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: QAˢ = 11 + 2PA - 3PB + aPc QBˢ = 9 + 6PB - PA - 3Pc QCˢ = 2PC - 3PA - 3PB₁ where a is a parameter. The demand is fixed: QAᵈ = QBᵈ = 10; QCᵈ=q. (a) 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, Pc]ᵀ. (b) Compute M. Show that the rank of M is 2 if a = 9/7 and 3 otherwise. (c) Assume a = 9/7. Find the value of q at which the system admits an infinity of solutions. (d) Suppose a ≠ 9/7. Use Cramer's rule to find the equilibrium value of PA. (e)Find δPA/δq. Show that δPA/δq < 0 if a < 0. Explain the economic intuition behind this sign.