The Multiplier

The "multiplier" tells us the change in equilibrium GDP from a given change in desired investment. Mathematically, we desire a formula for .

Suppose GDP is given by the formula Y = C + I_g where C takes the linear formula C = a + bY. As in earlier math notes, a is autonomous consumption spending and b is the MPC. Solving for equilibrium GDP, we obtain the general result Y_e = x (a + I_g).

If I_g changes by some amount D I_g, then Y_e will change by some amount D Y_e: Y_e + D Y_e = x (a + I_g + D I_g) = x (a + I_g) + x D I_g. We can simplify this by subtracting Y_e and its equivalent x (a + I_g) from both sides to obtain D Y_e = x D I_g. Now divide D I_g on both sides to obtain our desired result: = .

Since b is the MPC and (1 - b) is the MPS, this can be expressed alternatively as = = . For example, if the MPC is .75, the multiplier is = = 4, so that if D I_g = $5 billion, D Y_e = $20 billion.

Incidentally, the term has an alternate interpretation. Suppose we have an infinite sum of the form 1 + b + b² + b³ + b⁴ + .... We don’t know what, if anything, this sum is equal to, but suppose it converges to some value Z. That is, Z = 1 + b + b² + b³ + b⁴ + .... If we multiply each side of this equation by b, we would have bZ = b + b² + b³ + b⁴ + b⁵ + .... Next, subtract bZ from Z: Z - bZ = 1, as every term except the first term of Z cancels out. We can factor out Z from the term on the left and as long as b does not equal 1, divide both sides by 1 - b to obtain Z = . That is, if the infinite sum converges at all, it will converge to . Note that for b > 1 the sum is infinite. However, if b < 1 it can be shown that the sum converges to the value which you will recognize as the multiplier. This has an economic interpretation: the change in equilibrium GDP from a given change in investment can be seen as successive "rounds" of additional spending, each a constant proportion (equal to the MPC) of the prior round.