<p>I am given a following dynamic programming problem; 
$$
\sup_{k_{t+1}}\sum_{t=0}^{t=\infty}\delta^t(ak_t-\frac{b}{2}k_t^2-\frac{c}{2}(k_{t+1}-k_t)^2)
$$</p>

<p>where $f(k_t) = ak_t-\frac{b}{2}k_t^2$ is the production function of a firm and $\frac{c}{2}(k_{t+1}-k_t)^2$ is its cost function.</p>

<p>I am supposed to show that $\lim_{t\to\infty}\sum_{t=0}^{t=\infty}\delta^t(ak_t-\frac{b}{2}k_t^2-\frac{c}{2}(k_{t+1}-k_t)^2)$, exists by showing that $\lim_{t\to\infty}\sum_{t=0}^{t=\infty}\delta^t(ak_t-\frac{b}{2}k_t^2-\frac{c}{2}(k_{t+1}-k_t)^2) &lt; \frac{1}{1-\delta}\frac{a^2}{2b}$</p>

<p>I have spent hours in trying to think about this problem but i remain clueless. Any hints on how to go about this problem?</p>

CC BY-SA 3.0
2015-11-01T11:52:18.500Z
By "limit exists" you mean "limit is finite"? Also, more context is needed. Do you have a terminal condition on $k_t$? Is there a fixed point for the $k_t$ process? What is the expression for the $k_t$ process? In general the result does not hold.  If for example we specify that $k_{t+1} = (1+\delta^{-t})k_t$, the sum will go to minus infinity.
Alecos Papadopoulos
CC BY-SA 3.0
2015-11-01T15:04:56.023Z
@AlecosPapadopoulos Yes, finite. I'm sorry I'm just starting to learn dynamic programming, so my responses could be stupid. In the objective function given above t is from 0 to infinity, doesn't that mean that there is no terminal point (infinite horizon)? Do I need to derive the expression for $k_t$ in order to answer this question? If it helps above problem is the firm's profit maximization problem where $f(k_t) = ak_t-\frac{b}{2}k_t^2$ is the technology facing the firm and $\frac{c}{2}(k_{t+1}-k_t)^2$ is the cost where $k_t\varepsilon\mathbb{R}$ (all real numbers, not just positive).
Sher Afghan
CC BY-SA 3.0
2015-11-01T16:04:21.807Z
When the horizon is infinite, a "terminal condition" will take the form of a limiting condition.
Alecos Papadopoulos
CC BY-SA 3.0
2015-11-01T16:28:07.313Z
I see, so i guess the question is asking about calculating that limiting condition.
Sher Afghan
CC BY-SA 3.0
2015-11-01T16:31:03.520Z
There are two more places where you have to add the missing term. Guidance: As I have already mentioned the finiteness of the limit does not hold in general. So solve the profit maximization problem and obtain the optimal rule for $k_t$. Calculate its fixed point (the inequality you were given has something to do with it, trust me), and examine whether it is asymptotically stable. If it is, the process will converge there. What does that tell us? etc
Alecos Papadopoulos
CC BY-SA 3.0
2015-11-06T09:38:44.393Z
Hi I'm still stuck at this problem. Can you tell me how do we calculate fix point?
Sher Afghan
CC BY-SA 3.0
2015-11-06T10:36:18.673Z
Have derived the first-order condition for a maximum with respect to $k_{t+1}$?
Alecos Papadopoulos
CC BY-SA 3.0
2015-11-06T11:08:37.157Z
I differentiated the objective function with respect to $k_{t+1}$ and got the second difference equation in $k_t$. And then I don't know what to do with it. I'm trying to understand these concepts from SLP, but it is giving me a tough time and I remain lost. Do you know of any beginner level introduction for dynamic programming especially on euler equations, bellman equations, fixed point, contraction mapping etc?
Sher Afghan
CC BY-SA 3.0
2015-11-06T11:35:37.830Z
This foc-obtained difference equation in $k_t$ is an _optimal rule_. In it, set $k_t=k_{t+1}=k^*$ to find the fixed-point (that satisfies also the optimal rule). As for "beginner level dynamic programming"... I am not sure that such a thing exists.
Alecos Papadopoulos
<dynamic-programming>
Taking limit of a sequence
Sher Afghan
# Taking limit of a sequence

(Asked by: username_0 on 2015-11-01)

I am given a following dynamic programming problem;
$$
\sup\_{k\_{t+1}}\sum\_{t=0}^{t=\infty}\delta^t(ak\_t-\frac{b}{2}k\_t^2-\frac{c}{2}(k\_{t+1}-k\_t)^2)
$$

where $f(k\_t) = ak\_t-\frac{b}{2}k\_t^2$ is the production function of a firm and $\frac{c}{2}(k\_{t+1}-k\_t)^2$ is its cost function.

I am supposed to show that $\lim\_{t\to\infty}\sum\_{t=0}^{t=\infty}\delta^t(ak\_t-\frac{b}{2}k\_t^2-\frac{c}{2}(k\_{t+1}-k\_t)^2)$, exists by showing that $\lim\_{t\to\infty}\sum\_{t=0}^{t=\infty}\delta^t(ak\_t-\frac{b}{2}k\_t^2-\frac{c}{2}(k\_{t+1}-k\_t)^2) < \frac{1}{1-\delta}\frac{a^2}{2b}$

I have spent hours in trying to think about this problem but i remain clueless. Any hints on how to go about this problem?

**Tags:** dynamic-programming

> **Comments:**
> **username_1**: By "limit exists" you mean "limit is finite"? Also, more context is needed. Do you have a terminal condition on $k_t$? Is there a fixed point for the $k_t$ process? What is the expression for the $k_t$ process? In general the result does not hold.  If for example we specify that $k_{t+1} = (1+\delta^{-t})k_t$, the sum will go to minus infinity.  
> **username_0**: @AlecosPapadopoulos Yes, finite. I'm sorry I'm just starting to learn dynamic programming, so my responses could be stupid. In the objective function given above t is from 0 to infinity, doesn't that mean that there is no terminal point (infinite horizon)? Do I need to derive the expression for $k_t$ in order to answer this question? If it helps above problem is the firm's profit maximization problem where $f(k_t) = ak_t-\frac{b}{2}k_t^2$ is the technology facing the firm and $\frac{c}{2}(k_{t+1}-k_t)^2$ is the cost where $k_t\varepsilon\mathbb{R}$ (all real numbers, not just positive).  
> **username_1**: When the horizon is infinite, a "terminal condition" will take the form of a limiting condition.  
> **username_0**: I see, so i guess the question is asking about calculating that limiting condition.  
> **username_1**: There are two more places where you have to add the missing term. Guidance: As I have already mentioned the finiteness of the limit does not hold in general. So solve the profit maximization problem and obtain the optimal rule for $k_t$. Calculate its fixed point (the inequality you were given has something to do with it, trust me), and examine whether it is asymptotically stable. If it is, the process will converge there. What does that tell us? etc  
> **username_0**: Hi I'm still stuck at this problem. Can you tell me how do we calculate fix point?  
> **username_1**: Have derived the first-order condition for a maximum with respect to $k_{t+1}$?  
> **username_0**: I differentiated the objective function with respect to $k_{t+1}$ and got the second difference equation in $k_t$. And then I don't know what to do with it. I'm trying to understand these concepts from SLP, but it is giving me a tough time and I remain lost. Do you know of any beginner level introduction for dynamic programming especially on euler equations, bellman equations, fixed point, contraction mapping etc?  
> **username_1**: This foc-obtained difference equation in $k_t$ is an _optimal rule_. In it, set $k_t=k_{t+1}=k^*$ to find the fixed-point (that satisfies also the optimal rule). As for "beginner level dynamic programming"... I am not sure that such a thing exists.  

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## 0 answers

