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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:

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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Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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