Question

3. (An application to Economics: Demand, Supply and Price)

Let Qd, Qs and P be the quantity demanded, quantity supplied and price, respectively, of a certain 1 commodity. We assume that:

Qs = c1 + w1P + u1P' + v1P''

Qd = c2 + w2P + u2P'' + v2P''

where the primes denote derivatives with respect to t, time.

Now let c = c1 − c2; u = u1 − u2; v = v1 − v2; w = w1 − w2.

(a) Using the typical Economics assumption that demand equals supply i.e. Qs = Qd, derive a non-homogenous O.D.E for P.

(b) Given that w > 0 and c < 0, what is the equilibrium price? (Hint: The equilibrium price is a constant. In fact it is a particular solution to the O.D.E. above)

(c) If v > 0, w > 0 and c < 0, what extra conditions on u, v, w guarantee that the price oscillates, with a constant amplitude, about the equilibrium price?

(d) If v > 0, w > 0 and c < 0, what other conditions on u, v, w guarantees that the price approaches the equilibrium price?

Answer 1

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