For each output level Y, the IS curve defines the interest rate r at which the goods market clears:
Y(1-b)-G=I^0-ar,
where b is the marginal propensity to consume, G is the government spending, I^0 is the maximum investment level, and a is the responsiveness of investment to interest rates. The LM curve defines the interest rate at which the money market clears:
mY + M^0 - hr = M^8,
Where m is the responsiveness of the transactions demand for money to output, M^0 is the maximum liquidity demand, h is the responsiveness of liquidity demand to interest rates, and M^8 is the money supply.
a) Write down this system of equations in matrix form. Under what condition on the exogenous parameters can this system of two equations be solved for Y and r?
b) Using Cramer's rule, solve the system for Y and r when the condition in (a) is met.
c) What happens to the equilibrium interest rate r if government spending increases by ?G?