Give step-by-step solution with explanation and final answer:
9. Suppose the economy is characterized by the following behavioral equations:
C=C_0 +c_1Y_D
Y_D=Y-T
I= b_0 + b_1Y
where c_0=480, c_

Question

Give step-by-step solution with explanation and final answer: 
9. Suppose the economy is characterized by the following behavioral equations:
C=C_0 +c_1Y_D
Y_D=Y-T
I= b_0 + b_1Y
where c_0=480, c_1=0.5,t_0=80,t_1=0.25,G=120,b_0=200, and b_1=0.25. What is the
(income) multiplier for this economy?
a. 4
b. 3
c. 2
d. 1
e. All of the answers here are incorrectFieti! Atk LE EE A REE
dit 9. Suppose the economy is characterized by the following behavioral equat 1
Br C=cyrely | FE
Yo=Y-T ! hi Bc
; I'=by+b,y ha
where co = 430, ¢1 = 0.5, to = 80, ti = 0.25, G = 120, bo= 200, and by = 0.25, What is
(income) multiplier for this economy? i HN I -
a. 4 BER
b. 3 LH
i (TE
2 a
e. All of the answers here are incorrect i i

VIPSolutions · Accepted Answer

Let's solve the question step by step:

**Given:**
- $ C = c_0 + c_1 Y_D $
- $ Y_D = Y - T $
- $ T = t_0 + t_1 Y $
- $ I = b_0 + b_1 Y $
- $ G = 120 $
- $ c_0 = 480 $, $ c_1 = 0.5 $, $ t_0 = 80 $, $ t_1 = 0.25 $, $ b_0 = 200 $, $ b_1 = 0.25 $

### 1. **Aggregate Demand (AD) Equation:**

$ Y = C + I + G $

Substitute all the components:

#### **a. Consumption ($C$):**
$ C = c_0 + c_1 Y_D $
$ Y_D = Y - T = Y - (t_0 + t_1 Y) = Y - t_0 - t_1 Y = (1-t_1)Y - t_0 $
So,
$ C = c_0 + c_1 [(1-t_1)Y - t_0] $
$ C = c_0 + c_1(1-t_1)Y - c_1 t_0 $

#### **b. Investment ($I$):**
$ I = b_0 + b_1 Y $

#### **c. Government Spending ($G$):**
Given as $ G = 120 $

#### **d. Put all together:**
$ Y = C + I + G $
$ Y = [c_0 + c_1(1-t_1)Y - c_1 t_0] + [b_0 + b_1 Y] + G $

Combine like terms:
$ Y = c_0 - c_1 t_0 + b_0 + G + [c_1(1-t_1) + b_1]Y $

$ Y - [c_1(1-t_1) + b_1]Y = c_0 - c_1 t_0 + b_0 + G $
$ Y[1 - c_1(1-t_1) - b_1] = c_0 - c_1 t_0 + b_0 + G $

### 2. **Multiplier ($k$) Formula:**

The multiplier is the reciprocal of the coefficient on $Y$:

$$
k = \frac{1}{1 - c_1(1-t_1) - b_1}
$$

### 3. **Plug in values:**

- $ c_1 = 0.5 $
- $ t_1 = 0.25 $
- $ b_1 = 0.25 $

$$
k = \frac{1}{1 - 0.5(1-0.25) - 0.25}
$$
$$
= \frac{1}{1 - 0.5 \times 0.75 - 0.25}
$$
$$
= \frac{1}{1 - 0.375 - 0.25}
$$
$$
= \frac{1}{1 - 0.625}
$$
$$
= \frac{1}{0.375}
$$
$$
= 2.6667 \approx 2.67
$$

### 4. **Check the options:**

a. 4  
b. 3  
c. 2  
d. 1  
e. All of the answers here are incorrect

**None of the options match 2.67, so the correct answer is:**

---

**Final Answer:**
> **e. All of the answers here are incorrect**

VIPSolutions · Answer

Let's analyze why the other options are incorrect based on our calculations:

### Summary of the calculation:
- The derived income multiplier $k \approx 2.67$.
- The options provided are 4, 3, 2, 1, or an assertion that all answers are incorrect.

### Why the other options are incorrect:

1. **Option a: 4**  
   - The calculated multiplier (~2.67) is significantly less than 4.  
   - Therefore, **option a is incorrect**.

2. **Option b: 3**  
   - 3 is higher than the calculated 2.67, so **option b is incorrect**.

3. **Option c: 2**  
   - 2 is less than 2.67, so **option c is incorrect**.

4. **Option d: 1**  
   - The multiplier is much higher than 1; thus, **option d is incorrect**.

### Why the correct answer is **e**:  
Since none of the options match the calculated multiplier (~2.67), the correct choice is:

**e. All of the answers here are incorrect**

---

### Final note:
- The key step was identifying the correct formula for the multiplier in an economy with taxes and investment.
- The derived formula and values confirm that the multiplier is approximately 2.67, which isn't listed among the options.

**Hence, the correct answer is:**

### **e. All of the answers here are incorrect**

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