# Time-dependent market-clearing equilibrium price

I'm working my way through Chiang and Wainwright's Fundamental Methods of Mathematical Economics (4th ed) while holed up at home. On p. 532, in exercise 16.4 2. (b), the authors ask you to find the intertemporal equilibrium price and the market-clearing equilibrium price in the following model:

\begin{align} Q_d & = \alpha - \beta P + m P' + n P'' \\ Q_s & = -\gamma + \delta P \\ P' & = j(Q_d - Q_s) \end{align}

where $$\alpha, \beta, \gamma, \delta > 0$$, and $$n \ne 0$$. Reducing this to a single second-order differential equation yields

$$P'' + \frac{ mj - 1 }{ n } P' - \frac{ \beta + \delta }{ n } P = -\frac{ \alpha + \gamma }{ n }$$

from which the intertemporal equilibrium price

$$\bar P = \frac{ \alpha + \gamma }{ \beta + \delta }$$

can immediately be inferred. Now, for the market-clearing equilibrium price, my approach was to set $$Q_s = Q_d$$ and solve for $$P$$, observing that for $$Q_s = Q_d$$, $$P' = 0$$ by the third model equation; this yields

$$P^*(t) = \frac{ \alpha + \gamma + n P''(t) }{ \beta + \delta }$$

However, looking at the solution for this exercise, I'm finding that the market-clearing equilibrium price is also supposed to be

$$P^* = \frac{ \alpha + \gamma }{ \beta + \delta }$$

according to the authors. This would seem to require that $$P''(t) = 0$$ when the market is cleared, and I cannot see how this must necessarily follow from the model equations.

Plugging my solution into the model equations indicates that at the price $$P^*(t)$$, the market is indeed cleared, though if $$P''(t) \ne 0$$, this is merely a coincidence. That said, surely the authors are correct, and the error is on my part.

Unfortunately I cannot see where I went wrong. I'd appreciate any help, pointers or tips. Thank you, and stay safe!