The economy has three goods: A, B. and C. The prices of the goods are PA, PB and PC. The supply functions are as follows: QA 11+2PA-3PB + aPc QB = 9+6PB-PA-3Pc Qo = 2PC-3PA-3PB, where a is a parameter. The demand is fixed: Q = Q = 10; 2 = 9- (a) [10 marks] Using the equilibrium conditions that supply must equal demand, write down three equations that determine the equilibrium prices PA, PB and Pc. Write this system of equations in matrix form MP = N, where P = [PA, PB, PCT. (b) [10 marks] Compute M. Show that the rank of M is 2 if a = and 3 otherwise. (c) [10 marks] Assume a = . Find the value of q at which the system admits an infinity of solutions. (d) [10 marks] Suppose a . Use Cramer's rule to find the equilibrium value of PA (e) [10 marks] Find OPA/Oq. Show that OPA/Oq < 0 if a < 0. Explain the economic intuition behind this sign.