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Jul 18, 2010, 3:10:43 PM7/18/10

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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 problem is from 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.

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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