数学exam代考   Question A1 from 2019/20 exam paper:

Question A1 from 2019/20 exam paper:

数学exam代考 Mark has the following utility function: u(w) = ln(w), where w measures his terminal wealth and ln is the natural logarithm function.

Mark has the following utility function: u(w) = ln(w), where w measures his terminal wealth and ln is the natural logarithm function. Mark is considering the following prospects:

P1(.2, £2700, £400) P2(.4, £1400, £500)

• Use the Expected Utility Theory (EUT) to compute the expected utility of each  数学exam代考

[5 marks]

• Compute the expected value of each

[5 marks]

• According to the EUT, which prospect should Mark choose?

[5 marks]

• Compute the certainty equivalent for each

[5 marks]

Answer: a)  数学exam代考

U(P) = probability-weighted average of the possible utility levels U(P1) = 20% × u(£2700) + 80% × u(£400)

Since u(w) = ln(w), this becomes:

= 20% × ln(£2700) + 80% × ln(£400)

= 20% × 7.901 + 80% × 5.992 =   6.37

U(P2) = 40% × u(£1400) + 60% × u(£500)

= 40% × ln(£1400) + 60% × ln(£500)

= 40% × 7.244 + 60% × 6.215   = 6.63

b)

E(P) = probability-weighted average of the possible wealth levels

E(P1) = 20% × £2700 + 80% × £400 = £860 E(P2) = 40% × £1400 + 60% × £500 = £860

c)

Mark should choose the prospect with the higher expected utility, which in this case is prospect P2.  数学exam代考

d)

Certainty equivalent for P is the sure wealth level X such that U(P)=u(X).

What’s the amount of utility Mark would derive from prospect P1? The answer is U(P1) = 6.37

What sure level of wealth would give Mark an amount of utility equal to 6.37? Let’s set up this equation to find out:

u(X) = ln(X) = 6.37      Solve the equation for X:

X = e6.37 = £584.06 = certainty equivalent for P1

What’s the amount of utility Mark would derive from prospect P2? The answer is U(P2) = 6.63

What sure level of wealth would give Mark an amount of utility equal to 6.63? Let’s set up this equation to find out:

u(X) = ln(X) = 6.63      Solve the equation for X:

X = e6.63 = £757.48 = certainty equivalent for P2  数学exam代考

Remember to pay attention to the following:

EUT: arguments of the utility function are terminal wealth levels.

CPT: arguments of the value function are wealth changes (i.e. gains/losses) relative to a reference point, which is often the initial level of wealth.

• You may need to calculate the wealth changes if initial wealth and terminal wealth levels are given in the  数学exam代考

• g. Initial wealth = $100

• P1(0.5, $250, $50) (in terms of terminal wealth levels)

o z1  = +$150 (=$250-$100)         z2 = -$50 (=$50-$100)

• z1 and z2 are the wealth changes to be used in the valuefunction

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