微观经济分析代写 GR5211代写 ECON代写

ECON GR5211

Microeconomic Analysis I

PS 1 – Due: Wednesday September 18 @ 6pm

微观经济分析代写 If the answers are “no” and “yes”, then you should be able to rationalize C by complete and transitive preferences.

Question 1 Three proofs from class  微观经济分析代写

1. Show that a decision maker who makes choices by maximizing a utility function must satisfy properties α and β
2. Wesay that choices satisfy the Weak Axiom of Revealed Preference (WARP) if the following is true: for any x and y in X, if {x, y} ∈ A ∩ B and x ∈ C(A), and y ∈ C(B) then x ∈ C(B). Show that a choice correspondence satisfies WARP if and only if it satisfies the properties α and β.  微观经济分析代写

3. Let“ be a preference  Prove that u : X → R represents “ if and only if:

x “ y ⇒ u(x) “ u(y); and x > y ⇒ u(x) > u(y)

Question 2 Let P be a preference relation on a set X. Assume that P is complete, reflexive and transitive.

Let  the  binary  relation  “_P  represent  P.  Define  two  new  binary  relations  on  X, denoted >_P , by

x ∼_P  y ⇔ (x “_P  y) ∧ (y “_P  x)

x >_P  y ⇔ (x “_P  y) ∧ ¬(y “_P  x)  微观经济分析代写

Prove the following:

1. ∼_Pis symmetric: If x ∼_P y then y ∼_P 
2. ∼_Pis reflexive: x ∼_P 
3. ∼_Pis transitive: If x ∼_P y and y ∼_P z then x ∼_P z. Together, the first three parts have shown that ∼_P is an equivalence
4. Showthat if x >_P  y  and y “_P  z  then x >_P  z

Question 3 Consider a choice problem with choice set X = {x, y, z} and the following choice structures:  微观经济分析代写

1.(Bj,C(·)) , in which Bj = {{x, y}, {y, z}, {x, z}, {x}, {y}, {z}} and C({x, y}) = {x}, C({y, z}) ={y}, C({x, z}) = {z}, C({x}) = {x}, C({y}) = {y}, C({z}) = {z}

2.(Bjj, C(·)) , in which Bjj = {{x, y, z}, {x, y}, {y, z}, {x, z}, {x}, {y}, {z}} and C({x, y, z}) ={x}, C({x, y}) = {x}, C({y, z}) = {z}, C({x, z}) = {z}, C({x}) = {x}, C({y}) ={y}, C({z}) = {z}

3.(Bjjj, C(·)) , in which Bjjj = {{x, y, z}, {x, y}, {y, z}, {x, z}, {x}, {y}, {z}} and C({x, y, z}) ={x}, C({x, y}) = {x}, C({y, z}) = {y}, C({x, z}) = {x}, C({x}) = {x}, C({y}) ={y}, C({z}) = {z}

Note that any set B is a set of subsets of X: B ⊂ 2X For each choice structure  check whether properties α and β (or equivalently WARP) are satisfied and if there exists a rational preference relation ≤ that rationalizes C(.) relative to its B. If such a rationalization is possible, write it down. Comment on your results.

Question 4  Show  that  if  preferences “ on  RL+  satisfy  continuity,  and x “ z  “ y,  then  there  is  a w ∈ RL+  in the line segment connecting x and y  such that w ∼ z  .

Question 5 Consider the following choice procedure over a finite set of n alternatives X ={x₁, . . . , x_n}, called the “satisficing” procedure (Herbert Simon).  微观经济分析代写

The agent evaluates each alternative using a function v(x). That is, alternative x has a “level”v(x). v can be any function. The agent has a “reference” level v^∗.

Given any set of m alternatives A ⊂ X, denote A = {x_i1 , . . . , x_im } with 1 ™ i₁ < . . . < i_m ≤ n.

Theagent considers in turn alternatives x_i1 , x_i2 , . . . until she finds one that has a level v(x_i) “ v^∗. If she gets to the end of the set A without finding such an alternative, shechooses an alternative in A that has the highest v-level (this last part means C(A) = {x ∈ A : v(x) “ v(y), ∀y ∈ A}). 微观经济分析代写

1. Whydoes it make sense to call this choice procedure the “satisficing” procedure?
2. Can C(A) be empty? does it satisfyWARP?
3. If the answers are “no” and “yes”, then you should be able to rationalize C by complete and transitive preferences. What do they look like? that is, when do we have x ≤ y? and what about a utilityrepresentation?

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