Finding out financial theories and strategies requires an understanding of mathematical symbols. These symbols are used as shorthand to explain theoretical ideas in economics textbooks, theories, and analysis papers.

As such, economists have to have some quantitative expertise. The extra comfy they’re with math and econometrics, the higher for his or her analysis and profession alternatives.

However, economists will not be mathematicians. So, when college students of economics take Bachelor’s or Grasp’s courses whereas learning for his or her economics degree, they may be offered with symbol-heavy lecture notes. With out understanding these symbols, college students will in all probability wrestle to learn them, leaving their that means unclear.

This text goals to fill that hole. What are essentially the most generally used math symbols in economics, and what do they imply?

## Symbols and definitions

*Math symbols*

- (pm) – this image means “plus or minus”. It’s most frequently used when describing an interval of some kind, corresponding to a confidence interval.
- (preceq) – this image, and others prefer it, are used typically in economics to explain choice relations. Preferences are the constructing blocks of the demand curve. This image is seldom used outdoors of microeconomics programs that particularly deal with preferences.
- (propto) – this image signifies that the variable on the left facet is proportional to the variable on the suitable facet.
- (sim) – this image means “just like”.
- (approx) – this image means “roughly”, which implies one thing is nearly, however not fairly, equal to one thing else.
- (parallel) – parallel. In economics, that is used most frequently in econometrics when describing how issues relate to one another geometrically.
- (perp) – this implies perpendicular or “orthogonal”. Perpendicular means two issues intersect at a 90 diploma angle in geometry. In econometrics, orthogonal signifies that one variable has zero energy to clarify one other variable, as a result of they’ve a covariance equal to zero.
- (in) – this implies “in”, {that a} variable on the left facet is contained inside a variable, group, or idea on the suitable facet. It may be reversed to imply the alternative. For instance, typically in economics, a principle will describe which values of
*x*are attainable. If*x*can solely tackle actual quantity values, the outline may begin with*x*(in) (mathbb{R}). Which means that*x*can solely be an actual quantity, as a result of it’s “in” the set of (mathbb{R}). - (otimes) – this image is used to explain the Kronecker product, which is a matrix algebra idea that comes up often in econometrics.
- (partial) means “partial”, and is used to indicate a partial spinoff.
*I*is commonly used to indicate the id matrix with dimension_{N}*N*; it’s used ceaselessly in statistics and econometrics.- (cdot) this image is positioned above a variable to indicate the spinoff with respect to time of that variable. This image is often utilized in macroeconomics, however seldom outdoors of that context.

*Greek letters*

These are sometimes used to indicate variables, however some carry particular that means due to how typically they’ve been utilized in sure contexts.

- (alpha) – the letter alpha, typically used with Cobb-Douglas features to indicate the exponents. Additionally used when discussing p-values in speculation testing; the “alpha” stage units the quantity that you just examine the speculation check’s outcome to, to be able to decide if the result’s statistically important (corresponding to 0.05 for five% significance, which is commonly the default).
- (beta) – the letter beta, additionally used as an exponent in Cobb-Douglas features. However, it’s most famously used when describing regression formulae. In that context, this letter normally has a subscript and is the shorthand means of claiming “the regression coefficient on the variable it’s being multiplied with”.
- (epsilon) – the letter epsilon. This letter can also be generally utilized in regression evaluation, and normally denotes the error time period of the regression. Many introductory econometrics courses, and every other course that showcases regression evaluation, will make frequent use of this image. Lots of the assumptions that enable economists to justify utilizing a regression mannequin contain statements made in regards to the error time period.
- (delta) – the letter delta. This image is used typically to indicate the change in a variable, or depreciation in macroeconomics. It’s typically confused with the image for partial derivatives, which appears to be like very comparable!
- (lambda) – the letter lambda. It’s used most frequently when fixing constrained optimization issues utilizing the Lagrangian method; on this case it stands for the shadow value of the funds constraint. This letter can be seen fairly often by microeconomics college students.
- (mu) – the letter mu. This letter is used typically in statistics and normally denotes the imply, or common, of one thing.
- (rho) – the letter rho. Typically used to point the correlation coefficient between two variables.
- (Sigma) – the capital letter sigma. Used ceaselessly to indicate the sum of issues.
- (sigma) – the decrease case sigma. Used typically in regression evaluation and statistics to indicate normal deviation; when squared, it’s the variance of the topic in query.
- (Phi) – the capital letter phi. Normally used to indicate a likelihood distribution in statistics, mostly the cumulative distribution perform (cdf) of the traditional distribution.
- (phi) – the decrease case phi. Typically used to indicate the likelihood distribution perform (pdf) of the traditional distribution in statistics. However, this letter is ceaselessly used outdoors of this context too, not like the capital model described above.
- (chi) – the letter chi. Utilized in statistics to indicate the chi-squared distribution, which is a likelihood distribution that’s used quite a bit in regression evaluation.
- (Omega) – the capital letter omega. Used typically in econometrics to be the image for a matrix.

The next letters normally don’t carry particular that means on their very own, however are sometimes utilized in economics regardless. They’re generally used as placeholders for a variable when discussing a principle 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 decrease case letter psi.
- (omega) – the decrease case letter omega.

*Different statistics symbols*

The next symbols are sometimes utilized in financial statistics contexts. Notice that the Greek letters above comprise many symbols utilized in these contexts as properly.

- | means “on condition that” in a statistics sense, i.e.,
*X*|*Y*= 2 means “*X*given the truth that*Y*equals 2”. - (mathbb{E}) means “the expectation of”. Usually there’s a outlined variable, matrix, and so on. positioned in brackets subsequent to it (for instance,
*X*); then, this implies the expectation, or anticipated worth, of*X*. This image is used extraordinarily typically in statistics. - (mathbb{V}) means “the variance of”. It’s utilized in the identical means because the expectation image above, so when one thing is positioned in brackets subsequent to this, it means the variance of that variable, matrix, and so on. To not be confused with (sigma)
^{2}, which stands for a particular quantity. - (sigma)
^{2}is the image for the variance of a variable, matrix, and so on. This stands for the precise quantity that’s the variance of one thing. - (overline{X}): the bar above a variable denotes the typical. This image is used fairly often in financial formulae, for instance when describing the sum of least squares.

*Set principle symbols*

Set principle is used to explain how teams of issues relate to at least one one other. They’re just like a Venn diagram, however within the language of math.

- (subset) – this image exhibits that the variable or group on the left facet of it’s a subset of (included in) the group on the suitable facet of it. With a slash by way of it, it means the alternative.
- (subseteq) – the identical because the above image, besides that the subset it describes can be equal to the variable or group on the suitable facet. It doesn’t need to be utterly contained throughout the second group.
- (supset) – this image denotes a superset, which is the alternative of a subset. It signifies that the group on the left facet of this image incorporates the group on the suitable facet.
- (cap) – that is the image for the intersection of two units, or the issues they each comprise which might be the identical.
- (cup) – that is the image for the union of two units, or all the pieces that’s contained in at the least one of many units.
- (emptyset) – that is the “empty set” image, which is solely a set that incorporates nothing.

*Units of numbers*

The next symbols are nonetheless technically units as they describe teams of numbers, like “rational numbers” or “pure numbers”. These are used fairly often in financial formulae even when different set symbols won’t seem.

- (mathbb{N}) is the image for pure numbers.
- (mathbb{Z}) is the image for the set together with all integers.
- (mathbb{Q}) is the image for the set of all rational numbers.
- (mathbb{R}) is the image for the set of all actual numbers.

*Different helpful symbols*

- (exists) means “there exists at the least one”. It’s generally seen in proofs, which are usually utilized in econometrics- or statistics-heavy programs.
- (exists!) is a variation on the image above meaning “there exists one and just one”.
- (forall) means “for all”, normally within the context of stating some fact in an econometric context, i.e., stating {that a} explicit equation is true for all i in some set of noticed values N.
- (implies) means “implies”; the perform, formulation, or variable on the left facet implies some relationship written on the suitable facet.
- (iff) means “if and provided that”.

## Studying financial formulae

Now that the generally used symbols have been launched, we are able to use them to learn and interpret an financial formulation. The beneath instance is the Goldsmith equation, utilized in macroeconomics to explain the expansion of capital inventory over time:

start{equation*}

dot{K_t} = mathit{I_t}-deltamathit{K_t} = mathit{s}F(mathit{K_t},,mathit{A_t},mathit{L_t}) – deltamathit{K_t}

finish{equation*}

The primary variable on this formulation stands for the spinoff of capital with respect to time, signified by the dot above the *Ok _{t}*. Recall from the checklist of math symbols above that the dot means the derivative of one thing with respect to time.

That is equal to the funding (which right here is outlined as *I _{t}*) at a time limit minus the depreciation instances the capital inventory at a time limit. Recall that the image (delta) from above is commonly used to indicate depreciation.

Then, on the far right-hand facet, It’s damaged down additional. It’s multiplied by *s*, which stands for the financial savings price. That is multiplied by a perform of capital, labor, and technological progress. The perform’s actual type will not be outlined for us but, so it’s merely written as *F*(*Ok _{t}*,

*A*,

_{t}*L*), which is frequent apply in math.

_{t}start{equation*}

mathit{YED}_{A} = frac{Deltamathit{q}_{A}}{Deltamathit{Y}} = frac{frac{partial{q}_{A}}{mathit{q}_{A}}}{frac{partial{Y}}{Y}}

finish{equation*}

This equation, in the meantime, describes the income elasticity of demand with respect to good A. It tells us precisely what this elasticity is and tips on how to calculate it. The revenue elasticity of demand *YED _{A}* is the same as the change within the amount demanded divided by the change in revenue.

start{equation*}

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

finish{equation*}

This formulation is a foundational theorem in econometrics, referred to as the Regulation of Iterated Expectations. Recall that the (mathbb{E}) image means “expectation”. This formulation states the next in mathematical shorthand.

Our expectation of *X* on the whole is the same as our expectation of the worth of *X* that’s anticipated given some worth that the variable *Y* may tackle. In different phrases, we now have some common thought of what *X* must be over the entire vary of attainable *Y* values (which haven’t been outlined but). This expectation is identical as what we typically count on *X* to be.

In additional exact statistical language, this formulation signifies that the anticipated worth of *X* is the same as the expectation of the conditional expectation of *X* given *Y*.

It took many phrases to explain what a easy formulation may convey with only some symbols. Clearly, utilizing these symbols may help economists talk concepts far more successfully than by simply utilizing phrases!