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Sagot :
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
### 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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