Marginal value

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.

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

Marginal Value is the maximum amount of one good you would give up to get one more unit of a different good. (Purdue University, 2008)

Mathematical formulation

In a functional relationship like $y=f(x)$, where $x$ is the independent variable and $y$ is the dependent variable the marginal value of $y$ is given by $M={\frac {dy}{dx}}$. 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 $x$.