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Investment Multiplier
The investment multiplier model presented below answers the questions: is there a definite relationship between ∆Y and ∆I? if yes, what determines this relationship? The model given below provides an algebraic method of working out the investment multiplier.
Let us recall that the equilibrium level of income is given by:
Y = C + I (eq.1)
Now, let investment increase by ∆I. when ∆I takes place it results in ∆Y and ∆Y induces ∆C. thus, the post-∆I equilibrium level of income can be expressed as follows:
Y = ∆Y = C + ∆C + I + ∆I (eq.2)
Subtracting eq. 1 from eq.2, we get
∆Y = ∆C + ∆I (eq.3)
Given the consumption function as C = a + bY
C + ∆C = a + bY +b∆Y
Therefore, ∆C = b∆Y (eq.4)
By substituting eq. 4 for ∆C in eq.3, we get
∆Y = b∆Y + ∆I (eq.5)
∆Y (1 – b) = ∆I
∆Y = (1/1 – b) × ∆I
∆Y/∆I = (1/1 – b) = m (eq. 6)
Thus, the term 1/1 – b gives the value of the investment multiplier (m).
Recall that, in eq.6, b = MPC and 1 – MPC = MPS. Therefore, multiplier (m) can also be expressed as:
m = ∆Y/∆I = 1 – 1/b = 1/1 – MPC = 1/MPS (eq.7)
The last term in eq. 7 indicates that m = reciprocal of MPS.
An alternative method of working out the multiplier
The multiplier can be alternatively worked out by using the expanded form of aggregate demand equations at the points of national income equilibria before and after ∆I takes place. As pre-∆I national income equilibrium takes place at point E1, where,
Y1 = C + I
Since C = a + bY1, the pre-∆I equilibrium level of income (Y1) may be rewritten as:
Y1 = a + bY1 + I (eq.1)
= (1/1 – b) (a +1)
Similarly, at post-∆I equilibrium point E2,
Y2 = C + I + ∆I (eq.2)
= a + bY2 + I + ∆I
= (1/1 – b) (a + I + ∆I)
By subtracting eq. 1 from eq.2, we get,
∆Y = (1/1 – b) (a + I + ∆I) - (1/1 – b) (a +I)
∆Y = (1/1 – b) ∆I (eq. 3)
Eq.3 yields the relationship between ∆Y and ∆I, that is, ∆Y equals 1/(1 – b) times ∆I. therefore, 1/(1 – b) is the investment multiplier (m). thus,
Investment multiplier (m) = (1/1 – b)
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Let us recall that the equilibrium level of income is given by:
Y = C + I (eq.1)
Now, let investment increase by ∆I. when ∆I takes place it results in ∆Y and ∆Y induces ∆C. thus, the post-∆I equilibrium level of income can be expressed as follows:
Y = ∆Y = C + ∆C + I + ∆I (eq.2)
Subtracting eq. 1 from eq.2, we get
∆Y = ∆C + ∆I (eq.3)
Given the consumption function as C = a + bY
C + ∆C = a + bY +b∆Y
Therefore, ∆C = b∆Y (eq.4)
By substituting eq. 4 for ∆C in eq.3, we get
∆Y = b∆Y + ∆I (eq.5)
∆Y (1 – b) = ∆I
∆Y = (1/1 – b) × ∆I
∆Y/∆I = (1/1 – b) = m (eq. 6)
Thus, the term 1/1 – b gives the value of the investment multiplier (m).
Recall that, in eq.6, b = MPC and 1 – MPC = MPS. Therefore, multiplier (m) can also be expressed as:
m = ∆Y/∆I = 1 – 1/b = 1/1 – MPC = 1/MPS (eq.7)
The last term in eq. 7 indicates that m = reciprocal of MPS.
An alternative method of working out the multiplier
The multiplier can be alternatively worked out by using the expanded form of aggregate demand equations at the points of national income equilibria before and after ∆I takes place. As pre-∆I national income equilibrium takes place at point E1, where,
Y1 = C + I
Since C = a + bY1, the pre-∆I equilibrium level of income (Y1) may be rewritten as:
Y1 = a + bY1 + I (eq.1)
= (1/1 – b) (a +1)
Similarly, at post-∆I equilibrium point E2,
Y2 = C + I + ∆I (eq.2)
= a + bY2 + I + ∆I
= (1/1 – b) (a + I + ∆I)
By subtracting eq. 1 from eq.2, we get,
∆Y = (1/1 – b) (a + I + ∆I) - (1/1 – b) (a +I)
∆Y = (1/1 – b) ∆I (eq. 3)
Eq.3 yields the relationship between ∆Y and ∆I, that is, ∆Y equals 1/(1 – b) times ∆I. therefore, 1/(1 – b) is the investment multiplier (m). thus,
Investment multiplier (m) = (1/1 – b)
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