Solutions to Problems Chapter 1 DecisionMaking Under Uncertainty

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Solutions to Problems Chapter 1 Decision-Making
Under Uncertainty

• Exercise 1
• (a) There are 27 different functions
• (b) One can draw a 3-dimensional space for f(s1),
  f(s2), f(s3) which each three cases
• (c) Let (p?,p?,p?) be a probability distribution
  over the consequences. See the next slide for a
  plot.

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Probability distribution
p?
1
2/3
1/3
1/3
2/3
1
p?
3
Exercise 3

• U(x1,x2)pu(x1)(1-p)u(x2). u(.) is concave.
• The MRS(x1,x2)p/(1-p). u(x1)/u(x2)

x2
U(x1,x2)
p/1-p
x1
0
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Exercise 3 (continued)

• (b) V(p1,p2,p3)p1u(x1)p2u(x2)p3u(x3)
• pi?0 and p1p2p31
• For constant utility levels k we can rewrite
• p3(k-u(x2))/(u(x3)-u(x2))
  (u(x1)-u(x2))/(u(x3-u(x2)).p1

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Exercise 3 (continued)
p3
1
Higher levels of utility
p1
0
1
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Exercise 4(a)

• A decision maker ranks lotteries according to
  V(?,?2)?(10-?)
• (a) Lottery p pays 10 with probability 1 Lottery
  q pays 10 or 20 wit probability ½
• ?xpx1(1-p)x2, ?x2p(x1- ?x)2(1-p)(x2- ?x)2, so
  ?p10, ?p20, ?q15, and ?q225, so Vp100 and
  Vq75

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Exercise 5(b)

• (i) Ra(x) -b lnA
• (ii) Ra(x) ?/x
• (iii) Ra(x) 1/x
• (iv) Ra(x)-2a/(2ax1)

• Rr(x) -x b lnA
• Rr(x) ?
• Rr(x) 1
• Rr(x) -2ax/(2ax1)

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Exercise 6

• u(x)x-0.05 x2 and initial wealth W0lt10.
• Two assets a bond has a price 1 and returns 1
  unit, a stock costs qlt1 and returns 2 in state 1
  and 0 in state 2. The probability of state 1 (2)
  is equal to 0.5
• Amount invested in stock is a, in bonds b
• Pay-off in state 1 is 2ab, in state 2 b

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Exercise 6 (continued)

• Max V(a,b)0.5u(2ab)0.5u(b)
• Subject to qab?W0lt10
• V(a,b)0.5(2ab)-0.025(2ab)20.5b-0.025b2ab-0.0
  25 (4a24abb2b2)ab-0.1a2-0.1ab-0.05b2
• La(W0-qa)-0.1a2-0.1a(W0-qa)-0.05(W0-qa)2
• ?L/?a1-q-0.2a-0.1W00.2qa0.1q(W0-qa)0
• (0.2q-0.1q2-0.2)q-10.1W0-0.1qW0
• a(10-W0)(1-q)/(q2-2q2)