Marginal value

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Marginal Value is a term widely used in economics, to refer to the change in economic value associated with a unit change in output, consumption or some other economic choice variable.

Mathematically, the marginal value is the derivative of a differentiable function, or to related concepts such as the arc derivative (slope of a secant line for more general functions.

Mathematical formulation

In a functional relationship like $y=f(x)$ , x is the independent variable and y is the dependent variable, i.e. the value of y will change in accordance with the change in the value of x.

Now, in case x is a discrete variable, "Marginal Value of y" will be the "change in the value of y for a one unit change in the value of x". In case x is a continuous variable, "Marginal Value of y" will be the ratio of two changes. In the ratio, the denominator will be the "infinitesimal change in x" and in the numerator, there will be the "change in y".

For example, the utility function, in its simplest form, is provided by $U=f(x)$ , where U: the level of utility a consumer attains and x: the quantity of a good the consumer consumes. Here, "Marginal Value of U" will be called Marginal Utility (MU) and be expressed as MU = (Change in U)/(Change in x). In case, "change in x" is "+ one unit", MU = Change in U; for "change in x" is "- one unit", MU = -(Change in U).

Again, the consumption function, in its simplest form, is provided by $c=f(y)$ , where c: level of consumption and y: level of income. In economic terms, "Marginal Value of Consumption" is called Marginal Propensity to Consume (MPC). This will be given by MPC = (Change in consumption)/(Change in income).

For a linear functional relationship like $y=a+bx$ , the "Marginal Value of y" will be the co-efficient of x (in this case, b) and this will not change as the x changes and takes different values within its domain of definition. However, in case the functional relationship is non-linear, say $y=a.b^{x}$ , the "Marginal Value of y" will be different for different values of x. Say, the initial values of x are $(x1,x3)$ and the corresponding final values are $(x2,x4)$ . The linear function will provide same "Marginal value of y" for both the changes $x2-x1$ and $x4-x3$ ; but for the same two changes in x, "Marginal value of y" will be different in case of the non-linear function.