Exercise:
1. Let the demand and supply be
\(Y^d=\alpha-\beta P+\sigma \dot{P}\)
\(Y^s=-\gamma+\delta P\)
All the parameters are positive.
QUESTION A
\(\dot{P}_t=\theta(Y^d - Y^s)\)
\(\dot{P}_t=\theta(\alpha-\beta P_t+\sigma \dot{P}_t +\gamma-\delta P_t)\)
\((1-\theta\sigma)\dot{P}_t+\theta(\beta+\delta)P_t = \theta (\alpha + \gamma)\)
From there we have:
\((1-\theta\sigma)\dot{P}_t+\theta(\beta+\delta)P_t =0\)
\((1-\theta\sigma) \frac{dP_t}{dt} = -\theta(\beta+\delta)P_t\)
\(dP_t = \frac{-\theta(\beta+\delta)}{1-\theta\sigma} P_t dt\)
\(\int\frac{1}{P_t} dP_t = \frac{-\theta(\beta+\delta)}{1-\theta\sigma} \int dt\)
\(\ln P_t+c_a=\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t+c_b\)
\(e^{\ln P_t}=e^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t} e^c\)
with: \(A = e^c\)
So:
\(P_t=Ae^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t}\)
\(\dot{P_t}=\frac{-\theta(\beta+\delta)}{1-\theta\sigma}Ae^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t}\)
Substitute \(\dot{P}_t\) in:
\((1-\theta\sigma)\dot{P}_t+\theta(\beta+\delta)P_t = \theta (\alpha + \gamma)\)
\((1-\theta\sigma)[\frac{-\theta(\beta+\delta)}{1-\theta\sigma}Ae^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}0}]+\theta(\beta+\delta)P_0 = \theta (\alpha + \gamma)\)
\((1-\theta\sigma)\frac{-\theta(\beta+\delta)}{1-\theta\sigma}A = \theta (\alpha + \gamma)-\theta(\beta+\delta)P_0\)
\(-\theta(\beta+\delta)A = \theta (\alpha + \gamma)-\theta(\beta+\delta)P_0\)
\(A = - \frac{\alpha + \gamma}{\beta+\delta}+P_0\)
We substitute again \(\dot{P}\) in:
\((1-\theta\sigma)\dot{P}_t+\theta(\beta+\delta)P_t = \theta (\alpha + \gamma)\)
\(P= \frac{\alpha+\gamma}{\beta+\delta} - \frac{1-\theta\sigma}{\theta(\beta+\delta)}\dot{P}\)
\(P_t= \frac{\alpha+\gamma}{\beta+\delta} - \frac{1-\theta\sigma}{\theta(\beta+\delta)}[\frac{-\theta(\beta+\delta)}{1-\theta\sigma}Ae^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t}]\)
\(P_t= \frac{\alpha+\gamma}{\beta+\delta} + [P_0 - \frac{\alpha + \gamma}{\beta+\delta}]e^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t}\)
QUESTION B
The intertemporal equilibrium price is the value of the time path \(P\left(t\right)\) when \(\lim_{t\to\infty} P(t)\) since the rate of change of price over time is directly proportional to excess demand.
\(\lim_{t\to\infty} P_t = \frac{\alpha+\gamma}{\beta+\delta} + [P_0 - \frac{\alpha + \gamma}{\beta+\delta}]e^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t}\)
\(\lim_{t\to\infty} \frac{\alpha+\gamma}{\beta+\delta} = \frac{\alpha+\gamma}{\beta+\delta}\)
\(\lim_{t\to\infty} [P_0 - \frac{\alpha + \gamma}{\beta+\delta}]e^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t} = 0\)
so we have:
\(\lim_{t\to\infty} P_t = \frac{\alpha+\gamma}{\beta+\delta} \)
To find the market-clearing equilibrium price \(\bar{P}\) we must solve:
\(Y^d=Y^s\)
\(-\gamma+\delta P=\alpha-\beta P+\sigma \dot{P}\)
\((\beta+\delta)P=\alpha+\gamma + \sigma\dot{P}\)
\(P=\frac{\alpha+\gamma}{\beta+\delta} + \frac{\sigma\dot{P}}{\beta+\delta}\)
\(\dot{P}=\theta(\alpha-\beta P+\sigma \dot{P} +\gamma-\delta P)\)
\((1-\theta\sigma)\dot{P}+\theta(\beta+\delta)P = \theta (\alpha + \gamma)\)
\(\dot{P} = \frac{\theta(\alpha + \gamma)}{(1-\theta\sigma)}- \frac{\theta(\beta + \delta)}{(1-\theta\sigma)}P\)
Now we substitute:
\(P=\frac{\alpha+\gamma}{\beta+\delta} + \frac{\sigma}{\beta+\delta}[\frac{\theta(\alpha + \gamma)}{(1-\theta\sigma)}- \frac{\theta(\beta + \delta)}{(1-\theta\sigma)}P]\)
\(P(\frac{\sigma\theta}{1-\sigma\theta}+1)=\frac{\alpha+\gamma}{\beta+\delta} + \frac{\sigma\theta(\alpha+\gamma)}{(1-\theta\sigma)(\beta+\delta)}\)
\(P(\frac{\sigma\theta +(1-\theta\sigma)}{1-\sigma\theta})=\frac{\alpha+\gamma}{\beta+\delta} + \frac{\sigma\theta(\alpha+\gamma)}{(1-\theta\sigma)(\beta+\delta)}\)
\(P(\frac{1}{1-\sigma\theta})=\frac{\alpha+\gamma}{\beta+\delta} + \frac{\sigma\theta(\alpha+\gamma)}{(1-\theta\sigma)(\beta+\delta)}\)
\(P=(1-\theta\sigma)\frac{\alpha+\gamma}{\beta+\delta} + \sigma\theta\frac{\alpha+\gamma}{\beta+\delta}\)
\(\bar{P} = \frac{\alpha+\gamma}{\beta+\gamma}\)
QUESTION C
We have:
\(P_t= \frac{\alpha+\gamma}{\beta+\delta} + [P_0 - \frac{\alpha + \gamma}{\beta+\delta}]e^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t}\)
To insure dynamic stability, we must have:
\(\lim_{t\to\infty} [P_0 - \frac{\alpha + \gamma}{\beta+\delta}]e^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t}=0\)
Which implies that we must have:
\(\lim_{t\to\infty} e^{\frac{-\theta(\beta+\delta)}{1-\theta\sigma}t}=0\)
Since:
\(\lim_{X\to -\infty} e^X=0\)
And \(-\theta(\beta+\delta) <0\) since \(\theta, \beta, \delta > 0\) then to insure dynamic stability we must have \((1-\theta\sigma) >0\) or