Question

3. The quantity of cabbage supplied to a market depends on the price during the previous season, while the quantity demanded depends on the present price ls, =-4+2Pt-1 Qdy=80-2p, a. Write the equilibrium condition as a difference equation in P and solve it. [2 marks] b. Solve the equation to obtain an expression for price in terms of t, given ps = 10.5. Hence, determine whether prices will stabilize. [2 marks] C. Solve the difference equation to find the equilibrium price and quantity. [5 marks]

Answer 1

9 a) at Eqm Ost = Qdt = – 4+2 Pt 1= 80-2Pt. 2 Pt + q Pt +=84. 80 Pt = (84 – 2Pt-1/2 I TPt = 42- Pt-1 difference Eqn solving of xt = axt-i tb then x= (ast (Xo- b ) + (6/12). b=49, a=-1. 80. Pt = (-1)+ (Po - 42) + 42 Pe= tot (Po - 21) + 21 | 80l? Ang b.) Pt = (-1 t Apo-21) +21. t=1, Pt = -Po +21+21= 42-lo. t=2 Pg= Po-21+21 = Po. thus for t= odd value , Pt = 42-po. for even value Pt = Po. when t=5 lt=Pg=4Q-Po=1005 P=31-5, 80 prices will not stabilize. Price will oscilliate around P=21, with constant amplitude. c). Difference Eq" 801", Pt = f 42-Po t= odd Po t= even pp. 21- Q* = 580-2 (42-Po) t=odd 1 = 2P0-4. t=1 t=2. t=3 80-2Po , t= even Po ? ! 42607