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Graph {(x, y) ∈ R+² : X ≥ X’ >0, Y ≥ Y’ > 0}

  1. Jan 19, 2014 #1

    939

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    1. The problem statement, all variables and given/known data

    The consumption set of a consumer is: {(x, y) ∈ R+² : X ≥ X’ >0, Y ≥ Y’ > 0}. Graph it.

    I am only wondering how it looks, don't need a copy of the graph.

    2. Relevant equations

    {(x, y) ∈ R+² : X ≥ X’ >0, Y ≥ Y’ > 0}

    3. The attempt at a solution

    This first part, (x, y) ∈ R+², tells me it will be on the top right part of a Cartesian graph. The points can touch 0. But I am not sure what X ≥ X’ > 0, Y ≥ Y’ > 0 means. We are doing sets and convexity.

    The first x value is greater than the second, which are both greater than zero? And the same for Y? 0_o
     
  2. jcsd
  3. Jan 20, 2014 #2
    Hi, 939

    I'll try to cast light on this question ;)

    First, let us consider a point in R² (X',Y'), such as X’ >0, Y’ > 0. Therefore, our graph will be in the first quadrant.

    Second, What does mean: X ≥ X' and Y ≥ Y’?. It represent an angle in the space. The vertex of the angle is the point (X',Y'), and the angles lines will be the straight lines X=X' and Y=Y' accordingly with the constraints.

    I need draw it, but, now I can't.

    Is this useful?

    Regards
     
  4. Jan 20, 2014 #3
    From the conditions stated, the (0,0) would be represented with an open point. the graph approaches this point but never touches it.

    As for the conditions, [itex]X \geq X' > 0 [/itex], I think this means that for every X defined, there is an X' between that number and 0. It says something about the "neighborhood" near the point x = 0. Say, if you are at the point x = 0.0001, there is still a smaller point that is greater than 0. Yet, you never have x =0.

    In reality this might not make sense, because when would you say that you have 0.0000001 dollars?

    That's my take on it. I hope it helps.
     
  5. Jan 20, 2014 #4

    939

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    Thanks! My final question is would you be able to say if the set is convex, closed or bounded?
     
  6. Jan 20, 2014 #5
    The set is convex due to every pair of points can be linked with a continuous line. Also is closed, because the condition greater or EQUAL. Accordingly with the definition, the complementary set is open. And finally the set is not bounded, it's wonder, there isn't any contidion about the maximum of X or Y.

    Regards!
     
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