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 must have some quantitative abilities. The extra snug they’re with math and econometrics, the higher for his or her analysis and profession alternatives.
However, economists usually are not mathematicians. So, when college students of economics take Bachelor’s or Grasp’s programs whereas learning for his or her economics diploma, they is likely to be offered with symbol-heavy lecture notes. With out understanding these symbols, college students will in all probability battle 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 type, comparable to a confidence interval.
(preceq) – this image, and others prefer it, are used usually in economics to explain choice relations. Preferences are the constructing blocks of the demand curve. This image is seldom used exterior of microeconomics programs that particularly concentrate on preferences.
(propto) – this image implies that the variable on the left facet is proportional to the variable on the proper facet.
(sim) – this image means “just like”.
(approx) – this image means “roughly”, which implies one thing is sort of, 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 implies that one variable has zero energy to elucidate 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 proper facet. It may be reversed to imply the alternative. For instance, usually in economics, a concept will describe which values of x are potential. If x can solely tackle actual quantity values, the outline would possibly 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 sometimes in econometrics.
(partial) means “partial”, and is used to indicate a partial spinoff.
IN is commonly used to indicate the identification matrix with dimension N; it’s used steadily in statistics and econometrics.
(cdot) this image is positioned above a variable to suggest the spinoff with respect to time of that variable. This image is often utilized in macroeconomics, however seldom exterior of that context.
Greek letters
These are sometimes used to indicate variables, however some carry particular that means due to how usually they’ve been utilized in sure contexts.
(alpha) – the letter alpha, usually used with Cobb-Douglas capabilities to indicate the exponents. Additionally used when discussing p-values in speculation testing; the “alpha” degree units the quantity that you just evaluate the speculation take a look at’s outcome to, so as to decide if the result’s statistically important (comparable to 0.05 for five% significance, which is commonly the default).
(beta) – the letter beta, additionally used as an exponent in Cobb-Douglas capabilities. However, it’s most famously used when describing regression formulae. In that context, this letter often has a subscript and is the shorthand method 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 often denotes the error time period of the regression. Many introductory econometrics lessons, and another course that showcases regression evaluation, will make frequent use of this image. Lots of the assumptions that permit 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 usually to indicate the change in a variable, or depreciation in macroeconomics. It’s usually confused with the image for partial derivatives, which seems to be very comparable!
(lambda) – the letter lambda. It’s used most frequently when fixing constrained optimization issues utilizing the Lagrangian methodology; on this case it stands for the shadow value of the price range constraint. This letter shall be seen fairly often by microeconomics college students.
(mu) – the letter mu. This letter is used usually in statistics and often denotes the imply, or common, of one thing.
(rho) – the letter rho. Usually used to point the correlation coefficient between two variables.
(Sigma) – the capital letter sigma. Used steadily to indicate the sum of issues.
(sigma) – the decrease case sigma. Used usually 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. Often used to indicate a likelihood distribution in statistics, mostly the cumulative distribution perform (cdf) of the traditional distribution.
(phi) – the decrease case phi. Usually used to indicate the likelihood distribution perform (pdf) of the traditional distribution in statistics. However, this letter is steadily used exterior 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 so much in regression evaluation.
(Omega) – the capital letter omega. Used usually in econometrics to be the image for a matrix.
The next letters often 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 concept 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. Word that the Greek letters above include many symbols utilized in these contexts as effectively.
| 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 many others. positioned in brackets subsequent to it (for instance, X); then, this implies the expectation, or anticipated worth, of X. This image is used extraordinarily usually in statistics.
(mathbb{V}) means “the variance of”. It’s utilized in the identical method 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 many others. To not be confused with (sigma)2, which stands for a selected quantity.
(sigma)2 is the image for the variance of a variable, matrix, and many others. 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 concept symbols
Set concept is used to explain how teams of issues relate to 1 one other. They’re just like a Venn diagram, however within the language of math.
(subset) – this image reveals that the variable or group on the left facet of it’s a subset of (included in) the group on the proper facet of it. With a slash via 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 proper facet. It doesn’t need to be fully contained throughout the second group.
(supset) – this image denotes a superset, which is the alternative of a subset. It implies that the group on the left facet of this image comprises the group on the proper facet.
(cap) – that is the image for the intersection of two units, or the issues they each include which can be the identical.
(cup) – that is the image for the union of two units, or all the things that’s contained in at the least one of many units.
(emptyset) – that is the “empty set” image, which is just a set that comprises 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 may not 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 typically 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”, often 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, method, or variable on the left facet implies some relationship written on the proper 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 method. The under 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 method stands for the spinoff of capital with respect to time, signified by the dot above the Kt. Recall from the checklist of math symbols above that the dot means the spinoff of one thing with respect to time.
That is equal to the funding (which right here is outlined as It) 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 charge. That is multiplied by a perform of capital, labor, and technological progress. The perform’s precise type isn’t outlined for us but, so it’s merely written as F(Kt,At,Lt), which is frequent observe in math.
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 revenue elasticity of demand with respect to good A. It tells us precisely what this elasticity is and how one can calculate it. The revenue elasticity of demand YEDA 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 method is a foundational theorem in econometrics, generally known as the Regulation of Iterated Expectations. Recall that the (mathbb{E}) image means “expectation”. This method states the next in mathematical shorthand.
Our expectation of X usually is the same as our expectation of the worth of X that’s anticipated given some worth that the variable Y might tackle. In different phrases, now we have some common thought of what X needs to be over the entire vary of potential Y values (which haven’t been outlined but). This expectation is similar as what we typically count on X to be.
In additional exact statistical language, this method implies 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 method might convey with only some symbols. Clearly, utilizing these symbols might help economists talk concepts far more successfully than by simply utilizing phrases!
