3) So far we have assumed that the fiscal policy variables G and T are independent of the levels of income. In the real world, however, this is not the case. Taxes typically depend on the level of income, and so tend to be higher when income is higher. In this problem we examine how this automatic response of taxes can help reduce the impact of changes in autonomous spending on output. Consider the following behavioral equations: C=C₁+C₂YD+ T = t₁ +t₁Y+ YD=Y-T< e G and I are both constant. Assume that t₁ is between zero and one. a. Solve for equilibrium output. b. What is the multiplier? Does the economy respondmore to changes in autonomous spending when t, is zero or when t, is positive? Explain. c. Why is fiscal policy in this case called an automatic stabilizer?

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### Understanding Fiscal Policy Variables and Their Impact

**3)** Thus far, we have assumed that the fiscal policy variables \(G\) (government spending) and \(T\) (taxes) are independent of income levels. However, in real-world scenarios, this is not the case. Taxes typically depend on the level of income and thus tend to be higher when income is higher. In this problem, we examine how this automatic response of taxes can help reduce the impact of changes in autonomous spending on output.

Consider the following behavioral equations:

\[ C = C_0 + C_1 Y_D \]

\[ T = t_0 + t_1 Y \]

\[ Y_D = Y - T \]

- \(G\) and \(I\) are both constant.
- Assume that \(t_1\) is between zero and one.

**Questions:**

**a. Solve for equilibrium output.**

**b. What is the multiplier? Does the economy respond more to changes in autonomous spending when \(t_1\) is zero or when \(t_1\) is positive? Explain.**

**c. Why is fiscal policy in this case called an automatic stabilizer?**

---
**Explanations:**

- **Consumption Function \( (C) \)**:
  \( C = C_0 + C_1 Y_D \)
  - \(C_0\): Autonomous consumption
  - \(C_1\): Marginal propensity to consume
  - \(Y_D\): Disposable income
  
- **Tax Function \( (T) \)**:
  \( T = t_0 + t_1 Y \)
  - \(t_0\): Autonomous taxes
  - \(t_1\): Marginal tax rate
  - \(Y\): National income
  
- **Disposable Income \( (Y_D) \)**:
  \( Y_D = Y - T \)
  - The income left after taxes

### Steps to Solve:

**a. Solve for equilibrium output**:
  - Substitute the tax function into the disposable income equation.
  - Substitute the disposable income into the consumption function.
  - Set up the equilibrium condition where aggregate demand equals aggregate output.

**b. Determining the Multiplier**:
  - Calculate the multiplier in different scenarios for \(t_1\).
  - Compare the responsiveness of the economy to changes in autonomous spending under different \(t_1\) values.
Transcribed Image Text:### Understanding Fiscal Policy Variables and Their Impact **3)** Thus far, we have assumed that the fiscal policy variables \(G\) (government spending) and \(T\) (taxes) are independent of income levels. However, in real-world scenarios, this is not the case. Taxes typically depend on the level of income and thus tend to be higher when income is higher. In this problem, we examine how this automatic response of taxes can help reduce the impact of changes in autonomous spending on output. Consider the following behavioral equations: \[ C = C_0 + C_1 Y_D \] \[ T = t_0 + t_1 Y \] \[ Y_D = Y - T \] - \(G\) and \(I\) are both constant. - Assume that \(t_1\) is between zero and one. **Questions:** **a. Solve for equilibrium output.** **b. What is the multiplier? Does the economy respond more to changes in autonomous spending when \(t_1\) is zero or when \(t_1\) is positive? Explain.** **c. Why is fiscal policy in this case called an automatic stabilizer?** --- **Explanations:** - **Consumption Function \( (C) \)**: \( C = C_0 + C_1 Y_D \) - \(C_0\): Autonomous consumption - \(C_1\): Marginal propensity to consume - \(Y_D\): Disposable income - **Tax Function \( (T) \)**: \( T = t_0 + t_1 Y \) - \(t_0\): Autonomous taxes - \(t_1\): Marginal tax rate - \(Y\): National income - **Disposable Income \( (Y_D) \)**: \( Y_D = Y - T \) - The income left after taxes ### Steps to Solve: **a. Solve for equilibrium output**: - Substitute the tax function into the disposable income equation. - Substitute the disposable income into the consumption function. - Set up the equilibrium condition where aggregate demand equals aggregate output. **b. Determining the Multiplier**: - Calculate the multiplier in different scenarios for \(t_1\). - Compare the responsiveness of the economy to changes in autonomous spending under different \(t_1\) values.
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