Malin84

Monotone Utility and Junk Food

I've checked the past exams and this is not a prelim question. Must be a homework question then. The question reads like this:

Assume your preference over food is described by the following function U(t,h), where "t" stands for the tastiness of the food and "h" stands for the healthiness of the food. Your utility is strictly increasing in both dimensions of food: you like the food better if it is more tasty(t higher) or if it is more healthy(h higher).

Prove: If you want to sustain a constant level of utility, then among the food that you eat, junk food must be tasty.

Well.... this is embarrassingly easy to prove. We want U(t,h) to be a constant and it is strictly increasing in both arguments, therefore:

$\forall <t,h>$ such that $U(t,h)={\bar {U}}$,

we must have

$h_{1}\leq h_{2}\implies t_{1}\geq t_{2}$.

This is really just a complicated way to say, 鱼与熊掌不可得兼。

The implication is actually quite interesting. It is not that we don't enjoy something both tasty and healthy ---- suppose such food exists, then we can always twist it so that it becomes a little bit more a tastier and a little bit less healthier ---- yet we enjoy it the same. Repeat the twisting many times, then at the end of the day, among the food that we choose to eat, the really tasty ones must be the least healthy ones as well.

Average Utility and Marginal Utility

只有在边际效用是常数的时候，平均效用才会和边际效用相等。如果你要衡量"单位物品在价格为P的时候的每一美元的平均效用",那么公式应该是：

${\frac {\int _{0}^{q}MU(s)ds}{P\cdot q}}$

这里面分子是消费量为q的时候商品带给你的总效用，分母是为q单位商品支付的总财富。

上面这个式子只有在

$\int _{0}^{q}MU(s)ds=MU(q)\cdot q$

的时候才会等于${\frac {MU(q)}{P}}$。而上式说明边际效用为常数？

Weak Axiom Implies x(p,w) Homogenous of Degree Zero

假设x(p,w)满足弱公理，则有：

if $p\cdot x(p',w')\leq w\cap x(p,w)\neq x(p',w')$ then $p'\cdot x(p,w)>w'$

设 $p=p_{0},w=w_{0},p'=\alpha p_{0},w'=\alpha w_{0}$,直接代如弱公理，我们有：

if $p_{0}\cdot x(\alpha p_{0},\alpha w_{0})\leq w_{0}\cap x(p_{0},w_{0})\neq x(\alpha p_{0},\alpha w_{0})$ then $\alpha p_{0}\cdot x(p_{0},w_{0})>\alpha w_{0}$

这个then后面的推论显然是不成立的，因为对于任何的p,w我们必须有$p\cdot x(p,w)\leq w$。因此，if 部分也必然不成立，我们因此有if部分的否命题成立：

$p_{0}\cdot x(\alpha p_{0},\alpha w_{0})>w_{0}\cup x(p_{0},w_{0})=x(\alpha p_{0},\alpha w_{0})$

到这一步我们还是没法推论$x(p_{0},w_{0})=x(\alpha p_{0},\alpha w_{0})$，因为有前面那个“或”的命题。 现在考虑“或”前面的那个命题。这个命题也是不可能成立的，因为如果在不等式两边同时乘$\alpha$，我们就有：

$\alpha p_{0}\cdot x(\alpha p_{0},\alpha w_{0})>\alpha w_{0}$

跟then 命题必然不成立的道理一样。“或”前面的命题必然不成立，而整个命题成立，那么我们就必然有

$x(p_{0},w_{0})=x(\alpha p_{0},\alpha w_{0})$

成立了。x(p,w)零阶齐次证毕.

Is It Possible to Have a Decreasing Contract Curve in Exchange Economy?

考虑这个问题，假设每个人的效用函数分别是 $u_{1}(x,y),u_{2}(x,y)$,两种商品的禀赋分别是$w_{x},w_{y}$,那么契约曲线在内点一定满足

${\frac {u_{1x}(x,y)}{u_{1y}(x,y)}}={\frac {u_{2x}(w_{x}-x,w_{y}-y)}{u_{2y}(w_{x}-x,w_{y}-x)}}$

重新整理得：

$F(x,y)=u_{1x}(x,y)u_{2y}(w_{x}-x,w_{y}-x)-u_{1y}(x,y)u_{2x}(w_{x}-x,w_{y}-y)$

其中$u_{ix}$是第i个人对商品x的边际效用。因为我们考虑的是扭曲的问题，所以应该只需要讨论内点解。上式定义了一个$y$对$x$的隐含数，如果这个曲线左上到右下扭曲的话，那么也就是说这个隐含数是递减的。假定这个函数可导（纯粹为了省事，当然可能不可导），那么用隐含数定理可知:

${\frac {dy}{dx}}=-{\frac {\partial F/\partial x}{\partial F/\partial y}}$

$={\frac {u_{1xx}u_{2y}+u_{2xx}u_{1y}-u_{2yx}u_{1x}-u_{1yx}u_{2x}}{u_{2yy}u_{1x}+u_{1yy}u_{2x}-u_{1xy}u_{2y}-u_{2xy}u_{1y}}}$

现在我们就可以讨论了：

如果我们要求偏好是单调的且凸的话，那么我们就有$u_{ix},u_{iy}\geq 0,\;u_{ixx},u_{iyy}\leq 0$。但我们不知道$u_{ixy}$的符号，在这种情况下我们没法确定$dy/dx$的符号，那么契约曲线也就有可能递减。如果$u_{ixy}\geq 0$,比如Cobb-Douglass，或者任何x y可分离的情况，那么我们可以确定$dy/dx\geq 0$，这样也就不可能单调递减。除此之外，似乎在这个general form下没有什么办法可以确定dy/dx的符号。

如果我们进一步假定两个人的效用函数相同，那么：

${\frac {dy}{dx}}={\frac {u_{xx}u_{y}-u_{xy}u_{x}}{u_{yy}u_{x}-u_{xy}u_{y}}}$

这样的话这个问题就可以有一个确定的答案。如果$u_{xy}\geq 0$，那么如前所述，必有$dy/dx\geq 0$，不可能出现扭曲;所以我们只需要讨论$u_{xy}<0$。给同样的假定，即凸性和单调性，我们现在有：

$u_{xx}u_{yy}-u_{xy}^{2}\leq 0$

这个地方有点不严谨，因为我直接用了海赛阵而没有用加边海赛；结果应该差不多的，就偷懒了:)

这个条件等价于:

$u_{yy}\geq {\frac {u_{xy}^{2}}{u_{xx}}}$

这里面不等号变换了方向，因为每一项都是负的。那么，如果在dy/dx那个式子里面的分子大于零，那么一定有分母也大于零；换句话说，上下同号，故我们有$dy/dx\geq 0$,不可能出现扭曲：

$u_{xx}u_{y}-u_{xy}u_{x}\geq 0\implies u_{yy}u_{x}-u_{xy}u_{y}\geq 0$

最后这个不等式是这样的推导的：

$u_{yy}\geq {\frac {u_{xy}^{2}}{u_{xx}}}$

${\frac {u_{yy}}{u_{y}}}\geq {\frac {u_{xy}^{2}}{u_{xx}u_{y}}}$

从分子大于零我们知道：

${\frac {u_{xy}}{u_{xx}}}>{\frac {u_{y}}{u_{x}}}$

将这个式子带入上式，我们有：

${\frac {u_{yy}}{u_{y}}}\geq {\frac {u_{xy}^{2}}{u_{xx}u_{y}}}\geq {\frac {u_{y}}{u_{x}}}{\frac {u_{xy}}{u_{y}}}={\frac {u_{xy}}{u_{x}}}$

最后这个式子也就等价于分母大于零了。

总结一下：

如果两个人的效用函数不相同，那么没有扭曲的充分条件是偏好的凸性，单调性，以及$u_{ixy}\geq 0$。我觉的最后这个条件应该是有个名字的，跟complementary 有没有关系？不知道了，那位大哥指导一下。。。。。因为最后这个条件并不是最基本的条件，那么有可能出现左上到右下的扭曲

如果两个人的效用函数相同，那么只要偏好满足凸性和单调性（这点不太严谨，我证明的是效用函数满足凹性和单调性，关于凸性的问题，我觉的应该是可以成立的，不过懒得弄加边海赛了……），那么就不可能出现扭曲的情况。

Will Monopoly Produce When Demand is Inelastic?

一个垄断厂商解决这个最优化问题:

$\max _{q}p(q)\cdot q-c(q)$

First Order Condition:

${\frac {dp}{dq}}q+p=c'(q)$

$p({\frac {dp}{dq}}{\frac {q}{p}}+1)=c'(q)$

$p({\frac {1}{e}}+1)=c'(q)$

如果需求缺乏弹性，那么$e>-1$，故我们有：

$p({\frac {1}{e}}+1)<0$

但是如果生产可能性集合是凸的，那么我们总有 $c'(q)>0$，所以这里就矛盾了，故垄断厂商不会在需求缺乏弹性的时候生产。

WikiProject Economics census

Hello there. Sorry to bother you, but you are (titularly at least) a member of WP:WikiProject Economics, as defined by this category. If you don't know me, I'm a Wikipedia administrator, but an unqualified economist. I enjoy writing about economics, but I'm not very good at it, which is why I would like to support in any way I can the strong body of economists here on Wikipedia. I'm only bothering you because you are probably one of them. Together, I'd like us to establish the future direction of WikiProject Economics, but first, we need to know who we've got to help.

Whatever your area of expertise or level of qualification, if you're interested in helping with the WikiProject (even if only as part of a larger commitment to this wonderful online encyclopedia of ours), would you mind adding your signature to this page? It only takes a second. Thank you.

Message delivered on behalf of User:Jarry1250 by LivingBot.

Firstly, thank you for signing the census, and an apology if you are one of those editors who dislike posts such as this one for messaging you again in this way. I've now got myself organised and you can opt-out of any future communication at WP:WikiProject Economics/Newsletter. Just remove your name and you won't be bothered again.

Secondly, and most importantly, I would like to invite your comments on the census talk page about the project as a whole. I've given my own personal opinion on a range of topics, but my babbling is essentially worthless without your thoughts - I can't believe for one moment that everyone agrees with me in the slightest! :)

All your comments are welcomed. Thanks, - Jarry1250 ^{[Humorous? Discuss.]} 17:56, 21 April 2010 (UTC)

WikiProject Economics Newsletter (Issue I)

Positively Economics The Economics WikiProject Newsletter Issue I (May 2010)
Use the show/hide button to display or hide this newsletter. From the (self-appointed) editor Hello to all the economists who signed up on the census (covered in more detail below) and whom I have boldly signed up for this, the first in hopefully a long line of WikiProject Economics newsletters. The aim is simple: to distribute the project's news and developments in order that every member gets the opportunity to contribute to their fullest. It mirrors the ideals of any WikiProject: to support and assist. In creating this, on a whim, I have picked a name and stolen the layout of the Military History WikiProject's newsletter. If you have any comments or suggestions, feel free to leave them on the main newsletter talk page. The non-talk version of that page is also the place to unsubscribe yourself from the newsletter, which, for the moment, I anticipate will be monthly. Enjoy! - Jarry1250 [Humorous? Discuss.] 11:59, 25 April 2010 (UTC) Articles of note New featured articles: Economy of the Han Dynasty New featured lists: None New featured topics: None New good articles: Price elasticity of demand Production-possibility frontier Monetary Policy Committee Confederate war finance Redlinks to be squashed Skysmith has posted an updated list of missing economics topics. Most will need redirecting or new articles written to blueify them, so, if you can lend a hand with the effort, please do. There are some from virtually all areas of economics and hence to suit all tastes. Project news Census completed A big thank you to everyone who signed the census, an attempt to renew the project's membership list. Jarry1250 explained its purpose: "once we know how many active participants we have... we can make sure no interested parties get left out of discussions; ...to get everyone included. Not just to make them feel involved, but also to actively try to make them more involved [in the supportive atmosphere of the WikiProject]". Around forty editors, with a good range of specialisms and interests, have currently signed to show their activity, and it is still open for more. Project-wide discussions commence: A series of discussions have been commenced on the census talkpage about what, if anything, should be done to strengthen the Project. All comments are welcomed on a range of topics: the economic style guide, weight guidelines, notability and the creation of this newsletter. Other discussions, on categorisation, tagging and compiling an accurate list of unreferenced BLPs, are currently active. Your comments are invited in the following other ongoing project discussions: The normative economics of central banking. What article might this fall under? Does one exist already? Does it need a separate article? Inflation. Thoughts invited on the inflation article and how best to word the lead. 2010 European sovereign debt crisis - comments requested.
To start/stop receiving this newsletter, please add/remove your name from the list here. Thank you. This newletter was delivered to you by Jarry1250 at around 10:45, 1 May 2010 (UTC)

WikiProject Economics Newsletter (Issue II)

Positively Economics The Economics WikiProject Newsletter Issue II (June 2010)
Use the show/hide button to display or hide this newsletter. From the editor Sticking to my promise of sending these out monthly, it's time for the June edition of Positively Economics, covering the events of May, which has passed relatively quietly for the project. Nevertheless, Wikipedia is a wiki, and things can change quickly and often without much notice. All contributions can be, and often are, important, regardless of how trivial they may seem. I, for example, have been working on our new list of unsourced BLPs, having added a significant number of economists to the project earlier in the month. But more on that in the newsletter itself. - Jarry1250 [Humorous? Discuss.] 14:28, 27 May 2010 (UTC) Articles of note Note: this week's "new" FAs, and so forth, are ones that previously weren't included in the project but now are. New featured articles: Akhtar Hameed Khan New featured lists: None New featured topics: None New good articles: Abram Lincoln Harris George Lee (journalist) Stephen Harper Yulia Tymoshenko Redlinks to be squashed Skysmith has posted an updated list of missing economics topics. Most will need redirecting or new articles written to blueify them, so, if you can lend a hand with the effort, please do. There are some from virtually all areas of economics and hence to suit all tastes. Project news Unassessed articles: A whole swath of articles have been added to the WikiProject recently, mainly on notable economists. They are yet to be assessed on their talk pages; this also provides a good opportunity to check and improve some of our least monitored articles. Unsourced BLPs: Before the articles about economists were added, the project had very few unsourced biographies of living people; now there are a little over a hundred. If we can clear them, however, then it is unlikely that there will be many more. Again, they represent some of our least cared-for articles, and often present good targets for cleaning up. Your comments are invited in the following other ongoing project discussions and requests for help: Too big to fail Slowflation Jobless recovery Small Business Lending Index
To start/stop receiving this newsletter, please add/remove your name from the list here. Thank you. This newletter was delivered to you by User:Jarry1250 at around 14:50, 1 June 2010 (UTC)

WikiProject Economics Newsletter (Issue III)

Positively Economics The Economics WikiProject Newsletter Issue III (July 2010)
Use the show/hide button to display or hide this newsletter. From the editor Sticking to my promise of sending these out monthly, it's time for the July edition of Positively Economics, covering the events of June. It's been another constructive month for the project. Wikipedia remains, as ever, a very popular source of information about basic economic theory, as well as for biography and more specialised topics, and we shouldn't forget that contributions to these core articles remains our greatest strength. Onwards and upwards, as they say, for another month. - Jarry1250 [Humorous? Discuss.] 14:28, 27 May 2010 (UTC) Articles of note New featured articles: None New featured lists: None New featured topics: None New good articles: Brander–Spencer model Redlinks to be squashed Skysmith has posted an updated list of missing economics topics. Most will need redirecting or new articles written to blueify them, so, if you can lend a hand with the effort, please do. There are some from virtually all areas of economics and hence to suit all tastes. Project news Unassessed articles: A whole swath of articles have been added to the WikiProject recently, mainly on notable economists. They are yet to be assessed on their talk pages; this also provides a good opportunity to check and improve some of our least monitored articles. Unsourced BLPs: Before the articles about economists were added, the project had very few unsourced biographies of living people; now there are a little over a hundred. If we can clear them, however, then it is unlikely that there will be many more. Again, they represent some of our least cared-for articles, and often present good targets for cleaning up. Your comments are invited in the following other ongoing project discussions and requests for help: Featherbedding Herbert Hoover Intelligence-related citations
To start/stop receiving this newsletter, please add/remove your name from the list here. Thank you. This newletter was delivered to you by User:Jarry1250 at around 19:48, 1 July 2010 (UTC)

WikiProject Economics Newsletter (Issue IV)

Positively Economics The Economics WikiProject Newsletter Issue IV (September 2010)
Use the show/hide button to display or hide this newsletter. From the editor After a brief intermission over the Northern hemisphere summer season and the Southern hemisphere's mid-winter, Positively Economics is back. In recent weeks it has been my privilege to work on countering systemic bias with a couple of articles on the economic history of Uganda. Projects like these help, little by little, to open up new areas on knowledge to Western audiences. Indeed, with English in the ascendancy when it comes to second languages, many other audiences too. Both of the new creations made the front page via the Did you know? section for newly-created or significantly expanded articles. I would encourage other editors, particularly those who like badges, to try to get more economics articles onto the front page through this and other sections. Those with a particularly competitive bent might also wish to consider signing up for the 2011 Wikicup (great for focussing the mind). - Jarry1250 [Humorous? Discuss.] 18:55, 1 September 2010 (UTC) Example chart New bot User:LivingBot has just been approved on Wikimedia Commons to compile and upload SVG graphs based on monthly time series of, for example, UK interest rates (illustrated). It will then revisit these on a monthly basis to update them with the latest information. If you would like to bot to create and update a graph for you, get in touch with its operator. Articles of note New featured articles: None New featured lists: None New featured topics: None New good articles: Jevons paradox Redlinks to be squashed Skysmith has posted an updated list of missing economics topics. Most will need redirecting or new articles written to blueify them, so, if you can lend a hand with the effort, please do. There are some from virtually all areas of economics and hence to suit all tastes. Project news Unassessed articles: A whole swath of articles have been added to the WikiProject recently, mainly on notable economists. They are yet to be assessed on their talk pages; this also provides a good opportunity to check and improve some of our least monitored articles. Unsourced BLPs: Before the articles about economists were added, the project had very few unsourced biographies of living people; now there are a little over a hundred. If we can clear them, however, then it is unlikely that there will be many more. Again, they represent some of our least cared-for articles, and often present good targets for cleaning up. Your comments are invited in the following other ongoing project discussions and requests for help: Vanderford-Riley well-being schedule New Economic Indicator for Infoboxes Beth Hayes
To start/stop receiving this newsletter, please add/remove your name from the list here. Thank you. This newletter was delivered to you by User:Jarry1250 at around 19:25, 1 September 2010 (UTC)

Michigan Wikipedians

Greetings Malin84! I noticed that you made mention of the University of Michigan or Ann Arbor on your userpage. If you are a current student, faculty, or other affiliate at the University of Michigan, I would like to welcome you, on behalf of the Michigan Wikipedians, to our next weekly meeting on Monday September 30 (and every Monday thereafter). The meetings are held at 8:00 PM (EDT) in the University of Michigan Shapiro Library, room 4041. New and experienced editors alike are most welcome. Do not hesitate to leave me a message if you have any questions, and feel free to stop by the MWiki talk page. The Michigan Wikipedians are excited to meet you! Arbitrarily0 ^(talk) 00:20, 24 September 2013 (UTC)

Asian 10,000 Challenge invite

Hi. The Wikipedia:WikiProject Asia/The 10,000 Challenge has recently started, based on the UK/Ireland Wikipedia:The 10,000 Challenge and Wikipedia:WikiProject Africa/The 10,000 Challenge. The idea is not to record every minor edit, but to create a momentum to motivate editors to produce good content improvements and creations and inspire people to work on more countries than they might otherwise work on. There's also the possibility of establishing smaller country or regional challenges for places like South East Asia, Japan/China or India etc, much like Wikipedia:The 1000 Challenge (Nordic). For this to really work we need diversity and exciting content and editors from a broad range of countries regularly contributing. At some stage we hope to run some contests to benefit Asian content, a destubathon perhaps, aimed at reducing the stub count would be a good place to start, based on the current Wikipedia:WikiProject Africa/The Africa Destubathon which has produced near 200 articles in just three days. If you would like to see this happening for Asia, and see potential in this attracting more interest and editors for the country/countries you work on please sign up and being contributing to the challenge! This is a way we can target every country of Asia, and steadily vastly improve the encyclopedia. We need numbers to make this work so consider signing up as a participant! Thank you. --Ser Amantio di Nicolao^{Che dicono a Signa?}_{Lo dicono a Signa.} 05:20, 20 October 2016 (UTC)