Tax multiplier in IS-LM model

Question

I should consider a following modification of IS-LM model:
IS curve is standard: Y = C(Y-T) + I(r) + G
In LM curve the demand for money depends now on after tax income: M/P = L(r, Y-T)
Price level is fixed in the short run
I need to solve for tax multiplier.
What I have done is that I took the full differential of two equations and got the following:

dY = C'(Y-T)dY - C'(Y - T) + I'dr + dG

1/pdM - M/p^2 * dP = Lr * dr + Ly * dy - Lt dT

Lr, Ly, Lt are partial derivatives
Next the book gives a hint to use Cramer's rule to solve for tax multiplier, but I am not sure how I can do it. Because to me it seems that I have 2 equations and 3 unknown variables

1muflon1 · Accepted Answer

Hint:
The $T$ is variable itself so the total differential should read as:
$$dY = C'dY - C'dT + I'dr + dG implies dY = frac{1}{1-C'}I 'dr + frac{1}{1-C'}dG - frac{C'}{1-C'}dT$$
Furthermore, by Fisher equation $r=i-pi$. Hence we have:
$$dY = frac{1}{1-C'}I 'di - frac{1}{1-C'}I 'dpi + frac{1}{1-C'}dG - frac{C'}{1-C'}dT$$
$$(1/P)dM - (M/P^2) dP = L_r'dr + L_Y' dY - L_T'dT$$
again use the Fisher so:
$$(1/P)dM - (M/P^2) dP = L_i'di - L_{pi}'d pi + L' dY - L'dT$$
This is IS-LM model so you are looking for equilibrium output and nominal interest rate $dY$ and $di$. You have two equations and two unknowns. You can solve it even with substitution but you should probably follow the textbook's advice to use Cramer's rule.

Tax multiplier in IS-LM model

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