<p>So, the IS curve is a set of equilibria: all combinations of income and interest rate that achieve macroeconomic equilibrium are represented by the IS curve. After all, that's why they call it IS: Investment = Savings!!!</p>
<p>Hence, let's take off from here:</p>
<p>Investment = Savings = Public Sector Savings + Private Sector Savings</p>
<p>What's Public savings? Remember: savings is all income left after expenses. Thus:
T - G = Public Savings: Government's income (taxes) minus expenses (gov' purchases, G).</p>
<p>What's Private savings? Well, the people earned an income (Y), paid taxes on it (T), and spent on their consumption (C): Y - T - C</p>
<p>There we have it: Savings = S = (Y-T-C) + (T-G) = Private + Public savings</p>
<p>Question: how do we know S = I?</p>
<p>Put together your first 3 equations, and let's do one bold assumption: Y = Z (i.e., actual expenditure = planned expenditure. Not that bold, is it?)</p>
<pre><code>Y = C_0 + C_1*(Y−T) + I_0 - I_1*r + G
</code></pre>
<p>Solving for this equation (beware: Y appears on both sides of it, which means you need to to solve for it), goes like this:</p>
<pre><code>Y - C_1*Y = C_0 - C_1*T + I_0 - I_1*r + G
Y*[1 - C_1] = C_0 - C_1*T + I_0 - I_1*r + G
</code></pre>
<p><strong>Y = [C_0 - C_1<em>T + I_0 - I_1</em>r + G]/[1 - C_1]</strong></p>
<p>And that's your IS curve, exhibiting a negative relationship between your two key variables, income and interest rate.</p>

CC BY-SA 4.0
2021-01-21T23:53:51.080Z
2021-01-21T23:53:51.080Z
Oscar
<p>An Economy has a GDP described by the following:</p>
<p><span class="math-container">$Z=C(Y −T)+G+I(r)$</span></p>
<p><span class="math-container">$C(Y −T)=C_0 +C_1(Y −T)$</span></p>
<p><span class="math-container">$I(r) = I_0 − I_1r$</span></p>
<p>where Z is planned expenditure, Y is GDP, T is tax, G is public consumption, I is investment, r is interest. <span class="math-container">$C_0$</span>, <span class="math-container">$C_1$</span>, <span class="math-container">$I_0$</span>, <span class="math-container">$I_1&gt; 0$</span> are all parameters and <span class="math-container">$C_1 &lt;1$</span>. T and G are exogenous variables. r is also exogenous .</p>
<p><strong>How does one derive the IS-Curve Y as a function from r,G,T?</strong></p>
<p>Im not sure what is meant by this because my understanding is that the IS-Curve is just <span class="math-container">$Y = C(Y-T)+ I(r) + G$</span></p>
<p>Any help is appreciated.</p>
<p><strong>I have now solved this</strong></p>

<macroeconomics><gdp><supply-and-demand>
Derive IS-Curve (Y)
Marcus F
# Derive IS-Curve (Y)

(Asked by: username_0 on 2021-01-21)

An Economy has a GDP described by the following:

$Z=C(Y −T)+G+I(r)$

$C(Y −T)=C\_0 +C\_1(Y −T)$

$I(r) = I\_0 − I\_1r$

where Z is planned expenditure, Y is GDP, T is tax, G is public consumption, I is investment, r is interest. $C\_0$, $C\_1$, $I\_0$, $I\_1> 0$ are all parameters and $C\_1 <1$. T and G are exogenous variables. r is also exogenous .

**How does one derive the IS-Curve Y as a function from r,G,T?**

Im not sure what is meant by this because my understanding is that the IS-Curve is just $Y = C(Y-T)+ I(r) + G$

Any help is appreciated.

**I have now solved this**

**Tags:** macroeconomics, gdp, supply-and-demand

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## 1 answers

### Answer by username_1 on 2021-01-21. Score: 0 | Accepted answer

So, the IS curve is a set of equilibria: all combinations of income and interest rate that achieve macroeconomic equilibrium are represented by the IS curve. After all, that's why they call it IS: Investment = Savings!!!
Hence, let's take off from here:
Investment = Savings = Public Sector Savings + Private Sector Savings
What's Public savings? Remember: savings is all income left after expenses. Thus:
T - G = Public Savings: Government's income (taxes) minus expenses (gov' purchases, G).
What's Private savings? Well, the people earned an income (Y), paid taxes on it (T), and spent on their consumption (C): Y - T - C
There we have it: Savings = S = (Y-T-C) + (T-G) = Private + Public savings
Question: how do we know S = I?
Put together your first 3 equations, and let's do one bold assumption: Y = Z (i.e., actual expenditure = planned expenditure. Not that bold, is it?)
Y = C_0 + C_1*(Y−T) + I_0 - I_1*r + G

Solving for this equation (beware: Y appears on both sides of it, which means you need to to solve for it), goes like this:
Y - C_1*Y = C_0 - C_1*T + I_0 - I_1*r + G
Y*[1 - C_1] = C_0 - C_1*T + I_0 - I_1*r + G

Y = [C_0 - C_1T + I_0 - I_1r + G]/[1 - C_1]
And that's your IS curve, exhibiting a negative relationship between your two key variables, income and interest rate.


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