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Question: A consumer has wealth ß0 > 0 and wants to find the best way of consuming this wealth. Preferences over consumption c(t) are given by $\int$$\infty$0 e-pt((c(t)1-$\sigma$-1)/(1-$\sigma$))dt where 0 < $\rho\ < 1$ is an intertemporal discounting and $\sigma$ > 0, ß $\neq$ 1. Let ß(t) denote the amount of wealth at time t. ß does not spoil over time, that is, its value does not decrease if it is not consumed. On the other hand, it does not receive interest. A unit of consumption decreases wealth by the same unit. You can view ß0 as a cake and the problem of the consumer as the problem of eating the cake over time in the optimal way. Formulate the problem and find the optimal path of consumption over time.

(My queries in bold)

Solution:

The problem is max $\int$$\infty$0 e-pt((c(t)1-$\sigma$-1)/(1-$\sigma$))dt subject to $\int$$\infty$0 c(t).dt = ß0

Or $.{B}$ = -c with ß0 given. The Hamiltonian of this problem is:

H (c, ß, $\mu$) = ((c(t)1-$\sigma$-1)/(1-$\sigma$))-$\mu$c

1) What do we mean by the Hamiltonian? Is it a constraint of some sort?

where $\mu$ is the costate variable attached to the constraint .{B} = -c. The conditions for the optimum are $\partial$u/$\partial$c = 0 and .$\mu$ = $\rho$$\mu$ - $\partial$H/$\partial$B together with the constraint. The transversality condition is:

limt$\longrightarrow$$\infty$+e-$\rho$t$\mu$(t)ß(t)=0.

2) When we use a transversality condition, does this define the boundaries of consumption in this case?

These conditions imply:

c-$\sigma$ = $\mu$, .$\mu$ = $\rho$$\mu$

Notice that H does not depend on the state variable and so the second optimality condition is very simple.

3) Why are we using a second optimality condition?

c-$\sigma$ = $\mu$ implies -$\sigma$(.c/c) = $\rho$ $\Rightarrow$ .$\mu$/$\mu$. Together with the second equation:

-$\sigma$(.c/c) = $\rho$ $\Rightarrow$ .c/c = -($\rho$/$\sigma$)

Therefore, the optimum path involves setting consumption to the path c(t) = c0e($\rho$/$\sigma$)*t. To find c0, we can use the transversality condition. Another way of finding c0, more direct and more intuitive, is by the constraint $\int$$\infty$0 c(t).dt = ß0. According to this constraint, given the optimal path of c:

$\int$$\infty$0 c0e($\rho$/$\sigma$)*tdt = ß0

$\Rightarrow$ -(c0e(-$\rho$/$\sigma$)*t)/($\rho$/$\sigma$)|$\infty$$0$

As a result, the path for consumption, now fully characterised, is given by:

c(t) = ($\rho$/$\sigma$).ß0.e-($\rho$/$\sigma$).t

We see that consumption decreases gradually over time, that consumption is higher if ß0 is higher, and that $\rho$ and $\sigma$ affect the optimal path for consumption.

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