<p>I have the following dynamic system in discrete time </p>

<p>For p is price, d is demand and s is supply.</p>

<p><span class="math-container">$$p_{t+1}-p_t= a(d_t-s_t)$$</span>
<span class="math-container">$$s_{t+1}-s_t=bp_ts_t-ws_t$$</span>
<span class="math-container">$$d_t= k-gp_t$$</span></p>

<p>I have to linearize this system around the nontrivial optimal equilibrium point.</p>

<p>—- </p>

<p>What I did attempt is  </p>

<p>In the equilibrium, <span class="math-container">$p_{t+1}=p_t=p^*$</span> and <span class="math-container">$s_{t+1}=s_t=s^*$</span> and <span class="math-container">$d_t=d^*$</span></p>

<p>So the first equation implies that <span class="math-container">$d_t=s_t$</span>. Thus, <span class="math-container">$d^*=s^*=Q^*$</span></p>

<p>So, by the second equation, <span class="math-container">$p^*=w/b$</span> </p>

<p>And the third equation implies that <span class="math-container">$s^*=d^*=Q^*=k-(gw/b)$</span></p>

<p>Now let’s linearize the second question because first and third equations are already linear. </p>

<p><span class="math-container">$$s_{t+1}=[(bp^*-w+1)s^*]+[(bp^*-q+1)(s_t-s^*)]+[bs^*(p_t-p^*)]$$</span></p>

<p><span class="math-container">$$s_{t+1}=s_t+(bk-gw)p_t -kw+(gw^2/b)$$</span></p>

<p>However, the supply curve is defined in general as <span class="math-container">$s_t=\alpha+\beta p_{t-1}$</span></p>

<p>But in my solution,  <span class="math-container">$s_{t+1}$</span> depends on <span class="math-container">$ s_t$</span>. I think this is wrong. Where is my mistake? </p>

<p>According to the suggestion from @1muflon1</p>

<p><span class="math-container">$$s_{t+1}=s_t+(bk-gw)p_t -kw+(gw^2/b)$$</span></p>

<p><span class="math-container">$$s_{t+2}=s_{t+1}+(bk-gw)p_{t+1} -kw+(gw^2/b)$$</span></p>

<p><span class="math-container">$$s_{t+3}=s_{t+2}+(bk-gw)p_{t+2} -kw+(gw^2/b)$$</span></p>

<p>...</p>

<p><span class="math-container">$$s_{t+n}=s_{t+(n-1)}+(bk-gw)p_{t+(n-1)} -kw+(gw^2/b)$$</span></p>

<p>When we add them, we get that</p>

<p><span class="math-container">$$s_{t+n}=s_t+\sum_{i=t}^{t+n-1}(bk-gw)p_{i} -\sum_t^{t+n} (kw-(gw^2/b))$$</span></p>

<p>And how to proceed that?</p>

CC BY-SA 4.0
2020-01-10T09:57:32.783Z
I did not have time to go over it in detail but to me it seem you could solve the last equation by forward induction if s t+1= s t + something then also s t+2 must be s t+ 1 + other terms and so forth in all future periods so you could get rid of the this by substitution forward.
1muflon1
CC BY-SA 4.0
2020-01-10T10:08:37.637Z
Dear @1muflon1 I am sorry, I don’t exactly understand what you suggest. Please can you solve it by hand on a paper, and share it in the below answer part, I will type it. In this way, your time may not be taken too much. If you help me its solution, I will happy. I am really stack at this point. Many thanks :)
studentp
CC BY-SA 4.0
2020-01-10T10:30:39.063Z
but to fully solve it on paper would also take a lot of time but I can change the comment into hint where I can be more explicit in what I think could help you in this case once I get to computer, I know it won’t be much but if you think it would still help you I can do that
1muflon1
CC BY-SA 4.0
2020-01-10T10:31:22.703Z
I added what you said as an adit in the question part dear @1muflon1
studentp
CC BY-SA 4.0
2020-01-10T12:29:22.610Z
But please can you help me to proceed this? @1muflon1
studentp
CC BY-SA 4.0
2020-01-10T12:38:13.187Z
if you are looking for a model solution I think this would be it unless there is some arithmetic error because the last expression states that the supply in the end will be the initial supply at t and then sum of changes in prices and then the last term would just take into account the parameters that determine S-D, since none of the parameters there depends on t that’s just some constant multiplied by n. To be honest I am not totally sure what you are looking for if you would put st on other side then this would give you the expression for cumulative change in supply that would be linear in P
1muflon1
CC BY-SA 4.0
2020-01-10T12:44:38.263Z
I Need to find linearized system because I would like to draw phase diagram of the linearized system and determine its asymptotically stability conditions. @1muflon1
studentp
CC BY-SA 4.0
2020-01-10T12:46:09.667Z
Then, by using cumulative change of supply curve in p, is it possible to draw the phase diagram and determine the stability conditions? @1muflon1
studentp
CC BY-SA 4.0
2020-01-10T13:10:27.150Z
you can check the asymptotic stability of the system by checking how the expected st+n-st equation changes as t-> infinity, but from your comment I have feeling you need to find all local stable points and my previous advice was wrong. For that you actually don’t need to do the forward substitution, you have to turn the equation St-1=St+... into differential equation of form x’(t)=F(xt) where all equilibria are given by F(xt)=0 and they are locally stable if F’<0 and not stable if F’>0 once you have the differential equation set up you can build phase diagram from that
1muflon1
CC BY-SA 4.0
2020-01-10T13:19:00.957Z
Wow that’s what I need to do!! You are the best! Well, how to turn it to the form $x’(t)=F(x_t)$ I have no idea about that! Please show me the way please I need you in this case:) @1muflon1
studentp
CC BY-SA 4.0
2020-01-10T16:36:59.097Z
Will you help me dear @1muflon1 ? I will be really glad ☺️
studentp
<macroeconomics><mathematical-economics><dynamic-programming><dynamic-optimization>
Linearization of the dynamic system (I did it, but I have a mistake that I cannot catch. Help me please)
studentp
# Linearization of the dynamic system (I did it, but I have a mistake that I cannot catch. Help me please)

(Asked by: username_0 on 2020-01-09)

I have the following dynamic system in discrete time

For p is price, d is demand and s is supply.

$$p\_{t+1}-p\_t= a(d\_t-s\_t)$$
$$s\_{t+1}-s\_t=bp\_ts\_t-ws\_t$$
$$d\_t= k-gp\_t$$

I have to linearize this system around the nontrivial optimal equilibrium point.

—-

What I did attempt is

In the equilibrium, $p\_{t+1}=p\_t=p^\*$ and $s\_{t+1}=s\_t=s^\*$ and $d\_t=d^\*$

So the first equation implies that $d\_t=s\_t$. Thus, $d^\*=s^\*=Q^\*$

So, by the second equation, $p^\*=w/b$

And the third equation implies that $s^\*=d^\*=Q^\*=k-(gw/b)$

Now let’s linearize the second question because first and third equations are already linear.

$$s\_{t+1}=[(bp^\*-w+1)s^\*]+[(bp^\*-q+1)(s\_t-s^\*)]+[bs^\*(p\_t-p^\*)]$$

$$s\_{t+1}=s\_t+(bk-gw)p\_t -kw+(gw^2/b)$$

However, the supply curve is defined in general as $s\_t=\alpha+\beta p\_{t-1}$

But in my solution, $s\_{t+1}$ depends on $ s\_t$. I think this is wrong. Where is my mistake?

According to the suggestion from @username_1

$$s\_{t+1}=s\_t+(bk-gw)p\_t -kw+(gw^2/b)$$

$$s\_{t+2}=s\_{t+1}+(bk-gw)p\_{t+1} -kw+(gw^2/b)$$

$$s\_{t+3}=s\_{t+2}+(bk-gw)p\_{t+2} -kw+(gw^2/b)$$

...

$$s\_{t+n}=s\_{t+(n-1)}+(bk-gw)p\_{t+(n-1)} -kw+(gw^2/b)$$

When we add them, we get that

$$s\_{t+n}=s\_t+\sum\_{i=t}^{t+n-1}(bk-gw)p\_{i} -\sum\_t^{t+n} (kw-(gw^2/b))$$

And how to proceed that?

**Tags:** macroeconomics, mathematical-economics, dynamic-programming, dynamic-optimization

> **Comments:**
> **username_1**: I did not have time to go over it in detail but to me it seem you could solve the last equation by forward induction if s t+1= s t + something then also s t+2 must be s t+ 1 + other terms and so forth in all future periods so you could get rid of the this by substitution forward.  
> **username_0**: Dear @username_1 I am sorry, I don’t exactly understand what you suggest. Please can you solve it by hand on a paper, and share it in the below answer part, I will type it. In this way, your time may not be taken too much. If you help me its solution, I will happy. I am really stack at this point. Many thanks :)  
> **username_1**: but to fully solve it on paper would also take a lot of time but I can change the comment into hint where I can be more explicit in what I think could help you in this case once I get to computer, I know it won’t be much but if you think it would still help you I can do that  
> **username_0**: I added what you said as an adit in the question part dear @username_1  
> **username_0**: But please can you help me to proceed this? @username_1  
> **username_1**: if you are looking for a model solution I think this would be it unless there is some arithmetic error because the last expression states that the supply in the end will be the initial supply at t and then sum of changes in prices and then the last term would just take into account the parameters that determine S-D, since none of the parameters there depends on t that’s just some constant multiplied by n. To be honest I am not totally sure what you are looking for if you would put st on other side then this would give you the expression for cumulative change in supply that would be linear in P  
> **username_0**: I Need to find linearized system because I would like to draw phase diagram of the linearized system and determine its asymptotically stability conditions. @username_1  
> **username_0**: Then, by using cumulative change of supply curve in p, is it possible to draw the phase diagram and determine the stability conditions? @username_1  
> **username_1**: you can check the asymptotic stability of the system by checking how the expected st+n-st equation changes as t-> infinity, but from your comment I have feeling you need to find all local stable points and my previous advice was wrong. For that you actually don’t need to do the forward substitution, you have to turn the equation St-1=St+... into differential equation of form x’(t)=F(xt) where all equilibria are given by F(xt)=0 and they are locally stable if F’<0 and not stable if F’>0 once you have the differential equation set up you can build phase diagram from that  
> **username_0**: Wow that’s what I need to do!! You are the best! Well, how to turn it to the form $x’(t)=F(x_t)$ I have no idea about that! Please show me the way please I need you in this case:) @username_1  
> **username_0**: Will you help me dear @username_1 ? I will be really glad ☺️  

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## 0 answers

