Hotelling rule and the shadow prices in a peg

Question

Hotelling rule states that at equilibrium

$$\frac{\dot p(t)}{p(t)} = i$$

Meaning that rate of change of the price at time $t$ should be equal to $i$, the interest rate. Assume that the government pegs the price at the level $P^{*}$. Assume further that the shadow price at time t is given by: $Q_t =( \sigma i S_t)^\frac{-1}{\sigma}$

Show that if $\dot q(t)= i$ then $Q_t = P^{*}$

My questions:

(1) what exactly is $\dot q(t)$ or $\dot p(t)$. Is it the derivative? I couldn't make sense out of it.

(2) Is the hoteling condition given in logarithms?

(3) And what method to go with after understanding (1)-(2)

EDIT:

Yes, this is a homework question. However I never asked you to solve the question. I asked what kind of a method would be used in this case. Since the problem doesn't make sense.

$$q(t) = log (Q_t) = \frac{-1}{\sigma} log(\sigma i S_t) \\ \frac{dq(t)}{dt}=\dot q(t) = \frac{- \sigma i S_t}{\sigma^2 i} = \frac{- S_t}{\sigma }$$

So condition on $q(t)$ implies that $S_t = - \sigma i$. Which I believe there is a problem since $S_t <0$ Anyway :

since $S_t <0$ at this specified time $t$ all the resources have varied off.

If I ended up finding $S_t = 0$ then I would conclude that at this time $t$ all the speculators bought up the resource and since the resource was sold at the price $P^*$ somehow this would imply $Q_t = P^*$

But $S_t <0$ is the result we get. Which simply says that all the goods were bought plus government bought some goods. I think that implies a price $q(t)>P^*$

Answer 1

Partial Answer

I can give you an answer for parts (1) and (2). As for (3), I can't give an answer unless you show some work as per our site policy on homework questions.

---

(1)These variables are interpreted as the change in $q$ or $p$ over time. Mathematically this is the derivative, giving us the results: $$\dot q(t)=\frac{dq(t)}{dt}$$ and $$\dot p(t)=\frac{dp(t)}{dt}$$

(2) Hotelling's rule is not given in logarithms, as explained in as the notation is explained for (1). if it was it would be indicated by the notation of $\hat p(t)$. Written concisely as:

$$\hat p(t)=\ln(p(t))$$