019999999999982,33.829090909090894,33.583636363636344,33.28363636363634,32.929090909090888,32.519999999999975,32.056363636363628,31.538181818181801,30.965454545454534,30.338181818181809,29.656363636363611,28.919999999999973,28.129090909090891,27.283636363636354,26.383636363636345,25.429090909090881,24.419999999999973,23.356363636363618,22.238181818181801,21.065454545454532,19.838181818181798,18.556363636363606,17.219999999999985,15.829090909090905,14.383636363636356,12.883636363636338,11.32909090909089,9.7199999999999847]
[5.5,8.25,9.8590909090909093,11.413636363636364,12.913636363636362,14.359090909090909,15.75,17.086363636363636,18.368181818181817,19.595454545454544,20.768181818181816,21.886363636363633,22.950000000000003,23.959090909090904,24.913636363636364,25.813636363636363,26.659090909090907,27.449999999999996,28.186363636363634,28.868181818181817,29.495454545454542,30.068181818181817,30.58636363636364,31.049999999999997,31.459090909090907,31.813636363636355,32.11363636363636,32.359090909090909,32.549999999999997,32.686363636363637,32.768181818181816,32.795454545454547,32.768181818181816,32.68636363636363,32.549999999999997,32.359090909090909,32.11363636363636,31.813636363636355,31.459090909090907,31.04999999999999,30.58636363636364,30.068181818181817,29.49545454545455,28.868181818181824,28.186363636363627,27.449999999999989,26.659090909090907,25.81363636363637,24.91363636363636,23.959090909090897,22.949999999999989,21.886363636363633,20.768181818181816,19.595454545454547,18.368181818181814,17.086363636363622,15.75,14.35909090909092,12.913636363636371,11.413636363636353,9.8590909090909058,8.25]
[5.6,6.7200000000000131,8.3290909090909224,9.8836363636363771,11.383636363636375,12.829090909090922,14.220000000000013,15.556363636363647,16.83818181818183,18.065454545454557,19.238181818181829,20.356363636363646,21.420000000000016,22.429090909090917,23.383636363636377,24.283636363636376,25.12909090909092,25.920000000000009,26.656363636363647,27.33818181818183,27.965454545454556,28.53818181818183,29.056363636363653,29.52000000000001,29.92909090909092,30.283636363636369,30.583636363636376,30.829090909090922,31.020000000000007,31.15636363636365,31.238181818181825,31.265454545454556,31.238181818181829,31.156363636363643,31.020000000000014,30.829090909090922,30.583636363636376,30.283636363636369,29.92909090909092,29.520000000000003,29.056363636363653,28.53818181818183,27.965454545454563,27.338181818181837,26.65636363636364,25.92,25.12909090909092,24.283636363636383,23.383636363636374,22.42909090909091,21.42,20.356363636363646,19.238181818181829,18.065454545454561,16.838181818181827,15.556363636363635,14.220000000000013,12.829090909090933,11.383636363636384,9.8836363636363664,8.3290909090909189,6.7200000000000131]
[5.7,5.1300000000000097,6.739090909090919,8.2936363636363737,9.7936363636363719,11.239090909090919,12.63000000000001,13.966363636363644,15.248181818181827,16.475454545454554,17.648181818181826,18.766363636363643,19.830000000000013,20.839090909090913,21.793636363636374,22.693636363636372,23.539090909090916,24.330000000000005,25.066363636363644,25.748181818181827,26.375454545454552,26.948181818181826,27.466363636363649,27.930000000000007,28.339090909090917,28.693636363636365,28.993636363636373,29.239090909090919,29.430000000000003,29.566363636363647,29.648181818181822,29.675454545454553,29.648181818181826,29.56636363636364,29.43000000000001,29.239090909090919,28.993636363636373,28.693636363636365,28.339090909090917,27.93,27.466363636363649,26.948181818181826,26.375454545454559,25.748181818181834,25.066363636363636,24.33,23.539090909090916,22.693636363636379,21.79363636363637,20.839090909090906,19.829999999999998,18.766363636363643,17.648181818181826,16.475454545454557,15.248181818181823,13.966363636363631,12.63000000000001,11.23909090909093,9.7936363636363808,8.293636363636363,6.7390909090909155,5.1300000000000097]
[5.8,3.4799999999999898,5.0890909090908991,6.6436363636363538,8.143636363636352,9.5890909090908991,10.97999999999999,12.316363636363624,13.598181818181807,14.825454545454535,15.998181818181807,17.116363636363623,18.179999999999993,19.189090909090893,20.143636363636354,21.043636363636352,21.889090909090896,22.679999999999986,23.416363636363624,24.098181818181807,24.725454545454532,25.298181818181806,25.816363636363629,26.279999999999987,26.689090909090897,27.043636363636345,27.343636363636353,27.589090909090899,27.779999999999983,27.916363636363627,27.998181818181802,28.025454545454533,27.998181818181806,27.91636363636362,27.77999999999999,27.589090909090899,27.343636363636353,27.043636363636345,26.689090909090897,26.27999999999998,25.816363636363629,25.298181818181806,24.725454545454539,24.098181818181814,23.416363636363617,22.679999999999978,21.889090909090896,21.043636363636359,20.14363636363635,19.189090909090886,18.179999999999978,17.116363636363623,15.998181818181804,14.825454545454537,13.598181818181803,12.316363636363612,10.97999999999999,9.5890909090909098,8.1436363636363609,6.643636363636344,5.0890909090908956,3.4799999999999898]
[5.9,1.769999999999996,3.3790909090909049,4.93363636363636,6.4336363636363583,7.8790909090909045,9.269999999999996,10.60636363636363,11.888181818181813,13.115454545454542,14.288181818181814,15.406363636363631,16.47,17.4790909090909,18.43363636363636,19.333636363636359,20.179090909090903,20.969999999999992,21.70636363636363,22.388181818181813,23.015454545454539,23.588181818181813,24.106363636363636,24.569999999999993,24.979090909090903,25.333636363636352,25.633636363636359,25.879090909090905,26.06999999999999,26.206363636363633,26.288181818181808,26.315454545454539,26.288181818181812,26.206363636363626,26.069999999999997,25.879090909090905,25.633636363636359,25.333636363636352,24.979090909090903,24.569999999999986,24.106363636363636,23.588181818181813,23.015454545454546,22.38818181818182,21.706363636363623,20.969999999999985,20.179090909090903,19.333636363636366,18.433636363636356,17.479090909090893,16.469999999999985,15.406363636363631,14.28818181818181,13.115454545454543,11.88818181818181,10.606363636363618,9.269999999999996,7.8790909090909169,6.433636363636368,4.9336363636363503,3.3790909090909014,1.769999999999996]
[6,0,1.6090909090909089,3.1636363636363636,4.6636363636363622,6.1090909090909085,7.4999999999999991,8.8363636363636342,10.118181818181817,11.345454545454546,12.518181818181818,13.636363636363635,14.700000000000001,15.709090909090905,16.663636363636364,17.563636363636363,18.409090909090907,19.199999999999996,19.936363636363634,20.618181818181817,21.245454545454542,21.818181818181817,22.33636363636364,22.799999999999997,23.209090909090907,23.563636363636355,23.863636363636363,24.109090909090909,24.299999999999994,24.436363636363637,24.518181818181812,24.545454545454543,24.518181818181816,24.43636363636363,24.3,24.109090909090909,23.863636363636363,23.563636363636355,23.209090909090907,22.79999999999999,22.33636363636364,21.818181818181817,21.24545454545455,20.618181818181824,19.936363636363627,19.199999999999989,18.409090909090907,17.56363636363637,16.66363636363636,15.709090909090897,14.699999999999987,13.636363636363635,12.518181818181814,11.345454545454547,10.118181818181814,8.8363636363636218,7.4999999999999991,6.1090909090909209,4.663636363636372,3.1636363636363543,1.6090909090909054,0]

{"sheet":"2xBC","index":5,"n_rows":80,"n_cols":39,"rows_kept":80,"cut":false,"hidden":true,"merged":[]}
["We extend the model again, this time adding a BUDGET CONSTRAINT.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"A",null,null,null,"x",null,"b",null,"A-1",null,null,null,"A-1b = solution"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"-2*_a1","v":-6},0,-1,null,"x0",null,{"$f":"-a0","v":-18},null,{"$f":"MINVERSE(AA2:AC4)","v":-8.7301587301587283E-2},8.7301587301587311E-2,0.47619047619047622,null,{"$f":"MMULT(AI2:AK4,AG2:AG4)","v":1.0952380952380949}]
["Before, the consumer could buy any amount of x, from zero to infinity. Now, they have stay within their budget.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,{"$f":"-2*_a1/(1+dr)","v":-5.4545454545454541},{"$f":"-1/(1+ir)","v":-1},null,"x1","=",{"$f":"-a0/(1+dr)","v":-16.363636363636363},null,8.7301587301587311E-2,-8.7301587301587297E-2,0.52380952380952384,null,0.90476190476190443]
["We allow saving (consuming less in the present, then more in the future) or borrowing (the reverse), but the overall, lifetime budget must be balanced.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1,{"$f":"1/(1+ir)","v":1},0,null,"l",null,{"$f":"m+(1+mgr)*m/(1+ir)","v":2},null,-0.47619047619047622,-0.52380952380952384,-2.8571428571428572,null,11.428571428571427]
[]
["Utility is the same as before:"]
[null,null,null,null,"a0",18]
[null,null,null,null,"a1",3]
[]
["The price of x is $1/unit in both time periods (no inflation) and the consumer gets $1 of income each year."]
["That means that the 3,3 optimal solution that we got before is not feasible -- it would cost $6 and the consumer only has $2 of lifetime income."]
["Before you say, \"Fine. The answer is 1,1,\" think about it a little. Remember that you get more utility from present consumption. "]
["Wouldn't it make sense to buy a little more in the present than in the future? Let's find out."]
[]
["The constrained optimization problem looks like this:"]
[null,null,null,null,null,null,"interest rate (ir)",0,"This is the interest rate which we will use to present value future income."]
[null,null,null,"discount rate (dr)",0.1,null,"m growth rate (mgr)",0,"This allows us to increase income in future years if we wish."]
[null,null,null,null,null,null,"yearly m",1]
[null,null,"Time","x","U",null,null,"constraint"]
[null,null,0,3,{"$f":"(a0*D20-_a1*D20^2)/((1+dr)^C20)","v":27},null,null,{"$f":"m-D20","v":-2}]
[null,null,1,3,{"$f":"(a0*D21-_a1*D21^2)/((1+dr)^C21)","v":24.545454545454543},null,null,{"$f":"(1+mgr)*m/(1+ir)^C21-D21/(1+ir)^C21","v":-2}]
[null,null,null,"sum U",{"$f":"E20+E21","v":51.545454545454547},null,"sum constraint",{"$f":"H20+H21","v":-4}]
[]
["As noted above, the 3,3 unconstrained solution is not allowed because cell H22 is negative. Click to see the formula."]
["Thus, this consumer cannot afford the 3,3 combination and must buy less x to stay within the budget."]
["This problem can be solved with math, but run Solver to find the optimal solution."]
[]
["When you call Solver, notice how a constraint has been added -- Solver must find the values of x in cells E20 and E21 that maximize U (E22), while meeting the constraint in H22."]
[]
["As expected, the optimal solution has the consumer buying a little more in the present."]
["Change the m growth rate (cell H17) to 0.1. Now the consumer can afford a little more x (look at H22), but how will the extra income be allocated? Run Solver to find out."]
[]
["By the way, you might notice that cell H22 displays something like"]
[4.8095280424931301E-8]
["This is simply scientific notation. It is read, \"4.8095 times 10 to the minus 8\" and is the same as 0.000000048095 -- a really small number, which means the constraint is basically zero and has been met."]
["Excel defaults to scientific notation if the number has so many zeroes that the column is not wide enough to display the number."]
["You can change the number formatting in the Number group of the Home tab."]
[]
["We can graph the solution by showing the highest level curve we can reach, just tangent to the budget constraint -- exactly like you learned in Intermediate Micro."]
["The solution is at the red diamond in the chart below."]
[]
["We can extend the model"]
["again, adding another",null,null,null,null,null,null,null,null,null,null,"step",0.36]
["good.",null,null,null,null,null,null,null,null,null,"Time","x","U"]
[null,null,null,null,null,null,null,null,null,null,0,{"$f":"AM2","v":1.0952380952380949},{"$f":"($F$7*L45-$F$8*L45^2)/((1+dr)^K45)","v":16.115646258503396}]
[null,null,null,null,null,null,null,null,null,null,1,{"$f":"AM3","v":0.90476190476190443},{"$f":"($F$7*L46-$F$8*L46^2)/((1+dr)^K46)","v":12.572665429808284}]
[null,null,null,null,null,null,null,null,null,null,null,"U bar",{"$f":"SUM(M45:M46)","v":28.688311688311678},"<--this is the level curve being graphed",null,null,null,"This is the diagonal line (budget constraint)"]
[null,null,null,null,null,null,null,null,null,null,null,"x0","x1",null,null,null,null,"x0","x1"]
[null,null,null,null,null,null,null,null,null,null,null,0.25,{"$f":"(-a0/(1+dr)-SQRT((a0/(1+dr))^2-4*(-_a1/(1+dr))*(-$M$47+a0*L49-_a1*L49^2)))/(2*(-_a1/(1+dr)))","v":3.2494040515957727},null,null,null,null,0,{"$f":"(1+mgr)*m+m*(1+ir)","v":2}]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"L49+$M$43","v":0.61},4.4483239903255027,null,null,null,null,{"$f":"R49+$M$43","v":0.36},{"$f":"(1+mgr)*m+(m-R50)*(1+ir)","v":1.6400000000000001}]
[null,null,null,null,null,null,null,null,null,null,null,0.97,4.9616223849029621,null,null,null,null,{"$f":"R50+$M$43","v":0.72},1.28]
[null,null,null,null,null,null,null,null,null,null,null,1.33,5.3050298004477918,null,null,null,null,{"$f":"L45","v":1.0952380952380949},0.9047619047619051]
[null,null,null,null,null,null,null,null,null,null,null,1.69,5.5481841340359193,null,null,null,null,{"$f":"R52+$M$43","v":1.4552380952380948},0.54476190476190522]
[null,null,null,null,null,null,null,null,null,null,null,2.0499999999999998,5.7181247912765869,null,null,null,null,1.8152380952380947,0.18476190476190535]
[null,null,null,null,null,null,null,null,null,null,null,2.4099999999999997,5.8280810421471987,null,null,null,null,2.1752380952380945,-0.17523809523809453]
[null,null,null,null,null,null,null,null,null,null,null,2.7699999999999996,5.8849198222745098]
[null,null,null,null,null,null,null,null,null,null,null,3.1299999999999994,5.8917749533724759]
[null,null,null,null,null,null,null,null,null,null,null,3.4899999999999993,5.8490072623551503]
[null,null,null,null,null,null,null,null,null,null,null,3.8499999999999992,5.7543061523643999]
[null,null,null,null,null,null,null,null,null,null,null,4.2099999999999991,5.6020073752686388]
[null,null,null,null,null,null,null,null,null,null,null,4.5699999999999994,5.3810842868223681]
[null,null,null,null,null,null,null,null,null,null,null,4.93,5.0696768783924684]
[null,null,null,null,null,null,null,null,null,null,null,5.29,4.6163051633130374]
[null,null,null,null,null,null,null,null,null,null,null,5.65,3.8100631956535116]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"L49","v":0.25},{"$f":"(-a0/(1+dr)+SQRT((a0/(1+dr))^2-4*(-_a1/(1+dr))*(-$M$47+a0*L65-_a1*L65^2)))/(2*(-_a1/(1+dr)))","v":2.7505959484042282}]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"L50","v":0.61},1.5516760096744975]
[null,null,null,null,null,null,null,null,null,null,null,0.97,1.0383776150970379]
[null,null,null,null,null,null,null,null,null,null,null,1.33,0.69497019955220818]
[null,null,null,null,null,null,null,null,null,null,null,1.69,0.45181586596408124]
[null,null,null,null,null,null,null,null,null,null,null,2.0499999999999998,0.28187520872341365]
[null,null,null,null,null,null,null,null,null,null,null,2.4099999999999997,0.17191895785280137]
[null,null,null,null,null,null,null,null,null,null,null,2.7699999999999996,0.11508017772549098]
[null,null,null,null,null,null,null,null,null,null,null,3.1299999999999994,0.10822504662752436]
[null,null,null,null,null,null,null,null,null,null,null,3.4899999999999993,0.15099273764485013]
[null,null,null,null,null,null,null,null,null,null,null,3.8499999999999992,0.24569384763560037]
[null,null,null,null,null,null,null,null,null,null,null,4.2099999999999991,0.39799262473136177]
[null,null,null,null,null,null,null,null,null,null,null,4.5699999999999994,0.6189157131776315]
[null,null,null,null,null,null,null,null,null,null,null,4.93,0.93032312160753217]
[null,null,null,null,null,null,null,null,null,null,null,5.29,1.3836948366869626]
[null,null,null,null,null,null,null,null,null,null,null,5.65,2.1899368043464884]

{"sheet":"2xy","index":6,"n_rows":25,"n_cols":8,"rows_kept":25,"cut":false,"hidden":true,"merged":[]}
["Another extension: we add another good, y."]
[]
["The functional form of utility is still quadratic (it has a squared term), but now we have x and y in the present and future time periods."]
[null,null,null,null,"a0",18]
[null,null,null,null,"a1",3]
[null,null,null,null,"b0",8]
[null,null,null,null,"b1",1]
[]
["The price of x and y are both $1/unit in both time periods and the consumer gets $2 of income each year (the income growth and interest rate are zero in cells H12 and H13)."]
[]
["The constrained optimization problem looks like this:"]
[null,null,null,null,null,null,"interest rate (ir)",0]
[null,null,null,"discount rate (dr)",0.1,null,"m growth rate (mgr)",0]
[null,null,null,null,null,null,"yearly m",2]
[null,"Time","x","y","U",null,null,"constraint"]
[null,0,1,1,{"$f":"(a0*C16-_a1*C16^2+b0*D16-_b1*D16^2)/((1+dr)^B16)","v":22},null,null,{"$f":"m-C16-D16","v":0}]
[null,1,1,1,{"$f":"(a0*C17-_a1*C17^2+b0*D17-_b1*D17^2)/((1+dr)^B17)","v":20},null,null,{"$f":"((1+mgr)^B17)*m/(1+ir)^B17-C17/(1+ir)^B17-D17/(1+ir)^B17","v":0}]
[null,null,null,"sum U",{"$f":"E16+E17","v":42},null,"sum constraint",{"$f":"H16+H17","v":0}]
[]
["When you call Solver to find the optimal solution, notice how the changing cells are now C16 and C17 for x and D16 and D17 for y."]
["The consumer has to choose four variables, making sure the constraint is satisfied and utility is maximized."]
[]
["We don't have an easy way to display this result in a graph because we need four dimensions, one for each choice variable, but that's OK, we know we are maximizing utility subject to a constraint."]
[]
["It should be pretty clear that we can keep extending the model by adding other goods, but we're going to do something different -- and critical for the rational addiction model."]

{"sheet":"5xyS","index":7,"n_rows":48,"n_cols":55,"rows_kept":48,"cut":false,"hidden":true,"merged":[]}
["We have arrived at the key idea in the rational addiction model: utility depends on x and y and the stock of x.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"A (matrix of coefficients from first order conditions)",null,null,null,null,null,null,null,null,null,null,null,"x (vector var)",null,"b (vector constants)",null,"A-1",null,null,null,null,null,null,null,null,null,null,null,"A-1b = solution"]
["This makes the problem quite complicated, but we can use Solver and Excel to do the heavy lifting.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"-2*_a1-2*_c1/((1+dr)^1)-2*_c1*(1-dep)^2/((1+dr)^2)-2*_c1*(1-dep)^4/((1+dr)^3)-2*_c1*(1-dep)^6/((1+dr)^4)","v":-17.075947723857659},{"$f":"-2*_c1*(1-dep)/((1+dr)^2)-2*_c1*(1-dep)^3/((1+dr)^3)-2*_c1*(1-dep)^5/((1+dr)^4)","v":-7.8311411447305481},{"$f":"-2*_c1*(1-dep)^2/((1+dr)^3)-2*_c1*(1-dep)^4/((1+dr)^4)","v":-4.9375213441704791},{"$f":"-2*_c1*(1-dep)^3/((1+dr)^4)","v":-2.342394645174509},0,0,0,0,0,0,-1,null,"x0",null,{"$f":"-(a0+(-2*_c1*(1-dep)*B29+c0)/((1+dr)^1)+(-2*_c1*(1-dep)^3*B29+c0*(1-dep))/((1+dr)^2)+(-2*_c1*(1-dep)^5*B29+c0*(1-dep)^2)/((1+dr)^3)+(-2*_c1*(1-dep)^7*B29+c0*(1-dep)^3)/((1+dr)^4))","v":-136.31500580561433},null,{"$f":"MINVERSE(AA2:AK12)","v":-8.2387913199231472E-2},3.962090522537981E-2,1.8591871736511747E-2,8.6290652012332818E-3,3.7665538797790354E-3,1.9294552351849432E-3,2.1224007587034382E-3,2.3346408345737819E-3,2.5681049180311601E-3,2.8249154098342762E-3,1.5435641881479546E-2,null,{"$f":"MMULT(AQ2:BA12,AO2:AO12)","v":5.6022372710562367}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"-2*_c1*(1-dep)/((1+dr)^2)-2*_c1*(1-dep)^3/((1+dr)^3)-2*_c1*(1-dep)^5/((1+dr)^4)","v":-7.8311411447305481},{"$f":"-2*_a1/((1+dr)^1)-2*_c1/((1+dr)^2)-2*_c1*(1-dep)^2/((1+dr)^3)-2*_c1*(1-dep)^4/((1+dr)^4)","v":-13.697851922682872},{"$f":"-2*_c1*(1-dep)/((1+dr)^3)-2*_c1*(1-dep)^3/((1+dr)^4)","v":-5.1973908886005038},{"$f":"-2*_c1*(1-dep)^2/((1+dr)^4)","v":-2.4656785738679043},0,0,0,0,0,0,{"$f":"-1/(1+ir)","v":-1},null,"x1",null,{"$f":"-(a0/((1+dr)^1)+(-2*_c1*(1-dep)^2*B29+c0)/((1+dr)^2)+(-2*_c1*(1-dep)^4*B29+c0*(1-dep))/((1+dr)^3)+(-2*_c1*(1-dep)^6*B29+c0*(1-dep)^2)/((1+dr)^4))","v":-102.62823577624476},null,3.962090522537981E-2,-0.10985207663765455,3.4188623166891724E-2,1.519569810858969E-2,5.050844298957728E-3,2.587345962856564E-3,2.8460805591422208E-3,3.1306886150564424E-3,3.4437574765620884E-3,3.7881332242182962E-3,2.0698767702852512E-2,null,2.9738392275310428]
["The way it works is that you get utility from current consumption of x and y AND your built up, past consumption of x, which is the stock of x.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"-2*_c1*(1-dep)^2/((1+dr)^3)-2*_c1*(1-dep)^4/((1+dr)^4)","v":-4.9375213441704791},{"$f":"-2*_c1*(1-dep)^1/((1+dr)^3)-2*_c1*(1-dep)^3/((1+dr)^4)","v":-5.1973908886005038},{"$f":"-2*_a1/((1+dr)^2)-2*_c1/((1+dr)^3)-2*_c1*(1-dep)^2/((1+dr)^4)","v":-10.429615463424627},{"$f":"-2*_c1*(1-dep)^1/((1+dr)^4)","v":-2.5954511303872678},0,0,0,0,0,0,{"$f":"-1/(1+ir)^2","v":-1},null,"x2",null,{"$f":"-(a0/((1+dr)^2)+(-2*_c1*(1-dep)^3*B29+c0)/((1+dr)^3)+(-2*_c1*(1-dep)^5*B29+c0*(1-dep))/((1+dr)^4))","v":-70.883136397787013},null,1.859187173651174E-2,3.4188623166891717E-2,-0.1261175507058806,3.3074907826353403E-2,9.7548473383076498E-3,4.9970189903222025E-3,5.4967208893544218E-3,6.0463929782898643E-3,6.6510322761188546E-3,7.3161355037307391E-3,3.997615192257762E-2,null,1.8254531259555562]
["So, say you play the piano and get better at it and it is more enjoyable the next time you play -- you are getting utility not just from playing now, but also from your previous practice.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"-2*_c1*(1-dep)^3/((1+dr)^4)","v":-2.342394645174509},{"$f":"-2*_c1*(1-dep)^2/((1+dr)^4)","v":-2.4656785738679043},{"$f":"-2*_c1*(1-dep)^1/((1+dr)^4)","v":-2.5954511303872678},{"$f":"-2*_a1/((1+dr)^3)-2*_c1/((1+dr)^4)","v":-7.2399426268697464},0,0,0,0,0,0,{"$f":"-1/(1+ir)^3","v":-1},null,"x3",null,{"$f":"-(a0/((1+dr)^3)+(-2*_c1*(1-dep)^4*B29+c0)/((1+dr)^4))","v":-40.844204630831214},null,8.62906520123328E-3,1.5195698108589695E-2,3.3074907826353403E-2,-0.14576018583642744,2.152942152560848E-2,1.1028663441162732E-2,1.2131529785279005E-2,1.3344682763806907E-2,1.4679151040187601E-2,1.614706614420636E-2,8.8229307529301859E-2,null,1.4908320459603799]
["We model this with a new variable, S, that keeps track of your past consumption of x and with a utility function that includes a quadratic component for the stock.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,0,0,0,{"$f":"-2*_a1/((1+dr)^4)","v":-4.0980807321904233},0,0,0,0,0,{"$f":"-1/(1+ir)^4","v":-1},null,"x4",null,{"$f":"-a0/((1+dr)^4)","v":-12.294242196571268},null,3.7665538797790341E-3,5.0508442989577271E-3,9.7548473383076532E-3,2.1529421525608477E-2,-0.19461146124884032,2.5308315049088044E-2,2.7839146553996851E-2,3.0623061209396535E-2,3.3685367330336197E-2,3.7053904063369815E-2,0.20246652039270435,null,1.8146623163809312]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,0,0,0,0,{"$f":"-2*_b1","v":-8},0,0,0,0,-1,null,"y0","=",{"$f":"-b0","v":-8},null,1.929455235184943E-3,2.5873459628565636E-3,4.9970189903222034E-3,1.1028663441162731E-2,2.5308315049088044E-2,-0.11203556021664092,1.4260883761694989E-2,1.5686972137864488E-2,1.7255669351650942E-2,1.8981236286816035E-2,0.10371551826687264,null,0.3927988097026831]
["But it's not as simple as just adding up past consumption. We will also include a depreciation parameter that reflects how fast we use up the stock of previous x.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,0,0,0,0,0,{"$f":"-2*_b1/((1+dr)^1)","v":-7.2727272727272725},0,0,0,{"$f":"-1/(1+ir)","v":-1},null,"y1",null,{"$f":"-b0/((1+dr)^1)","v":-7.2727272727272725},null,2.1224007587034373E-3,2.8460805591422199E-3,5.4967208893544235E-3,1.2131529785279004E-2,2.7839146553996851E-2,1.4260883761694989E-2,-0.12181302786213552,1.7255669351650935E-2,1.8981236286816038E-2,2.087935991549764E-2,0.11408707009355991,null,0.33207869067295182]
["The depreciation rate can be anywhere from 0 to 100% -- zero means no depreciation and 100% means you use up all of the stock.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,0,0,0,0,0,0,{"$f":"-2*_b1/((1+dr)^2)","v":-6.6115702479338836},0,0,{"$f":"-1/(1+ir)^2","v":-1},null,"y2",null,{"$f":"-b0/((1+dr)^2)","v":-6.6115702479338836},null,2.3346408345737814E-3,3.1306886150564424E-3,6.0463929782898669E-3,1.3344682763806905E-2,3.0623061209396535E-2,1.5686972137864488E-2,1.7255669351650939E-2,-0.13226876371318397,2.0879359915497643E-2,2.2967295907047405E-2,0.1254957771029159,null,0.26528655974024651]
["It's a complicated process, but it does go step-by-step. You have to read carefully and slowly. Walk through each step in the model below and you will understand it.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,0,0,0,0,0,0,0,{"$f":"-2*_b1/((1+dr)^3)","v":-6.0105184072126203},0,{"$f":"-1/(1+ir)^3","v":-1},null,"y3",null,{"$f":"-b0/((1+dr)^3)","v":-6.0105184072126203},null,2.5681049180311596E-3,3.4437574765620871E-3,6.6510322761188546E-3,1.4679151040187599E-2,3.3685367330336197E-2,1.7255669351650942E-2,1.8981236286816035E-2,2.087935991549764E-2,-0.14340770409295264,2.5264025497752151E-2,0.13804535481320754,null,0.19181521571427074]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,0,0,0,0,0,0,0,0,{"$f":"-2*_b1/((1+dr)^4)","v":-5.4641076429205642},{"$f":"-1/(1+ir)^4","v":-1},null,"y4",null,{"$f":"-b0/((1+dr)^4)","v":-5.4641076429205642},null,2.8249154098342758E-3,3.7881332242182958E-3,7.3161355037307399E-3,1.6147066144206357E-2,3.7053904063369815E-2,1.8981236286816035E-2,2.087935991549764E-2,2.2967295907047401E-2,2.5264025497752151E-2,-0.1552220719524727,0.15184989029452828,null,0.11099673728569881]
["The model for five time periods is set up below. It has the same framework as before, with two additions: another good, the stock of x, S, in the utility function and the depreciation rate in cell D21.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1,{"$f":"1/(1+ir)","v":1},{"$f":"1/(1+ir)^2","v":1},{"$f":"1/(1+ir)^3","v":1},{"$f":"1/(1+ir)^4","v":1},1,{"$f":"1/(1+ir)","v":1},{"$f":"1/(1+ir)^2","v":1},{"$f":"1/(1+ir)^3","v":1},{"$f":"1/(1+ir)^4","v":1},0,null,"l",null,{"$f":"m+((1+mgr)^1)*m/((1+ir)^1)+((1+mgr)^2)*m/((1+ir)^2)+((1+mgr)^3)*m/((1+ir)^3)+((1+mgr)^4)*m/((1+ir)^4)","v":15},null,-1.5435641881479544E-2,-2.0698767702852509E-2,-3.9976151922577627E-2,-8.8229307529301845E-2,-0.20246652039270435,-0.10371551826687264,-0.11408707009355991,-0.1254957771029159,-0.13804535481320754,-0.15184989029452828,-0.82972414613498113,null,4.8576095223785352]
["Here are the utility function parameters:"]
[null,null,null,"a0",18,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Below is the analytical optimal solution given parameter values: ",null,null,null,null,null,null,null,null,"Check analytical and Solver (after running Solver, of course):"]
[null,null,null,"a1",3,"a0 and a1 are parameters for x in the quadratic utility function",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Time","S of x","x","y","U","depS","DS","constraint",null,"Time","S of x","x","y","U"]
[null,null,null,"b0",8,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,0,{"$f":"B29","v":0},{"$f":"BC2","v":5.6022372710562367},{"$f":"BC7","v":0.3927988097026831},{"$f":"(a0*AC16-_a1*AC16^2+b0*AD16-_b1*AD16^2+c0*AB16-_c1*AB16^2)/((1+dr)^AA16)","v":9.2103104133834606},{"$f":"$D$22*AB16","v":0},{"$f":"AC16-AF16","v":5.6022372710562367},{"$f":"((1+mgr)^AA16)*$H$22/((1+dr)^AA16)-AC16-AD16","v":-2.9950360807589198},null,0,null,{"$f":"AC16-C29","v":4.1022372710562367},-0.6072011902973169,-15.039689586616539]
[null,null,null,"b1",4,"b0 and b1 are parameters for y in the quadratic utility function",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,1,{"$f":"AB16+AG16","v":5.6022372710562367},{"$f":"BC3","v":2.9738392275310428},{"$f":"BC8","v":0.33207869067295182},{"$f":"(a0*AC17-_a1*AC17^2+b0*AD17-_b1*AD17^2+c0*AB17-_c1*AB17^2)/((1+dr)^AA17)","v":173.2116702726955},{"$f":"$D$22*AB17","v":0.28011186355281187},{"$f":"AC17-AF17","v":2.6937273639782311},{"$f":"((1+mgr)^AA17)*$H$22/((1+dr)^AA17)-AC17-AD17","v":-0.57864519093126754},null,1,{"$f":"AB17-B30","v":4.1022372710562367},{"$f":"AC17-C30","v":1.4738392275310428},{"$f":"AD17-D30","v":-0.66792130932704818},{"$f":"AE17-E30","v":100.7116702726955}]
[null,null,null,"c0",40,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,2,{"$f":"AB17+AG17","v":8.2959646350344673},{"$f":"BC4","v":1.8254531259555562},{"$f":"BC9","v":0.26528655974024651},{"$f":"(a0*AC18-_a1*AC18^2+b0*AD18-_b1*AD18^2+c0*AB18-_c1*AB18^2)/((1+dr)^AA18)","v":180.90465331794834},{"$f":"$D$22*AB18","v":0.41479823175172337},{"$f":"AC18-AF18","v":1.4106548942038328},{"$f":"((1+mgr)^AA18)*$H$22/((1+dr)^AA18)-AC18-AD18","v":0.38859915727940364},null,2,5.3709646350344675,0.3254531259555562,-0.73471344025975349,78.310645053485544]
[null,null,null,"c1",2,"c0 and c1 are parameters for S (the stock of x) in the quadratic utility function",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3,{"$f":"AB18+AG18","v":9.7066195292382993},{"$f":"BC5","v":1.4908320459603799},{"$f":"BC10","v":0.19181521571427074},{"$f":"(a0*AC19-_a1*AC19^2+b0*AD19-_b1*AD19^2+c0*AB19-_c1*AB19^2)/((1+dr)^AA19)","v":166.32790494068138},{"$f":"$D$22*AB19","v":0.48533097646191498},{"$f":"AC19-AF19","v":1.005501069498465},{"$f":"((1+mgr)^AA19)*$H$22/((1+dr)^AA19)-AC19-AD19","v":0.57129714103008178},null,3,5.4278695292382997,-9.1679540396201187E-3,-0.80818478428572926,47.030687153303518]
[null,null,null,null,null,null,"interest rate (ir)",0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,4,{"$f":"AB19+AG19","v":10.712120598736764},{"$f":"BC6","v":1.8146623163809312},{"$f":"BC11","v":0.11099673728569881},{"$f":"(a0*AC20-_a1*AC20^2+b0*AD20-_b1*AD20^2+c0*AB20-_c1*AB20^2)/((1+dr)^AA20)","v":152.04520526509157},{"$f":"$D$22*AB20","v":0.53560602993683826},{"$f":"AC20-AF20","v":1.2790562864440931},{"$f":"((1+mgr)^AA20)*$H$22/((1+dr)^AA20)-AC20-AD20","v":0.12338131242858164},null,4,5.1473080987367643,0.31466231638093123,-0.88900326271430119,25.750400484210886]
[null,null,"discount rate (dr)",0.1,null,null,"m growth rate (mgr)",0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"SUM(AE16:AE20)","v":681.69974420980031},null,null,{"$f":"SUM(AH16:AH20)","v":-2.4904036609521203},null,null,null,null,null,236.763713377079]
[null,null,"depreciation rate (dep)",0.05,null,null,"yearly m",3]
["In the table below, we start with zero for the stock of x, then the chosen values of x and y (and zero S) determine U. "]
["There is zero depreciation in cell F29 because there was no stock to wear out in time=0, so \"delta S\", the change in the stock of S is simply 1.5, the amount of x consumed in time=0."]
["The constraint is 0.5 because we have m=3 and we only used up 2.5 (1.5 for x and 1 for y)."]
["In time=1, the stock of x is 1.5 (cell B30) because that was our \"delta S\" value in cell G29. The process then proceeds as before: x, y, S --> U, then DS = S of x - depS."]
["Walk yourself through row 30. Why is U 72.5? Why is depS 0.075? Why is DS 1.425? "]
["Time","S of x","x","y","U","depS","DS","constraint","<-- Notice the little red triangles -- they are comments. Put your cursor over the cell to read the comment."]
[0,0,1.5,1,{"$f":"(a0*C29-_a1*C29^2+b0*D29-_b1*D29^2+c0*B29-_c1*B29^2)/((1+dr)^A29)","v":24.25},{"$f":"dep*B29","v":0},{"$f":"C29-F29","v":1.5},{"$f":"m-C29-D29","v":0.5},"Click on the cells to the see the formulas -- this is a HUGE advantage of a spreadsheet."]
[1,{"$f":"B29+G29","v":1.5},1.5,1,{"$f":"(a0*C30-_a1*C30^2+b0*D30-_b1*D30^2+c0*B30-_c1*B30^2)/((1+dr)^A30)","v":72.5},{"$f":"dep*B30","v":7.5000000000000011E-2},{"$f":"C30-F30","v":1.425},{"$f":"((1+mgr)^A30)*m/((1+ir)^A30)-C30/((1+ir)^A30)-D30/((1+ir)^A30)","v":0.5},"For complicated ones, use the trace arrows. Click on cell E30, then in the Formulas tab, click the Trace Precedents and Trace Dependents buttons."]
[2,{"$f":"B30+G30","v":2.9249999999999998},1.5,1,{"$f":"(a0*C31-_a1*C31^2+b0*D31-_b1*D31^2+c0*B31-_c1*B31^2)/((1+dr)^A31)","v":102.5940082644628},{"$f":"dep*B31","v":0.14624999999999999},{"$f":"C31-F31","v":1.35375},{"$f":"((1+mgr)^A31)*m/((1+ir)^A31)-C31/((1+ir)^A31)-D31/((1+ir)^A31)","v":0.5},"Click Remove Arrows to, um, remove the arrows."]
[3,{"$f":"B31+G31","v":4.2787499999999996},1.5,1,{"$f":"(a0*C32-_a1*C32^2+b0*D32-_b1*D32^2+c0*B32-_c1*B32^2)/((1+dr)^A32)","v":119.29721778737786},{"$f":"dep*B32","v":0.2139375},{"$f":"C32-F32","v":1.2860624999999999},{"$f":"((1+mgr)^A32)*m/((1+ir)^A32)-C32/((1+ir)^A32)-D32/((1+ir)^A32)","v":0.5}]
[4,{"$f":"B32+G32","v":5.5648124999999995},1.5,1,{"$f":"(a0*C33-_a1*C33^2+b0*D33-_b1*D33^2+c0*B33-_c1*B33^2)/((1+dr)^A33)","v":126.29480478088068},{"$f":"dep*B33","v":0.27824062499999996},{"$f":"C33-F33","v":1.221759375},{"$f":"((1+mgr)^A33)*m/((1+ir)^A33)-C33/((1+ir)^A33)-D33/((1+ir)^A33)","v":0.5}]
[null,null,null,"sum U",{"$f":"SUM(E29:E33)","v":444.93603083272131},null,"sum constraint",{"$f":"SUM(H29:H33)","v":2.5}]
[]
["The consumer's task is tough one: choose x and y for all five time periods that maximize the sum of discounted (lifetime) utility."]
[null,null,null,null,null,null,null,null,"This chart shows the path of the variables over time:"]
["The x=1.5 and y=1 solution is obviously wrong because the constraint isn't zero."]
["Change C29 to 2 and fill down so all five values of x are 2."]
["The constraint is now met, but is this really the best answer? The max U?"]
[]
["Once again, we can call Excel's Solver and it will give us the optimal solution."]
[]
["Wow -- that is a pretty complicated optimal path!"]
["For S of x, it's rising, but flattening -- the consumer is building an addiction to x!"]
[]
["What would happen if we had more time periods?"]
["Let's find out!"]

{"sheet":"50xyS","index":8,"n_rows":66,"n_cols":34,"rows_kept":66,"cut":false,"hidden":true,"merged":[]}
["This is the full-blown, complete rational addiction model. We simulate a 50 period case to see how the S of x optimal path behaves under different situations."]
[null,null,null,null,null,null,null,null,"This chart is LIVE (it updates if the cells in columns B, C, and D change).",null,null,null,null,null,null,null,null,"This chart is DEAD (it is a picture). Change a parameter and run Solver to determine its effect by comparing the LIVE chart to this one."]
["Instructions and explanations for this sheet are below the charts."]
[]
[null,null,null,"a0",18]
[null,null,null,"a1",3]
[null,null,null,"b0",8]
[null,null,null,"b1",4]
[null,null,null,"c0",40]
[null,null,null,"c1",2]
[null,null,null,null,null,null,"interest rate (ir)",0]
[null,null,"discount rate (dr)",0.1,null,null,"m growth rate (mgr)",0]
[null,null,"depreciation rate (dep)",0.05,null,null,"yearly m",3]
[]
[null,null,null,"sum U",{"$f":"SUM(E17:E66)","v":2099.9171226917142},null,"sum constraint",{"$f":"SUM(H17:H66)","v":-6.6100575679683971E-8}]
["Time","S","x","y","U","depS","DS","constraint"]
[0,0,5.5953353025069257,1.0416558667528146,{"$f":"(a0*C17-_a1*C17^2+b0*D17-_b1*D17^2+c0*B17-_c1*B17^2)/((1+dr)^A17)","v":10.785763157744135},{"$f":"$D$13*B17","v":0},{"$f":"C17-F17","v":5.5953353025069257},{"$f":"m-C17-D17","v":-3.6369911692597405}]
[1,{"$f":"B17+G17","v":5.5953353025069257},2.9108726829779581,1.0460100941487034,174.69596277667594,0.27976676512534632,{"$f":"C18-F18","v":2.6311059178526119},{"$f":"((1+mgr)^A18)*m/((1+ir)^A18)-C18/((1+ir)^A18)-D18/((1+ir)^A18)","v":-0.95688277712666148},null,"Solver has already been run and the chart above shows the optimal path."]
[2,{"$f":"B18+G18","v":8.2264412203595381},1.6489311635310859,1.0506761147216352,181.17565721758257,0.41132206101797691,{"$f":"C19-F19","v":1.2376091025131091},0.30039272174727882,null,null,"You can safely ignore the end--weird stuff happens as the terminal period approaches and most models assume infinite lives to avoid these complications."]
[3,{"$f":"B19+G19","v":9.464050322872648},1.0553698075104105,1.0556783296953947,164.58929814046371,0.4732025161436324,{"$f":"C20-F20","v":0.58216729136677814},0.8889518627941948,null,"Addiction"]
[4,{"$f":"B20+G20","v":10.046217614239426},0.77636323972279897,1.061253488546765,147.63133568735955,0.50231088071197127,{"$f":"C21-F21","v":0.2740523590108277},1.1623832717304361,null,null,"The red series, optimal S, clearly shows a build up and then a persistently high accumulation that is the hallmark of addiction."]
[5,{"$f":"B21+G21","v":10.320269973250253},0.64527930026738156,1.0673727560227917,132.96568739977042,0.51601349866251267,{"$f":"C22-F22","v":0.12926580160486889},1.2873479437098267,null,null,"The consumer is using this high stock of x to gain a lot of utility."]
[6,10.449535774855121,0.58365949523358829,1.0743144238267046,120.26548356786559,0.52247678874275605,6.1182706490832239E-2,1.3420260809397071,null,"Steady state"]
[7,10.510718481345954,0.55478921163240713,1.0813906855107003,109.0536271746288,0.52553592406729777,2.9253287565109365E-2,1.3638201028568926,null,null,"The flat part is a key characteristic of the optimal solution. "]
[8,10.539971768911064,0.54123270223026765,1.0899222843265259,99.015215065868148,0.52699858844555325,1.4234113784714397E-2,1.3688450134432064,null,null,"Notice that delta S, in column G, gets close to zero after a few time periods and stays at DS=0."]
[9,10.554205882695779,0.53471073912951772,1.0986945366375289,89.956955628509036,0.52771029413478898,7.0004449947287384E-3,1.3665947242329535,null,null,"This is called the steady-state solution, analogous to an equilibrium point, but it is a path."]
[10,10.561206327690508,0.5321479074109472,1.1084144183766569,81.755302070433117,0.52806031638452544,4.0875910264217552E-3,1.3594376742123957,null,null,"Look closely at x and depS starting at row 10. They are almost equal. That is why delta S is almost zero.",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"This is shock of S=0 at time=10."]
[11,10.565293918716929,0.53105358555368998,1.1196619349978718,74.310497011886838,0.52826469593584646,2.7888896178435152E-3,1.3492844794484382,null,null,"This balance in additions and subtractions from the stock is indicative of a stable position that we call the steady state."]
[12,10.568082808334772,0.53030304118320282,1.1314572531748537,67.545664631166488,0.52840414041673867,1.8989007664641511E-3,1.3382397056419437,null,"Shock"]
[13,10.569981709101237,0.5300051647691173,1.1446575235267924,61.398395978917684,0.52849908545506186,1.5060793140554418E-3,1.3253373117040901,null,null,"Suppose there is a negative shock, for example, an alcoholic gets arrested for DUI, how can we model that?"]
[14,10.571487788415292,0.53024933795479356,1.1586633778018101,55.812296255051642,0.52857438942076462,1.6749485340289416E-3,1.3110872842433965,null,null,"One way is to say that S is suddenly changed. It is not as valuable."]
[15,10.573162736949321,0.53063342298227012,1.1751991536736901,50.733609187199129,0.5286581368474661,1.9752861348040218E-3,1.2941674233440399,null,null,"Change cell B27 to 0 (zero). What will the consumer do?"]
[16,10.575138023084126,0.53080461685421421,1.1919686384948076,46.115667662213468,0.52875690115420626,2.047715700007946E-3,1.2772267446509782,null,null,"We have to re-optimize for the remaining 40 years. Change cells E15 and H15 so they sum from 27:66 (change 17 to 27)."]
[17,10.577185738784134,0.53086181700668056,1.2117967962601963,41.916232496836855,0.5288592869392067,2.0025300674738622E-3,1.2573413867331229,null,null,"Call Solver and adjust the changing cells from 17 to 27 --> $C$27:$D$66"]
[18,10.579188268851608,0.53112945569382497,1.2328022135172376,38.098827226856201,0.52895941344258046,2.1700422512445083E-3,1.2360683307889375,null,null,"Run Solver to see that the consumer responds by getting re-addicted!"]
[19,10.581358311102852,0.53169210511563381,1.2558705783229125,34.628463711965239,0.52906791555514265,2.6241895604911525E-3,1.2124373165614535,null,"Another Shock"]
[20,10.583982500663343,0.53228234541541253,1.281689444152375,31.472559410279874,0.52919912503316713,3.0832203822453996E-3,1.1860282104322124,null,null,"But what if the shock also affected satisfaction?"]
[21,10.587065721045589,0.53259094907421156,1.3098085864556817,28.602069059550686,0.52935328605227949,3.237663021932069E-3,1.1576004644701068,null,null,"Suppose the shock is breaking news that playing football causes CTE, how can we model that?",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"This adds shock of c0=4."]
[22,10.590303384067521,0.53261672064119581,1.340773763917348,25.991092149676906,0.52951516920337605,3.1015514378197606E-3,1.1266095154414564,null,null,"Easy. Change c0 (cell E9) from 40 to 4 and run Solver."]
[23,10.59340493550534,0.53270515992852785,1.3748995841232836,23.616681755132952,0.52967024677526708,3.0349131532607654E-3,1.0923952559481886,null,null,"Wow. This time, the agent has effectively quit!"]
[24,10.596439848658601,0.53326768139870995,1.4122345864891479,21.457888407559903,0.52982199243293004,3.4456889657799028E-3,1.0544977321121423]
[25,10.599885537624381,0.53442434607359413,1.4533159403031277,19.494862574443481,0.52999427688121903,4.4300691923750968E-3,1.012259713623278,null,"Obviously, many shocks are possible. Figuring out how the optimal paths of x, y, and S respond to shocks"]
[26,10.604315606816757,0.53587836569948666,1.4987687225792696,17.708987393014979,0.53021578034083783,5.6625853586488306E-3,0.96535291172124382,null,"is called comparative statics analysis. This is how economists explain changes in observed behaviors and make predictions."]
[27,10.609978192175406,0.53711809668422617,1.5488067426782859,16.08343474130394,0.53049890960877033,6.6191870754558346E-3,0.91407516063748773]
[28,10.616597379250862,0.53779013429244882,1.6036455390709456,14.60333712183003,0.53082986896254314,6.9602653299056794E-3,0.85856432663660542]
[29,10.623557644580767,0.53794130191832468,1.6638980555282519,13.255555552234878,0.53117788222903839,6.7634196892862875E-3,0.79816064255342356]
[30,10.630321064270053,0.53797891578441115,1.7304133362967884,12.028305007409401,0.5315160532135027,6.4628625709084453E-3,0.73160774791880034]
[31,10.636783926840961,0.53844313495645257,1.8038559547585147,10.910844605570597,0.53183919634204802,6.6039386144045586E-3,0.65770091028503264]
[32,10.643387865455365,0.53966079637188513,1.8845902258615777,{"$f":"(a0*C49-_a1*C49^2+b0*D49-_b1*D49^2+c0*B49-_c1*B49^2)/((1+dr)^A49)","v":9.8931754449691027},{"$f":"$D$13*B49","v":0.53216939327276824},7.491403099116889E-3,0.575748977766537]
[33,10.650879268554482,0.54159241078850839,1.972964963554698,8.965915217851947,0.53254396342772414,9.0484473607842508E-3,{"$f":"((1+mgr)^A50)*m/((1+ir)^A50)-C50/((1+ir)^A50)-D50/((1+ir)^A50)","v":0.48544262565679341}]
[34,10.659927715915266,0.54390096289909107,2.0696552571734927,8.1203129296800434,0.53299638579576336,1.0904577103327706E-2,0.3864437799274163]
[35,10.670832293018593,0.54607517955454687,2.1758112983921736,7.34828083681324,0.53354161465092964,1.2533564903617234E-2,0.27811352205327955]
[36,10.68336585792221,0.54791443441808152,2.2929300647777797,6.6426196368727695,0.53416829289611056,1.3746141521970956E-2,0.15915550080413876]
[37,10.69711199944418,0.54981993819078456,2.4224872585590669,5.9970694931164719,0.53485559997220899,1.4964338218575568E-2,2.7692803250148668E-2]
[38,10.712076337662756,0.55129766137436687,2.5655949544850842,5.4056052394451228,0.53560381688313785,1.5693844491229014E-2,-0.11689261585945099]
[39,10.727770182153986,0.55534217347101222,2.723034717268352,4.8641956929499814,0.53638850910769931,1.8953664363312916E-2,-0.27837689073936422]
[40,10.746723846517298,0.55971045459516622,2.8956724433463075,4.3669624978757291,0.53733619232586494,2.2374262269301282E-2,-0.4553828979414738]
[41,10.769098108786599,0.56824561187459421,3.0848716595767067,3.9105999865541405,0.53845490543932994,2.9790706435264269E-2,-0.65311727145130094]
[42,10.798888815221863,0.58412965146247542,3.2926026857595265,3.4911841066626863,0.53994444076109316,4.418521070138226E-2,-0.87673233722200195]
[43,10.843074025923245,0.61652522151549827,3.521265174059526,3.1060469652198419,0.54215370129616225,7.4371520219336018E-2,-1.1377903955750241]
[44,10.917445546142581,0.68796431316327356,3.7733608967399617,2.7543421479346866,0.54587227730712906,0.14209203585614449,-1.4613252099032352]
[45,11.059537581998725,0.8514522280561122,4.0511703208329966,2.4374570911109354,0.55297687909993631,0.29847534895617589,-1.9026225488891089]
[46,11.358012930954901,1.2291761634195519,4.3567022255918548,2.1555919176152303,0.56790064654774508,0.66127551687180686,-2.5858783890114068]
[47,12.019288447826709,2.1111570712065126,4.6920770923370103,1.8815626042220157,0.60096442239133552,1.5101926488151771,-3.8032341635435229]
[48,13.529481096641886,4.1629211631882779,5.0600800519174447,1.4027545834937005,0.67647405483209433,3.4864471083561837,-6.2230012151057226]
[49,17.015928204998069,8.9557290017728448,5.4644823627213013,-0.50211352767646122,0.85079641024990349,8.1049325915229407,-11.420211364494147]

{"sheet":"End","index":9,"n_rows":28,"n_cols":15,"rows_kept":28,"cut":false,"hidden":true,"merged":[]}
["The end!"]
[]
["You can enjoy a musical performance (either symphony or rock concert), but if you know a lot about the music or the artist, from years of following them or the genre,"]
["you will enjoy it much more. You can use your built up stock of consumption capital to remember other performances or artists and savor this one. "]
["This is the core idea behind the rational addiction model."]
[]
["It takes consumption each time period and accumulates it (subtracting for depreciation), using the stock of x as an input into utility."]
["Solving this problem is quite difficult, but Excel's Solver does all of the hard work for us so we can focus on the properties of the model."]
[]
["Of critical importance is the idea that consumer chooses an optimal path, one that may result in \"addiction,\" with a high stock of x."]
["This outcome is determined by the parameters in the model -- the utility function, discount rate, and other exogenous variables."]
["According to the rational addiction model,  addiction is not something you fall into by mistake, it is the result of choices made by a forward-looking, rational agent."]
[]
["Another key idea is that a shock may lead to different outcomes. Some people, again depending on the parameters, might quit; while others simply increase x again and stay addicted."]
["The model helps us understand these different patterns."]
["It does not say that the people who quit are \"good,\" and those that revert to past choices are \"bad.\" Both are optimizing."]
[]
["The rational addiction model is not without its detractors and doubters. "]
["To see just one example, click here:"]
["Rogeberg (2004)",null,"Accessed May 20, 2019, if the link is broken, find Rogeberg's paper on \"Taking Absurd Theories Seriously\" from here -->",null,null,null,null,null,null,null,null,null,null,null,"Rogeberg search"]
[]
["Of course, there are many other extensions and modifications to the model, like "]
["time inconsistent preferences. "]
["This is really just the tip of the iceberg."]
[]
["But if you gained mastery of the idea of an optimal path, that is a substantial accomplishment!"]
[]
["Hope you learned a little economics and good luck out there!"]