# Linearization of the dynamic system (I did it, but I have a mistake that I cannot catch. Help me please)

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.

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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 @1muflon1

$$s_{t+1}=s_t+(bk-gw)p_t -kw+(gw^2/b)$$

$$s_{t+2}=s_{t+1}+(bk-gw)p_{t+1} -kw+(gw^2/b)$$

$$s_{t+3}=s_{t+2}+(bk-gw)p_{t+2} -kw+(gw^2/b)$$

...

$$s_{t+n}=s_{t+(n-1)}+(bk-gw)p_{t+(n-1)} -kw+(gw^2/b)$$

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?

• 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 Jan 10 at 9:57
• 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 :) – B11b Jan 10 at 10:08
• 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 Jan 10 at 10:30
• I added what you said as an adit in the question part dear @1muflon1 – B11b Jan 10 at 10:31
• But please can you help me to proceed this? @1muflon1 – B11b Jan 10 at 12:29