The quantity supplied of some commodity in a given week is obviously an increasing

Chapter 1, Problem 10

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The quantity supplied of some commodity in a given week is obviously an increasing function of its price the previous week, while the quantity demanded in a given week is a function of its current price. Let 4 and Ll,. denote, respectively, the quantities supplied and demanded in the jth week, and let 4 denote the price of the commodity in the jth week. We assume that there exist positive constants a, b, and c such that 2$=aPj-, and D,=b-cP,. (a) Show that 5 = b/(a + c) + (- a/c)j(Po - b/(a + c)), if supply always equals demand. (b) Show that 4 approaches b/(a + c) as j approaches infinity if a/c < 1. (c) Show that P = b/(a + c) represents an equilibrium situation. That is to say, if supply always equals demand, and if the price ever reaches the level b/(a + c), then it will always remain at that level.