https://file.aiquickdraw.com/mathbot/file/1727785262527gaq63c6o.jpg

Let's break down the two questions provided in the image.

---

### **Question ONE: National Income Model**
The given national income model is:

\[
Y = C + G + I + X - M
\]
Where:
- \( C = a + bY_d \)
- \( I = r_0 + r_1IR \)
- \( T = t_0 + t_1Y \)
- \( G = G_0 \)
- \( X = X_0 \)
- \( M = m_0 + m_1Y \)

#### a. Find the equilibrium income.
Equilibrium income can be derived by setting aggregate demand equal to output (Y), so:
\[
Y = C + I + G + X - M
\]
Substitute the respective formulas for consumption, investment, government expenditure, exports, and imports.

#### b. Explain what a, b, t1, m0, and m1 stand for.
- \( a \): Autonomous consumption (the consumption level when disposable income is zero)
- \( b \): Marginal propensity to consume (MPC)
- \( t_1 \): The rate at which taxes increase with income
- \( m_0 \): Autonomous imports (imports when national income is zero)
- \( m_1 \): The marginal propensity to import (how much imports change with income)

#### c. Find the rate of change of equilibrium income with respect to \( G \) and \( R \).
To find the sensitivity of income with respect to government expenditure and interest rates, we differentiate the equilibrium equation with respect to \( G \) and \( R \).

#### d. Calculate equilibrium income with given parameters.
Given:  
- \( a = 10 \), \( b = 0.5 \), \( t_0 = 100 \), \( t_1 = 0.15 \),  
- \( r_0 = 4 \), \( r_1 = 2 \), \( IR = 0.2 \),  
- \( G_0 = 2000 \), \( X_0 = 500 \), \( m_0 = 4 \), \( m_1 = 0.4 \).

Substitute these into the national income model to compute the equilibrium income.

#### e. Find the value of the multiplier.
The multiplier is calculated as:
\[
\text{Multiplier} = \frac{1}{1 - b(1 - t_1) + m_1}
\]

#### f. Calculate the change in equilibrium income when \( G_0 \) increases.
If government expenditure increases from 2000 to 3000, use the multiplier to find the change in equilibrium income.

---

### **Question TWO: Demand and Total Cost Functions**
Given:
- \( P = 10 - 7Q \)
- \( TC = 20 + 3Q + 4Q^2 + 5Q^3 \)

#### a. Find the level of \( Q \) and \( P \) that maximizes profits.
To maximize profits, we first need to compute the total revenue (TR) and marginal revenue (MR), then equate \( MR = MC \) (marginal cost) to find \( Q \).

#### b. Find the level of \( Q \) that maximizes total revenue.
Maximize \( TR = PQ = (10 - 7Q)Q \) by differentiating with respect to \( Q \) and setting it equal to zero.

#### c. Find the level of \( Q \) that maximizes \( MC \) and \( AVC \).
For maximizing marginal cost (MC) and average variable cost (AVC), derive MC and AVC from the total cost function and optimize.

---

Would you like to proceed with solving these parts in more detail, or do you need help with specific sections?

Here are some follow-up questions you might consider:
1. How can the multiplier effect be interpreted in the context of fiscal policy?
2. What economic factors influence the values of \( m_1 \) and \( b \)?
3. How does the interest rate affect investment and equilibrium income in this model?
4. What is the role of government expenditure in driving national income growth?
5. Why is maximizing profits different from maximizing total revenue?

**Tip:** Understanding the role of multipliers in fiscal policy can help predict the impact of government spending changes on the overall economy.

Using the national income model and the given functions, solve for equilibrium income, multiplier, and various economic scenarios. Then solve for the demand and total cost functions in terms of profit maximization, revenue, and costs.

Learn how to solve a national income model for equilibrium income and the multiplier effect. Also, solve for profit maximization, revenue maximization, and cost functions using demand and total cost equations. Suitable for undergraduate economics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation