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From: Xian Chi on 18 Jul 2010 14:54 Ladies and Gentleman, Dudes and Dudettes! I have got a question, for most of you I guess it's pretty trivial. So I would really appreciate it if you could help me out a little bit, as I am not the biggest math genius. There is a consumer maxmization problem \max_{c(t),k(t)} \int_{0}^{\infty} e^{-pt}u(c(t))dt s.t. c+\dot{k}=wL^(c)+(r-\delta)(1-\tau)k+T^{c} ok, fine the problem is easy to solve by using a Hamiltonian H=e^{-pt}u(c(t)) +\lambda[wL^(c)+(r-\delta)(1-\tau)k+T^{c}-c] and solving for the first order conditions \partial H / \partial c: e^{-pt}u'(c)=\lambda (1) \partial H / \partial k: \lambda(r-\delta)(1-\tau)=-\dot{\lambda} (2) \partial H / \partial \lambda: ... (3) \partial \lambda / \partial t: .... (4) Okay this way is clear. I merge (4) and (1) and (2). Toegether with two this gives me the two equilibrium equations. I know that (2) could be interpreted as a sort of arbitrage condition or fisher equation. Merging (1) and (2) would therefore give the following arbitrage condition: -\dot{\lambda}=e^{-pt}u'(c)(r-\delta)(1-\tau) (5) And this is where my problem starts. The author denotes the arbitrage condition as: u'(c) \equiv \int_{t}^{\infty}e^{-pt}u'(c)(r-\delta)(1-\tau) Ok this looks very similar. But where does the integral come from? How is it possible to define it as u'(c) ? Can I just take the integral from (5) to write \lambda instead of /dot{\lambda} and interpret \lambda as the shadow price of consumption? It would be so nice and helpful from you if you could take a look at this problem and give me some ideas! Best regards, yours Chi
From: Ray Vickson on 18 Jul 2010 14:59 On Jul 18, 11:54 am, Xian Chi <t16...(a)simonews.com> wrote: > Ladies and Gentleman, Dudes and Dudettes! > > I have got a question, for most of you I guess it's pretty trivial. So > I would really appreciate it if you could help me out a little bit, as > I am not the biggest math genius. > > There is a consumer maxmization problem > > \max_{c(t),k(t)} \int_{0}^{\infty} e^{-pt}u(c(t))dt > s.t. > c+\dot{k}=wL^(c)+(r-\delta)(1-\tau)k+T^{c} > > ok, fine the problem is easy to solve by using a Hamiltonian > > H=e^{-pt}u(c(t)) +\lambda[wL^(c)+(r-\delta)(1-\tau)k+T^{c}-c] > > and solving for the first order conditions > > \partial H / \partial c: e^{-pt}u'(c)=\lambda (1) > \partial H / \partial k: \lambda(r-\delta)(1-\tau)=-\dot{\lambda} (2) > \partial H / \partial \lambda: ... (3) > \partial \lambda / \partial t: .... (4) > > Okay this way is clear. I merge (4) and (1) and (2). Toegether with two > this gives me the two equilibrium equations. I know that (2) could be > interpreted as a sort of arbitrage condition or fisher equation. > Merging (1) and (2) would therefore give the following arbitrage > condition: > > -\dot{\lambda}=e^{-pt}u'(c)(r-\delta)(1-\tau) (5) > > And this is where my problem starts. The author denotes the arbitrage > condition as: Are you reading a book or a paper? If so, what are its title and author? That information could be helpful in case we are familiar with the source of your problem. R.G. Vickson > > u'(c) \equiv \int_{t}^{\infty}e^{-pt}u'(c)(r-\delta)(1-\tau) > > Ok this looks very similar. But where does the integral come from? How > is it possible to define it as u'(c) ? Can I just take the integral > from (5) to write \lambda instead of /dot{\lambda} and interpret > \lambda as the shadow price of consumption? > > It would be so nice and helpful from you if you could take a look at > this problem and give me some ideas! Best regards, > > yours Chi
From: Xian Chi on 18 Jul 2010 15:06 On 2010-07-18 20:59:40 +0200, Ray Vickson said: > g a book or a paper? If so, what are its title and > author? That information could be helpful in case we are familiar with > the source of your problem. > > R.G. Vickson ahhh ok of course, it's the classic Kenneth Judd paper from 1985. Here is the free link to the working paper version www.kellogg.northwestern.edu/research/math/papers/572.pdf It is about equation (2) in the text, page 4. I mean I wouldn't take this to serious, the equilibrium conditions can be found by the Hamiltonian anyway, but I need this arbitrage to show later that the solution is bounded. Thank you very much for your help! Best regards, Chi
From: Ray Vickson on 18 Jul 2010 16:37 On Jul 18, 12:06 pm, Xian Chi <t16...(a)simonews.com> wrote: > On 2010-07-18 20:59:40 +0200, Ray Vickson said: > > > g a book or a paper? If so, what are its title and > > author? That information could be helpful in case we are familiar with > > the source of your problem. > > > R.G. Vickson > > ahhh ok of course, it's the classic Kenneth Judd paper from 1985. Here > is the free link to the working paper version > > www.kellogg.northwestern.edu/research/math/papers/572.pdf > > It is about equation (2) in the text, page 4. I mean I wouldn't take > this to serious, the equilibrium conditions can be found by the > Hamiltonian anyway, but I need this arbitrage to show later that the > solution is bounded. > > Thank you very much for your help! > > Best regards, > > Chi When I click on this link I get the message "not found". So, I went to the web page of Kenneth Judd and found citations to several of his papers from 1985. What is the title of the paper? Of course, it might have a publication date >= 1986, making the search even harder. (That is why I asked you for a title!) R.G. Vickson
From: Xian Chi on 18 Jul 2010 16:58
On 2010-07-18 22:37:01 +0200, Ray Vickson said: > I click on this link I get the message "not found". So, I went to > the web page of Kenneth Judd and found citations to several of his > papers from 1985. What is the title of the paper? Of course, it might > have a publication date >= 1986, making the search even harder. (That > is why I asked you for a title!) Hello! Strange, here the link works. But anyway the title of the paper is "Redistributive Taxation in a simple perfect foresight Model" Kenneth L. Judd, 1982. "Redistributive Taxation in a Simple Perfect Foresight Model," Discussion Papers 572, Northwestern University, Center for Mathematical Studies in Economics and Management Science. published as: Judd, Kenneth L., 1985. "Redistributive taxation in a simple perfect foresight model," Journal of Public Economics, Elsevier, vol. 28(1), pages 59-83, October. Thanks for your effort and patience with me! Best regards, Chi |