DIAGRAMMATICAL EXPOSITION
of the
FIXED-PRICE KEYNESIAN MODEL

 S:            saving (S =  -a + (1-b) (Y-T), where a and b are parametric)  T:            taxes (modeled as a lump-sum tax)  I:             investment spending (Shifts of the curve can swamp movements along it.)  G:           government spending (a policy tool used to offset shifts in investment spending)  Mtrans    transactions demand for money (in accordance with the equation of exchange)  Mspec    speculative demand for money (based on expected movements in the interest rate)  i:             the interest rate (monetary phenomenon for Keynes; real phenomenon for the classicals)  Y:           income (which moves in lockstep with labor income and is a measure of output)

ISLM analysis builds upon the simple Keynesian Income-Expenditure relationships by adding interest-rate considerations. Using this analysis, we see that the multiplier effect is sometimes not as great as the simple multipliers imply, owing to a change in the rate of interest and hence a movement along the demand for investment funds.
In a number of applications, however, the simple multipliers do apply. That is, DY = [1/(1 - b)] DI; DY = [1/(1 - b)] DG; or DY = [1/(1 - b)] DEo, where DEo is the net change (DG - DI) in autonomous expenditures.

Examples of conditions or instances in which the simple Keyneisan spending multiplier applies include:

1. An economy mired in the liquidity trap, in which case the interest rate does not change.
2. An economy with a perfectly inelastic demand for investment funds, in which case the changing interest rate has no effect on investment.
3. An instance where fiscal policy is fully accommodated by monetary policy, in which case any movement in the rate of interest is arrested by a suitable adjustment in the supply of money.
4. An instance where the initial round of spending is pre-adjusted for the expected "crowding out" of investment. This is the application, mentioned above, where the simple multiplier is applied to the net change in autonomous expenditures.
5. An instances where the issue is the extent of the shift of the IS curve in response to a given shift in investment demand or increase in government spending. Of course, the increase in income, DY, may not be as great as the actual shift in IS, owing the interest-rate effect on investment.
6. An instance where an increase in the money supply lowers the interest rate and stimulates investment. Here, the DY (associated with a movement along the unshifted IS curve) is related to the DI (associated with a movement along the unshifted investment demand curve) by the simple Keynesian spending multiplier.

A similar list could be compiled to identify the conditions or instances in which the simple Keynesian tax multiplier applies.

The question "Can I use the simple Keynesian multiplier to calculate the effect of X on income" resolves itself into a sequence of subsidiary questions:

1. Does X affect the interest rate?
If no, then use the simple Keynesian multiplier.
If yes, then go on to question 2.

2. Does the change in the interest rate affect investment?
If no, then use the simple Keynesian multiplier.
If yes, then go on to question 3.

3. Is the interest-rate-induced change in investment taken into account?
If yes, then use the simple Keynesian multiplier.