11. Consider the following linear model of demand for and supply of a good. The endogenous variables are: Qd – quantity demanded; Qs – quantity supplied; P price of the good; the exogenous variables are P* - price of inputs; Y – income. The terms a, b, f, c, d, and g are constant coefficients where b<0, f>0, d>0, g<0.
Qd = a + bP + fY (1)
Qs = c + dP + gP* (2)
Qd = Qs (3)
a) Arrange equations (1)-(3) to express the system in form AX=B where A is a 3x3 coefficient matrix, X a column vector of the three endogenous variables given by X = [Qd, Qs, P], and B is a vector of constants and the exogenous variables. (Use the specific ordering of the endogenous variables given in X).
b)
i. State the condition for the inverse of A to exist in terms of the coefficients of the model.
ii. Use the information provided to determine whether this condition is satisfied. Or, if it is unclear, explain if there are any further conditions that would ensure the existence of a solution.
c)
i. Use Cramer’s rule to obtain solutions for each endogenous variable in terms of the coefficients and the exogenous variables.
ii. Compare and contrast the impact of an increase in input prices on the equilibrium values of each of the endogenous variables.