1. Consider the National Income Model: Y = C + Io + Go C = a +B(Y-T) T = y + SY a. Define and interpret each of the components of the model. Identify parameters and variables. b. Impose reasonable assumptions on the sings and values of the parameters. c. Solve for the equilibrium income. d. Obtain and discuss the six comparative-static derivatives.
1. Consider the National Income Model: Y = C + Io + Go C = a +B(Y-T) T = y + SY a. Define and interpret each of the components of the model. Identify parameters and variables. b. Impose reasonable assumptions on the sings and values of the parameters. c. Solve for the equilibrium income. d. Obtain and discuss the six comparative-static derivatives.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Please solve the remaining questions. From part d to part f. Thank you!
![### National Income Model
#### 1. Consider the National Income Model:
\[ Y = C + I_0 + G_0 \]
\[ C = \alpha + \beta(Y - T) \]
\[ T = \gamma + \delta Y \]
#### Tasks:
a. **Define and interpret each of the components of the model. Identify parameters and variables.**
- **\(Y\)**: National income (output)
- **\(C\)**: Consumption
- **\(I_0\)**: Autonomous investment
- **\(G_0\)**: Government spending
- **\(\alpha\)**: Autonomous consumption
- **\(\beta\)**: Marginal propensity to consume
- **\(T\)**: Taxes
- **\(\gamma\)**: Autonomous taxes
- **\(\delta\)**: Tax rate
b. **Impose reasonable assumptions on the signs and values of the parameters.**
The parameters should reflect typical economic relationships, such as:
- \(\alpha > 0\)
- \(0 < \beta < 1\)
- \(\gamma > 0\)
- \(0 \leq \delta < 1\)
c. **Solve for the equilibrium income.**
To find the equilibrium income, substitute the consumption \(C\) and taxes \(T\) equations into the national income equation and solve for \(Y\).
d. **Obtain and discuss the six comparative-static derivatives.**
Calculate how changes in each parameter (\(\alpha, \beta, \gamma, \delta, I_0, G_0\)) affect the equilibrium income by differentiating the income equation with respect to each parameter. Discuss the economic implications of these changes.
This model demonstrates the relationships between national income, consumption, investment, government spending, and taxes, highlighting fundamental concepts in macroeconomics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F362a762f-914f-42d1-b0fa-542c2b1764ea%2Fecacc4a0-63bd-407c-9312-3720cd93f4ea%2Fbeg80wd_processed.png&w=3840&q=75)
Transcribed Image Text:### National Income Model
#### 1. Consider the National Income Model:
\[ Y = C + I_0 + G_0 \]
\[ C = \alpha + \beta(Y - T) \]
\[ T = \gamma + \delta Y \]
#### Tasks:
a. **Define and interpret each of the components of the model. Identify parameters and variables.**
- **\(Y\)**: National income (output)
- **\(C\)**: Consumption
- **\(I_0\)**: Autonomous investment
- **\(G_0\)**: Government spending
- **\(\alpha\)**: Autonomous consumption
- **\(\beta\)**: Marginal propensity to consume
- **\(T\)**: Taxes
- **\(\gamma\)**: Autonomous taxes
- **\(\delta\)**: Tax rate
b. **Impose reasonable assumptions on the signs and values of the parameters.**
The parameters should reflect typical economic relationships, such as:
- \(\alpha > 0\)
- \(0 < \beta < 1\)
- \(\gamma > 0\)
- \(0 \leq \delta < 1\)
c. **Solve for the equilibrium income.**
To find the equilibrium income, substitute the consumption \(C\) and taxes \(T\) equations into the national income equation and solve for \(Y\).
d. **Obtain and discuss the six comparative-static derivatives.**
Calculate how changes in each parameter (\(\alpha, \beta, \gamma, \delta, I_0, G_0\)) affect the equilibrium income by differentiating the income equation with respect to each parameter. Discuss the economic implications of these changes.
This model demonstrates the relationships between national income, consumption, investment, government spending, and taxes, highlighting fundamental concepts in macroeconomics.
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