Approximation around steady state:
All equations here are direct copies from a textbook.
In steady state, we have $$i = rho + pi + sigma y.$$
The goal is then to do a first-order Taylor approximation around this steady state of the function $$f(i_t) = expleft( i_t - sigma Delta c_{t+1} - pi_{t+1} + Delta x_{t+1}right)$$
Doing this, my textbook gets the approximation $$f(i_t) approx 1 + (i_t - i) - sigma(Delta c_{t+1} - y) - (pi_{t+1} - pi_t) + Delta x_{t+1}.$$
But how does it do this?
Note that evaluation at the steady state gives $$f(i) = exp( - sigma(Delta c_{t+1} - y) - (pi_{t+1} - pi_t) + Delta x_{t+1})$$ which is the first term in the Taylor approximation, yet this term doesn't appear anywhere?
taylor-expansion
asked Nov 27 '18 at 21:17
SAK
1012
add a comment |
All equations here are direct copies from a textbook.
In steady state, we have $$i = rho + pi + sigma y.$$
The goal is then to do a first-order Taylor approximation around this steady state of the function $$f(i_t) = expleft( i_t - sigma Delta c_{t+1} - pi_{t+1} + Delta x_{t+1}right)$$
Doing this, my textbook gets the approximation $$f(i_t) approx 1 + (i_t - i) - sigma(Delta c_{t+1} - y) - (pi_{t+1} - pi_t) + Delta x_{t+1}.$$
But how does it do this?
Note that evaluation at the steady state gives $$f(i) = exp( - sigma(Delta c_{t+1} - y) - (pi_{t+1} - pi_t) + Delta x_{t+1})$$ which is the first term in the Taylor approximation, yet this term doesn't appear anywhere?
taylor-expansion
asked Nov 27 '18 at 21:17
SAK
1012
add a comment |
All equations here are direct copies from a textbook.
In steady state, we have $$i = rho + pi + sigma y.$$
The goal is then to do a first-order Taylor approximation around this steady state of the function $$f(i_t) = expleft( i_t - sigma Delta c_{t+1} - pi_{t+1} + Delta x_{t+1}right)$$
Doing this, my textbook gets the approximation $$f(i_t) approx 1 + (i_t - i) - sigma(Delta c_{t+1} - y) - (pi_{t+1} - pi_t) + Delta x_{t+1}.$$
But how does it do this?
Note that evaluation at the steady state gives $$f(i) = exp( - sigma(Delta c_{t+1} - y) - (pi_{t+1} - pi_t) + Delta x_{t+1})$$ which is the first term in the Taylor approximation, yet this term doesn't appear anywhere?
taylor-expansion
asked Nov 27 '18 at 21:17
SAK
1012
All equations here are direct copies from a textbook.
In steady state, we have $$i = rho + pi + sigma y.$$
The goal is then to do a first-order Taylor approximation around this steady state of the function $$f(i_t) = expleft( i_t - sigma Delta c_{t+1} - pi_{t+1} + Delta x_{t+1}right)$$
Doing this, my textbook gets the approximation $$f(i_t) approx 1 + (i_t - i) - sigma(Delta c_{t+1} - y) - (pi_{t+1} - pi_t) + Delta x_{t+1}.$$
But how does it do this?
Note that evaluation at the steady state gives $$f(i) = exp( - sigma(Delta c_{t+1} - y) - (pi_{t+1} - pi_t) + Delta x_{t+1})$$ which is the first term in the Taylor approximation, yet this term doesn't appear anywhere?
taylor-expansion
taylor-expansion
asked Nov 27 '18 at 21:17
SAK
1012
asked Nov 27 '18 at 21:17
SAK
1012
asked Nov 27 '18 at 21:17
SAK
1012
asked Nov 27 '18 at 21:17
SAK
1012
asked Nov 27 '18 at 21:17
SAK
1012
1012
add a comment |
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016309%2fapproximation-around-steady-state%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3016309%2fapproximation-around-steady-state%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
