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The function below hypothetically represents the model of the Liberian economy for fiscal year 2020:

[tex]\[ Y = C + I + G \][/tex]
[tex]\[ C = a + c Y^d, \text{ where } Y^d = Y - T \][/tex]
[tex]\[ T = T_0 + t Y \][/tex]
[tex]\[ I = I_0 \][/tex]
[tex]\[ G = G_0 \][/tex]

Given:
[tex]\[ a = 50, \][/tex]
[tex]\[ I_0 = 73, \][/tex]
[tex]\[ T_0 = 40, \][/tex]
[tex]\[ c = 0.2, \][/tex]
[tex]\[ t = 0.6 \][/tex]

Tasks:
1. Determine the equilibrium level of national income. (8 points)
2. Determine the equilibrium level of consumption. (4 points)
3. Determine the equilibrium level of saving. (4 points)
4. Calculate both the Government multiplier and the Tax multiplier. (3 points each)



Answer :

GinnyAnswer
Certainly! Let's solve each part step-by-step using the given economic parameters:

### Given Parameters:
- Autonomous consumption (a): 50
- Investment (I_0): 73
- Autonomous taxes (T_0): 40
- Marginal propensity to consume (c): 0.2
- Tax rate (t): 0.6
- Government spending (G_0): 0
- National income (Y): Here we need to determine equilibrium.

### 1. Determine the Equilibrium Level of National Income (Y):
To find the equilibrium level of national income, we use the fact that at equilibrium:
[tex]\[ Y = C + I + G \][/tex]

From the given consumption function:
[tex]\[ C = a + c \cdot (Y - T) \][/tex]
Where [tex]\( T \)[/tex] is the total tax and given by:
[tex]\[ T = T_0 + t \cdot Y \][/tex]

Thus,
[tex]\[ C = a + c \cdot (Y - (T_0 + t \cdot Y)) \][/tex]
[tex]\[ C = a + c \cdot (Y - T_0 - t \cdot Y) \][/tex]
[tex]\[ C = a + c \cdot (1 - t) \cdot Y - c \cdot T_0 \][/tex]

Now substitute this back into the national income equation:
[tex]\[ Y = a + c \cdot (1 - t) \cdot Y - c \cdot T_0 + I_0 + G \][/tex]
Since [tex]\(G = G_0 = 0\)[/tex], we simplify the equation:
[tex]\[ Y = a + c \cdot (1 - t) \cdot Y - c \cdot T_0 + I_0 \][/tex]

Rearranging terms to solve for [tex]\(Y\)[/tex]:
[tex]\[ Y - c \cdot (1 - t) \cdot Y = a - c \cdot T_0 + I_0 \][/tex]
[tex]\[ Y \cdot (1 - c + c \cdot t) = a - c \cdot T_0 + I_0 \][/tex]
[tex]\[ Y = \frac{a - c \cdot T_0 + I_0}{1 - c + c \cdot t} \][/tex]

Substitute the given values:
[tex]\[ Y = \frac{50 - 0.2 \cdot 40 + 73}{1 - 0.2 + 0.2 \cdot 0.6} = \frac{50 - 8 + 73}{1 - 0.2 + 0.12} = \frac{115}{0.92} = 125.0 \][/tex]

So, the equilibrium level of national income [tex]\(Y\)[/tex] is 125.0.

### 2. Determine the Equilibrium Level of Consumption (C):
Using the equilibrium [tex]\(Y\)[/tex], we determine the equilibrium consumption [tex]\(C\)[/tex]:
[tex]\[ T_{eq} = T_0 + t \cdot Y_{eq} = 40 + 0.6 \cdot 125.0 = 40 + 75 = 115.0 \][/tex]

[tex]\[ C_{eq} = a + c \cdot (Y_{eq} - T_{eq}) = 50 + 0.2 \cdot (125.0 - 115.0) = 50 + 0.2 \cdot 10 = 50 + 2 = 52.0 \][/tex]

Thus, the equilibrium level of consumption [tex]\(C\)[/tex] is 52.0.

### 3. Determine the Equilibrium Level of Saving (S):
Savings [tex]\(S\)[/tex] can be calculated using:
[tex]\[ S = Y - C - T \][/tex]
[tex]\[ S_{eq} = 125.0 - 52.0 - 115.0 = 125.0 - 167.0 = -42.0 \][/tex]

So, the equilibrium level of saving [tex]\(S\)[/tex] is -42.0.

### 4. Calculate the Government and Tax Multipliers:
Government Multiplier ([tex]\( k_G \)[/tex]):
[tex]\[ k_G = \frac{1}{1 - c + c \cdot t} \][/tex]
[tex]\[ k_G = \frac{1}{1 - 0.2 + 0.2 \cdot 0.6} = \frac{1}{1 - 0.2 + 0.12} = \frac{1}{0.92} \approx 1.0869565217391304 \][/tex]

So, the government multiplier [tex]\( k_G \)[/tex] is 1.0869565217391304.

Tax Multiplier ([tex]\( k_T \)[/tex]):
[tex]\[ k_T = \frac{-c}{1 - c + c \cdot t} \][/tex]
[tex]\[ k_T = \frac{-0.2}{1 - 0.2 + 0.2 \cdot 0.6} = \frac{-0.2}{0.92} \approx -0.21739130434782608 \][/tex]

Thus, the tax multiplier [tex]\( k_T \)[/tex] is -0.21739130434782608.

In summary:
- Equilibrium Level of National Income (Y): 125.0
- Equilibrium Level of Consumption (C): 52.0
- Equilibrium Level of Saving (S): -42.0
- Government Multiplier (k_G): 1.0869565217391304
- Tax Multiplier (k_T): -0.21739130434782608

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