Instructions. You are encouraged to work in groups, but everybody must write their own

solution to the problem that is for grade. Good Luck!

1. (For Grade) There are finitely many states s 2 S. The set of outcomes is [0; 1), the

amount of consumption. Consider an expected utility maximizer with utility function

u (c) = p

c. Suppose that for each state s 2 S, there is an asset As that pays 1

unit of consumption if the state is s and 0 otherwise (these are called Arrow-Debreu

securities). Suppose also that we know the preference of the decision maker among

these assets and the constant consumption levels; e.g., we know how he compares an

asset As to consuming c at every state.

(a) Derive the decision maker's preference relation among all acts from the above

information.

(b) Assume that the decision maker has a fixed amount of money M, which he cannot

consume unless he invests in the Arrow-Debreu securities above, assuming that

these securities are perfectly divisible, and the price of a unit of As is some ps > 0.

Derive the demand of the decision maker for these securities as a function of the

price vector p = (ps)

s2S

:

2. Ann is an expected utility maximizer, but she does not know her preferences, which

she can learn by costly contemplation. To model this situation, take S = [0; 1], and let

Z R be a finite set of consequences with at leat two elements. Assume that Anns

von Neumann utility function is

u (z) = z 8z 2 Z;

and her belief on S is represented by uniform distribution. For any n and some fixed

c > 0, by spending cn utilities, Ann can obtain a partition

Pn = f[0; 1=2

n

];(1=2

n

; 2=2

n

); : : : ;(k=2

n

;(k + 1) =2

n

); : : : ; [(2n1) =2

n

; 1]g

and observe the cell In (s) 2 Pn in which the true state s lies. After the observation,

she assigns uniform distribution on In (s) and can choose an act f : S ! Z under the

new belief. Her eventual payoff is u (f (s))cn. Now imagine that, given any two acts

f and g, Ann first chooses n and, after observing the cell in which s lies, she chooses

one of the acts f and g. She does so in order to maximize her expected payo minus

the cost cn, knowing all along that she will choose one of the acts f and g optimally

based on her observation. Write f s g if Ann may end up choosing f when the true

state happens to be s. Check which of the postulates P1-P5 of Savage is satisfied by

s for any fixed s.

3. Under the assumptions P1-P5, prove or disprove the following statement

(a) For any partition A1; : : : ; An of S, and for any acts f; g 2 F,

[f g given Ak for all Ak] ) f g:

(b) If A _

1B1, A _

2B2, and A1 \ A2 = ?, then A1 [ A _

2B1 [ B2.

(c) For any given event D, define "_ given D" by A_ B given D if A \ D_ B \ D.

The relation _ given D is a qualitative probability.

Evaluate the improper integral or state that it is divergent 8 312 - 21 O A. - In2 OB. 2In 3 OC. , In 3 OD. - - In 3Find the general solution of the equation y'(1) + 2y = 15 O A. y = F + Ce 21 O B. Y= 2 + Ce - 21 O C. y = 15 + Ce 21 15 OD. y= +e21 + Ce - 21EXERCISES 1. Returns to scale in production: Do the following production functions exhibit increasing, constant, or decreasing returns to scale in K and L? (Assume A is some fixed positive number.) (a) Y = K1/21 1/2 CRS (b) Y = K2/312/3 (c) Y = K1/3/1/2 1) Y = K+L )Y=K+ K1/311/3 ")Y = K1/312/3 + APPS g) Y = K1/31213 - AIR Write each produUse the graph to answer the following questions. 350 300 250 200 Real GDP 150- 100 50 100 200 300 400 500 600 Capital If capital is increasing, then O the production function will shift upward. the production function will shift downward. there is an upward movement along the production function. there is a downward movement along the production function. there is an upward shift of the production function and an upward movement along the production function.Orange Coast Coll... 5) Apple 5) iCloud S) Yahoo Question 32 1 pts Consider the production function shown in the figure below. An increase in capital will result in Output 50 45 40 0 10 20 30 40 50 60 70 80 90 100 Labor Input a downward shift in the production function. the slope of the production function becoming flatter. a leftward movement along the production function a rightward movement along the production function itan upward shift in the production function. 1 pts