To find the equilibrium level of income in this Keynesian model, we can use the aggregate demand (AD) equation, which includes consumption (C), investment (I), government spending (G), exports (X), and imports (M). The basic equation for aggregate demand is:
\[ AD = C + I + G + X - M \]
Where:
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Consumption (C) is given by: \[ C = C_0 + MPC \cdot (1 - t) \cdot Y \] Where:
- \(C_0\) = Autonomous consumption = 20 million
- \(MPC\) = Marginal Propensity to Consume = 0.8
- \(t\) = Tax rate = 0.2
- \(Y\) = Level of income (what we want to find)
Therefore, the consumption function will be: \[ C = 20 + 0.8(1 - 0.2)Y = 20 + 0.8 \cdot 0.8Y = 20 + 0.64Y \]
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Investment (I) = 25 million
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Government Spending (G) = 30 million
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Exports (X) = 10 million
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Imports (M), which is influenced by Y: \[ M = M_0 + MPM \cdot Y \] Where:
- \(M_0\) = Autonomous imports = 15 million
- \(MPM\) = Marginal Propensity to Import = 0.25
Thus, the import function will be: \[ M = 15 + 0.25Y \]
Now we combine these elements into the aggregate demand equation:
\[ AD = (20 + 0.64Y) + 25 + 30 + 10 - (15 + 0.25Y) \]
Now simplify this:
\[ AD = 20 + 0.64Y + 25 + 30 + 10 - 15 - 0.25Y \] \[ AD = 70 + (0.64Y - 0.25Y) \] \[ AD = 70 + 0.39Y \]
In equilibrium, aggregate demand (AD) equals income (Y):
\[ Y = 70 + 0.39Y \]
To solve for Y, we rearrange the equation:
\[ Y - 0.39Y = 70 \] \[ 0.61Y = 70 \] \[ Y = \frac{70}{0.61} \approx 114.75 \text{ million} \]
Therefore, the level of income \(Y\) is approximately 114.75 million.
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