Question:
Consider the following equations for a closed economy:
Consider the following equations for a closed economy:
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In dynamic macroeconomic models, first impose the equilibrium condition \(S=I\), then derive the recursive relation connecting current and previous income levels.
Updated On: Jun 5, 2026
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Correct Answer: 3600
Solution and Explanation
Step 1: Use the equilibrium condition for a closed economy.
In a closed economy without government and foreign trade, equilibrium requires
\[ S_t=I_t \]
Given,
\[ S_t=aY_t \] and
\[ I_t=g(Y_t-Y_{t-1}) \]
Therefore, equilibrium implies
\[ aY_t=g(Y_t-Y_{t-1}) \]
Step 2: Substitute the values of \(a\) and \(g\).
We are given
\[ a=0.5 \] and
\[ g=0.6 \]
Substitute into the equilibrium equation:
\[ 0.5Y_t=0.6(Y_t-Y_{t-1}) \]
Expand the right-hand side:
\[ 0.5Y_t=0.6Y_t-0.6Y_{t-1} \]
Step 3: Rearrange the equation.
Bring the \(Y_t\)-terms to one side:
\[ 0.6Y_t-0.5Y_t=0.6Y_{t-1} \]
\[ 0.1Y_t=0.6Y_{t-1} \]
Divide both sides by \(0.1\):
\[ Y_t=6Y_{t-1} \]
Thus, income follows a multiplier process where each period’s income is six times the previous period’s income.
Step 4: Calculate \(Y_1\).
Given,
\[ Y_0=100 \]
Using
\[ Y_t=6Y_{t-1} \] for \(t=1\),
\[ Y_1=6Y_0 \] \[ Y_1=6(100) \] \[ Y_1=600 \]
Step 5: Calculate \(Y_2\).
Again using
\[ Y_t=6Y_{t-1} \] for \(t=2\),
\[ Y_2=6Y_1 \] \[ Y_2=6(600) \] \[ Y_2=3600 \]
Step 6: Final conclusion.
Hence, the value of
\[ Y_2 \] is
\[ \boxed{3600} \]
In a closed economy without government and foreign trade, equilibrium requires
\[ S_t=I_t \]
Given,
\[ S_t=aY_t \] and
\[ I_t=g(Y_t-Y_{t-1}) \]
Therefore, equilibrium implies
\[ aY_t=g(Y_t-Y_{t-1}) \]
Step 2: Substitute the values of \(a\) and \(g\).
We are given
\[ a=0.5 \] and
\[ g=0.6 \]
Substitute into the equilibrium equation:
\[ 0.5Y_t=0.6(Y_t-Y_{t-1}) \]
Expand the right-hand side:
\[ 0.5Y_t=0.6Y_t-0.6Y_{t-1} \]
Step 3: Rearrange the equation.
Bring the \(Y_t\)-terms to one side:
\[ 0.6Y_t-0.5Y_t=0.6Y_{t-1} \]
\[ 0.1Y_t=0.6Y_{t-1} \]
Divide both sides by \(0.1\):
\[ Y_t=6Y_{t-1} \]
Thus, income follows a multiplier process where each period’s income is six times the previous period’s income.
Step 4: Calculate \(Y_1\).
Given,
\[ Y_0=100 \]
Using
\[ Y_t=6Y_{t-1} \] for \(t=1\),
\[ Y_1=6Y_0 \] \[ Y_1=6(100) \] \[ Y_1=600 \]
Step 5: Calculate \(Y_2\).
Again using
\[ Y_t=6Y_{t-1} \] for \(t=2\),
\[ Y_2=6Y_1 \] \[ Y_2=6(600) \] \[ Y_2=3600 \]
Step 6: Final conclusion.
Hence, the value of
\[ Y_2 \] is
\[ \boxed{3600} \]
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