Proof of U(pie(x)+(1-pie)y) > pie*U(x)+(1-pie)U(y) using sqrt property

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Homework Help Overview

The discussion revolves around proving the inequality U(π(x) + (1-π)y) > πU(x) + (1-π)U(y) for the utility function U(w) = sqrt(w). The subject area involves concepts of utility functions and properties of concavity.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the approach of squaring both sides of the inequality to simplify the proof. There is a discussion on the implications of squaring and the properties of the square root function, particularly its concavity.

Discussion Status

Some participants have offered guidance on manipulating the inequality, while others express uncertainty about the correctness of their steps and seek clarification on the implications of their findings. Multiple interpretations of the steps taken are being explored.

Contextual Notes

There is a clarification that "π" refers to a variable between 0 and 1, not the mathematical constant. This context is important for understanding the nature of the proof being discussed.

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Homework Statement



For U(w)=sqrt(w), prove that U(pie(x)+(1-pie)y) > pie*U(x)+(1-pie)U(y)



Homework Equations



sqrt(x)=x^(1/2)

The Attempt at a Solution



I have:

sqrt(pie(x)+(1-pie)y) > pie*sqrt(x)+(1-pie)sqrt(y) so...

(pie(x)+(1-pie)y)^(1/2) > pie*(x)^1/2+(1-pie)(y)^1/2

From here I don't know where to go. I don't have much experience with proofs so can anyone give some guidance? Thanks for your help!
 
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Try squaring both sides.
 


alright, then (pi(x)+(1-pi)y) > pi^2*(x)+(1-pi)^2*(y). Therefore pi+(1-pi)>(pi^2)+((1-pi)^2)
=1 > (pi^2)+((1-pi)^2)
=1-(pi^2)>(1-pi)^2

Is this sufficient? What I don't quite get is that if you then square both sides, don't you get sqrt((1-(pi^2))) > sqrt((1-pi)^2)

=sqrt(1)-pi > (1-pi)
=1-pi > 1-pi ?? This obviously can't be right, so where did I go wrong? Thanks again.
 


Here "pi" is NOT the usual "pi", it is a number between 0 and 1, right? This is essentially a proof that the square root function is concave.
 


yes exactly, sorry for not making that clear before.
 

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