The demand and supply functions of shoes are as follows: Qa = a - bp Qs = c + dP where a = 50, b = 1, c = 20, and d = 2. There is a possibility that the initial price P(0) does not equal to long-term equilibrium price P* which is attainable in the long term-due to process of time. In other words, all variables can change overtime or be functions of time (t). Based on the information, answer the following questions a. Find the market long-term equilibrium price P*! (5 points) b. Given sufficient time for the adjustment process to work, what is the price time path P(t) by assuming that the rate of price change with respect to time (at) at any moment is directly proportional to excess demand (Qa -Qs)? (hint: use nonhomogeneous first order linear differential equation to find the complete solution P(t) that consists of complementary function (Pc) and particular integral (Pp)) (8 points) c. Based on your solution in part b, will the price path over time P(t) converge to long-term equilibrium price P* as t - co given current price is P(0) = 8? Use the qualitative diagram to support your answer! (7 points) d. Give economic interpretation of complementary function (P) and particular integral (Pp) in attaining the long-term equilibrium price P*! (5 points)