Marginal value

A marginal value is

1. a value that holds true given particular constraints,
2. the change in a value associated with a specific change in some independent variable, whether it be of that variable or of a dependent variable, or
3. when underlying values are quantified, the ratio of the change of a dependent variable to that of the independent variable.

(This third case is actual a special case of the second).

In the case of differentiability, at the limit, a marginal change is a mathematical differential, or the corresponding mathematical derivative.

These uses of the term “marginal” are especially common in economics, and result from conceptualizing constraints as a border or margin. The sorts of marginal values most common to economic analysis are those associated with unit changes of resources and, in mainstream economics, those associated with instantaneous changes. Marginal values associated with units are considered because many decisions are based upon decisions made by unit, and marginalism explains unit price in terms of such marginal values. Mainstream economics uses instantaneous values in much of its analysis for reasons of mathematical tractability.

Mathematical formulation

In a functional relationship such as $y=f(x)$, where $x$ is the independent variable and $y$ is the dependent variable the marginal value of $y$ is given by $dy/dx$. In the case where $x$ is a discrete variable, the marginal value of $y$ will be the change in the value of $y$ for a one unit change in the value of $x$.

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 the marginal value of U will be called marginal utility (MU) and be expressed as MU = (Change in U)/(Change in x). In this case, the change in x represents a discrete one unit increase in consumption.

As another example consider the consumption function. In its simplest form, it is given by $c=f(y)$, where $c$: level of consumption and $y$: level of income. In economic terms the marginal value of consumption is called the 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 simply be the co-efficient of $x$ (in this case, $b$) and this will not change as $x$ changes. However, in the case where the functional relationship is non-linear, say $y=ab^{x}$, the marginal value of $y$ will be different for different values of $x$.

See also

• Marginal concepts