Question

Pcoer IS approximately ES0 2. Consider the following model of Supply and Demand. where P is the price of the good, Q is quantity demanded and QS is quantity supplied. (i) what condition should δ satisfy in order for the second equation to be a reasonable supply function. (ii) What condition should B and 6 satisfy in order for this system to have a unique equilibrium. Ģi Assuming a unique equilibrium exists express the system in matrix form and use matrix algebra to find the equilibrium values of Q and P Using calculus derive an expression for how the equilibrium price changes when the supply curve shifts (iv)

Answer 1

Q^alpha = alpha - beta p, Q^delta = gamma + delta p, (i) delta should be +ve delta elementof (0, infinity), as curve is upward sloping so dQ/dp > 0 (ii) at eq^m Q^d = Q^s Rightarrow alpha - beta p = gamma + delta p alpha - gamma = p (beta + delta) Rightarrow p = alpha - gamma/beta + delta so for unique eq^m, alpha - gamma > 0, and beta + delta notequalto 0, (iii) Now in matrix from rightarrow Q + beta p = alpha Q - delta p = gamma Rightarrow [1 beta 1 - delta] [Q P] = [alpha gamma] Ax = B so X = A^-1 B A^-1 = [-delta - beta -1 1] 1/(delta - beta) = 1/(delta + beta) [delta beta 1 -1] [Q P] = 1/(delta + beta) [delta beta 1 -1] [alpha gamma] = 1/(delta + beta) [alpha delta + beta gamma alpha gamma] so Q^x = alpha delta + beta gamma/(delta + beta), P^x = alpha - gamma/beta + delta