Mixing
understanding the proof using the mean value theorem in the economic paper.
$begingroup$
This is the economic paper: Hopkins and Kornienko (2004): Running to Keep in the same place. The picture shows the proof of the proposition 1 in this paper. In this proof, I don't understand how the mean value theorem is applied.
Here is my attempt to understand the proof and I am considering the first inequality:
By mean value theorem,
$$V(x(z), z-px(z))(alpha + G(z)) - V(x(check{z}), z - px(check{z}))(alpha + G(check{z})) - [(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(z))=0, $$
where $tilde{z} in (z, hat{z})$. Then, I think they try to make this bigger or equal than 0:
$$ V(x(z), z-px(z))(alpha + G(z)) - V(x(check{z}), z - px(check{z}))(alpha + G(check{z})) le V(x(z), z-px(z))[G(z) - G(hat{z})], $$
and
$$-[(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(z)) le -[(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(hat{z})).$$
But, both are wrong because by assumption, $ V(x(z), z-px(z)) ge V(x(check{z}), z - px(check{z}))$, and $G(z)$ is strictly increasing.
If this is wrong, I don't understand what's going on here. Am I missing something in MVT?
I appreciate if you give any help.
real-analysis derivatives economics
asked Jan 31 at 6:02
DaeseonDaeseon
855311
$endgroup$
add a comment |
$begingroup$
This is the economic paper: Hopkins and Kornienko (2004): Running to Keep in the same place. The picture shows the proof of the proposition 1 in this paper. In this proof, I don't understand how the mean value theorem is applied.
Here is my attempt to understand the proof and I am considering the first inequality:
By mean value theorem,
$$V(x(z), z-px(z))(alpha + G(z)) - V(x(check{z}), z - px(check{z}))(alpha + G(check{z})) - [(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(z))=0, $$
where $tilde{z} in (z, hat{z})$. Then, I think they try to make this bigger or equal than 0:
$$ V(x(z), z-px(z))(alpha + G(z)) - V(x(check{z}), z - px(check{z}))(alpha + G(check{z})) le V(x(z), z-px(z))[G(z) - G(hat{z})], $$
and
$$-[(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(z)) le -[(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(hat{z})).$$
But, both are wrong because by assumption, $ V(x(z), z-px(z)) ge V(x(check{z}), z - px(check{z}))$, and $G(z)$ is strictly increasing.
If this is wrong, I don't understand what's going on here. Am I missing something in MVT?
I appreciate if you give any help.
real-analysis derivatives economics
asked Jan 31 at 6:02
DaeseonDaeseon
855311
$endgroup$
add a comment |
$begingroup$
This is the economic paper: Hopkins and Kornienko (2004): Running to Keep in the same place. The picture shows the proof of the proposition 1 in this paper. In this proof, I don't understand how the mean value theorem is applied.
Here is my attempt to understand the proof and I am considering the first inequality:
By mean value theorem,
$$V(x(z), z-px(z))(alpha + G(z)) - V(x(check{z}), z - px(check{z}))(alpha + G(check{z})) - [(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(z))=0, $$
where $tilde{z} in (z, hat{z})$. Then, I think they try to make this bigger or equal than 0:
$$ V(x(z), z-px(z))(alpha + G(z)) - V(x(check{z}), z - px(check{z}))(alpha + G(check{z})) le V(x(z), z-px(z))[G(z) - G(hat{z})], $$
and
$$-[(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(z)) le -[(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(hat{z})).$$
But, both are wrong because by assumption, $ V(x(z), z-px(z)) ge V(x(check{z}), z - px(check{z}))$, and $G(z)$ is strictly increasing.
If this is wrong, I don't understand what's going on here. Am I missing something in MVT?
I appreciate if you give any help.
real-analysis derivatives economics
asked Jan 31 at 6:02
DaeseonDaeseon
855311
$endgroup$
This is the economic paper: Hopkins and Kornienko (2004): Running to Keep in the same place. The picture shows the proof of the proposition 1 in this paper. In this proof, I don't understand how the mean value theorem is applied.
Here is my attempt to understand the proof and I am considering the first inequality:
By mean value theorem,
$$V(x(z), z-px(z))(alpha + G(z)) - V(x(check{z}), z - px(check{z}))(alpha + G(check{z})) - [(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(z))=0, $$
where $tilde{z} in (z, hat{z})$. Then, I think they try to make this bigger or equal than 0:
$$ V(x(z), z-px(z))(alpha + G(z)) - V(x(check{z}), z - px(check{z}))(alpha + G(check{z})) le V(x(z), z-px(z))[G(z) - G(hat{z})], $$
and
$$-[(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(z)) le -[(x(z) - x(hat{z})][V_1(x(tilde{z}), z - px(tilde{z})) - pV_2(x(tilde{z}), z - px(tilde{z}))](alpha + G(hat{z})).$$
But, both are wrong because by assumption, $ V(x(z), z-px(z)) ge V(x(check{z}), z - px(check{z}))$, and $G(z)$ is strictly increasing.
If this is wrong, I don't understand what's going on here. Am I missing something in MVT?
I appreciate if you give any help.
real-analysis derivatives economics
real-analysis derivatives economics
asked Jan 31 at 6:02
DaeseonDaeseon
855311
asked Jan 31 at 6:02
DaeseonDaeseon
855311
asked Jan 31 at 6:02
DaeseonDaeseon
855311
asked Jan 31 at 6:02
DaeseonDaeseon
855311
asked Jan 31 at 6:02
DaeseonDaeseon
855311
855311
add a comment |
add a comment |
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