<p><strong>Hint</strong>:</p>
<p>The <span class="math-container">$T$</span> is variable itself so the total differential should read as:</p>
<p><span class="math-container">$$dY = C'dY - C'dT + I'dr + dG \implies dY = \frac{1}{1-C'}I 'dr + \frac{1}{1-C'}dG - \frac{C'}{1-C'}dT$$</span></p>
<p>Furthermore, by Fisher equation <span class="math-container">$r=i-\pi$</span>. Hence we have:</p>
<p><span class="math-container">$$dY = \frac{1}{1-C'}I 'di - \frac{1}{1-C'}I 'd\pi + \frac{1}{1-C'}dG - \frac{C'}{1-C'}dT$$</span></p>
<p><span class="math-container">$$(1/P)dM - (M/P^2) dP = L_r'dr + L_Y' dY - L_T'dT$$</span></p>
<p>again use the Fisher so:</p>
<p><span class="math-container">$$(1/P)dM - (M/P^2) dP = L_i'di - L_{\pi}'d \pi + L' dY - L'dT$$</span></p>
<p>This is IS-LM model so you are looking for equilibrium output and nominal interest rate <span class="math-container">$dY$</span> and <span class="math-container">$di$</span>. 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.</p>

CC BY-SA 4.0
2020-10-13T21:42:15.017Z
2020-10-14T00:17:58.220Z
2020-10-14T00:17:58.220Z
1muflon1
<p>I should consider a following modification of IS-LM model:</p>
<p>IS curve is standard: Y = C(Y-T) + I(r) + G</p>
<p>In LM curve the demand for money depends now on after tax income: M/P = L(r, Y-T)</p>
<p>Price level is fixed in the short run</p>
<p>I need to solve for tax multiplier.</p>
<p>What I have done is that I took the full differential of two equations and got the following:</p>
<ol>
<li><p>dY = C'(Y-T)dY - C'(Y - T) + I'dr + dG</p>
</li>
<li><p>1/p<em>dM - M/p^2 * dP = Lr * dr + Ly * dy - Lt</em> dT</p>
</li>
</ol>
<p>Lr, Ly, Lt are partial derivatives</p>
<p>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</p>

<macroeconomics><mathematical-economics>
Tax multiplier in IS-LM model
Anya Pilipentseva
# Tax multiplier in IS-LM model

(Asked by: username_0 on 2020-10-13)

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/p*dM - 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

**Tags:** macroeconomics, mathematical-economics

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## 1 answers

### Answer by username_1 on 2020-10-13. Score: 1 | 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:

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

$$(1/P)dM - (M/P^2) dP = L\_r'dr + L\_Y' dY - L\_T'dT$$

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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