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Sample Final

Duration: 90 min

Out of 40 pts

1. a) Let  X  = {x ∈  R+ :  x  ≤  5}. Prove that  X   is compact.

b)Let f   : X  → R, such that for each  x ∈  X ,  f (x) = 2x2. Prove that  f   is

continuous.

c) Argue whether or not  f  attains its maximum on set  X .

2. Think of a consumer who has a current wealth of  I . This agent would

live and consume for two periods. He should decide on how to choose  c1,  c2,

subject to  c1  ≥  0,   c2  ≥  0. He invests his money in a bank with an interest

rate of  r, that is if you invest $2 today you would get $2(1+ r) tomorrow. He

has the following utility function over the consumption flow to be maximized:

u(c1, c2) = 2lnc1 + 3lnc2

a) Formulate the constrained optimization problem of this agent.

b) Check if the objective function is concave and constraint functions are

convex.

c) Write down the Kuhn-Tucker conditions for the above problem.

d) Find the solution of this problem by solving for Kuhn-Tucker condi-

tions.

Now assume that our agent lives for 3 periods (0, 1, 2). So he has to

optimize for  c0, c1, c2. Suppose he has the following utility function over the

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consumption flow to be maximized:

u(c0, c1, c2) = c1/20   + 2lnc1 + 3lnc2

e) Formulate the Bellman equation that links period 0 to period 1.

f) Notice that you have alraedy solved the problem for periods 1 and 2 at

part (c).

Now suppose that a wealth of  w0 is avaliable at the very beginning (t = 0).Given your solution from (c), write an expression for the value functions you

specified at (e) in terms of only  w0

g) Now, by using your answer to part (f), also solve the first period prob-

lem and obtain an expression for  c0,  c1  and  c2  only in terms of  I  and  r.

3. Let   M   =  {1, 2, 3}  and   W   =  {a,b,c}. Let the preference profile   R  be

as follows:

R1   R2   R3

a b c

c a b

b c a

Ra   Rb   Rc

3 1 2

2 3 1

1 2 3

Find the men-optimal and women-optimal stable matchings via Gale-Shapey Algorithm and check that those are indeed stable.

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