What is a 'Multiplier'
In economics, a multiplier is the factor by which gains in total output are greater than the change in spending that caused it. It is usually used in reference to the relationship between investment and total national income. The multiplier theory and its equations were created by British economist John Maynard Keynes.
BREAKING DOWN 'Multiplier'
By “investment,” Keynes meant government spending. He believed that any injection of government spending created a proportional increase in overall income for the population, since the extra spending would carry through the economy.The Mathematics of the Multiplier
In his 1936 book "The General Theory of Employment, Interest, and Money," Keynes wrote following equation to describe the relationship between income (Y), consumption (C) and investment (I): Y = C + I. For any level of income, people spend a fraction and save/invest the remainder.
Keynes represented the marginal propensity to save (MPS) as 1b, and the marginal propensity to consume (MPC) as b. This creates the equations C = bY, and I = Y  C. In other words, bY determines how much remains for investments. Keynes rearranged the equations to solve for income, or Y = I / (1b).
Here, Keynes redefined Y as k, writing k = 1 / (1b). This allowed Keynes to assert Y = k*I. Since investment was multiplied by k, Keynes referred to k as the multiplier. For any new injection of government spending, k showed the relationship between change in income (dY) and change in investment (dI), or dY = k*dI.
Given an MPC = 0.8, for example, then k = 1 / (1  0.8) = 5. This suggested any change in income will be five times the change in new investment, or new government expenditures.
Problems in Multiplier Math
In Keynes' first derivation, income (Y) is treated as an independent variable, or a cause which drives other changes in the economy; after introducing the concept of k, but investment (I) and the multiplier (k) were suddenly independent variables. Income became a dependent variable, or an effect.
Keynes' multiplier reversed cause and effect after k was introduced. While still mathematically true, this reversal only demonstrated a necessary accounting relationship in Keynes’ equation, not any meaningful causal relationship.
For an analogy, consider the equation for converting Celsius into Fahrenheit: F = 32 + 1.8C. Here, an increase in 10 degrees Celsius implies an increase of 18 degrees Fahrenheit. This can be expressed mathematically as dF/dC = k, where k is a Celsius multiplier. (Mathematically, this is identical to the Keynesian multiplier.)
It would not make much sense to claim a 10degree rise in Celsius causes an 18degree rise in Fahrenheit. The two may be mathematically related in a fixed equation, but there is no sensible causal link involved. The same goes for Keynesian multipliers.
In fact, a derivation of the Keynesian multiplier can be written dY / dC = 1/b. With an MPC = 0.8, a change in income is only 1.25 times the change in new expenditures, not five times.

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