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    Linear Algebra: Interest Rates and Cramer's Rule

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    For each output level Y, the IS curve defines the interest rate r at which the goods market clears:


    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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    Please see the attached file.

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    Solution Summary

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