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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