ECON600 2021
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1. In this problem, we consider a problem in which a decision maker (DM) must compare the optimal values in two optimization problems in the presence of uncertainty. The DMís payo§ depends on an action x chosen from a Önite set A and a nonnegative parameter y . That is, the decision makerís payo§ is a function u : A R+ ! R+ . The value of the parameter is the realization of a random variable Y . The DM does not observe the realization of Y but can observe the realization z of one of two random variables Z1 or Z2 where each Zi is correlated with Y . We interpret Z1 and Z2 as "experiments". For each nonnegative number z, suppose that f(. jz) is a density function, i.e., f(yjz) 0 and R01 f(yjz)dy = 1. Furthermore, suppose that z 7! f(yjz) is convex on R+ for each y e R+ : Upon observing z, the DM solves the problem
maximize Z0 1 u(x;y)f(yjz)dy st x e A
For each z 0, let v(z) denote the optimal value for this problem. For each experiment Zi ; we deÖne the DMís ex ante expected payo§ as
E [v(Zi )] :
If Z1 %C Z2 (the concave order), show that experiment Z2 is more valuable than experiment Z1 ; i.e., show that E[v(Z1 )] < E[v(Z2 )]: You may assume that all relevant optimal solutions exist and that all relevant expectations exist and
are Önite.
Solution: It su¢ces to show that v : R+ ! R is convex. To see this, choose z1 ;z2 e R+ and let
xi e arg x2A(max) Z0 1 u(x;y)f(yjzi )dy;i = 1; 2
Choose t e [0; 1], let z = tz1 + (1 一 t)z2 and choose
x e arg x2A(max) Z0 1 u(x;y)f(yjz)dy
Then
v(z) = Z0 1 u(x;y)f(yjz)dy
< Z0 1 u(x;y)[tf(yjz1 ) + (1 一 t)f(yjz1 )]dy
= t Z0 1 u(x;y)f(yjz1 )dy + (1 一 t) Z0 1 u(x;y)f(yjz2 )dy < t Z0 1 u(x1 ;y)f(yjz1 )dy + (1 一 t) Z0 1 u(x2 ;y)f(yjz2 )dy
= tv(z1 ) + (1 一 t)v(z2 ):
2. A private production economy with L goods and n consumers and m Örms is deÖned by the following objects:
(i) For each i, a utility function ui : R ! R
(ii) For each i, an initial endowment !i e R
(iii) For each i, a closed production set Yi RL satisfying R \ Yi = f0g:
Let N = f1;::;ng and let S N: An allocation (xi ;yi )i2S is S-feasible if xi e R and yi e Yi for each i e S and
X xi = X !i +X yi :
i2S i2S i2S
An N-feasible allocation (xi ; yi )i2S is a core allocation for the private pro- duction economy is there does not exist a coalition S N and an S-feasible allocation (x;y)i2S such that ui (x) > ui (xi ) for each i e S:
A Walrasian equilibrium for the private production economy is a collection (x1 ; y 1 );::; (xn ; yn ) and p e R such that yi e Yi for each i and
(i)
n n n
X xi = X !i +Xyi
i=1 i=1 i=1
(ii) For each i, xi solves the problem
maximize ui (xi ) st p . xi < p . !i + p . yi and xi e R
(iii) For each i,
p . yi p . yi for all yi e Yi :
Show that a Walrasian equilibrium allocation in a private production econ- omy is a core allocation.
Solution: Suppose that (x;y)i2S satisÖes (a) x e R and y e Yi for each
i e S, (b) Pi2S x = Pi2S !i +Pi2S y and (c) ui (x) > ui (xi ) for each i e S .
Then for each i e S, it follows that
p . x > p . !i + p . yi
Therefore
p . x > p . !i + p . y
implying that
p . "i2S(X)x 一 i2S(X)!i 一 i2S(X)y# > 0
an impossibility.
3. Consider an n player "generalized public goods game" with nonempty, Önite strategy sets A1 ;::;An : Let g : A1 . . . An ! R , h1 : A1 ! R;:::;hn : An ! R be functions and deÖne the payo§ function ui : A1 . . . An ! R for each i as
ui (x1 ;::;xn ) = g(x1 ;::;xn ) 一 hi (xi ):
Suppose that (x1 ;::; xn ) e A1 . . . An solves the problem
n
maximize g(x1 ;::;xn ) 一 X hi (xi ) subject to (x1 ;::;xn ) e A1 . . . An
i=1
Show that (x1 ;::; xn ) is a Nash equilibrium of the game with strategy sets Ai and payo§ functions ui :
Solution: Suppose that (x1 ;::; xn ) e A1 . . . An solves the problem
n
maximize g(x1 ;::;xn ) 一 X hi (xi ) subject to (x1 ;::;xn ) e A1 . . . An
i=1
If (x1 ;::; xn ) is not a NE, then there exists an i and yi e Ai such that ui (x —i ;yi ) > ui (x): Therefore,
g(x —i ;yi ) 一 hi (yi ) > g(x) 一 hi (xi )
implying that
g(x —i ;yi ) 一 hi (yi ) 一 X hj (xi ) > g(x) 一 hi (xi ) 一 X hj (x i )
j j
:ji :ji
This is impossible since (x —i ;yi ) e A1 . . . An and (x1 ;::; xn ) solves the optimization problem.
2023-06-30