11.5
Modelling with Difference Equations (The Cobweb Model)
This market model assumes that everything that is produced is sold (no inventory) at the price
that was established in the preceding period (lagged). In other words, it is assumed that:
a)
In the supply function the quantity to be brought to market depends on the price in the
previous period (lagged)
b)
Everything that is produced is sold (no inventory)
The assumption implies that the commodity is perishable or that simply no inventory is ever
kept.
The Cobweb Model
Cobweb models explain irregular fluctuations in prices and quantities.
Both 𝑞𝑑𝑡 and 𝑞𝑠𝑡 are linear functions of price 𝑝𝑡 but the supply function 𝑞𝑠𝑡 is lagged.
Example 1
Find the equilibrium price and the equilibrium quantity for the following market.
𝑞𝑑𝑡
𝑞𝑠𝑡
=
=
−
220
20
−
+
10𝑝𝑡
2𝑝𝑡−1
2
Example 2
If the supply and the demand functions are given by:
𝑞𝑑𝑡 = 50 − 5𝑝𝑡 and 𝑞𝑠𝑡 = −5 + 2𝑝𝑡−1 . Assume 𝑞𝑑𝑡 = 𝑞𝑠𝑡 and 𝑝0 = 10.
a) Find 𝑞𝑠1 , 𝑝1 , 𝑞𝑠2 , 𝑝2 , 𝑞𝑠3 , 𝑝3 , 𝑞𝑠4 , 𝑝4 .
b) Determine the difference equation for 𝑝𝑡 and solve it.
3
c) Does the system converge to the equilibrium and why? What are the equilibrium price
and equilibrium quantity?
Example 3
Given the demand and supply functions for the cobweb model as follows, find the equilibrium
price, determine whether the equilibrium is stable and solve the difference equation. Assume
that the market is at equilibrium.
a)
𝑞𝑑𝑡 = 30 − 5𝑝𝑡 and 𝑞𝑠𝑡 = 20 + 3𝑝𝑡−1,
𝑝0 = 2
b)
𝑞𝑑𝑡 = 40 − 10𝑝𝑡 and 𝑞𝑠𝑡 = 2 + 9𝑝𝑡−1 ,
Do WS #20
𝑝0 = 3