Course2: Question on Government Multiplier

[Raiden]

Can anyone explain to me how the formula for Government expenditure multiplier derived? In the notes, it says do a total diffferentialof Y = C+I+G+NS with respect to G (DY/DG).

Thanks a lot

[bg23516]

This one will be tough on the computer, but here goes:

We know:
Y = C[Y-T(Y), R] + I[R] + G +NX[Y, e]

Take a total differential of both sides:
dY = Cy*(dY-T'dY) + CrdR +IrDR + dG + NXydY + NXeDe

You take a total differential by implicitly differenitiating each term with respect to each exogenous and endogenous variable, and then mark everything with a dVAR to show which variable you took a derivative with respect to. If you want more info on actual differentials and how to take them, I would direct you to a multivariable calculus text.

We then put all the dY stuff on the left, and factor out a dY.

dY * (1 - Cy*(1-T') - NXy) = CrdR + IrDR + dG + NXeDe

Since we want to look at the dY/dG, which is the CHANGE in Y given a CHANGE in g, holding all else constant, we want to set dR = dE = 0.
Our equation becomes:

dY * (1 - Cy*(1-T') - NXy) = dG
Rearrange by division to get:
dY/dG = 1 / (1- Cy*(1-T') - NXy)

Thats all there is to it. Now, we identify want we are looking at. It usually helps to look at this earlier, but oh well.

Cy is the partial of Consumption with restpect to Income, aka the Marginal Prepensity to Consume (MPC). 0 < MPC < 1
T' is the derivative of the tax function. 0 < T' < 1
NXy is the the effect of an increase in income on Net Exports. Expect this to be negative, as when income rises, IMPORTS will rise, so NX will decline.

Thats really the derivation. Realistically, we could have rearranged the equation to find dR/dY, dY/de, or anything else. Always remember, a partial holds ALL other variables constant, so your dVARS = 0, for no change.

HTH.
Marc