Proof of Lemma 2 Notice from (4.14) that if and only if
This is impossible because the coefficient of variation Thus, Next, observe from (4.15) that is increasing in x and The second statement of the Lemma follows immediately from these two observations.
Proof of Proposition 4.4 Since it is easy to check from (4.14) that For any x that has Lemma 2 implies that
In this case, one can show from (4.17) that Next, since Lemma 2 implies that and that It follows from the Mean Value Theorem that there exists an optimal that satisfies the first order condition.
Next, when it is easy to check from (4.14) that In this case, we have Hence, it can be shown from (4.17) that
and that It follows from the Mean Value Theorem that there exists an optimal that satisfies the first order condition. This completes the proof.
Before we present the proof of Proposition 4.5, let us prove the following Lemma that will be useful.
Lemma A.1 Consider the function given in (4.15) and the function where The term that has has the following properties:
1 When the term
2 When the term when In addition, when
the term when is sufficiently small and the
term when is sufficiently large.
Proof of Lemma A.1 Observe from (4.15) and (4.14) that Hence, First, when the first statement of Proposition 4.4 implies that The term
1) is non-negative because Next, when these
cond statement of Proposition 4.4 implies that and In this case, the term is non-negative when We finally consider the case when By substituting the expression for given in (4.14) into the term
It can be shown that this term is non-negative (negative) when is sufficiently small (large).
Proof of Proposition 4.5 Let us consider the case in which
First, considering the fact that satisfies we can use the implicit function theorem to differentiate the function with respect to By considering (4.17) and (4.15), it can be shown that:
By applying the result from Lemma A.1 to the above expression, one can see that Thus, is decreasing in
Next, let us differentiate the function with respect to By considering (4.17) and (4.15), it can be shown that:
Since Lemma A.1 implies that the denominator of the above equation is non-negative. In this case, if and only if the numerator
is positive. The numerator is positive when It follows from the definition of that Therefore, we can conclude that the numerator is positive when and is negative when This proves the first statement.
For the case when we can apply the same approach to prove the second statement. We omit the details.
Finally, when we can determine the expression for by setting We omit the details. This completes the proof.
Proof of Lemma 4.6 Since the term and the function are analogous to and we can use the same approach presented in the proof of Lemma 2 to prove Lemma 4.6. We omit the details.
Proof of Proposition 4.7 Observe that the term and the function are anal- ogous to the terms and Also, notice that the first derivative of given in (4.24) is analogous to the first derivative of given in (4.17). In this case, we can use the same approach presented in the proof of Proposition 4.4 to prove Proposition 4.7. We omit the details.
Before we present the proof of Proposition 4.8, let us prove the following Lemma that will be useful.
Lemma A.2 Consider the function given in (4.15) and the function where The term that has has the following properties:
1 When the term
2 When the term when In addition, when
the term when is sufficiently small and the term when is sufficiently large.
Proof of Lemma A.2 Since the term and the function are analogous to and we can use the same approach presented in the proof of Lemma A.1 to prove Lemma A.2.We omit the details.
Proof of Proposition 4.8 First, differentiate the function with respect to and getting:
Then we can prove Proposition 4.8 by using the same approach as presented in the proof for Proposition 4.5. We omit the details.
Proof of Proposition 4.9 Let us compare the function given in (4.24) and the function given in (4.17). It is easy to check from (4.22) and (4.15) that the key difference between and lies with the term v given in (4.20). Essentially, reduces to when Since we can prove the Proposition by showing that is decreasing in Let us differentiate the function with respect to By considering (4.24) and (4.22), it can be shown that :
By applying the result from Lemma A.2 to the above expression, we prove the Propo- sition. This completes the proof.
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