Finite Horizon Dynamic programming optimization (consumption-savings problem)
$begingroup$
I am trying to solve this finite horizon dynamic problem (consumption-savings) using backward induction.
Maximize $sum_{t=0}^{T}u(c_{t})$
subject to $w_{0}>0, cin [0,w],w(t+1)=(w_{t}-c_{t})(1+r)$
and $u(c_{t})=c^{alpha},alphain(0,1)$
where, u=utility from consumption, w=wealth of consumer, r=rate of interest, c=consumption.
I have solved this question so far.
Maximize the value function at each stage: $c^{alpha}+V((w-c)(1+r))$
Backward induction gives optimal strategy at t=T-2:
Optimal strategy=$c_{T-2}(w)=dfrac{w(1+(1+r)^{2alpha})^{1/(alpha-1)}(r+2)}{1+r+(1+(1+r)^{2alpha})^{1/(alpha-1)}(2+r)}$
If I plug it into the value function to get the maximized value function, I get a complicated expression which I am not able to solve to get the general form of value function across all periods.
Am I doing anything wrong or is there any simpler way to solve this? Would be grateful for any hint regarding this.
optimization education economics dynamic-programming
asked Dec 13 '18 at 14:58
Hillary DiazHillary Diaz
62
$endgroup$
add a comment |
$begingroup$
I am trying to solve this finite horizon dynamic problem (consumption-savings) using backward induction.
Maximize $sum_{t=0}^{T}u(c_{t})$
subject to $w_{0}>0, cin [0,w],w(t+1)=(w_{t}-c_{t})(1+r)$
and $u(c_{t})=c^{alpha},alphain(0,1)$
where, u=utility from consumption, w=wealth of consumer, r=rate of interest, c=consumption.
I have solved this question so far.
Maximize the value function at each stage: $c^{alpha}+V((w-c)(1+r))$
Backward induction gives optimal strategy at t=T-2:
Optimal strategy=$c_{T-2}(w)=dfrac{w(1+(1+r)^{2alpha})^{1/(alpha-1)}(r+2)}{1+r+(1+(1+r)^{2alpha})^{1/(alpha-1)}(2+r)}$
If I plug it into the value function to get the maximized value function, I get a complicated expression which I am not able to solve to get the general form of value function across all periods.
Am I doing anything wrong or is there any simpler way to solve this? Would be grateful for any hint regarding this.
optimization education economics dynamic-programming
asked Dec 13 '18 at 14:58
Hillary DiazHillary Diaz
62
$endgroup$
add a comment |
$begingroup$
I am trying to solve this finite horizon dynamic problem (consumption-savings) using backward induction.
Maximize $sum_{t=0}^{T}u(c_{t})$
subject to $w_{0}>0, cin [0,w],w(t+1)=(w_{t}-c_{t})(1+r)$
and $u(c_{t})=c^{alpha},alphain(0,1)$
where, u=utility from consumption, w=wealth of consumer, r=rate of interest, c=consumption.
I have solved this question so far.
Maximize the value function at each stage: $c^{alpha}+V((w-c)(1+r))$
Backward induction gives optimal strategy at t=T-2:
Optimal strategy=$c_{T-2}(w)=dfrac{w(1+(1+r)^{2alpha})^{1/(alpha-1)}(r+2)}{1+r+(1+(1+r)^{2alpha})^{1/(alpha-1)}(2+r)}$
If I plug it into the value function to get the maximized value function, I get a complicated expression which I am not able to solve to get the general form of value function across all periods.
Am I doing anything wrong or is there any simpler way to solve this? Would be grateful for any hint regarding this.
optimization education economics dynamic-programming
asked Dec 13 '18 at 14:58
Hillary DiazHillary Diaz
62
$endgroup$
I am trying to solve this finite horizon dynamic problem (consumption-savings) using backward induction.
Maximize $sum_{t=0}^{T}u(c_{t})$
subject to $w_{0}>0, cin [0,w],w(t+1)=(w_{t}-c_{t})(1+r)$
and $u(c_{t})=c^{alpha},alphain(0,1)$
where, u=utility from consumption, w=wealth of consumer, r=rate of interest, c=consumption.
I have solved this question so far.
Maximize the value function at each stage: $c^{alpha}+V((w-c)(1+r))$
Backward induction gives optimal strategy at t=T-2:
Optimal strategy=$c_{T-2}(w)=dfrac{w(1+(1+r)^{2alpha})^{1/(alpha-1)}(r+2)}{1+r+(1+(1+r)^{2alpha})^{1/(alpha-1)}(2+r)}$
If I plug it into the value function to get the maximized value function, I get a complicated expression which I am not able to solve to get the general form of value function across all periods.
Am I doing anything wrong or is there any simpler way to solve this? Would be grateful for any hint regarding this.
optimization education economics dynamic-programming
optimization education economics dynamic-programming
asked Dec 13 '18 at 14:58
Hillary DiazHillary Diaz
62
asked Dec 13 '18 at 14:58
Hillary DiazHillary Diaz
62
edited Dec 13 '18 at 18:03
Hillary Diaz
asked Dec 13 '18 at 14:58
Hillary DiazHillary Diaz
62
asked Dec 13 '18 at 14:58
Hillary DiazHillary Diaz
62
asked Dec 13 '18 at 14:58
Hillary DiazHillary Diaz
62
62
add a comment |
add a comment |
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