differential equation in economic application

Question

We have $q^d=-3pt-1+130$ and $q^s=4pt+25$. I solved the system and I found out that there is stability and equilibrium with $p*=15$ and $q*=85$. If price grow 30% will we have again equilibrium and in which time? I thought that the part with time I could find it with Excel, but what should I do to find an equation to solve it?

Answer 1

Your nomenclature confused me a bit. So I'll assume that you meant to write

$$q_d =-3p -t+130$$

for the demand function (note that the demand is dropping with time!) and for the supply function,

$$q_s =4p +25$$

At $t=0$, this gives you the equilibrium values of $p_*=15$ and $q_*=85$ that you calculate.

Now if we perturb $p$ by setting $p=Kp_*$, ($K=1.3$ in your case for a 30% price increase) all you do is make this substitution above, set $q_d = q_s$ and solve for $t$ which in this case is negative implying that equilibrium is not possible with a price increase if the demand drops with time.

If however, you have $+t$ instead of $-t$ (demand growing with time) in the demand equation, then the equilibrium time is $t=105(K-1)$, which for 30% increase in price is $t=31.5$

Please let me know if I'm misinterpreting anything in your post.