<p>A firm accumulates useful knowledge <span class="math-container">$k$</span> by investing in R&amp;D activities. Specifically, if the firm invests <span class="math-container">$r &gt; 0$</span> dollars into R&amp;D, the stock of useful knowledge grows by about <span class="math-container">$2\sqrt{r}$</span> units, minus the forgotten knowledge <span class="math-container">$\delta k$</span> where <span class="math-container">$\delta \in (0, 1)$</span>.
Suppose that the firm operates with the so-called Ak production function, with A > 0 being the fixed productivity, and let the price of output be normalized to 1.</p>

<p>The objective function is <span class="math-container">$$\int_0^{\infty}(Ak-r)e^{-it}dt$$</span></p>

<p><span class="math-container">$k_0&gt;0$</span> and <span class="math-container">$i&gt;0$</span> </p>

<p>I just want you to help me writing CONSTRAINT (LAW OF MOTION). </p>

<p>I don’t want you to solve this question. Just help me how to write its constraint according to above info in order to solve for maximization problem with hamiltonian function. </p>

CC BY-SA 4.0
2019-12-01T16:11:39.650Z
This seems really straightforward. Could you explain where exactly your difficulty is?
Giskard
CC BY-SA 4.0
2019-12-01T16:16:56.517Z
Indeed, I write $$ \dot{k}=Ak-2\sqrt{r}-\delta k$$ but I am not sure. Is it true? And my another small question is which is control variable and which is state variable here?@Giskard
studentp
CC BY-SA 4.0
2019-12-01T19:59:52.010Z
"the firm operates with the ... Ak production function": is it producing knowledge, or some other output?
Adam Bailey
CC BY-SA 4.0
2019-12-01T20:08:30.907Z
@AdamBailey I guess only knowledge. All question is this. I don’t know more or less. What is your opinion for constraint?
studentp
CC BY-SA 4.0
2019-12-02T08:21:22.320Z
@Giskard I would like to learn your opinion please 
studentp
CC BY-SA 4.0
2019-12-02T08:54:25.303Z
It is impossible to give a definite answer if "All question is this". My guess is $$\dot{k} = Ak+2\sqrt{r}−\delta k$$ (You had a sign error.) My other guess would be $$\dot{k} = 2\sqrt{r}−\delta k,$$ this is if the output of the production function is measured in dollars, not knowledge.
Giskard
CC BY-SA 4.0
2019-12-02T10:10:46.737Z
I have considered something. The question says useful knowledge grows $2\sqrt{r}$ units, so in the first equation, should we write $=Ak+ 2\sqrt{r}k-\delta k$. I would like to ask you. Thanks. Dear @Giskard
studentp
CC BY-SA 4.0
2019-12-02T10:28:31.263Z
This is not the place for back and forth...please ask your professor or teacher's assistant if you find the question unclear.
Giskard
CC BY-SA 4.0
2019-12-02T19:05:49.210Z
@user315 It's hard to be sure, but I suspect the production function is about production of some output other than knowledge, and knowledge is just an input into that production function.  If so, then Ak should not appear within the equation for $\dot{k}$.
Adam Bailey
CC BY-SA 4.0
2019-12-02T19:16:58.177Z
I understand. Then, please can you write all possible constraints which you suppose ?@AdamBailey btw, thank you so much for your answer
studentp
<macroeconomics><mathematical-economics><self-study><dynamic-optimization>
Writing constraint
studentp
# Writing constraint

(Asked by: username_0 on 2019-12-01)

A firm accumulates useful knowledge $k$ by investing in R&D activities. Specifically, if the firm invests $r > 0$ dollars into R&D, the stock of useful knowledge grows by about $2\sqrt{r}$ units, minus the forgotten knowledge $\delta k$ where $\delta \in (0, 1)$.
Suppose that the firm operates with the so-called Ak production function, with A > 0 being the fixed productivity, and let the price of output be normalized to 1.

The objective function is $$\int\_0^{\infty}(Ak-r)e^{-it}dt$$

$k\_0>0$ and $i>0$

I just want you to help me writing CONSTRAINT (LAW OF MOTION).

I don’t want you to solve this question. Just help me how to write its constraint according to above info in order to solve for maximization problem with hamiltonian function.

**Tags:** macroeconomics, mathematical-economics, self-study, dynamic-optimization

> **Comments:**
> **username_1**: This seems really straightforward. Could you explain where exactly your difficulty is?  
> **username_0**: Indeed, I write $$ \dot{k}=Ak-2\sqrt{r}-\delta k$$ but I am not sure. Is it true? And my another small question is which is control variable and which is state variable here?@username_1  
> **username_2**: "the firm operates with the ... Ak production function": is it producing knowledge, or some other output?  
> **username_0**: @AdamBailey I guess only knowledge. All question is this. I don’t know more or less. What is your opinion for constraint?  
> **username_0**: @username_1 I would like to learn your opinion please   
> **username_1**: It is impossible to give a definite answer if "All question is this". My guess is $$\dot{k} = Ak+2\sqrt{r}−\delta k$$ (You had a sign error.) My other guess would be $$\dot{k} = 2\sqrt{r}−\delta k,$$ this is if the output of the production function is measured in dollars, not knowledge.  
> **username_0**: I have considered something. The question says useful knowledge grows $2\sqrt{r}$ units, so in the first equation, should we write $=Ak+ 2\sqrt{r}k-\delta k$. I would like to ask you. Thanks. Dear @username_1  
> **username_1**: This is not the place for back and forth...please ask your professor or teacher's assistant if you find the question unclear.  
> **username_2**: @user315 It's hard to be sure, but I suspect the production function is about production of some output other than knowledge, and knowledge is just an input into that production function.  If so, then Ak should not appear within the equation for $\dot{k}$.  
> **username_0**: I understand. Then, please can you write all possible constraints which you suppose ?@AdamBailey btw, thank you so much for your answer  

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## 0 answers

