I have the following problem. Given this function E [ π ] = ( 1 − r ) [ α b − ( 1 − p ) C − K ] + TI would like to find the maximum w.r.t. r given this constraint: U = ( 1 − r ) b − T ≥ 0It is an economic problem that I am formalizing, but this is not relevant to its solution, I am only interested in the mathematical resolution of the problem. I provide some background. Our variable r is a number between 0 and 1, p is some probability, α , b , C , K and T are all positive constants. If it is useful for the resolution of the problem, we can also assume that α is between 0 and 1. An important assumption (namely, assumption &amp;) is that ( 1 − p ) C + K &gt; α b. Readers used to economics may recognize that the objective function is a sort of expected profit and the constraint is a sort of expected utility. I tried with Kuhn-Tucker, but with miserable results since deriving w.r.t. r does not yield any expression with r.Now I'm following a more intuitive approach. I start by assuming that U = 0 is the constraint, then I can find an expression for r from the constraint and I substitute it in the target function. After easy steps, I get this T α b − ( 1 − p ) C − K b + TAt this point, I can use assumption &amp; to conclude that the first T above will be negative, but I'm not able to conclude whether it will be lower, equal or higher to/than the second T because it is multiplied by some constant.At this point I'm stuck. I do not know if my approach can work. Could you please suggest a nicer way to proceed? I am on the right track or it's a dead end?

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I have the following problem. Given this function  E  [  π  ]  =  (  1  −  r  )  [  α  b  −  (  1  −  p  )  C  −  K  ]  +  TI would like to find the maximum w.r.t.   r given this constraint:  U  =  (  1  −  r  )  b  −  T  ≥  0It is an economic problem that I am formalizing, but this is not relevant to its solution, I am only interested in the mathematical resolution of the problem. I provide some background. Our variable   r is a number between 0 and 1,   p is some probability,   α  ,  b  ,  C  ,  K and   T are all positive constants. If it is useful for the resolution of the problem, we can also assume that   α is between 0 and 1. An important assumption (namely, assumption   &amp;amp;) is that   (  1  −  p  )  C  +  K  &amp;gt;  α  b. Readers used to economics may recognize that the objective function is a sort of expected profit and the constraint is a sort of expected utility. I tried with Kuhn-Tucker, but with miserable results since deriving w.r.t.   r does not yield any expression with   r.Now I&#039;m following a more intuitive approach. I start by assuming that   U  =  0 is the constraint, then I can find an expression for   r from the constraint and I substitute it in the target function. After easy steps, I get this  T            α      b      −      (      1      −      p      )      C      −      K        b    +  TAt this point, I can use assumption   &amp;amp; to conclude that the first   T above will be negative, but I&#039;m not able to conclude whether it will be lower, equal or higher to/than the second   T because it is multiplied by some constant.At this point I&#039;m stuck. I do not know if my approach can work. Could you please suggest a nicer way to proceed? I am on the right track or it&#039;s a dead end?

Brendon Fernandez · Accepted Answer

Calling&amp;nbsp;ϕ&amp;nbsp;=&amp;nbsp;(&amp;nbsp;1&amp;nbsp;−&amp;nbsp;p&amp;nbsp;)&amp;nbsp;C&amp;nbsp;+&amp;nbsp;K&amp;nbsp;−&amp;nbsp;α&amp;nbsp;b&amp;nbsp;&amp;gt;&amp;nbsp;0&amp;nbsp;and consideringU&amp;nbsp;=&amp;nbsp;(&amp;nbsp;1&amp;nbsp;−&amp;nbsp;r&amp;nbsp;)&amp;nbsp;b&amp;nbsp;−&amp;nbsp;T&amp;nbsp;≥&amp;nbsp;0&amp;nbsp;⇒&amp;nbsp;r&amp;nbsp;≤&amp;nbsp;1&amp;nbsp;−&amp;nbsp;T&amp;nbsp;b&amp;nbsp;The issue can be stated asmin&amp;nbsp;r&amp;nbsp;(&amp;nbsp;r&amp;nbsp;−&amp;nbsp;1&amp;nbsp;)&amp;nbsp;ϕ&amp;nbsp;+&amp;nbsp;T&amp;nbsp;,&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;s. t.&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;{&amp;nbsp;r&amp;nbsp;≥&amp;nbsp;0&amp;nbsp;r&amp;nbsp;≤&amp;nbsp;1&amp;nbsp;−&amp;nbsp;T&amp;nbsp;b&amp;nbsp;&amp;nbsp;now as&amp;nbsp;(&amp;nbsp;r&amp;nbsp;−&amp;nbsp;1&amp;nbsp;)&amp;nbsp;ϕ&amp;nbsp;+&amp;nbsp;T&amp;nbsp;is linear, and one of the set of solutions{&amp;nbsp;T&amp;nbsp;−&amp;nbsp;ϕ&amp;nbsp;,&amp;nbsp;T&amp;nbsp;−&amp;nbsp;ϕ&amp;nbsp;T&amp;nbsp;b&amp;nbsp;}&amp;nbsp;

I have the following problem. Given this function E [ π ] = (

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