#### Economics Math Help

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By Sean McClung

Studying economic theories and methods requires an understanding of mathematical symbols. These symbols are used as shorthand to describe theoretical concepts in economics textbooks, theories, and research papers.

As such, economists need to have some quantitative skills. The more comfortable they are with math and econometrics, the better for their research and career opportunities.

But, economists are not mathematicians. So, when students of economics take Bachelor’s or Master’s courses while studying for their economics degree, they might be presented with symbol-heavy lecture notes. Without understanding these symbols, students will probably struggle to read them, leaving their meaning unclear.

This article aims to fill that gap. What are the most commonly used math symbols in economics, and what do they mean?

## Symbols and definitions

*Math symbols*

- \(\pm\) - this symbol means “plus or minus”. It’s most often used when describing an interval of some sort, such as a confidence interval.
- \(\preceq\) - this symbol, and others like it, are used often in economics to describe preference relations. Preferences are the building blocks of the demand curve. This symbol is seldom used outside of microeconomics courses that specifically focus on preferences.
- \(\propto\) - this symbol means that the variable on the left side is proportional to the variable on the right side.
- \(\sim\) - this symbol means “similar to”.
- \(\approx\) - this symbol means “approximately”, which means something is almost, but not quite, equal to something else.
- \(\parallel\) - parallel. In economics, this is used most often in econometrics when describing how things relate to each other geometrically.
- \(\perp\) - this means perpendicular or “orthogonal”. Perpendicular means two things intersect at a 90 degree angle in geometry. In econometrics, orthogonal means that one variable has zero power to explain another variable, because they have a covariance equal to zero.
- \(\in\) - this means “in”, that a variable on the left side is contained within a variable, group, or concept on the right side. It can be reversed to mean the opposite. For example, often in economics, a theory will describe which values of
*x*are possible. If*x*can only take on real number values, the description might start with*x*\(\in\) \(\mathbb{R}\). This means that*x*can only be a real number, because it is “in” the set of \(\mathbb{R}\). - \(\otimes\) - this symbol is used to describe the Kronecker product, which is a matrix algebra concept that comes up occasionally in econometrics.
- \(\partial\) means “partial”, and is used to denote a partial derivative.
*I*is often used to denote the identity matrix with dimension_{N}*N*; it’s used frequently in statistics and econometrics.- \(\cdot\) this symbol is placed above a variable to signify the derivative with respect to time of that variable. This symbol is commonly used in macroeconomics, but seldom outside of that context.

*Greek letters*

These are often used to denote variables, but some carry special meaning because of how often they’ve been used in certain contexts.

- \(\alpha\) - the letter alpha, often used with Cobb-Douglas functions to denote the exponents. Also used when discussing p-values in hypothesis testing; the “alpha” level sets the number that you compare the hypothesis test’s result to, in order to determine if the result is statistically significant (such as 0.05 for 5% significance, which is often the default).
- \(\beta\) - the letter beta, also used as an exponent in Cobb-Douglas functions. But, it’s most famously used when describing regression formulae. In that context, this letter usually has a subscript and is the shorthand way of saying “the regression coefficient on the variable it’s being multiplied with”.
- \(\epsilon\) - the letter epsilon. This letter is also commonly used in regression analysis, and usually denotes the error term of the regression. Many introductory econometrics classes, and any other course that showcases regression analysis, will make frequent use of this symbol. Many of the assumptions that allow economists to justify using a regression model involve statements made about the error term.
- \(\delta\) - the letter delta. This symbol is used often to denote the change in a variable, or depreciation in macroeconomics. It’s often confused with the symbol for partial derivatives, which looks very similar!
- \(\lambda\) - the letter lambda. It is used most often when solving constrained optimization problems using the Lagrangian method; in this case it stands for the shadow price of the budget constraint. This letter will be seen very often by microeconomics students.
- \(\mu\) - the letter mu. This letter is used often in statistics and usually denotes the mean, or average, of something.
- \(\rho\) - the letter rho. Often used to indicate the correlation coefficient between two variables.
- \(\Sigma\) - the capital letter sigma. Used frequently to denote the sum of things.
- \(\sigma\) - the lower case sigma. Used often in regression analysis and statistics to denote standard deviation; when squared, it is the variance of the subject in question.
- \(\Phi\) - the capital letter phi. Usually used to denote a probability distribution in statistics, most commonly the cumulative distribution function (cdf) of the normal distribution.
- \(\phi\) - the lower case phi. Often used to denote the probability distribution function (pdf) of the normal distribution in statistics. But, this letter is frequently used outside of this context too, unlike the capital version described above.
- \(\chi\) - the letter chi. Used in statistics to denote the chi-squared distribution, which is a probability distribution that is used a lot in regression analysis.
- \(\Omega\) - the capital letter omega. Used often in econometrics to be the symbol for a matrix.

The following letters usually don’t carry special meaning on their own, but are often used in economics regardless. They are commonly used as placeholders for a variable when discussing a theory or equation.

- \(\gamma\) - the letter gamma.
- \(\eta\) - the letter eta.
- \(\kappa\) - the letter kappa.
- \(\nu\) - the letter nu.
- \(\tau\) - the letter tau.
- \(\upsilon\) - the letter upsilon.
- \(\Psi\) - the capital letter psi.
- \(\psi\) - the lower case letter psi.
- \(\omega\) - the lower case letter omega.

*Other statistics symbols*

The following symbols are often used in economic statistics contexts. Note that the Greek letters above contain many symbols used in these contexts as well.

- | means “given that” in a statistics sense, i.e.,
*X*|*Y*= 2 means “*X*given the fact that*Y*equals 2”. - \(\mathbb{E}\) means “the expectation of”. Typically there is a defined variable, matrix, etc. placed in brackets next to it (for example,
*X*); then, this means the expectation, or expected value, of*X*. This symbol is used extremely often in statistics. - \(\mathbb{V}\) means “the variance of”. It’s used in the same way as the expectation symbol above, so when something is placed in brackets next to this, it means the variance of that variable, matrix, etc. Not to be confused with \(\sigma\)
^{2}, which stands for a specific number. - \(\sigma\)
^{2}is the symbol for the variance of a variable, matrix, etc. This stands for the actual number that is the variance of something. - \(\overline{X}\): the bar above a variable denotes the average. This symbol is used very often in economic formulae, for example when describing the sum of least squares.

*Set theory symbols*

Set theory is used to describe how groups of things relate to one another. They’re similar to a Venn diagram, but in the language of math.

- \(\subset\) - this symbol shows that the variable or group on the left side of it is a subset of (included in) the group on the right side of it. With a slash through it, it means the opposite.
- \(\subseteq\) - the same as the above symbol, except that the subset it describes can also be equal to the variable or group on the right side. It does not have to be completely contained within the second group.
- \(\supset\) - this symbol denotes a superset, which is the opposite of a subset. It means that the group on the left side of this symbol contains the group on the right side.
- \(\cap\) - this is the symbol for the intersection of two sets, or the things they both contain that are the same.
- \(\cup\) - this is the symbol for the union of two sets, or everything that is contained in at least one of the sets.
- \(\emptyset\) - this is the “empty set” symbol, which is simply a set that contains nothing.

*Sets of numbers*

The following symbols are still technically sets as they describe groups of numbers, like “rational numbers” or “natural numbers”. These are used very often in economic formulae even when other set symbols might not appear.

- \(\mathbb{N}\) is the symbol for natural numbers.
- \(\mathbb{Z}\) is the symbol for the set including all integers.
- \(\mathbb{Q}\) is the symbol for the set of all rational numbers.
- \(\mathbb{R}\) is the symbol for the set of all real numbers.

*Other useful symbols*

- \(\exists\) means “there exists at least one”. It’s commonly seen in proofs, which tend to be used in econometrics- or statistics-heavy courses.
- \(\exists!\) is a variation on the symbol above that means “there exists one and only one”.
- \(\forall\) means “for all”, usually in the context of stating some truth in an econometric context, i.e., stating that a particular equation is true for all
*i*in some set of observed values*N*. - \(\implies\) means “implies”; the function, formula, or variable on the left side implies some relationship written on the right side.
- \(\iff\) means “if and only if”.

## Reading economic formulae

Now that the commonly used symbols have been introduced, we can use them to read and interpret an economic formula. The below example is the Goldsmith equation, used in macroeconomics to describe the growth of capital stock over time:

\begin{equation*}

\dot{K_t} = \mathit{I_t}-\delta\mathit{K_t} = \mathit{s}F(\mathit{K_t},\mathit{A_t},\mathit{L_t}) - \delta\mathit{K_t}

\end{equation*}

The first variable in this formula stands for the derivative of capital with respect to time, signified by the dot above the *K _{t}*. Recall from the list of math symbols above that the dot means the derivative of something with respect to time.

This is equivalent to the investment (which here is defined as *I _{t}*) at a point in time minus the depreciation times the capital stock at a point in time. Recall that the symbol \(\delta\) from above is often used to denote depreciation.

Then, on the far right-hand side, it is broken down further. It’s multiplied by *s*, which stands for the savings rate. This is multiplied by a function of capital, labor, and technological growth. The function’s exact form is not defined for us yet, so it’s simply written as *F*(*K _{t}*,

*A*,

_{t}*L*), which is common practice in math.

_{t}\begin{equation*}

\mathit{YED}_{A} = \frac{\Delta\mathit{q}_{A}}{\Delta\mathit{Y}} = \frac{\frac{\partial{q}_{A}}{\mathit{q}_{A}}}{\frac{\partial{Y}}{Y}}

\end{equation*}

This equation, meanwhile, describes the income elasticity of demand with respect to good A. It tells us exactly what this elasticity is and how to calculate it. The income elasticity of demand *YED _{A}* is equal to the change in the quantity demanded divided by the change in income.

\begin{equation*}

\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X|Y]]

\end{equation*}

This formula is a foundational theorem in econometrics, known as the Law of Iterated Expectations. Recall that the \(\mathbb{E}\) symbol means “expectation”. This formula states the following in mathematical shorthand.

Our expectation of *X* in general is equal to our expectation of the value of *X* that is expected given some value that the variable *Y* could take on. In other words, we have some general idea of what *X* should be over the whole range of possible *Y* values (which have not been defined yet). This expectation is the same as what we generally expect *X* to be.

In more precise statistical language, this formula means that the expected value of *X* is equal to the expectation of the conditional expectation of *X* given *Y*.

It took many words to describe what a simple formula could convey with only a few symbols. Clearly, using these symbols can help economists communicate ideas much more effectively than by just using words!

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