midterm

Question 1

free disposability property

Answer 1

the economy can rid itself of surpluses of commodities at no cost

Question 2

monotone transformation

Answer 2

A utility function v, where v is the set of N-vectors with nonnegative components, is a monotone transformation of u if v(x) = f(u(x)), for all x, where f is an increasing function.

Question 3

increasing function

Answer 3

A function f: R -> R is increasing if, f(r)> f(s)

Question 4

What is the suprenum of a set of numbers?

Answer 4

the smallest number bigger than or equal to any number in the set or is infinity if htere is no number exceeding every number in the set

Question 5

A sequence of N-vectors (x1, x2…) is Cauchy if:

Answer 5

?? Loosely, xn is Cauchy if its members are arbitrarily close together far enough out in the sequence.

Question 6

What is a subsequence?

Answer 6

Consists of a sequence of the form xn(1), xn(2), xn(3) … xn(k)… such that n(k) < n(k+1)

Question 7

A set of N-vectors is closed if:

Answer 7

lim n -> infinity xn belongs to A for any convergent sequence xn in A

Question 8

A set of N-vectors is bounded if:

Answer 8

there exists a positive number b such that |x| <= b, for all x in A.

Question 9

A set of N-vectors is compact if:

Answer 9

it is closed and bounded

Question 10

If A is a set of N-vectors, the closure of A is:

Answer 10

the set of all limits of sequences in A, including constant sequences always equal to the same point in A

Question 11

A set is closed IFF

Answer 11

it equals its closure.

Question 12

What is the Bolzano-Weierstrass Theorem?

Answer 12

Any sequence in a compact set of N-vectors, A, has a subsequence that converges to a point in A.

Question 13

What is the most important consequence of the Bolzano-Weierstrass theorem?

Answer 13

Any continuous function achieves a maximum and a minimum on a compact set.

Question 14

If the set of feasible allocations for an economy E is compact and nonempty and if its utility functions are continuous, then it:

Answer 14

has a Pareto optimal allocation

Question 15

what is the frontier of the utility possibility set?

Answer 15

this set consists of the vectors of utility levels that the economy can achieve

Question 16

what is the brouwer fixed point theorem?

Answer 16

any continuous function from a nonempty, compact, and convex subset of R^N to itself has a fixed point

Question 17

What is a fixed point?

Answer 17

If f: X -> X is a function, a fixed point of f is a point x element of X, such that x = f(x).

Question 18

What is Walras’ Law?

Answer 18

Walras law says that the market excess demand function is zero. That is for x element of delta p, and y element of n p, for all j and i, then p.((the sum of x - e) - the sum of y) = 0, for all price vectors p, whether it be an equilibrium price vector or not. The law applies if, for all i, the utility functions ui are locally non-satiated.

Question 19

What is an example of a Robinson Crusoe economy (u,e,Y) with no optimal allocation?

Answer 19

A RC economy has an optimal allocation if the utility function is continuous and the set of feasible allocations is nonempty and compact. For example if you have a strict inequality or parentheses rather than a strong/bracket, it might not be closed. Then you could get that say, (1,0) is a feasible consumption allocation, but is not feasible since it’s a dotted boundary line.

Question 20

A firm produces nothing. Could its adjoinment Pareto dominate a locally nonsatiated equilibrium allocation?

Answer 20

No; if the original utility fucntions are locally non-satiated, the original equilibrium allocation is Pareto optimal and therefore cannot be Pareto dominated.

Question 21

Give an example of an Edgeworth box economy that is concave and continuous utility functions and positive e, but has no equilibrium.

Answer 21

Make them bads, not goods. With an endowment, and people who get negative utility from consumption, then the price must be zero. But by the definition of equilibrium, no equilibrium price vector can be zero.

Question 22

Give an example of an economy with a competitive equilbrium that is not Pareto optimal

Answer 22

If some consumer gets no utility from consumption, but both have endowments of one, then a competitive equilibrium might be all ones even though the consumption allocation (0,0), (2,2) Pareto dominates.

Question 23

Give an example of an economy with continuous utility functions, production possibility sets that are closed, and that has no Pareto optimal allocation.

Answer 23

A PO allocation exists if the utility functions are continuous and the feasible allocations is compact. In this case, everything but boundedness (part of “compact”) is assumed. Thus, it must be unbounded. You can do this by making the input-output equation such that y1 <= 0 and y2 >= 0.

Question 24

Show that there exists a Pareto Optimal allocation, given a feasible allocation equation and continuous utility functions.

Answer 24

In order to show that a PO allocation exists, it is sufficient to show that there exists a feasible allocation that maximizes the sum of the utilities of the two consumers. Since the utility function of each consumer is continuous, a maximum exists if the feasible allocations set is compact and nonempty.

Question 25

Show that if each utility function is concave, then the maximum value is concave.

Answer 25

Recall the definition of concavity: f(ax + (1-a)x) > = af(x) + (1-a)f(x). Yeah, the proof is too difficult.

Question 26

Define an equilbrium.

Answer 26

1. for pure trade, consumption is less than endowment 2. the price vector is an element of the reals and is greater than zero 3. each xk solves the maximization problem V(x) subject to the feasibility constraint. 4. for n = 1, … , N, pn = 0 if the sum of consumption is less than the endowment. (excess supply)