Question  1.  Three individuals living in a household have well defined  (complete, transitive) preferences defined on a consumption set X of allocations for the household. The current allocation of the household  is denoted by x  . Let A, B and C  respectively denote the sets of consumption bundles that consist of allocations that each of the three individuals weakly prefer to x  . Using the notation of union and intersection, write down the expression for the following sets of consumption bundles.

1.  Set of allocations that are weakly preferred to x  by all of them.                           (2.5 pts)

2.  Set of allocations that are weakly preferred to x  by exactly one of them.           (2.5 pts)

Question 2. Provide an explicit example for each of the following.

1. A sequence in R that is bounded but  does not converge.                                         (1 pts)

2. Two sets that are not compact but their intersection is compact.                              (1 pts)

3. An intersection of open sets that is not open.                                                            (1 pts)

4. An example of a function where one of the conditions (specify which one) of the Weierstrass theorem fails and where a solution does not exist to the optimization problem. A simple picture is OK.                                                                                                             (1 pts)

Question 3.  Let d(x; y) denote the distance between any two elements x; y 2 Rm .   Given an infinite set A  Rm , answer the following.

1. What property of A (such as being open or closed or convex etc.) would ensure the existence of a number s  such that for every ">0, there exists x; y 2A such that s ¡"<d(x; y)  s  .

(2 pts)

2. Now, given an example of a set A that violates the condition you provided in part (1) for which no such s  as described above exists.                                                               (2 pts)

3. In Part 1 of your answer, does it necessarily follow that there exist x  ; y  2 A such that d(x  ; y  ) = s  ? Why?                                                                                              (1 pts)

Answer. Part 1. Expect answers to such as compact   or bounded .

Part 2. For example, take A = (¡1; 0).

From the definition of a supremum then, every " > 0, there exists r 2 T such that r > s  ¡ ". From the definition of T, there exist x; y 2 A such that d(x; y) = r > s  ¡ ".

Part 3. If compact , answer is yes. If bounded answer is no , take A = (0; 1) and s  = 0  A.

Question 4. Consider the square  X = [0; 100] [0; 100]R2 and let    be a  preference relation (i.e. complete and transitive relation) defined on X .

1. Name the appropriate Theorem and list only the missing assumptions to above environment to conclude that    admits a continuous utility representation.  (You get no points if you include an assumption that is already met).                                                             (2 pts)

2. Now present an additional assumption so that for any x = (x1 ; x2) 2X , there exists a unique   2 R such that (x1 ; x2)  (  ;   ).                                                                           (2 pts)

3.  State the (name) of the Theorem that would allow us to prove Part 2.                   (1 pts)

Question 5. Tom's preferences over consumption bundles on a convex set  X R are such that he  regards a pair of consumption bundles sufficiently close together as being similar and hence indifferent between them. That is, assume that there is a fixed " > 0 (but small). For all x 2 X , his preference relation  satisfies

y 2 B" (x)\X  )  x  y:

Assuming, as usual, that    complete and transitive,   prove that Tom is indifferent between every

pair  x; y 2 X .                                                                                                                        (5 pts)

Question 6. Let S Rm be convex and consider a real valued function  f : S ¡!R. Given  2R, V ()  =  fx 2 Sj f(x) g

is called  the upper-level set (of f at ).

1. Let m = 1 and  say S = [0; 1].  Draw a continuous function for which the upper level set is not convex.                                                                                                                 (3 pts)

2. Now consider the general case where  S could be multi-dimensional. Assume further that f is quasi-concave. Show that V () is a convex set for all   .                                 (2 pts)

Question 7. Two types of human skills, say A and B ,  may be used to  simultaneously produce two outputs, say X and Y . The technology itself is simple converting inputs into outputs in fixed proportions   that is, there exist coefficients vax; vay; vbx; vby  so that for a given level of inputs (a ; b) jointly determine output (x ; y)  of the two goods:  the resulting quantity of X is vax a+vbx b and  vay a+vby b of  Y .