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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 www.wiley.com/college/bradley Page 1 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. www.wiley.com/college/bradley Page 2 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 www.wiley.com/college/bradley Page 3