The marginal cost (MC) can be estimated by calculating the ratio of the change in total cost ($ΔC$) to the change in the quantity of output ($ΔQ$). The formula is expressed as: $MC \approx \frac{ΔC}{ΔQ}$.

Estimating Marginal Cost

The Greek letter delta (Δ) is a mathematical symbol used to represent 'the change in' a variable. For example, in economic formulas, ΔC denotes the change in total cost and ΔQ represents the change in quantity.

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In economics, a marginal change quantifies the impact of a small, incremental adjustment to a variable. When two variables, $x$ and $y$, are related by a function $y = f(x)$, the marginal change is represented graphically by the slope of the function. For continuous functions, this change is precisely measured using the function's derivative. [1, 2] This use of derivatives is a standard technique for measuring marginal changes throughout all mathematical extensions in economic analysis. [3]

Marginal Changes

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

CORE Econ - The Economy 2.0: Microeconomics

The Marginal Rate of Substitution (MRS) quantifies an individual's willingness to trade one good for another while maintaining the same level of utility. While the slope of an indifference curve is mathematically a negative number, the MRS is defined as a positive quantity representing the magnitude of this trade-off. Specifically, the MRS is the absolute value of the slope of the indifference curve. It is common to hear the phrase ‘the MRS is equal to the slope,’ but this is a simplification; what is strictly meant is that the MRS is equal to the absolute value of the slope. For two goods, y (on the vertical axis) and z (on the horizontal axis), the slope is written as $-\frac{dy}{dz}$, making the MRS a positive value.

Marginal Rate of Substitution (MRS)

The 'Introduction to Mathematical Extensions' provides foundational context for the mathematical techniques used in the course. It offers a brief explanation of calculus and outlines the specific mathematical level, notation, and conventions that are followed in the analyses.

Introduction to Mathematical Extensions for Economic Analysis

When a function is represented graphically by a curve rather than a straight line, its slope is not constant and will differ at various points. This principle is essential for understanding concepts that rely on non-linear relationships, such as the diminishing marginal rate of substitution along a convex indifference curve, where the slope changes continuously.

Variable Slope of a Curve

A derivative provides the precise rate of change for an infinitesimal adjustment in a variable. In practice, when dealing with small, discrete units, this value can be closely estimated using a finite difference. For a function $y = f(x)$, the derivative, denoted as $\frac{df}{dx}$, is approximately equal to the change observed from a one-unit increase in $x$, calculated as $f(x+1) - f(x)$. This approximation is particularly accurate when the unit of measurement is small.

Approximation of a Derivative by a Finite Difference

Marginal utility is the additional satisfaction or benefit resulting from a one-unit increase in the consumption of a good. More formally, it is the rate of change in total utility as the quantity of the good consumed increases. In economic models, marginal utility is often calculated using partial derivatives.

Marginal Utility

Implicit differentiation is a calculus technique often employed in economic models. Its method involves analyzing how one variable, such as consumption, would need to change in response to a small increase in another variable, like free time, in order to keep a third value, such as utility, constant.

Implicit Differentiation in Economics

The composite function rule, more commonly known as the chain rule, is a formula in calculus for computing the derivative of a composite function (a function created by composing two or more other functions). In the context of the work-leisure model, this rule is specifically applied to differentiate the feasible frontier equation in order to calculate the Marginal Rate of Transformation (MRT).

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