SM286A – Mathematics for Economics
Asst. Prof. Nelson Uhan
Fall 2015
Lesson 11. Dynamics of Market Price
1
Last time: a market equilibrium model with price dynamics
● Market with single commodity
● Variables:
Qd = quantity demanded
Qs = quantity supplied
P = unit price
● Equations:
Qd = α − βP
(α, β > 0)
(1)
Qs = −γ + δP
dP
= j(Qd − Qs )
dt
(γ, δ > 0)
(2)
( j > 0)
(3)
● What does (3) say about how the price changes over time?
○ Qd > Qs ⇒ dP/dt > 0 ⇒ price increases over time
○ Qd < Qs ⇒ dP/dt < 0 ⇒ price decreases over time
○ Qd = Qs ⇒ dP/dt = 0 ⇒ price stays the same over time
● Two types of equilibrium price:
○ market-clearing equilibrium price: demand equals supply, or Qd = Qs
○ intertemporal equilibrium price: price is constant over time, or dP/dt = 0
○ In this model, by (3), these two types of equilibrium price are the same
○ This may not be true in every model
● Substitute (1) and (2) into (3) to obtain the differential equation:
● This is a differential equation of the form
dy
+ ay = b
dt
with definite solution
● Therefore, we can find P(t):
1
b
b
y(t) = (y(0) − )e −at +
a
a
● In Lesson 1, we used (1) and (2) with the equilibrium condition Qd = Qs to find P ∗ =
α+γ
β+δ
● So, we can rewrite P(t) as
2
Dynamic stability
● As t → ∞, the price P(t) →
● The price adjustment process defined by the model (1)-(3) is dynamically stable: P(t) converges to a
constant P ∗ with enough time
● How does P(t) converge to P ∗ ?
P(t)
P∗
t
3
Ensuring dynamic stability
● Suppose we do not assume α, β, γ, δ > 0
● What restrictions do we need to impose on α, β, γ, δ to ensure dynamic stability?
● In order for P(t) → P ∗ , we need
● In words:
● When the demand curve is negatively sloped (
positively sloped (
) and the supply curve is
), this happens automatically
2