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From: Xian Chi on 19 Jul 2010 06:52
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?

The title of the paper where the problem is from 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.


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



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