In the 3-sector Keynesian model economy, if the proportional tax rate t=0, then a 1 unit increase in government spending and a one unit increase in exogenous taxes, what about the equilibrium output Y

In the 3-sector Keynesian model, which includes households, businesses, and the government, the equilibrium output \( Y \) is determined by the aggregate demand in the economy. The key components of aggregate demand in this model are consumption \( C \), investment \( I \), and government spending \( G \). The consumption function typically includes disposable income, which is affected by taxes.

### Given:
- Proportional tax rate \( t = 0 \) (so there are no taxes on income, only exogenous taxes)
- A 1-unit increase in government spending \( G \)
- A 1-unit increase in exogenous taxes

The equilibrium condition in the Keynesian model is:
\[
Y = C + I + G
\]

The consumption function \( C \) is usually expressed as:
\[
C = C_0 + c(Y_d)
\]
Where:
- \( C_0 \) is autonomous consumption
- \( c \) is the marginal propensity to consume (MPC)
- \( Y_d = Y - T \) is disposable income (but since \( t = 0 \), \( Y_d = Y \))

Now, let's consider the effects of the changes:

1. **Increase in Government Spending (\( \Delta G = +1 \))**:
   - Directly increases aggregate demand, leading to an increase in \( Y \).
   - The change in output \( \Delta Y \) due to \( \Delta G \) is given by the government spending multiplier:
     \[
     \Delta Y = \frac{1}{1 - c} \times \Delta G
     \]
     Since \( \Delta G = +1 \), we have:
     \[
     \Delta Y_G = \frac{1}{1 - c}
     \]

2. **Increase in Exogenous Taxes (\( \Delta T = +1 \))**:
   - Exogenous taxes reduce disposable income and thus consumption.
   - The change in output \( \Delta Y \) due to \( \Delta T \) is given by the tax multiplier:
     \[
     \Delta Y = -\frac{c}{1 - c} \times \Delta T
     \]
     Since \( \Delta T = +1 \), we have:
     \[
     \Delta Y_T = -\frac{c}{1 - c}
     \]

### Net Effect on Equilibrium Output \( Y \):
The total change in output \( \Delta Y \) is the sum of the effects of the change in government spending and the change in taxes:
\[
\Delta Y = \Delta Y_G + \Delta Y_T
\]
Substituting the values:
\[
\Delta Y = \frac{1}{1 - c} - \frac{c}{1 - c} = \frac{1 - c}{1 - c} = 1
\]

So, **the equilibrium output \( Y \) will increase by 1 unit**. 

This result makes intuitive sense because the 1-unit increase in government spending has the same magnitude (but opposite direction) as the 1-unit increase in taxes, but since the tax rate \( t = 0 \), the tax does not diminish the spending multiplier, leading to an overall increase in output by 1 unit.

Would you like more details or have any questions?

### Related Questions:
1. How does the value of the marginal propensity to consume (MPC) affect the Keynesian multipliers?
2. What happens to equilibrium output if the tax rate \( t \) is positive instead of zero?
3. How does the introduction of a proportional tax change the relationship between government spending and equilibrium output?
4. What is the impact of a change in investment \( I \) on equilibrium output in the 3-sector model?
5. How does the multiplier effect work in the presence of automatic stabilizers like unemployment benefits?

### Tip:
Understanding the relationship between different components of aggregate demand and how they interact with taxes and government spending is key to mastering Keynesian economics.

Explore the impact of fiscal policy changes in a 3-sector Keynesian model with no proportional tax rate. Learn how a 1 unit increase in government spending and exogenous taxes affects equilibrium output Y. Includes detailed formulas and explanations.

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