Download Chapter 10: The Cobweb Model

Chapter 10: The Cobweb Model
The supply of many agricultural products is based on the price of the product in the previous
year. If prices were high in year 1 more of the product will be planted in year 2 in the
expectation that prices will remain high. This may cause an oversupply, resulting in lower
prices.
Because of the low prices in year 2, less of the product will be planted in year 3, resulting in
higher prices. (This is based on the adaptive expectations hypothesis, where expectations are
based on past behaviour).
Due to the delay in production time, supply and demand do not interact to reach equilibrium
in the short term, and this process may continue for a number of years.
Given a demand curve Qdt = 150 -3Pt and a supply curve Qst = -50 + 2Pt-1,
and given P0 = 48 at time t = 0,
a) Find the equilibrium price and quantity.
b) Plot the price for the first 10 years.
Solution
a) At equilibrium, Qdt = Qst
150 – 3Pt = -50 + 2Pt-1
-3Pt - 2Pt-1 = -50 - 150
3Pt + 2Pt-1 = 200
The general solution has the form Pt = CF + PI
Where CF is called the Complementary Function and PI is called the Particular Integral
Complementary function (CF). The complementary function is the solution of the
corresponding homogeneous equation (in other words, equate all terms in P to zero)
3Pt + 2Pt-1 = 0
Substituting the general form of the solution: Pt.c = A(a)t, hence Pt-1.c = A(a)t-1 = A.at.a-1 into
the homogeneous form of the equation, 3Pt + 2Pt-1 = 0
3Aat + 2Aata-1 = 0
Aat(3 + 2a-1) = 0
Aat = 0 gives the trivial solution, which is of no practical use.
Hence 3 + 2a-1 = 0
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Chapter 10: The Cobweb Model
3+
=0
= -3
-3a = 2
a = -0.667
Therefore Pt..c = A(-0.667)t
Particular integral (PI)
The particular integral is a function that satisfies the full original difference equation:
3Pt + 2Pt-1 = 200.
Since the right hand side is a constant, the general form of the particular integral is also some
constant, say k
Substituting the general form of the PI, Pt.p = k and Pt-1.p = k, into the full difference equation
3k + 2k = 200
5k = 200
k = 40
Therefore, Pt.p = 40
The General Solution is
Pt = CF + PI
Pt = Pt.c + Pt.p
Pt = A(-0.667)t + 40
The value of (-0.667)t gets smaller as t increases, and oscillates between positive and
negative, depending on whether t is even or uneven. Pt converges to a stable value of 40.
At the equilibrium price of 40, the quantity demanded = 150 – 3(40) = 30,
and the quantity supplied = -50 + 2(40) = 30.
Use the information given: P0 = 48 (which means that P = 48 when t = 0) to find the value of
the constant, A.
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Chapter 10: The Cobweb Model
The general solution is Pt = A(-0.667)t + 40
Given P0 = 48 then substitute P = 48 and t = 0 into the general solution to find the value of
the constant A.
48 = A(-0.667)0 + 40
48 = A + 40
A=8
The particular solution is Pt = 8(-0.667)t + 40
b) Substituting the values 0 to 10 into Pt = 8(-0.667)t + 40, we get Table 10.1C
Table 10.1C. Pt = 8(-0.667)t + 40
t
0
1
2
3
4
5
6
7
8
9
10
Pt
48
34.664
43.559
37.626
41.583
38.944
40.704
39.53
40.313
39.791
40.139
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