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Linear Algebra: Interest Rates and Cramer's Rule

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?

Please see the attached file for full equations.

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Solution Summary

The interest rates and Cramer's rules are examined. The responsiveness of investments to interest rates are provided.

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