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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:

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

  2. 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

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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:

$$dY = \frac{1}{1-C'}I 'di - \frac{1}{1-C'}I 'd\pi + \frac{1}{1-C'}dG - \frac{C'}{1-C'}dT$$

$$(1/P)dM - (M/P^2) dP = L_r'dr + L_Y' dY - L_T'dT$$

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.

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