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Week 5 Video 1: Utility with risk aversion and expected value of income (VIDEO TRANSCRIPT BELOW): Why buy insurance? 0: hi welcome back we are on week five 0: chapter seven the demand for insurance 0: it's one of my favorite lessons so why 0: do people buy insurance essentially the 0: demand for insurance it's driven by fear 0: of the unknown it's driven by protecting 0: yourself against uncertainty hedging 0: against your risks so purchasing 0: insurance it means that you forfeit 0: income in the good times to get the 0: payout in the bad time so you're paying 0: your health insurance car insurance home 0: insurance premiums every month or every 0: year even those times when you don't 0: have a car accident or you don't really 0: get any payouts directly from it and the 0: purpose is just like that you're willing 0: to pay to protect you just in case 0: so risk aversion is this thing that Risk aversion 0: drives the demand for insurance and we
can model risk aversion through looking 0: at the utility from income so utility is 1: kind of the way that an individual might 1: value something so utility increases 1: with income as income increases you're 1: getting more and more total value but 1: each additional dollar is less and less 1: important to you so going from zero 1: dollars to ten or fifteen thousand can 1: be extremely valuable whereas going from 1: 100 to 110 wouldn't necessarily change 1: your life as much so utility increases 1: with income but it does so at a smaller 1: and smaller rate of increase this is 1: what that looks like on a graph you can 1: see on the x-axis is income and on the 1: y-axis is utility and so as income 1: increases utility is also getting higher 1: and higher but it's not a one-for-one 1: increase so at the higher levels of 1: income the change along the y-axis and 1: utility it's a much smaller and smaller
signing up for insurance you don't know 3: exactly whether or not you're going to 3: be sick during that year so you might 3: have some probability of sickness some 3: percentage between zero and one so we'll 3: just call the probability of sickness P 3: and the probability of staying healthy 3: would be one minus P so if your 3: probability of sickness is 25% your 3: probability of being healthy is 75% um 3: and so and when you get sick and that 3: 25% chance that you get sick you're 3: gonna lose money from just hospital 3: bills and medical services in addition 3: to potentially lost income depending on 3: how bad it is if you have to miss work 3: so the combination of those things will 3: reduce the kind of your net income 3: available for you to spend so I sub S is 3: income when you're sick I sub H is 3: income when you're healthy and income 4: when you're healthy is greater than 4:
income when you're sick so the expected 4: value is something both in insurance and 4: just in general and math it's when you 4: multiply outcomes by their probabilities 4: so let's say that you know you have a 4: lottery that you're playing and there 4: are four potential outcomes in one case 4: you 4: a lot of money you know maybe another 4: one you get nothing and the other two is 4: somewhere in between and they each have 4: a probability of happening so ahead of 4: time when you want to think about what's 4: the overall value of that ticket you can 4: multiply each of the probabilities times 4: there the value and that outcome and you 4: sum it and that is the expected value 4: across the potential outcomes in the 4: case of looking at a person who might be 4: purchasing insurance that would just be 4: their probability of being sick that was 4: P times the income when they're sick 5:
just a one hundred percent chance of 6: getting two hundred and fifty dollars so 6: technically the expected values are 6: exactly the same however most people People prefer certain outcomes 6: would prefer the two hundred and fifty 6: dollar check with certainty as opposed 6: to getting a lottery of the same 6: expected value and there's like a lot of 6: empirical studies that suggest that 6: people just prefer even at the same 6: expected value P 6: prefer certainty if you actually prefer 6: the uncertain option that says something 6: different about your utility function 6: but we'll get into that so first we're 6: going to define expected utility kind of 6: in a similar way as how we've defined 6: expected income for a lottery or a 7: lottery meeting an uncertain outcome
Week 5 Video 2: Expected Utility for Risk Averse Consumers (VIDEO TRANSCRIPT BELOW): 0: so expected utility expected utility is 0: very similar to expected income in the 0: way that it's calculated you're 0: weighting each of the outcomes by their 0: probability of occurrence so in our 0: example if a student preferred option B 0: which was certainty over option A that 0: means that the utility that they would 0: derive from option B would be greater 0: than that of a so that the expected 0: utility of B is greater than or at least 0: equal to the expected utility from 0: option A so the utility of two hundred 0: and fifty dollars is greater than or 0: equal to 0.5 times the utility of five 0: hundred plus 0.5 times the utility of 0: zero this means that the student is 0: acting in a risk-averse manner because 0: we know that the expected income is the 1: same in both options but they prefer 1: certainty so that's called risk aversion
plus one minus the probability of 2: illness times the utility of income when 2: you're healthy should be fairly 2: straightforward in terms of how to count 2: you'll eat it so graphically it's really 2: interesting the way that this plays out 2: so first of all this is you know that 2: graphs from the last video with utility 2: on the y-axis and income on the X and if 2: you plot along that income distribution 2: on the x axis at the higher end of the x 2: axis is income when you're healthy 2: that's when you have more of your income 2: and then on the lower end is income when 2: you're sick you have less money at that 2: point and the line between s and H is 2: all of the probabilities so if you know 2: for sure you're going to be sick P 2: equals one then your expected utility is 2: right there it's at Point s it's equal 3: to just having the certain income of 3: sickness because that is the outcome
regardless the same thing if you have a 3: hundred percent chance of being healthy 3: so if you have a zero percent chance of 3: being sick P equals zero then it's the 3: same thing except that your utility is 3: going to be up at point H now what 3: happens if you're somewhere in between 3: is where it gets interesting so if you 3: have some probability between zero and a 3: hundred percent then your expected 3: utility is going to lie on the straight 3: line segment between point s and point H 3: so this is a pretty critical distinction 3: but your expected utility is on this 3: straight line and then you can trace it 3: out to the y-axis to find out what the 3: utility value would be but it lies on 3: the straight-line segment so we can do 3: an example 3: if you have a 25 percent chance of being 4: sick P equals 0.25 then the expected 4: income would be 0.25 times being sick
the graph the way that you can show that 5: like that lottery example so the 5: expected utility is on the line segment 5: always the straight-line segment between 5: s and H and that's point a point a-prime 5: above that would be the utility of 5: having that income with certainty 5: so in those options a and B from the 5: example above for option A was the 5: uncertainty option B was the certainty 5: so in that case option A would be the 5: one higher up here and option B would be 5: the one lower down so expected utility 5: lies on the straight line and it's below 6: the curved line and the difference 6: between the two is the value of 6: certainty so risk-averse individuals 6: they prefer certain outcomes to 6: uncertain ones even when the expected 6: income is the same so they prefer they 6: have a greater utility from the expected 6: income then they do with the expected
utility of over the uncertain income so 6: that's what that means when the utility 6: of the expected income is greater than 6: the expected value of the utility of 6: income 6: essentially so a concave utility 6: function is when it's got that downward 6: hump right it's that downward hump with 6: the risk-averse individual concave so 6: yeah it looks like a cave and just like 6: in the last video when you have a 6: concave utility function that means 6: you're risk averse you prime of I is 7: greater than zero so your utility is 7: increasing as income increases but it's 7: doing so at a smaller and smaller rate 7: so you double prime of I is less than 7: zero so this is that concave utility 7: function with the risk-averse individual 7: whose expected utility is lower than the 7: utility of the expected income so we'll 7: stop there
through that so Q should be able to 1: compensate you for those losses so Income with insurance 1: income with insurance we're gonna note 1: as I sub H Prime and I sub S prime so 1: the prime is what notes having insurance 1: ideally for the risk-averse consumer you 1: want to avoid the uncertain possibility 1: of losing that money when you're sick so 1: optimally for the risk-averse consumer 1: insured income when healthy equals 1: insured income while sick so with full 1: insurance there is no income uncertainty 1: the risk-averse consumer would prefer 1: full insurance where your final income 1: is state independent meaning independent 1: of whether you're in the healthy state 1: or the six state especially given a 1: choice at the same price the risk-averse 1: consumer would prefer to have 1: uncertainty in the full insurance Full insurance payout 2: contract remember they are equal to each
other i prime i sub h prime and i sub s 2: prime are equal so from a couple of 2: slides back we saw that i sub h prime in 2: the healthy state your income would be 2: you know your healthy income plus a zero 2: payout - the premium are with full 2: insurance that should equal income when 2: you're sick plus Q that benefit that 2: payout - the premium R which you always 2: pay so what this is kind of showing you 2: when you get rid of the R from both 2: sides and get rid of the zero it's 2: simply the fact that Q the payout is 2: what you have to add to I sub s in order 2: for it to be equal to the healthy income 2: I sub H in other words the definition of 2: a full insurance contract is when the 2: payout q is equal to the difference 2: between IH and is so it should fully 2: compensate you for those two sources of 3: income loss in the six state so that was Actuarially fair insurance 3:
you know paying out exactly the expected 4: value of what they're getting in and on 4: the flip side the insurance company 4: isn't really making any profit or loss 4: from this from this scenario on the 4: consumer side the reason that they 4: benefit is simply because they get to 4: have a reduction in uncertainty although 4: the expect 4: value is equal so the reason that the Insurance and risk aversion 4: risk-averse consumer would ever purchase 4: insurance is that despite the fact that 4: it is the same especially under an 4: actual really fair contract despite the 4: fact that the expected benefit is equal 4: to the premium that you have to pay out 4: it's better to do to know with certainty 4: what the income is going to be so in 4: other words the risk of risk consumer 4: they pay the premium all the time even 4: when they're healthy but they're willing 4: to sacrifice that premium when they're
healthy in order to avoid or ease the 5: bad times in the 60 in the potential 5: outcome of a six date on the flip side Insurer profits 5: for the insurance company you know it 5: their experience mirrors the consumers 5: experience but from the other side from 5: the insurance company's perspective 5: they're receiving premiums as revenues 5: so premiums are still equal equal to our 5: the payout is equal to Q but from the 5: insurance company's perspective it's you 5: know the potential cost of them having 5: to pay out to the enrolled individual 5: and the probability of that payout is 5: equal to P you know the probability that 5: the individual is sick so their expected 5: profits where profits are denoted as pi 5: it's going to be the premiums that 5: they're receiving minus the prompt the 5: expected value of the payout so in Fair and unfair insurance 5: actually fair insurance their expected
