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a.

 


 

The likelihood probability of the discrete event will be

 

model

 


 

n possible events, i=1,2,?n

 

Pi is the probability of occasion a occurring

 

0<Pi<1

 

P1+P2+P3+?.Pn=1

 

These probability can either be objective or subjective depending on the

 


 

?The expected valuetherefore is ? ? Weighted average of possibilities, weight is

 


 

probability ? A total of the possibilities times probabilities ? x={x1,x2?xn} ? P={P1,P2,?

 

Pn} ? E(x) = P1X1 + P2X2 + P3X3 +?.PnXn

 

80% chance of earning $25/month

 

30% change of $16/month ? U(Y) = Y0.5 ? Expected utility ? E(U) = P1U(Y1) +

 

P2U(Y2) ? E(U) = 0.8(25)0.5 + 0.3(16)0.5 = 0.8(25) + 0.3(16) = 44 16 ? Note that expected

 

utility in this case is very different from expected income ? E(Y) = 0.8(25) + 0.6(16) =

 

1960

 

Ramsey discounting rate recipe, in which the rebate rate connected to net advantages at

 

time t, ?t,equals the whole of the utility rate of markdown discount rate (?) and the rate of

 

development in utilization between t what's more, the present (gt), weighted by the versatility

 

of minor utility of utilization (?)

 

U=ln(y)

 

? EUa = ln(25) = 10.82 ? EUb = 0.5 ln(16) = 10.31 ?

 

Job (a) is preferred

 

Based on income the best career to choose is Cardiac Surgeon

 

b.

 


 

Where:

 

Ct = net cash inflow during the period t

 

Co = total initial investment costs

 

r = discount rate, and

 

t = number of time

 

periods

 


 

Cardiac Surgeon -10+2+15+30=37

 

NPV= (37/1.8)+10 = 30.55

 

Professional Skier 5+7+7=8 = 27

 

NPV=(27/1.8)-5 = 10

 


 

c.

 


 

= Cardiac Surgeon -10+2+15+30=37

 

NPV= (37/2.8)+10 = 23.21

 

Professional Skier 5+7+7=8 = 27

 

NPV=(27/2.8)-5 = 4.6428

 

Additional Problem #2

 

a.

 

model

 


 

The likelihood probability of the discrete event will be

 

n possible events, i=1,2,?n

 

Pi is the probability of occasion a occurring

 

0<Pi<1

 

P1+P2+P3+?.Pn=1

 

These probability can either be objective or subjective depending on the

 


 

?The expected valuetherefore is ? ? Weighted average of possibilities, weight is

 


 

probability ? A total of the possibilities times probabilities ? x={x1,x2?xn} ? P={P1,P2,?

 

Pn} ? E(x) = P1X1 + P2X2 + P3X3 +?.PnXn

 

80% chance of earning $25/month

 

30% change of $16/month ? U(Y) = Y0.5 ? Expected utility ? E(U) = P1U(Y1) +

 

P2U(Y2) ? E(U) = 0.8(25)0.5 + 0.3(16)0.5 = 0.8(25) + 0.3(16) = 44 16 ? Note that expected

 


 

utility in this case is very different from expected income ? E(Y) = 0.8(25) + 0.6(16) =

 

1960

 

Ramsey discounting rate recipe, in which the rebate rate connected to net advantages at

 

time t, ?t,equals the whole of the utility rate of markdown discount rate (?) and the rate of

 

development in utilization between t what's more, the present (gt), weighted by the versatility

 

of minor utility of utilization (?)

 

U=ln(y)

 

? EUa = ln(25) = 10.82 ? EUb = 0.5 ln(16) = 10.31 ?

 

Job (a) is preferred

 

Based on income the best career to choose is Cardiac Surgeon

 

b.

 

In a Utility bad state ? U[Y- L + q ? pq - t]

 

? E(u) = (1-p)U[Y ? pq ? t] + pU[Y-L+q-pq-t]

 

? Solve this to, let t=0 (no loading costs)

 

? E(u) = (1-p)U[Y ? pq] + pU[Y-L+q-pq]

 

? simplify this by Maximizing the utility by choosing an optimal q

 

? dE(u)/dq = 0

 


 

c.

 


 

Now, assume that you are risk averse and that your utility from a certain

 


 

income amount is the square root of that income amount. E.g. utility from certain

 

income of 25 equals 5. Now, answer part A again but with this new (risk averse) utility

 

function, rather than assuming risk neutrality. ? E(u) = (1-p)U[Y ? pqk] + pU[Y-L+q-pqk]

 

? dE(u)/dq = (1-p) U' (y-pqk)(-pk)

 

? + pU'(Y-L+q-pqk)(1-pk) = 0

 

? p(1-pk)U'(Y-L+q-pqk) = (1-p)pkU'(Y-pqk)

 

? p terminate on each side 82

 


 

? (1-pk)U'(Y-L+q-pkq) = (1-p)kU' (Y-pkq)

 

? (a)(b) = (c)(d)

 

? Since k is more than 1, can illustrate that

 

? (1-pk) < (1-p)k

 

? Since (a) is less than (c), must be the case that

 

? (b) is more than (d)

 

? U'(Y-L+q-pkq) > U'(Y-pkq)

 

? Since U'(y1) is more than U'(y2), must be that y1that is less than y2

 

d.

 

E(u(I)) = u(25) ? 0.8 + u(16) ? 0.3 = 20 ? 0.5 + 4.8 ? 0.5 = 22.4.

 

Risk Premium = E(I) ? E(I ) = 20 ? 4.8 = 15.2.

 

0.5 * log(25) + 0.5 * log(16) = 1.15,

 

e. Take your answer to part D, the discount factor that equates both career paths

 

assuming the utility function in part C. Assume now that you can fully insurance your income

 

stream in period 1 for both careers, with a fair premium. At the discount rate that solves part D,

 

now with full insurance for both career paths which career will you choose? At this discount

 

rate, how much would you be willing to pay for the full insurance contract for the career you

 

choose?

 

Answer

 


 

The solution to this problem can be solved by entering the total yearly income flow plus

 

the discount rate into the cash flow book of a standard financial calculator and evaluate the net

 

present value. Assuming that a 30% discount rate, then the income flow should be valued at

 

$26. Alternatively, if a discounting rate is valued at 80%, the value of income flow should be $4.8

 

Additional Problem #3: Death Spiral in Graphical Framework (Week 4A)

 

a.

 


 

Given that the initial price the firm decides to set equals 10, what is the

 


 

initial quantity of insurance sold to employees? What are the average and marginal costs

 

at that quantity?

 

P = 20 - .8 Q

 

Qmax = 20

 

MC = 20 ? 1.2Q AC = 20 - .6Q

 

P=10

 

10=20-8Q

 

8Q=10

 

Substitute

 

Q=1.25

 

MC= 20-1.2(1.25)= 18.5

 

AC= 20-6(1.25)= 12.5

 

b.

 

Y

 


 

I

 


 

ear

 


 

ncome

 

1

 


 

$

 

3,000

 


 

2

 


 

$

 

4,000

 


 

3

 


 

$

 

6,000

 


 

4

 


 

$

 

1,000

 


 

c. The welfare is increasing, Assume we watch a protection market that contains an

 

arrangement of back up plans j = 1; 2; ::; J who each o?er one protection arrangement.

 

These arrangements shift in their premiums (pj ) and the liberality of scope o?ered (gj ) 5. In

 

this market, every buyer i has private data, i F(), about his inclinations, and utilizing this

 

privatedata, selects in the arrangement that amplifies expected utility:

 

EUij = igj pj + "ij where "ij is a stun to the utility i gets from arrangement j. Note that

 

heterogneity in suggests determination.

 

Since dEUij gj = i , high shoppers are all the more eager to exchange pay for benefitts

 

and will have a tendency to enlist in liberal protection arranges

 


 

 


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