econ 6100 5/14/2021 section 12
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ECON 6100 5/14/2021
Section 12
Lecturer: Larry Blume TA: Abhi Ananth
Problem 1 (2011 June V). Consider an economy in which there is one public good (x) and one private good
(y). There are I individuals, indexed i = 1, . . . , I (with I � 2). Individual i has an endowment ai > 0 of
the private good, and none of the public good. The total endowment of the private good, (a1 + · · ·+ aI),is denoted by a. The public good can be produced from the private good, using a production function,
h : R+ ! R+. Assume that h has the following form: h(z) = z for z 2 R+.
Each individual’s consumption set is R2+ and consumer i’s preferences are represented by a utility func-
tion:
ui(x, yi) = fi(x) + gi(yi) for (x, yi) 2 R2
+
For each i 2 {1, . . . , I}, the functions fi and gi are assumed to satisfy:
(A1) fi(0) = 0; fi is increasing, strictly concave and continuously differentiable on R+.
(A1) gi(0) = 0; gi is increasing, strictly concave and continuously differentiable on R+.
(A1) fi(a) < gi(ai) and f 0i (0) > g0i(ai)
(a) Let (x, y1, . . . , yI) � 0 be a Pareto Efficient allocation. Show that:
I
Âi=1
f 0i (x)g0i(yi)
= 1
(b) Let (c1, . . . , cI) be a voluntary contributions equilibrium, with ci 2 [0, ai] for each i 2 {1, . . . , I}. The
associated allocation (x, y1, . . . , yI) is defined by:
x =I
Âi=1
ci and yi = ai � ci for all i 2 {1, . . . , I}
(i) Show that we must have ci < ai for each i 2 {1, . . . , I}, and ÂIi=1
ci > 0.
(ii) Using (i), show that the allocation (x, y1, . . . , yI), associated with a voluntary contributions equi-
librium (c1, . . . , cI), cannot be Pareto Efficient.
(c) Let (c1, . . . , cI) be any voluntary contributions equilibrium, satisfying (c1, . . . , cI) � 0, with associated
allocation (x, y1, . . . , yI). Let (x0, y01, . . . , y0I) be any Pareto Efficient Allocation satisfying (x0, y0
1, . . . , y0I) �
0. Can x � x0?
Problem 2 (2009 Aug III). Consider an economy with two consumers, A and B and two assets, 1 and 2.
There are three units of asset 1 and three units of asset 2 in the economy. The initial endowment of A at
t = 0, is given by (eA1
, eA2) = (2, 1), and the initial endowment of B at t = 0 is (eB
1, eB
2) = (1, 2). The price of
asset 1 is q1, the price of asset 2 is q2 = 1.
At t = 1, there are two possible states S = {w1, w2}, which occur with equal probability. The payoff
matrix is given by:
A =
2 1
0 1
�
1
0
-
? / fiyai)EfiCgicai)
I- "
→ gilx)€> ¥ai
-
" →
,-
#¥0:=
wrath* =0 -
-
-
Consumers are both expected utility maximizer with utility for state-contingent wealth x given by
uA(x) = 5 ln x + 2
uB(x) = 13x
(a) At t = 0, the two consumers choose portfolios of assets so as to maximize their expected utility of
state-contingent consumption. State the optimization problems of the two consumers at t = 0.
(a’) Suppose q1 = 5
2. Draw the budget constraint of consumer A. What is the optimal choice of consump-
tion in state w2 for this consumer? Derive the set of values of q1 for which the budget sets of both
consumers are bounded.
(b) For the set of values of q1 derived in part (a’), solve the optimization problems of both consumers. Set
up the conditions for a market equilibrium and derive the equilibrium consumption and asset prices.
Illustrate the equilibrium in an Edgeworth box.
(c) Which of the two consumers is fully insured in equilibrium? Show that this consumers will be fully
insured in equilibrium for any distribution of initial endowments such that: eA1> 0, eA
2> 0, eB
1> 0,
eB2> 0, eA
1+ eB
1= 3, eA
2+ eB
2= 3, and eA
1+ eA
2 3.
(d) New research has uncovered a third state, w3 which can occur at t = 1 with probability 0.2. States w1
and w2 are still considered to be equally probable. The payoff matrix is now
A =
2
42 1
0 1
1 2
3
5
Is it possible to determine whether the equilibrium of this economy is Pareto-optimal without actually
computing it?
(d’) How would your answer to (d) change if there were a third asset and the payoff matrix, for some
r 2 R1+, is now:
A =
2
42 1 3
0 1 1
1 2 r
3
5
2
F- a -- Elna? + Yulntxv
↳#u=Xi+✗r--fÉ
/ ✓ Xu-b-
÷-
0 Let & endow = a, ,az . ; a.+94-3.
B endow =3 -91,3-92.
u"Ñ" (0,1 ,o) ① Rank (A) =L .
511--3
-
- + = tin ?HIE
t
9=33-9
.
✓
70
① a) PO allocation ⇒ solve social planners problem w/
Y, Pareto weight Q ,- . -
,OI : Ii Oi --1 .
max Ii Oiffifxtgifyi)]✗itSt .
✗ + Fyi E a -- - (A)
✗70 - . - (µ)Yi 70 - - - (µi)
kT¥s :(x) : Ii Oififx) - ✗ +µ -0
Hi Hi) : Oigifyi) - ✗ + µi=0'
Iueq .
const .
Ca -✗ - 7-iiyi)X=O , µ✗=O , µiYi=0 ,#
✗, µ , Mi70
we know ,y,>o ⇒ µ
- -0 , Hi
✗70 ⇒ µ=O ,Hi .
TOG : Ii Oififx) = A = Oigifyi) - - - -
Hi1 -
IiOifi¥# = 1
3=ÉE= Ii Oifitx) → Take sum
ojgjyj= 't
'
outside→ Orgoityk) ,
Hk .
⑨ UMPG) :
[email protected]_toc.fi'
/ 7¥,g- +a.) - gitai -a)- little' -0 .
Ineg . j C-s ,non - neg .
①"
a. <ai , Iti"
Suppose a- =ai → By CS → µi=0 ; ✗ i30
• e : t.IE#+E-i07-x:--0'
30
f- i '(¥g.g +a) Zgi'G) .
⇒ →fila > fiE¥ij- > fifty.
# g- +a:) > gild>8¥97I¥IiG"" \ ?¥g+ai ↳ concavity
" concavity glo) -0a-- Fai > ¥¥j+9i ffo)-0
Ith flag - to) >z.fi/z).::..-------#.-
¥,¥, > ¥¥i; 71 .
c
✓- r
fifa) > gilai) → contradiction"Ia . > 0
"
:
↳ Suppose 9--0 , Hi ⇒ Xi --0 , Hi.
Toc ⇒ f- i'G) -gita:) + ai =O-
T.IO Contradiction-
① From i :a. < ai iti ⇒ Xi --0
Some i : Ci > 0 ⇒µi=O .
(atleast1) .↳ Foc : f-
'
fIj±ig+ci) - g:(a) = 0For these i -
For others TOC : fill f) - giaaa.it/Ui--O .
⇒oefi¥=1 -11<-1gila:)
gil
ñE : ¥ ftp.YY?-a, = I"
+ I "
i.ci?0i:I--0y.
c- 1 .
7--1 . (In PO -12> (a) .
Contradiction.
(c) Shown above :
HE : Ii !÷g >1 .
Po :u = 1 .
Fi : tig.fi#, > fj¥¥, -1 .
•.
1- Concavity .
ti¥¥, >1 . ⇒II.
② a) 91--2.5
UMP(A) = Max
✗a.zahttxit) +
lnfx.rs/s.t.qzit-2EE2q+qzCOEXiAE22,A+zit✓
0<-42*-4 zit✗A 70
Bca = { 2A : 2.5-zi-2-i22.IT?;?#when oh --2-5
qz --1
t
\ 22
• 7 Thus,the BC is unbounded !
✓'
'•- ✗ Optimal choice of ⇐ is
loft ofi. -a .
. I [- 2,
- .
-5 \, slope of brown
> ¥.
- '& > -1£
1•⇒t
In read . ofbdd ofboth agents
' constraints
<→ affine transformb) UMP A : M" "
i. +a.is. -1 . q , 9A -122A 12g ,
-11 → 2.a-= 2g ,
-11 -qzt-
0 Exit 2- It
Max lu( 4+2%+11 -2g) >it) + bust
2)*
= Lq , -11*
i - i-*= &%¥!?
✗it't= 2
,
A- * and X,
☒= 22,A*+z*
Do the same for B to compute zB* . (Linear system withg.<2)
then,use market clearing to compute q, :
off = 1 , zit*=D
, 21=3 ; Xf*=X¥ =3
2 ,B*= 3 ; 2¥ = 0 ; x,B*= 6 ; ✗{*
=D .
Edgeworth representation : I
Uz = 2211-2<1-22
"""'"""-
= 2 (21+22) .
Optimum
1