Example of Application of Differential Equations:

1. Let the demand and supply be
\(Y^d = \alpha - \beta P + \sigma \dot{P}\)
\(Y^s = - \gamma + \delta P\)
All parameters are positive. 
a.  \(\dot P = \theta ( Y^d - Y^s)\)
\(\dot P = \theta (\alpha - \beta P + \sigma \dot P + \gamma - \delta P)\)
\(\dot P = \theta \alpha + \theta \gamma - \theta \beta P - \theta \delta P+ \theta \sigma \dot P\)
\(\theta P (\beta + \delta) + \dot P (1 - \theta \sigma) = \theta (\alpha + \gamma) \)
\(P (\beta + \delta) + \dot P (\frac{1 - \theta \sigma}{\theta}) = \alpha + \gamma\)
\(P + \dot P (\frac{1- \theta \sigma}{\theta \beta + \theta \delta}) = \frac{\alpha + \gamma}{\beta + \delta}\)
Similar to form: \(b = \dot y + ay\)
\(y_c = [P_0 - \frac{\alpha + \gamma}{\beta + \delta}] e^ {-\frac{\theta \beta + \theta \delta}{1 - \theta \sigma}t}\)
\(y_p = \frac {b}{a} = \frac{\alpha + \gamma}{\beta + \delta}\)
\(P_t = y_c + y_p = [P_0 - \frac{\alpha + \gamma}{\beta + \delta})] e^ {-\frac{\theta \beta + \theta \delta}{1 - \theta \sigma}t} + \frac {\alpha + \gamma}{\beta + \delta}\)
b. Intertemporal equilibrium price (when \(t \rightarrow \infty\)):
\(\lim_ {t \to \infty} P_t = [P_0 - \frac{\alpha + \gamma}{\beta + \delta})] e^ {-\frac{\theta \beta + \theta \delta}{1 - \theta \sigma}t} + \frac {\alpha + \gamma}{\beta + \delta}\)
Since we assumed that parameters are positive:
\(\lim_{ t\to \infty} P_1 = [P_0 - \frac{\alpha + \gamma}{\beta + \delta})] e^ {-\infty} + \frac {\alpha + \gamma}{\beta + \delta} = \frac {\alpha + \gamma}{\beta + \delta}\)
Market-clearing equilibrium price (when \(Y^d = Y^s\)):
\(\alpha - \beta P + \sigma \dot P = -\gamma + \delta P\)
Substituting in \(\dot P\) from problem a: 
\(\alpha - \beta P + \sigma(\frac {\theta(\alpha + \gamma) - \theta P (\beta + \delta)}{1-\theta \sigma}) = -\gamma + \delta P\)
\(\alpha + \frac{\sigma \theta \alpha + \sigma \theta \gamma}{1-\theta \sigma} + \gamma = \delta P + \beta P + P(\frac{\sigma \theta \beta + \sigma \theta \delta}{1-\theta\sigma})\)
\(\alpha + \frac{\sigma \theta \alpha + \sigma \theta \gamma}{1-\theta \sigma} + \gamma = P(\delta + \beta + \frac{\sigma \theta \beta + \sigma \theta \delta}{1-\theta\sigma})\)
\(\frac{\sigma \theta \alpha + \sigma \theta \gamma}{1-\theta \sigma} \cdot -\frac{1 - \theta \sigma}{\sigma \theta \beta + \sigma \theta \delta} = P(\delta + \beta) - \alpha - \gamma\)
\(P = \frac{\alpha + \gamma}{\beta + \delta}\)
c. In order for the intertemporal equilibrium to converge at the market-clearing equilibrium price, \(\sigma < 0\), so that the exponent is negative and the complementary function approaches 0. Thus, the particular function-- which is the market-clearing price-- remains.