Assume the following general specifications for the IS and LM equations:

Taking total differentials of each:

where differentials of the endogenous variables Q^D and i have been place on the left-hand side and all the exogenous variables on the right.

Similary, for LM,

To analyze the system it is convenient to write it in matrix form as:

To compute comparative static results for the model, we can apply Cramer’s Rule. For example, to solve for the government spending multiplier, dQ^D/dG, we can assume all other exogenous shocks are zero, yielding the simplified system:

Cramer’s Rule gives:

where is the determinant of the coefficient matrix,.

Or, simplifying,

Some things to think about:

1. How does the IS-LM multiplier compare with the "simple expenditure multiplier" of Keynesian income determination models? Why?
2. How might the multiplier change over the business cycle?
3. Would the multiplier differ depending on how dG is financed?
4. Can you derive the effect of dG on interest rates? Can you derive the multiplier for an increase in the real money supply?