Question1

Evaluate the problem below

We postulate the utility function

u! :L : u!(x) = a x - [1/2] x B x , (1.1)

where a = (a1,..., aL ) ++

L , and B is an LL symmetric, positive definite matrix.

1.1. Compute the gradient vector u!(x).

1.2. Compute the Hessian matrix D2

u!(x).

1.3. Is the preference relation represented by u! continuous? Homothetic? Quasilinear? No need to

justify your answers.

1.4. Is the preference relation represented by u! convex? Strictly convex? Locally nonsatiated?

Explain your answers.

1.5. In order to develop your intuition, graphically represent the indifference map for L = 2, for the

special case where a1 = a2 a and B = b c

c b

. Separately consider the cases c = 0 and c > 0.

Be as precise as possible.

1.6. You can assume that, for (p, w) >> 0, a solution to the UMAX[p, w] problem exists and is

unique. Without attempting at this point to explicitly solve the UMAX[p, w], show that for w

above a certain level, that you should made explicit, demand is insensitive to increases in wealth, 2

whereas for values of w below this level Walrasian demand is affine in wealth. (Hint: For the

second part, use the Implicit Function Theorem.)

1.7. We now consider a society with I consumers, each endowed with the preference relation

represented by u! in (1.1). We denote by P := ++

L+I

the domain of prices and individual wealth

vectors ( p;w

1

,...,w

I

). Is there a subset of P for which a positive representative consumer exists?

Explain.

1.8. We return to the one-consumer case. Solve the UMAX[p, w] problem for good 1 in the case

where L = 2, a1 = a2 = a and B = b c

c b

, for c > 0. Can one of the goods be inferior at a point

of the domain of the Walrasian demand function? (Hint. Consider the wealth expansion paths in

(x1, x2) space.)

1.9. We now consider a consumer who faces uncertainty. Her ex ante preference relation satisfies

the expected utility hypothesis with von Neumann-Morgenstern-Bernoulli utility function

u :[0,a / b ] , where a and b are positive parameters, and with coefficient of absolute risk

aversion equal to b

a bx . She is facing the contingent-consumption optimization problem of

maximizing her expected utility subject to a budget constraint. Let there be two states of the world,

s1 and s2, with probabilities and 1 , respectively

1.9.1. Comment on her ex ante preferences.

1.9.2. To what extent is her problem formally a special case of the UMAX problem of 1.6

above? Explain

Question2

Graphic, Inc. (Graphic), is a California corporation that sells office copying equipment. Its

Articles of incorporation prohibit Graphic's sale of paper products. Graphic's common stock is

registered for trading on a stock exchange. Frank, Graphic's president, recently signed a contract

with Papco on behalf of Graphic to buy a paper mill owned by Papco. Frank intends to reveal the

contract for the first time at a Board of Directors meeting next week.

Frank recently received a letter from Alice, who owns 9.2% of Graphic's common stock. Alice

has asked to "look at a list of Graphic's shareholders and all contracts signed by Graphic in the

past three months." Frank directed the corporate secretary to write to Alice denying her request,

which was done.

Graphic's accountants advised Frank that Graphic will report a $5 million loss for its current

fiscal year, which will be the only loss in its twenty year history. Frank then sold 100,000 shares

of his Graphic common stock through his broker for $25 per share. The sale included 20,000

shares he had purchased two months ago by exercising a stock option at $22 per share.

Frank called a press conference at which he stated that "Graphic has signed a major contract

and will have other news to announce after its Board of Directors meeting." Alice heard about the

press conference and purchased 5,000 additional shares of Graphic common stock at $28 per

share through her broker. When the news of Graphic's fiscal year loss became public, the price of

Graphic stock declined to $20 per share.

Alice wishes (1) to compel Graphic to make available for her inspection the shareholders' list

and all contracts signed in the last three months, (2) to recover her loss on her recent stock

purchase, (3) to force Frank to disgorge the profits on his stock sale, (4) and to have the Papco

contract declared invalid.

What are Alice's rights and remedies, if any, with regard to (1) through (4) above? Discuss.