# What are the mathematical constants used in economy formulas?

For the calculation of the taxes, the rate is the coefficient of proportionality. This coefficient is constant because

It is the value of the ratio between a quantity and another that is constantly repeated in a set of values. It is a constant value which is in direct proportional functions or reverse, quadratically proportional or inverse

Other examples of this type in macroeconomic, finance, commercial.. ?

P.S - I refer both type of constants: specific fixed numeric value and as constants in economic theory

• That's too broad to be answerable, and it also needs clarifications: are you referring to constants that have a specific fixed numeric value like in physics? Or to economic concepts that, in most cases or always, appear as constants in economic theory? Finally, the last sentence of the question is a totally different question than the one asked by the title. – Alecos Papadopoulos Nov 26 '16 at 13:54
• I don't think the coefficient of proportionality is a mathematical constant because there is more than one line. – Giskard Nov 26 '16 at 15:12
• I refer both type of constants: specific fixed numeric value and as constants in economic theory – user6791392 Nov 26 '16 at 16:28

## 2 Answers

Economics is not an exact science and so we don't have Physical Constants i.e. quantities with a universal fixed numerical value.

That said, there are many concepts in economics that predominantly appear as constants in economic models (although in principle they can vary between individuals or across time, and there exists research that explores what happen when they do vary).

Well-known examples include the depreciation rate of capital (usually $\delta$), population growth rate (usually $n$), the rate of pure-time preference (usually $\rho$).

In calibration studies (either purely simulated or combined with empirical data), such constants tend to be assigned widely used "benchmark values" (i.e. quasi- "fixed values") that come from empirical experience usually supported over some decades of data and many countries. For example when calibrated and not estimated we often see $\delta =0.05$, $n=0.01$, $\rho=0.02$.

On another front, Euler's $e$, the base of the natural logarithms, emerges naturally when calculating compound interest rate continuously (and not in discrete intervals). Assume we make a deposit $D$ at yearly interest rate $r$. If interest is calculated once per year, at the end of one year we will get $D(1+r)$. Assume now that we calculate interest monthly, add the interest to principal and then next month calculate interest again. After one year we will get

$$D\left (1+ \frac {r}{12}\right)^{12}$$

In general if we partition the year in $n$ periods we will have

$$D\left (1+ \frac {r}{n}\right)^{n}$$

and if we let $n$ go to infinity we get

$$\lim_{n\to \infty}\left (1+ \frac {r}{n}\right)^{n} = e^r$$

Finally, there are certain "stylized facts" , according to which various economic ratios appear to be "stable" (if not strictly "constant") over time, like the income share of production inputs, or the capital/output ratio. The seminal reference here is

Kaldor, Nicholas, “Capital Accumulation and Economic Growth,” in F.A. Lutz and D.C. Hague, eds., The Theory of Capital, St.Martins Press, 1961, pp. 177–222.

while see

Jones, C. I., & Romer, P. M. (2010). The new Kaldor facts: Ideas, institutions, population, and human capital. American Economic Journal: Macroeconomics, 2(1), 224-245.

for extensions and updates.

• There are some references, texts for this calibration studies about this concepts appears as constants in economic models ? by such examples you have obtained formulas that you used? – user6791392 Nov 28 '16 at 17:31
• @user6791392 Scattered I would say. For a seminal paper (Kaldor's "stylized facts") that validates the approach to treat certain economic magnitudes as constants, see Kaldor, Nicholas, “Capital Accumulation and Economic Growth,” in F.A. Lutz and D.C. Hague, eds., The Theory of Capital, St.Martins Press, 1961, pp. 177–222. – Alecos Papadopoulos Nov 28 '16 at 17:47

Any model that uses the normal distribution will, as you would expect, invoke the use of $e$ and $\pi$.

Here's a nice one: Black-Scholes-Merton Option Pricing Formula

The Golden ratio, $\phi$, also appears in Hörner, Mu, and Vieille's paper on Markovian Implementation. You need a strong background in microeconomic theory to read this, however.

• mmm.. interesting..you can have also more pdf, text or reference ? – user6791392 Nov 29 '16 at 0:46
• I don't have any easy references, since understanding Black-Scholes-Merton takes quite a lot of work. However, Shreve is the most readable text that I know of. – Theoretical Economist Nov 29 '16 at 3:40