# A dynamic problem with Pontryagin's Maximum Principle

• Dec 9th 2008, 01:53 PM
A dynamic problem with Pontryagin's Maximum Principle
Let the typical individual maximizes lifetime utility function be:

$\displaystyle U(t)= \int ^ \infty _t e^{-(p- \lambda )( \tau - t)} \log u( \tau )d \tau \ , \ \ \ \ p > \lambda \geq 0$

where $\displaystyle \log u = \log C + \gamma \log(1-l)+ \mu \log G , \ \ \ \gamma , \mu > 0$

Flow budget constraint is given by:

$\displaystyle \frac {dA}{dt} = \dot {A}= [r(1-t_A)- \lambda ]A+1- t_L)Wl-(1+t_E)E+T$

Question: Find the growth rate of expenditure $\displaystyle \frac {dE}{dt} \frac {1}{E} = \frac { \dot {E}}{E}$

Solution so far:

Now, I look ahead in the paper, and I see $\displaystyle \frac { \dot {E}}{E}= r(1-t_A)-p$

First, I set up the Hamiltonian for this problem as follows:

$\displaystyle J = e^{-(p- \lambda )( \tau - t)} \log (u) + v \{ [r(1-t_A)- \lambda ]A+1- t_L)Wl-(1+t_E)E+T \}$

I assume that $\displaystyle C = E$, consumption is the same as expenditure here.

So I have $\displaystyle J = e^{-(p- \lambda )( \tau - t) } \log (u) + v \{ [r(1-t_A)- \lambda ]A+1- t_L)Wl-(1+t_E)C+T \}$

By the Pontryagin's Maximum Principle, we have the following conditions:

i. $\displaystyle \frac { \partial J}{ \partial u } = 0$ and ii. $\displaystyle \frac { \partial J }{ \partial A } = - \dot {v}$

Now, by i., I know that $\displaystyle \frac {1}{C}e^{-(p- \lambda )( \tau - t)} -v(1+t_E) = 0$.

So I then have $\displaystyle v = \frac {e^{-(p- \lambda )( \tau - t)}}{C}( \frac {1}{1+t_E})$
$\displaystyle v = ( \frac {1}{1+t_E}) (e^{-(p- \lambda )( \tau - t)}C^{-1})$

Taking the derivative with respect to time, I have:

$\displaystyle \dot {v} = ( \frac {1}{1+t_E})(e^{-(p- \lambda )( \tau - t)}(-C^{-2} \dot {C} + e^ {p- \lambda } C^{-1} )$

$\displaystyle \dot {v} = ( \frac {1}{1+t_E})(e^{p- \lambda })( \frac {1}{C} - e^{- ( \tau - t)} \frac { \dot {C} }{C^2})$

by ii., I have $\displaystyle \frac { \partial J }{ \partial A} = v[r(1-t_A)- \lambda ] = - \dot {v}$

$\displaystyle v[r(1-t_A)- \lambda ] = -( \frac {1}{1+t_E})(e^{p- \lambda })( \frac {1}{C} - e^{- ( \tau - t)} \frac { \dot {C} }{C^2})$

$\displaystyle ( \frac {1}{1+t_E}) (e^{-(p- \lambda )( \tau - t)}C^{-1})[r(1-t_A)- \lambda ] = ( \frac {1}{1+t_E})(e^{p- \lambda })( e^{- ( \tau - t)} \frac { \dot {C} }{C^2} - \frac {1}{C})$

$\displaystyle (e^{-( \tau - t)}C^{-1})[r(1-t_A)- \lambda ] = ( e^{- ( \tau - t)} \frac { \dot {C} }{C^2} - \frac {1}{C})$

$\displaystyle (e^{-( \tau - t)})[r(1-t_A)- \lambda ] = ( e^{- ( \tau - t)} \frac { \dot {C} }{C} - 1)$

$\displaystyle r(1-t_A)- \lambda + e^{ \tau - t } = \frac { \dot {C} }{C}$

So I guess somehow $\displaystyle e^{ \tau - t } - \lambda = p$, the interest rate?

In the attachment I have a copy of the paper that this equation is on, they are on page 4 of the pdf file.

Thank you very much!!!