課程名稱︰個體經濟理論一
課程性質︰必修
課程教師︰王道一
開課學院：社科院
開課系所︰經濟所
考試日期（年月日）︰103.11.12
考試時限（分鐘）：1420-1720(-1930) (180+130mins)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Exam Time: 11/12 2:20pm-5:20pm. You have 3 hours; allocate your time wisely.
﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌﹌
Part A (42 + bonus 8%): Prediction Markets for Daibuck Mayoral Election
Zi Lu and Gee Lan both care only about their own consumption under different
states of the world. Gee Lan thinks GP will win the Daibuck mayoral election
this year with probability 0.9. Zi Lu thinks there is an equal chance for GP
to win or lose. The preference scaling function of each person is logarithmic:
(u(x)=ln(x)).
1.(2%) What is the expected utility function of Zi Lu and Gee Lan over the
states s=1 (GP wins) and s=2 (GP loses)?
2.(4%) What are the degree relative risk aversion, R(x)? Are they risk
averse, risk neutral or risk loving?
3.(3%) Can we use a representative agent to replace a group of people who
have the same expected utility function as Zi Lu (or Gee Lan)? Why or why
not?
4.(2%) Consider zFuture in Daibuck that offers event futures for each state.
Assuming Zi Lu and Gee Lan both have the same total wealth (W, W), write
down their consumer problems when facing state claim market price ratio
p_1/p_2.
5.(6%) Write down the Lagrangians and derive the Kuhn-Tucker conditions of
these consumer problems. (Hint: Define notation for subjective
probabilities and present your answer using such notation.)
6.(6%) What is the individual demand for each state claim? What is the
market demand for each state claim?
7.(5%) What is the Walrasian equilibrium state claim price ratio?
8.(8%) What is the equilibrium consumption of Zi Lu and Gee lan (c_1, c_2)?
How much can they lose under the worse case scenario compared to auturky
(consuming thier initial wealth)? Is your answer consistent with their
risk preference as found in question 2? Explain.
9.(6%) How would your equilibrium price ratio change if the beliefs of Zi Lu
and Gee Lan change? (Hint: How do your Kuhn-Tucker condition depend on the
beliefs of Zi Lu and Gee lan?)
10.(bonus 8%) Consider figures on the last page. What is required to apply
the above analysis to obtain "market beliefs" on candidates' winning
probabilities? Are inferred market beliefs consistant with other poll
results? Why or why not?
Part B (30%): Riz and Lucy Avery's Preferences
Consider Lucy and Riz Avery who both obey expected utility theory. Lucy
Avery has expected utility function u(x)=min{x-a, l(x-a)} where l≧0 and
a=constant, while Riz Avery has expected utility function satisfying
v'(x)=x^(-r) with r > 0. Both face the ten lottery choices of Holt and Laury
(2002) listed below:
You will roll a ten-sided die and get paid according to your decision
(choice A or B):
┌─────┬─────────┬─────────┬──────────┐
│Decision │Lottery A │Lottery B │Your Choice (A or B)│
├─────┼─────────┼─────────┼──────────┤
│Question 1│ 1 : Gain NT$200│ 1 : Gain NT$385│ │
│ │2~10 : Gain NT$160│2~10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question 2│1~2 : Gain NT$200│1~2 : Gain NT$385│ │
│ │3~10 : Gain NT$160│3~10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question 3│1~3 : Gain NT$200│1~3 : Gain NT$385│ │
│ │4~10 : Gain NT$160│4~10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question 4│1~4 : Gain NT$200│1~4 : Gain NT$385│ │
│ │5~10 : Gain NT$160│5~10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question 5│1~5 : Gain NT$200│1~5 : Gain NT$385│ │
│ │6~10 : Gain NT$160│6~10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question 6│1~6 : Gain NT$200│1~6 : Gain NT$385│ │
│ │7~10 : Gain NT$160│7~10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question 7│1~7 : Gain NT$200│1~7 : Gain NT$385│ │
│ │8~10 : Gain NT$160│8~10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question 8│1~8 : Gain NT$200│1~8 : Gain NT$385│ │
│ │9~10 : Gain NT$160│9~10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question 9│1~9 : Gain NT$200│1~9 : Gain NT$385│ │
│ │ 10 : Gain NT$160│ 10 : Gain NT$ 10│ │
├─────┼─────────┼─────────┼──────────┤
│Question10│1~10 : Gain NT$200│1~10 : Gain NT$385│ │
│ │ │ │ │
└─────┴─────────┴─────────┴──────────┘
1.(3%) Find the von Neumann-Morgenstern utility function v(.) of Riz Avery
and its corresponding degree of relative risk aversion R(x).
2.(2%) Show that for x>a, Lucy Avery exhibit both constant relative risk
aversion and constant absolute risk aversion.
3.(8%) Show that Lucy Avery's utility function is concave: If
x_k=kx_0+(1-k)x_1, then u(x_k)≧ku(x_0)+(1-k)u(x_1). Show that the
inequality is strict unless x_0 and x_1 are both greater/smaller than a.
4.(4%) Show that if Lucy chooses lottery B in Question k, she would also
choose lottery B in Question (k+1).
5.(5%) Show that if a person follows expected utility theory and choose
lottery B in Question k, he would also choose lottery B in Question (k+1).
What is the critical assumption required for the above statement to be
true.
6.(3%) If a<10, show that Lucy would choose lottery A for Question 1~4 and
lottery B otherwise.
7.(4%) Now suppose a=200, l=2, and r=0. Would Lucy choose more or less
lottery A's than Riz? Why?
Part C (28 + bonus 12%): The People-Salmon Problem
Consider the relationship between the People of Daiwan (P) and the returning
salmon A-Shin (A). The people in Daiwan have utility function
u(x)=min{x-a, l(x-a)} with l≧0 and a=constant. A-Shin decides whether it
wants use genuine (e=1) or suspicious (e=0) material to produce food oil, and
this affects whether the national health insurance surplus is y_1>0 (s=1) or
y_0<0 (s=0). π_1(e), the probabitlty of s=1, is π_1(1)=0.9 and π_1(0)=0.5.
The people of Daiwan care about the national health insurance surplus and
design a punishment/reward scheme to induce A-Shin to stop using suspicious
material. Assume the two parties have expected utility:
1 1
U_P=Σπ_s(e)u(y_s-w_s) U_A=Σπ_s(e)v(w_s)-C(e)
s=0 s=0
where v'(x)=x^(-r), r≧0, C(1)=c_1>C(0)=c_0, and A-Shin's outside option (not
being a salmon) yields reservation utility U_A=0. First assume the people of
Daiwan government has access to espionage and abuses it.
1.(2%) Show that the increasing likelihood ratio assumption holds. In other
words, show that for e'>e, π1(e')/π1(e)>π0(e')/π(e).
2.(16%) Write down the optimization problem of the people of Daiwan and
derive the Kuhn-Tucker conditions for its solution. (Hint: You can write
the minimum as a constraint to avoid having a discontinuous point in the
utility funcion, such as max{min(x,y)}=max{x|y>x}, and discuss different
cases depending on whether y_s-w_s>a.)
3.(4%) Suppose l=1 and r>0. Show that the optimal reward/punishment scheme
involves the A-Shin receiving the same reward/payment regardless of the
national health insurance is running a surplus or deficit (i.e. w_1=w_0).
4.(4%) Suppose r=0 and l>1. What is the optimal reward/punishment scheme for
the people of Daiwan?
Now suppose more realistically that the people of Daiwan cannot observe
whether salmon A-Shin use suspicious material.
5.(2%) If r=0 and l>1, what is the optimal reward/punishment scheme for
the people of Daiwan if we assume 0.4(y_1-y_0)≧c_1-c_0?
6.(bonus 10%) If l=1 anf r=1, when y_1 is sufficiently large, design an
optimal reward/punishment scheme for the people of Daiwan to induce e=1.
7.(bonus 2%) Did the people of Daiwan adopt such an optimal
reward/punishment scheme? Why or why not? (What assumption could fail in
real life?)
Figures for bonus question in Part A:
http://ppt.cc/BvcL
http://ppt.cc/alkT