Question:

In an equilibrium market, let the price, supply and demand of a certain product on day t be respectively, denoted as Pt, St and Dt, for t = 0, 1, 2, .... The supply on day t + 1 depends on the one-day lagged price Pr given by St+1 = 2Pt - 10, t = 0, 1, 2, . . . . However, the demand on day t + 1 depends on the current price Pt+1 given by Dt+1 = -3Pt+1 + 80, t = 0, 1, 2, ... . The market is always in equilibrium in the sense that St+1 = Dt+1, fort = 0, 1, 2, . . . . The initial price is Po = 6. (1) Show that the price sequence { Pr : t = 0, 1, 2, ...} satisfies the following dynamic equation Pt+1 = apt + c. Find the values of a and c. (2) Rewrite P in the form of a difference equation and show that as t goes to infinity, Pt converges to a constant (denoted by P.). (3) Explain why both the supply (S.) and the demand (D,) converge to a constant (de- noted by Co), as t goes to infinity. Compute the value of Co (rounded to 2 decimal places). (4) Express S, in the form of a difference equation. (5) What is the minimum number of waiting days, which is denoted as To, such that Co - 1 To