in a group. If you are found to have violated this ort a single answer, you will receive
a zero on the entire assignment and may receive a failing grade in the course.
\+ Refer to pages 1,68-1,69 in Bernell. This assignment has you calculate actuarially fai
", ,' premiums for 10 individuals and the maximum premium they are willing to pay.
premium for the group of l-0 as a whole, you must determine the premium andpremium for the group of l-0 as a whole, you must determine the premium and
which of the 10 individuals would opt to not be insured.
You are encouraged to set this assignment up in Excel. You can use the example in
the text to check if you have your formulas set up corectly.the text to check if you have your formulas set up corectl
This is.& ir€ividual assisnment. That me&$'Be*l8ffinot rThis isft"jffividual assignment. That me6p$@1€ffinot work in groups, share
ansvUffi@;herwise collaborate on wo$fting,f,ku problems. The intent is to
enffe ffqv6u51n do some basic econo&t#Akffilations, notwhetheryou can
get
anftgers b$working in a group. ,f,re
#
,%*1,
d*". Jf,
f,;, ,B[;" entering your answers int@odle in a'quiz' set up to take your&answ#s. gl'%#s
You have the following inro.,fu,ten individuals, A-|:
Pr(Sick)
0.25
Cost of
eing sick
55,ooo
$z5,ooo
s10,000
Slo,ooo
s5,ooo
s25,000
S2o,ooo
s25,000
Sls,ooo
s15,000
0.2s
.20
0.0s
0.05
0.0s
0.2s
0.20
0.10
0.05
s
x
*--
lndividual
A
B
C
D
E
F
G
Hffi
#rd
J-%
.l-
Each individual has the same utility function as given in the text as equation 8.2.
Please note that the 'cost of being sick' is not their wealth if they are sick. It is
much their wealth is reduced if they are sick. This is different from the examp
the text, where you are given wealth if they are sick. Wealth if sick = initial
Part 1: Fully informed insura
For each individual, calculate:
a.
.
The actuarialiy fai
ance company charged
Ievel of expected utility
The loading facto
give the individ
The above willbe
answers in total).
Part 2:
11.If the i
know
actua
ten i purchase insurance?
urance company [1) doesn't know the risk of each individual, but does
pected cost of the group's sickness, and [?] sets a single premium that is
ir for the group as a whole, what would be the premium, assuming all
as the answers to ..., 10b (that is 20
ance company
problem of adverse selection and why a
this continues, with individuals dropping thei
which single individual would be left buying
your answer to 11, which individuals would opt to NOT buy insurance? To
is, you need to compare the certain utility [with the premium) and
that to the expected utility without insurance.
13. If the individuals you've identified in question 12 didn't buy insurance and the
insurance company set a new premium for the group that did buy insurance, what
would that new premium
14. Questions 12 and
market for insurance
insurance as premi
insurance?
would
t insurance.
168 Health Economics: Core Concepts and EssentiatTools
Exmer 8.1
Diminishing
Marginal Utility
of Wealth
Expecled weolth
A function of the
probability of getting
sick, income when
sick, and income when
heatthy
Expecreo U
l
v AND RrsK
The relationship berween utiliry and the purchase of insurance is best illustrated with an
example.
Sarah is heaithy, rvith wealth of $150,000 and a utiliqv level oi400, She has a
75 percent chance ofbeing healthy, anda25 percent chance ofbeing sick. Ifshe gets sick,
her wealth decreases to $ 10,000 and her udliry level goes down to 100. The cost of treating
her illness is $150,000.
Sarah may choose not to purchase insurance and pa1. the full cost of medical care
out ofpocket. Alternatively, she could purchase insurance and pay an actuarially fair pre-
mium. This actuarially fair premium is a function of total weairh and expected wealth,
expressed mathematically in Equation 8.1:
Equation 8.1 Expected wealth (E!7) = [Pr(Healthy) x (Healthy wealth)]
+ [Pr(Sick) x (Sick wealth)]
pX7 = [0.75 x $ 160,000] XXXXXXXXXXx$1 0,0001
: $tzz,'oa
In our example, rhe acruarially fair premium is $37,500 ($160,000 - $122,500). (Pr is
short for probabiliry. )
Assume thar individuals have a utilirv Function that looks like the one in Erhibit
8.1. The utility function can be represented mathematically by Equation 8.2:
E,quatior.r 8.2 u- !7'0 5
Weatth
Chapter 8: The lnsurance Marl
In realiry utiliry is a function of wealth (tU(Wl) only when wealth can be viewed with
certainty. If there is a possibiliry of losing some wealth, economists turn to expected utility
(EIJ) (Equation 8.3). Similar to expected wealth, expected utiliry is a function of the prob-
abiliry ofbeing in a particular state (e.g., sick) and the utiliry ofeach state. In healthcare,
it is reasonable to assume that some level of uncertaintv exists (e.g., timing of sickness,
success of treatment).
Equation 8.3 Expected utiliry (EU) = [Pr(Healthy) x (Healthy utility)]
+ [Pr(Sick) x (Sick utility)]
EIJ =10.75xa00]+ [o.zlx rool
= 325
The actuariaily fair plan, which is based on probabilities, generates an expected wealth of
$122,500 (Equation 8.1) and an expected urlliry of 325.
The expected utility-maximizing consumer witl purchase heatth insurance
if the expected utitity with insurance is greater than the utitity without insurance:
E Urith in.u
n." ) E Uwthout insuranc . kssumPtion) '
In our exampie, an acruariaily fair plan provides a certain wealth of $ 122,500. Vith
wealth equal to $122,500, there is certain utiliry level o{ 350 (calculated by taking the
square roor of $122,500). The certain utiliry level, which is achieved by purchasing insur-
ance, is higher than the expected utiliry of risk taking (350 > 325). Hence, the risk-averse
individual will want to purchase insurance.
Lolort
c Facron
lnsurance is typically not sold at an actuarially fair price. Consumers are willing to Pay
more, and insurance companies need to cover administrative costs and make a profit.
Continuing with the earlier example, the expected udliry with risk taking is 325
(see Equation 8.3). An individual can obtain a certain uriligv of 325 if certain wealth is
$105, XXXXXXXXXX'). What this suggests is that a risk-averse person can be made better off
y paying an insurer an amount above the expected loss to be relieved ofthe risk associ-
ated with not having insurance. The total amount the person is willing to pay for insurance
is $54,375 ($160,000 - $ 105,625). The extra money paid to the insurance company above
the actuarially fair premium is the loading factor or risk premium.
Rrsr LovrR
In the case of a risk lover (an individual who em
aces risk), the marginal utiliry (change
in utiliry) of wealth increases with more wealth (Exhibit 8.2). fusk lovers have a highe
expected utility from the uncertainry ofwealth than from the certainry of wealth (U, > Ur).
Expected utility (EU)
A function ofthe
probabitity of being
n a particular state
(e.g., sick) and the
utility of each state
170 Health Economics: Core Concepts and EssentiaI Tools
Exxretr 8.2
Marginal Utility
of Wealth for a
Risk Love
Herfindahl-Hirschman
tndex (HHl)
A measure of how
equatty market share
is distributed across
companies in a markel
Utitity
Utitity tevel
ased on risk
Utitity level
ased on certainty
Wr W"
Wealth
Simply put, for a risk lover, rhe risk of uncertainq, is worth more (provides more happi-
ness) than certain wea.lth.
Measun
uG CoMpETrnoN tN THE IwsunlrucE MARKET
Economists measure the level of competition in a market with a tool called the Herfindahl-
Hirschman Index (HHI). The HHI is calculated as the sum of squares of firm-specific
market share (see Equation 8.4):
Equation 8.4 *'=*',
N is the number of firms in the market. Each firm, i, has a market share equal to s..
HHI va-lues vary from 0 to 10,000. Values close to 0 suggest a more comperirive
market, and values close to 10,000 indicate a less competitive market. If the market has only
one firm with a 100 percenr market share, then the HHI is 10, XXXXXXXXXXAs the numbe
of firms increases, such that each firm has a smaller market share, the HHI decreases and
moves closer to o.
To calculate the HHI, the researcher must set a threshold of marker control. Fo
example, only firms that control at least 5 percent of the market ma1, be included in the
HHI calculation. The threshold amount is chosen by the researcher.
u2
u1