<h1>Partial Answer</h1>
<p>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 <a href="https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions">site policy on homework questions</a>.</p>
<hr />
<p>(1)These variables are interpreted as the change in <span class="math-container">$q$</span> or <span class="math-container">$p$</span> over time. Mathematically this is the derivative, giving us the results:
<span class="math-container">$$\dot q(t)=\frac{dq(t)}{dt}$$</span> and <span class="math-container">$$\dot p(t)=\frac{dp(t)}{dt}$$</span></p>
<p>(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 <span class="math-container">$\hat p(t)$</span>. Written concisely as:</p>
<p><span class="math-container">$$\hat p(t)=\ln(p(t))$$</span></p>

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2017-10-04T03:06:29.987Z
2017-10-05T01:15:09.760Z
2020-06-17T09:40:54.643Z
EconJohn
<p>Hotelling rule states that at equilibrium </p>

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

<p>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: $Q_t =( \sigma i S_t)^\frac{-1}{\sigma}$</p>

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

<p>My questions:</p>

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

<p>(2) Is the hoteling condition given in logarithms?</p>

<p>(3) And what method to go with after understanding (1)-(2)</p>

<p>EDIT: </p>

<p>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.</p>

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

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

<p>since $S_t &lt;0$ at this specified time $t$ all the resources have varied off. </p>

<p>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 $Q_t = P^*$</p>

<p>But $S_t &lt;0$ 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 $q(t)&gt;P^*$</p>

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2017-10-04T02:55:02.570Z
can you show us what you did so far for part 3?
EconJohn
<macroeconomics><financial-economics><resource-economics>
Hotelling rule and the shadow prices in a peg
Kuantew
# Hotelling rule and the shadow prices in a peg

(Asked by: username_0 on 2017-10-04)

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 $q(t)>P^\*$

**Tags:** macroeconomics, financial-economics, resource-economics

> **Comments:**
> **username_1**: can you show us what you did so far for part 3?  

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## 1 answers

### Answer by username_1 on 2017-10-04. Score: 1 | Accepted answer

# 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](https://economics.meta.stackexchange.com/questions/1465/economics-se-policy-on-homework-questions).

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(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:

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


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