Where does multiplier collide?

The ratio between the total change in income or output and the initial change in spending that set it off: when someone spends, that outlay becomes another person's income, part of which is spent again, so a single injection circulates and adds up to more than itself. Kahn's 1931 The Relation of Home Investment to Unemployment worked out this ratio for public investment and employment, giving the idea that Keynes later placed at the centre of his theory of demand. The referent is a macroeconomic ratio measuring how far spending amplifies as it circulates, and it has nothing to do with an auxiliary variable used to solve a constrained problem or a circuit whose output is a product of its inputs.[1]

Unit

A dimensionless ratio: total change in output divided by the initial change in spending

Recovery means

Not applicable

Measured by

Estimated from national accounts and from natural experiments in spending, as the cumulative output response to a one-unit fiscal or monetary injection

An auxiliary variable introduced when finding the largest or smallest value of a function subject to a constraint: attaching one such variable to each constraint turns a restricted problem into an unrestricted one, and at the best point the variable's value measures how much the optimum would shift if the constraint were loosened by a unit. Everett's 1963 Generalized Lagrange Multiplier Method generalised this classical device of Lagrange to allocation problems and read each multiplier as the price of its resource. The referent is a mathematical bookkeeping variable that prices a constraint, not a ratio describing how spending circulates through an economy or a circuit that forms a product of two signals.[2]

Unit

One scalar variable per constraint, valued as the sensitivity of the optimum to that constraint

Recovery means

Not applicable

Measured by

Solved for as part of the stationarity conditions of the constrained problem, and read off as the shadow price of relaxing each constraint

A circuit whose output is the product of two input signals, used to mix, modulate, and detect signals or to raise a signal's frequency by combining it with itself: a four-quadrant version handles inputs of either polarity and returns their true product. Gilbert's 1968 A precise four-quadrant multiplier with subnanosecond response described the transistor arrangement, now known as the Gilbert cell, that made such multiplication fast and accurate on a chip. The referent is a piece of hardware that physically computes a product of voltages or currents, unrelated to a ratio for how spending amplifies or to an auxiliary variable that prices a constraint.[3]

Unit

One circuit whose output voltage or current equals a scaled product of its two inputs

Recovery means

Not applicable

Measured by

Characterised on the bench by linearity, gain, bandwidth, and feedthrough error across the range of both inputs