Question

In the market for Fante Kenley, the supply and demand functions respectively are Q^s = 0.25P+10 and  -0.5P+100

When there is excess demand, price adjusts according to the equation dp/dt = 0.5(Qd - Qs)

 

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)

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