Question

Determine if completeness and transitivity are satisfied for the following preferences defined on x = (x1, x2) and y = (y1, y2). (Hints: 1- You have to use z = (z1, z2) to prove or disprove transitivity. 2- You can disprove by a counter example) — x ≽y iff x1 > y1 or x1 = y1 and x2 > y2.

Answer #1

Determine if completeness and transitivity are satisfied for the
following preferences defined on x = (x1, x2) and y = (y1, y2).
(Hints: 1- You have to use z = (z1, z2) to prove or disprove
transitivity. 2- You can disprove by a counter example) — x ≽y iff
x1 > y1 or x1 = y1 and x2 > y2.

Determine if completeness and transitivity are satisfied for the
following preferences defined on x = (x1, x2)
and y = (y1, y2).
x ≽ y iff x1 > y1 or
x1 = y1 and x2 >
y2.
(Hints: 1- You have to use z = (z1, z2) to
prove or disprove transitivity. 2- You can disprove by a counter
example)

Consider a two-good economy c = (c1, c2) where the goods can
only be consumed in positive integer choices, that is c ∈ Z^2 and c
≥ 0. Consider the following three consumption bundles, x = (2,1), y
= (α, 2), z = (2, β).. These are the only three consumption bundles
Anne can choose from. Anne’s preferences are such that x ≻ y and y
≻ z, where “≻” means strict preference. Anne’s preferences are
complete and satisfy transitivity...

Consider the set V = (x,y) x,y ∈ R with the following two
operations: • Addition: (x1,y1)+(x2,y2)=(x1 +x2 +1, y1 +y2 +1) •
Scalarmultiplication:a(x,y)=(ax+a−1, ay+a−1). Prove or disprove:
With these operations, V is a vector space over R

Write vectors in R2 as (x,y). Define the relation on R2 by
writing (x1,y1) ∼ (x2,y2) iff y1 − sin x1 = y2 − sin x2 . Prove
that ∼ is an equivalence relation.
Find the classes [(0, 0)], [(2, π/2)] and draw them on the
plane. Describe the sets which are the equivalence classes for this
relation.

Let A = R x R, and let a relation S be defined as: “(x1 , y1 ) S
(x2 , y2 ) ⬄ points (x1 , y1 ) and (x2 , y2 ) are 5 units apart.”
Determine whether S is reflexive, symmetric, or transitive. If the
answer is “yes,” give a justification (full proof is not needed);
if the answer is “no” you must give a counterexample

4.4-JG1 Given the following joint density function in Example
4.4-1:
fx,y(x,y)=(2/15)d(x-x1)d(y-y1)+(3/15)d(x-x2)d(y-y1)+(1/15)d(x-x2)d(y-y2)+(4/15)d(x-x1)d(y-y3)
a) Determine fx(x|y=y1) Ans: 0.4d(x-x1)+0.6d(x-x2)
b) Determine fx(x|y=y2) Ans: 1d(x-x2)
c) Determine fy(y|x=x1) Ans: (1/3)d(y-y1)+(2/3)d(y-y3)
d) Determine fx(y|x=x2) Ans:
(3/9)d(y-y1)+(1/9)d(y-y2)+(5/9)d(y-y3)
4.4-JG2
Given fx,y(x,y)=2(1-xy) for 0 a) fx(x|y=0.5) (Point Conditioning)
Ans: (4/3)(1-x/2)
b) fx(x|0.5

N particles can move in plane (x,y).
Write down coordinates and momenta of all particles forming the
phase space and determine number of degrees of freedom
s.
(a) x1, px1,
y1, py1,
x2, px2,
y2, py2,…,
xN, pxN,
yN, pyN ,
s=2N
(b) x1, px1,
x2, px2, …,
xN, pxN ,
s=2N
(c) y1, py1,
y2, py2,…,
yN, pyN ,
s=2N
(d) x1, y1,
x2, y2, …,
xN, yN ,
s=2N
and why you choose

Let there be two goods; coco puffs and grits. Assume that the
preferences satisfy basic axioms and assumptions and they can be
represented by the utility function u(x1,x2)= x1x2.
Consider two bundles X= (1,20) and Y=(10,2). Which one do you
prefer?
Use bundles X and Y to illustrate that these tastes are in
fact convex.
What is the MRS at bundles X and Y respectively?
Now consider tastes that are instead defined by the function
u(x1,x2) = x1^2 + x2^2...

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