KEYNESIAN THEORY AND POLICY AT A GLANCE
DERIVATION OF THE TAX MULTIPLIER

When the level of taxation increases by some amount, DT, the equilibrium level of income will decrease by some amount, DY. The tax multiplier, first cousin to the investment multiplier, is negative and is ratio of DY to DT. It can be derived from the equilibrium condition (Y = C + I + G) together with the equation defining disposable (i.e., after-tax) income (Yd = Y - T) and the consumption equation as applied to disposable income (C = a + bYd).
 

 
I N S T R U C T I O N S
 
 
E Q U A T I O N S
1. Write the equilibrium condition letting it describe the initial equilibrium.
Y = C + I + G, where C = a + bYd and Yd = Y - T 
2. Replace the C in this equation with its algebraic equivalent, a + bYd; and Yd with its equivalent Y - T.
Y = a + bYd + I + G;  Y = a + b(Y - T) + I + G
3. Remove the parentheses algebraically.
Y = a + bY - bT + I + G
4. Rewrite equation 3 substituting Y + DY for Y and T +DT for T. (This equation describes the new equilibrium, once the economy has adjusted to the increase in the level of taxation. XXXY + DY = a + b(Y + DY) - b(T + DT) + I + G
5. Remove the parentheses algebraically. XXXY + DY = a + bY + bDY - bT - bDT + I + G
6. Rewrite equation 3 aligning the corresponding terms. XXXY + DY = a + bY + bDY - bT - bDT + I + G 
XXX___________________________________
7. Subtract equation 6 from equation 5. X +XY + DY = a + bY + bDY - bT - bDT + I + G
8. Transpose bDY to the left side of the equation.
DY - bDY = - bDT
9. Factor out the DY.
(1 - b)DY = - bDT
10. Now divide both sides of the equation by (1 - b). This equation tells us that if we know the level of taxation has increased by DT, we can multiply by -b/(1 - b) to determine the corresponding decrease (DY) in the level of income. 
 DY = [-b/(1 - b)] DT
11. Alternatively, divide both sides of this equation by DT to get the definitional statement of the tax multiplier. Note that the tax multiplier is negative of the ratio of the marginal propensity to consume to the marginal propensity to save.
DY/DT = -b/(1 - b)