Question

In the market for Fante Kenley, the supply and demand functions respectively are
and
When there is excess demand, price adjusts according to the equation

a) Find the long run equilibrium price, P* (that is, the price at which there is no excess demand or supply).
b) Formulate and solve he first order differential equation giving P as a function of time, t. Is this market dynamically stable or unstable?
c) If the initial price is P = 50, how close will the price be to its long run equilibrium value, when t = 10?

Answer 1

a)

$Qs = 0.25P + 10\\ Qd = -0.5p + 100$

If excess demand , price adjustment $\frac{∂p}{∂t} = 0.5(Q^d − Q^s)$

For Long run equilibrium put

$QD = Qd\\ 0.25P + 10 = -0.5p + 100\\ 0.25P + 0.5p = 100 - 10 0.30P = 90\\ P = \frac{90}{0.30}\\ P = 300$

Now put P = 300 in any equation to calculate the Equilibrium Quantity

Qs = 0.25P + 10

Qs = 0.25*300 + 10

Qs = 75 + 10

Qs = 80

$Qs = 0.25P + 10\\ Qs = 0.25\times300 + 10\\ Qs = 75 + 10\\ Qs = 80$

b)

The Given Differentiation is $\frac{∂p}{∂t} = 0.5(Q^d − Q^s)$

$\frac{∂p}{∂t} = 0.5(-.5p+100-0.25p-10)$

which can be arranged as

$\frac{∂p}{∂t} = 0.5(-.5p+100-0.25p-10)$

$\frac{∂p}{∂t} = -0.15p+45$ where b=-0.15 and c=45

$p=Ae^{bt}-\frac{c}{b}$ where $A=P(0)+\frac{c}{b}$

since b=-0.15 and c=45, we have $\frac{c}{b}=-p^*$ (i.e equilibrium price)

$P = (P(0) + P*)e^{bt}+p^*$

$P = (P(0) + 300)e^{0.15t} + 300$

 Accordingly as b is < 0 this implies it will increase monotonically

 through time, therefore market is Dynamically unstable

c)

$When\space P(0) = 50, A\\ P= (P(0) + 300)_e^{0.15t}− 300\\P = (50 + 300)_e^{0.15\times10} − 300\\P = (350)\times4.4816 − 300\\P = 820.422$