Why is the open economy autonomous expenditure multiplier smaller than the closed economy one?

In a closed economy, the Equilibrium Level of Income (Y) is given by –

Y = C + cY + I + G

Or, Y - cY = C + I + G

Or, Y (1 - c) = C + I + G

Therefore, Y = C + I + G / 1 - c

Let, C + I + G = A_{1}

Y = A_{1}/ 1 – c

∆Y/∆A_{1} = 1 / 1 – c (eq1)

In an open economy, the Equilibrium Level of Income (Y) is given by –

Y = C + cY + I + G + X - M – mY

Or, Y - cY + mY = C + I + G + X -M

Or, Y (1 - c + m) = C + I + G + X - M

Or, Y= (C+I+G+X-M) / 1-c+m

Let autonomous expenditure (A_{2}) =C + I + G + X - M

Or, Y= A_{2} / 1-c+m

∆Y/∆A_{2} = 1 / 1 – c + m (eq2)

Comparing Equation 1 and 2, we can conclude that multipliers in open economy is smaller than the multiplier in closed economy because the denominator in open economy is greater than the denominator in closed economy

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