Consider an economy described by the following functions: C = 20 + 0.80Y, I = 30, G = 50, TR = 100.

Find the equilibrium level of income and the autonomous expenditure multiplier in the model.

If government expenditure increases by 30, what is the impact on equilibrium income?

If a lump-sum tax of 30 is added to pay for the increase in government purchases, how will equilibrium income change?

Q: Consider an economy described by the following functions: C = 20 + 0.80Y, I = 30, G = 50, TR = 100. 1. Find the equilibrium level of income and the autonomous expenditure multiplier in the model. 2. If government expenditure increases by 30, what is the impact on equilibrium income? 3. If a lump-sum tax of 30 is added to pay for the increase in government purchases, how will equilibrium income change?

1. Equilibrium level of income $\text{Y}=\frac{1}{1-\text{c}}[\bar{\text{C}}+\text{cT}+\text{I}+\text{G}]$ $=\frac{1}{1-0.80}[20+0.80\times100+30+50]$ $\frac{1}{0.20}\times180$ $=\frac{180}{20}\times100$ $=900$ Expenditure multiplier $\frac{1}{1-\text{c}}=\frac{1}{1-0.80}$ $=\frac{1}{0.20}$ $=\frac{100}{20}=5$ 2. Increase in government expenditure $=\frac{1}{1-\text{c}}[\bar{\text{C}}+\text{cT}+\text{I}+\text{G}+\Delta\text{G}]$ $=\frac{1}{1-0.80}[20+0.80\times100+30+50+30]$ $=\frac{1}{0.20}\times[210]=\frac{210}{20}\times100=1050$ 3. $\text{Tax multiplier}=\frac{-\text{c}}{1-\text{c}}$ $\frac{\Delta\text{Y}}{\Delta\text{T}}=\frac{-\text{c}}{1-\text{c}}$ $\Delta\text{Y}=\frac{-\text{c}}{1-\text{c}}\times\Delta\text{T}$ $=\frac{-0.80}{1-0.80}\times30$ $=\frac{-80}{20}\times30$ $=-120$ New equilibrium level of income $=\text{Y}+\Delta\text{Y}$ $=900+(-120)$ $=780$

Question

Consider an economy described by the following functions: C = 20 + 0.80Y, I = 30, G = 50, TR = 100.

Find the equilibrium level of income and the autonomous expenditure multiplier in the model.
If government expenditure increases by 30, what is the impact on equilibrium income?
If a lump-sum tax of 30 is added to pay for the increase in government purchases, how will equilibrium income change?

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Solution

Equilibrium level of income

$\text{Y}=\frac{1}{1-\text{c}}[\bar{\text{C}}+\text{cT}+\text{I}+\text{G}]$
$=\frac{1}{1-0.80}[20+0.80\times100+30+50]$
$\frac{1}{0.20}\times180$
$=\frac{180}{20}\times100$
$=900$
Expenditure multiplier
$\frac{1}{1-\text{c}}=\frac{1}{1-0.80}$
$=\frac{1}{0.20}$
$=\frac{100}{20}=5$

Increase in government expenditure

$=\frac{1}{1-\text{c}}[\bar{\text{C}}+\text{cT}+\text{I}+\text{G}+\Delta\text{G}]$
$=\frac{1}{1-0.80}[20+0.80\times100+30+50+30]$
$=\frac{1}{0.20}\times[210]=\frac{210}{20}\times100=1050$

$\text{Tax multiplier}=\frac{-\text{c}}{1-\text{c}}$

$\frac{\Delta\text{Y}}{\Delta\text{T}}=\frac{-\text{c}}{1-\text{c}}$
$\Delta\text{Y}=\frac{-\text{c}}{1-\text{c}}\times\Delta\text{T}$
$=\frac{-0.80}{1-0.80}\times30$
$=\frac{-80}{20}\times30$
$=-120$
New equilibrium level of income $=\text{Y}+\Delta\text{Y}$
$=900+(-120)$
$=780$

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