<p>As has already been said, the MOST fundamental equation is surely:
$$\text{MB}=\text{MC}$$</p>

<p>EDIT: This equation is fundamental in terms of the way economists think. As pointed out in the comments below, in terms of fundamental equations of economic models, the most fundamental equations describe equivalences between the uses and supplies of items (money, goods, etc.). These provide the tension of the marginal cost side of this equation.</p>

<p>I would add equations related to comparative statics:</p>

<ul>
<li><strong><a href="http://en.wikipedia.org/wiki/Envelope_theorem">Envelope theorem</a></strong>
$$V^\prime(y)=f_y(x,y)$$</li>
<li><strong>"Delta" analysis</strong>, as described in Samuelson's Foundations of Economic Analysis:
$$\Delta p\Delta y-\Delta w\Delta x\geq0$$
(this examines responses of price-taking producers in terms of vectors of production $y$ and uses of inputs $x$, to their prices $p$ and $w$, essentially revealed preference for producers)</li>
<li><strong><a href="http://en.wikipedia.org/wiki/Revealed_preference">Revealed preference</a></strong></li>
</ul>

<p>If we can claim game theorists or mathematicians whose equations we use constantly:</p>

<ul>
<li><strong><a href="http://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions">Karush-Kuhn-Tucker conditions</a></strong>, especially complementary slackness. There's no single equation for linear programming, but I think econ has a claim to Kantorovich too. \
Stationarity:
$$\nabla f(x^*) = \sum_{i=1}^m \mu_i \nabla g_i(x^*) + \sum_{j=1}^l \lambda_j \nabla h_j(x^*)$$
Primal feasibility:
$$g_i(x^*) \le 0, \mbox{ for all } i = 1, \ldots, m$$
$$h_j(x^*) = 0, \mbox{ for all } j = 1, \ldots, l \,\!$$
Dual feasibility:
$$\mu_i \ge 0, \mbox{ for all } i = 1, \ldots, m$$
Complementary slackness:
$$\mu_i g_i (x^*) = 0, \mbox{for all}\; i = 1,\ldots,m.$$</li>
<li><strong><a href="http://en.wikipedia.org/wiki/Nash_equilibrium">Nash equilibrium</a></strong>
$$\theta_{i}^\star = \arg \max_{\theta_i} u_i(\theta_i ,\theta_{-i}^\star)$$</li>
<li><strong><a href="http://en.wikipedia.org/wiki/Revelation_principle">Revelation principle</a></strong>: which to be fair isn't so much an equation as a theorem...</li>
<li><strong><a href="http://en.wikipedia.org/wiki/Bellman_equation">Bellman equation</a></strong>
$$V(x)=\max_{c\in\Omega(x)} U(x,z)+\beta\left[V(x^\prime)\right]$$</li>
</ul>

CC BY-SA 3.0
2014-12-29T23:59:46.853Z
I would suggest that there are some inequalities which are even more fundamental than the first equation above. Unlike equations which represent approximations, some of the inequalities represent absolute.  For example, the total quantity of something people will be able to afford cannot exceed the total quantity that will exist.  If the number of people who would like to have something exceeds the quantity that exists, unless more of the thing are produced or some people stop wanting it, not everyone who wants one will get one, period, no matter what else is done.
supercat
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2014-12-30T00:01:48.480Z
That's fair. I suppose budget constraints are also "more fundamental" in that sense.
jayk
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2014-12-30T00:14:17.820Z
If someone proposes a policy which, if successful, would violate one of the normal equations associated with economics, such a person should be called upon to justify the expectation that the equation would not hold in that case, but since most equations don't hold 100% of the time it would be plausible that the policy might work despite the equation suggesting otherwise.  On the other hand, a policy which couldn't achieve its stated aims without violating some fundamental inequalities cannot reasonably be expected to achieve those aims; no wise person could plausibly expect otherwise.
supercat
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2014-12-30T00:22:04.927Z
Does my edit above get at what you're trying to express? I see this as a difference in framing of the term "fundamental". You seem to mean that physical constraints are the most fundamental element of any given *economic model*, with which I agree. But I see $\text{MB}=\text{MC}$ as the most fundamental element in an *economists toolkit* because it combines these constraints with notions of efficient use. I'm especially fond of it because it is a general equation, whereas physical constraints tend to be stated differently for different situations.
jayk
CC BY-SA 3.0
2015-01-01T17:55:56.237Z
If one imagines the state of an economic system as being a marble rolling over a hilly surface, the equations define grooves in which the marble will tend to roll, but the limiting inequalities define boundaries.  Merely knowing the boundaries in which the marble is constrained without knowing how it will behave within them isn't very useful, but likewise a prediction of the marble's behavior which ignores the existence of a boundary between its present position and expected future position is apt to be very wrong.  In a sense, though, I think the constraints are somewhat more foundational...
supercat
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2015-01-01T18:02:48.320Z
...since in their absence the natural "solution" would simply be to have everyone get everything they could possibly desire.  It is only the existence of resource limits that makes all the other compromises necessary.  Also, BTW, a concept I've not seen mentioned much is that it's *much* easier to turn \$1 into \$0.95 than \$1.05; if it seems like everyone in some particular field is turning \$1 into \$1.05 with little effort, odds are what's really happening is they're turning \$1 into \$0.95 (or less) but making it *look* like \$1.05.  The fact that the market price of something is \$1.05...
supercat
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2015-01-01T18:05:06.717Z
...doesn't mean that it will ever be possible to convert a million units of that commodity into \$1,050,000 or even \$1,000,000.  If people too-easily convert \$1,000,000 into 1,000,000 units of something, odds are good that the total liquidation value will end up being noticeably less than \$1,000,000.
supercat
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2014-11-19T05:22:36.393Z
2014-12-30T00:38:39.513Z
2014-12-30T00:38:39.513Z
jayk
<p>Most of intro econ is intersecting lines. Specifically,</p>
<p><span class="math-container">$$MB = MC$$</span>
<a href="https://ctaar.rutgers.edu/gag/NOTES/micnotes5.html" rel="nofollow noreferrer">* Equilibrium is achieved when Marginal Benefit is equal to Marginal Cost*</a></p>
<p><span class="math-container">$$\dfrac{MU_x}{p_x}=\dfrac{MU_y}{p_y}.$$</span>
<a href="http://dev.business.und.edu/Goenner/teaching/econ201/Resources/Book%20supplemental/07sg.pdf" rel="nofollow noreferrer"><em>Marginal Utility per unit cost should always be equal</em></a></p>
<p>Economics is about the logic of human behavior, how we make decisions in a world of scarcity. These equations describe constrained optimization under some usual assumptions like continuity, convex preferences, and no corner solutions. I'd also give prominence to consumer theory over producer. Most of undergrad producer theory can be understood with the same tools used in consumer theory.</p>

CC BY-SA 3.0
2014-11-19T11:51:33.927Z
It makes sense to assume that consumers are rational optimizers. That is a toothless statement (complete and transitive preferences). It's just much harder to know what a human's objective is. I think of producer theory as often being a special kind of consumer. They are risk neutral consumers who get utility from dollars.
Pburg
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2014-11-19T02:14:55.490Z
2022-03-19T05:25:09.650Z
2022-03-19T05:25:09.650Z
Pburg
<p>I once heard Roger Myerson talk about why he thought Economics has, as a Social Science, been so successful in applying (or has so readily incorporated) mathematics. He suggested that perhaps it was due to some of the fundamental linearities within the world. Two examples would be the flow-balance constraints of scarce goods (commodity constraints) and
no-arbitrage conditions. These are fundamentally linear constraints.</p>
<ul>
<li><p>It's important to emphasize the importance of these because we can get a surprising amount out of the two. For example, a lot of people think that the law of demand is a consequence of assuming rationality (specifically, preferences that exhibits a diminishing marginal rate of substitution). A result due to Gary Becker shows that the law of demand (albeit just a slightly weaker version) can be derived <em>from the budget constraint alone</em>. (See Becker 1962, &quot;<a href="https://www.journals.uchicago.edu/doi/pdf/10.1086/258584" rel="nofollow noreferrer">Irrational Behavior and Economic Theory</a>.&quot;) That is, this fundamental economic result can be derived from the reality of scarce resources alone---without assuming rationality.</p>
</li>
<li><p>The no-arbitrage condition is an application of the linear duality theorem (<a href="http://en.wikipedia.org/wiki/Farkas%27_lemma" rel="nofollow noreferrer">Farkas' lemma</a>). A lot of economics and finance (asset pricing) can be done just by the assumption that in economic equilibrium there is no arbitrage.</p>
</li>
</ul>
<p>Extra Notes:</p>
<p>Gary Becker made a lot of advances in the field by studying the way constraints affect human behavior. One famous quote, taken from his Nobel prize lecture, is the remark that &quot;different constraints are decisive for different situations, but the most fundamental constraint is limited time.&quot; (Some discussion <a href="http://conversableeconomist.blogspot.com/2014/05/gary-becker-and-time-constraint.html" rel="nofollow noreferrer">here</a>.) Some more resources about how his work in this regard can be found <a href="https://cehd.uchicago.edu/sites/heckman2013.uchicago.edu/files/uploads/Becker-slides_2014-10-31.pdf" rel="nofollow noreferrer">here</a> and <a href="http://bfi.uchicago.edu/video/impact-work-gary-s-becker" rel="nofollow noreferrer">here</a>.</p>
<p>Linear duality can be used to describe the no arbitrage condition. More generally, this theorem is typically proved with the <a href="http://en.wikipedia.org/wiki/Hyperplane_separation_theorem" rel="nofollow noreferrer">Hyperplane Separation Theorem</a>, which is mathematical tool that shows up a lot in economics textbooks.</p>
<p>Also, keep in mind that it's enough just to assume that in economic equilibrium, there
is approximately no arbitrage.</p>

CC BY-SA 4.0
2014-11-19T02:55:33.330Z
2022-03-19T19:11:59.100Z
2022-03-19T19:11:59.100Z
jmbejara
<p>I think one of the most important equations (at least within macroeconomics) is:</p>

<p>$$E\left[ m R \right] = 1$$</p>

<p>This equation has been used to derive many foundational results.  This equation motivated the <a href="http://en.wikipedia.org/wiki/Hansen%E2%80%93Jagannathan_bound">Hansen–Jagannathan bound</a>.  It is fundamental for asset pricing as well.</p>

<p>Also, something interesting I saw from once from Tom Sargent.  If you use the stochastic discount factor for a standard model $m = \beta E_t \left[ \frac{u'(c_{t+1})}{u'(c_t)} \right]$ then depending on which piece of the equation you allow to be exogenous you can get some fundamental results of macro:</p>

<ul>
<li>Permanent Income Hypothesis:  Let $\beta R = 1$ then we get $c_t = E [c_{t+1}]$</li>
<li>Lucas Asset Pricing Model:  Let the process for consumption be a given.  Then the price of an asset can be described by $R_t^{-1} = p_t = E \left[ \frac{u'(c_{t+1})}{u'(c_t)} \right]$</li>
</ul>

CC BY-SA 3.0
2014-11-22T21:55:29.013Z
2014-11-22T22:17:58.113Z
2014-11-22T22:17:58.113Z
cc7768
<p>Although I agree with Jyotirmoy Bhattacharya that the most interesting ideas in economics are not always best expressed through equations, I still want to mention the Slutsky or compensated law of demand from consumer theory</p>

<p>$$ (p' -p) \Big[ x\big(p', p' x(p,w)\big) – x\big(p,w\big)\Big]^T \leq 0,$$</p>

<p>where $p',p \in \mathbb{R}_{++}^n$ are any two price vectors, $w \in \mathbb{R}_+$ is any level of income, and $x(\cdot,\cdot) \in \mathbb{R}^n$ is the demand function. </p>

<p>The underlying relation is a couple of orders of certitude away from fundamental equations in other fields. Also, it does not ground the discipline, in the sense that it is not used all that often. </p>

<p>However, I tend to view it as fundamental because</p>

<ul>
<li>It is an <em>absolutely non-trivial</em> consequence of three simple and fundamental assumptions in consumer theory, namely, 

<ul>
<li>That the demand function $x(\cdot,\cdot)$ is homogenous of degree zero (no money illusion)</li>
<li>Walras' law (people do not burn money)</li>
<li>The weak axiom of revealed preferences (if you chose A when B is available “today”, you will not chose B “tomorrow” if A  remains available) </li>
</ul></li>
<li>Therefore testing the inequality is equivalent to testing these three assumptions jointly. </li>
<li>The three assumptions are used in the vast majority (maybe more than 90%?) of the models including consumers in economic theory. </li>
<li>Their validity (at least as approximations) is therefore crucial to the validity of most models in economic theory (at least as approximations).</li>
<li>Although it is not always obvious how to relate the notions of prices, goods and income  to observables, all the element in the equation are observable <em>in principle</em> (as opposed to utility levels for instance) and the validity of the inequality can therefore be tested empirically.</li>
</ul>

CC BY-SA 3.0
2014-12-30T00:59:55.573Z
I'd add that it gets even better: there are _three_ laws of demand, which are equivalent (and boil down to Slutsky negative semidefiniteness) in the infinitesimal case but are distinct in general. After the price change from $p$ to $p'$, you can either (1) adjust wealth such that it can purchase the old bundle, (2) adjust wealth such that utility is constant, or (3) adjust wealth such that the newly chosen bundle could have been purchased yesterday - in all cases you get a law of demand. (These are arguably the laws of overcompensated, compensated, and undercompensated demand, respectively.)
nominally rigid
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2014-11-19T05:18:00.853Z
2017-03-22T00:40:17.043Z
2017-03-22T00:40:17.043Z
Martin Van der Linden
<p>Whilst not as foundational as, for example, the Slutsky equation, the condition on the Lerner index that a profit maximising firm with price $p$, cost $c$, and price elasticity of demand $\eta$ has
$$\frac{p-c}{p}=-\frac{1}{\eta}$$
is an important equation in industrial organisation.</p>

<p>This is not only an elegant formulation of the solution of the firm's problem, but it is also practically useful:</p>

<ul>
<li>A firm that estimates its $\eta$ and knows its $c$ can use this formula to calculate the profit-maximising price.</li>
<li>A regulator that observes a $p$ and estimates $\eta$ can use the formula to calculate $c$—important in many forms of regulation.</li>
</ul>

CC BY-SA 3.0
2014-11-19T10:58:02.683Z
2014-11-19T10:58:02.683Z
Ubiquitous
<p>For me, one of the most important ones is the budget constraint. It might seem too obvious but a lot of laypersons (though maybe not physicist) don't get it!</p>

<p>$p⋅x \leq w$</p>

CC BY-SA 3.0
2015-04-13T13:08:37.483Z
It's not that foundational if you remember about borrowing.
user829438
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2014-11-19T14:40:38.533Z
2014-11-19T14:40:38.533Z
s_a
<p>It is already written but Euler equation in continous time yields </p>

<p>$$\frac{\dot{C}}{C}=\sigma(r-\rho)$$</p>

<p>where $\sigma$ is intertemporal elasticity of substitution, $r$ interest rate and $\rho$ is the discount rate (impatience level).</p>

CC BY-SA 3.0
2015-04-24T12:54:17.997Z
2015-04-24T12:54:17.997Z
optimal control
<p>The foundation of intertemporal economics is the <a href="http://en.wikipedia.org/wiki/Net_present_value" rel="noreferrer">net present value equation</a>. That is, the net present value of a future income stream is the yearly incomes divided by an appropriate discount factor, based on the prevailing interest rate, r, taken to the nth power, where n is the number of years.</p>

CC BY-SA 3.0
2014-11-25T20:39:55.917Z
NPV as described in the linked Wikipedia article doesn't seem as general and central to economics as does $E[mR] = 1$, for example.
jmbejara
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2014-11-25T21:31:03.607Z
@jmbejara: It's the foundation of finance, as it relates to the value of bonds, the mortgage on you house, etc.
Tom Au
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2014-11-25T21:43:08.280Z
I know. What I mean to point out is that, if we think of $E[mR] = 1$ more generally (e.g., drop the equilibrium interpretation), it can encompass NPV as you described it. But it can also do much more. If you write it as $E[m X] = P$ and you treat $X$ as a stream of future cash flows and $m$ as the appropriate discount factor, you can recover your definition of NPV.
jmbejara
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2014-11-25T21:47:01.070Z
http://en.wikipedia.org/wiki/Stochastic_discount_factor
jmbejara
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2014-11-22T19:01:50.807Z
2014-11-22T19:01:50.807Z
Tom Au
<p>I don't think there are any economics equations with the same status as, say, Maxwell's equations in physics. In its place we have concepts like the equimarginal principle, competitive equilibrium or Nash equilibrium which are at the core of the "economist's approach". But I think that the real worth of economics is not even in these ideas themselves but in what we know about concrete problems in specific areas of applications: for example what we know about business cycles in macro. In this economics may be more like medicine than physics.</p>

CC BY-SA 3.0
2014-12-30T03:11:50.233Z
The acknowledgment that the totality of activities has a scale limit is understandably slow because economic developments are evaluated in the conceptual and quantitative
terms of a system that repudiates the existence of such limits; tough, Maxwell can marginally be introduced to the core of the "economist's approach": [Entropy, limits to growth, and the prospects for weak sustainability](http://www.embl-hamburg.de/aboutus/science_society/discussion/discussion_2006/ref1-13July06.pdf) and the axiomatic base: [Ten Proofs of the Generalized Second Law](http://arxiv.org/pdf/0901.3865v2.pdf)
ikaerom
CC BY-SA 3.0
2014-11-19T02:06:56.440Z
2014-11-19T02:06:56.440Z
Jyotirmoy Bhattacharya
<p>A bit late to the game, but I'm surprised no one has named the equation to calculate OLS estimates:
$$
\hat\beta=(X'X)^{-1}X'y
$$</p>

CC BY-SA 4.0
2022-03-20T14:50:57.907Z
I don't know if that can be called economics. Or if it can, then what about other results from statistics and mathematics that are widely used in economics?
Richard Hardy
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2017-03-11T01:41:37.537Z
2017-03-11T01:41:37.537Z
Herr K.
<p>Well for microeconomics there are several, however they all follow the same pattern.  </p>

<blockquote>
  <p><em><strong>Here I'll attempt to teach an entire intermediate microeconomics course in one post.</strong></em></p>
</blockquote>

<h2>Most microeconomics problems follow this format:</h2>

<p>Though leaving out some minor details, if you do enough microeconomics practice sets the problems end up looking the same after a while. This is what I got to share.</p>

<h2>Production/Utility functions</h2>

<p>There are three main types of utility/production functions you will be exposed to in an intermediate microeconomics course<sup>1</sup>. They are:</p>

<ol>
<li>Cobb Douglas<br>
$$f(x_1,x_2)=x_1^ax_2^b$$</li>
<li>Leontif/ Perfect Complement<br>
$$f(x_1,x_2)=\min\{x_1,x_2\}$$</li>
<li>Perfect substitutes
$$f(x_1,x_2)=x_1+x_2$$</li>
</ol>

<h2>Budget lines and Cost functions</h2>

<p>In consumer theory, you have a budget line represented by the formula:</p>

<p>$$m=p_1x_1+p_2x_2$$</p>

<p>In producer theory we call it a cost function.
$$C(x_1,x_2)=w_1x_1+w_2x_2$$</p>

<p>we either want to maximize consumption given a budget/cost function or minimize costs holding your utility/output level constant. To do this we use another equation:</p>

<h2>The Lagrangian Multiplier:</h2>

<p>Though not exclusive to economics tool per say, its the primary tool of all intermediate microeconomics students.</p>

<p>$$\mathcal{L}=f(x_1,x_2)\pm\lambda(H-g(x_1,x_2))$$</p>

<p>where $H-g(x_1,x_2)$ is either a budget line/cost function or Utility/Production function when its equal to zero. </p>

<p>We use this for calculating utility/profit maximizing consumption bundles/inputs or Minimize Costs holding profit/utility constant.</p>

<p>And thats a wrap!* </p>

<hr>

<p><sub>*Though there is what to say on marshallian and hicksian demands I'll leave that for others to fill in.</sub></p>

CC BY-SA 3.0
2017-11-10T05:14:51.340Z
2017-11-10T16:55:52.090Z
2017-11-10T16:55:52.090Z
EconJohn
<p>Instead of proposing specific equations, I will point to two concepts that <em>lead</em> to specific equations for specific theoretical set ups:</p>

<p><strong>A) Equilibrium</strong><br>
The most fundamental and the most misunderstood concept in Economics. People look around and see constant movement -how more irrelevant can a concept be, than "equilibrium"? So the job here is to convey that Economics models the observation that things most of the time <em>tend</em> to "settle down" -so by characterizing this "fixed point", it gives us an anchor to understand the movements outside and around this equilibrium (which may be changing of course).</p>

<p>It is <em>not</em> the case that "<strong>quantity supplied equals quantity demanded</strong>" (here is a foundational equation)  </p>

<p>$$Q_d = Q_s$$</p>

<p>but it <em>is</em> the case that supply <em>tends to equal</em> demand (of <em>anything</em>) for reasons that any economist should be able to convincingly present to anyone interested in listening (and deep down they all have to do with finite resources).</p>

<p>Also, by determining the conditions for equilibrium, we can understand, when we observe divergence, which conditions were violated. </p>

<p><strong>B) Marginal optimization under constraints</strong><br>
In a <strong>static environment</strong>, it leads to the equation of marginal quantities/first derivatives of functions.<br>
Goods market: <strong>marginal revenue equals marginal cost</strong>.<br>
Inputs market: <strong>marginal revenue product equals marginal reward</strong> (rent, wage).<br>
Etc. (I left "utility maximization" out of the picture on purpose, because, here first one would have to present what this "utility index" is all about, and how crazy we are (<em>not</em>), by trying to model human "enjoyment" through the concept of utility).  </p>

<p>Perhaps you could cover it all under the umbrella "marginal benefit equal marginal cost" as other questions suggested:</p>

<p>$$MB = MC$$</p>

<p>Economists live in marginal optimization and most consider it self-evident. But if you try to explain it to an outsider, there is a respectable probability that he will object or remain unconvinced, instead usually proposing "average optimization" as "more realistic", since "people do not calculate derivatives" (we don't argue that they do, only that their thought processes can be modeled <em>as if</em> they were). So one has to get his story straight about marginal optimization, with convincing examples, and a discussion about "why not average optimization".</p>

<p>In an <strong>intertemporal setting</strong>, it leads to the discounted trade-off between "the present and the future", again "at the margin" -starting with the <strong>"Euler equation in consumption"</strong>, which in its discrete deterministic version reads</p>

<p>$$u'(c_{t})=\beta(1+r_{t+1})u'(c_{t+1})$$</p>

<p>...and one cannot avoid the theme of utility, after all: $u'()$ is marginal utility from consumption, $0&lt;\beta&lt;1$ is a discount rate and $r_{t+1}$ is the interest rate</p>

<p>(<em>don't</em> consult wikipedia article on Euler's equation in consumption, the concept behind it is much more generally applicable and foundational than the specific application that the wikipedia article discusses).  </p>

<p>Interestingly, although dynamic economics are more technically demanding, I find this more intuitively appealing since people seem to understand way better "what you save today will determine what you will consume tomorrow", than "your wage rate will be the marginal revenue product of all labor employed".  </p>

CC BY-SA 3.0
2015-05-28T06:43:29.317Z
-1 "for reasons that any economist should be able to convincingly present to anyone interested in listening" Unless of course those economists who actually tried to explain how this dynamics are supposed to work. See for example [this sobering survey](http://economics.mit.edu/files/6988) by Franklin M. Fisher, arguably the leading authority on the subject.
Michael Greinecker
CC BY-SA 3.0
2015-05-28T11:00:49.433Z
@MichaelGreinecker I am one of "those economists" and I never have trouble explaining it. Thanks for the link by the way, although this link refers to the "competitive general equilibrium" concept - which is a Platonian ideal that has not much  to do with the concept of "equilibrium" as I understand it... CONTD
Alecos Papadopoulos
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2015-05-28T11:02:53.217Z
@MichaelGreinecker CONTD...-and I understand it as a tendency and not as a situation in which we usually find ourselves. Because if we found ourselves in equilibrium, things wouldn't move -which is the opposite of what we observe... And this is exactly the distinction  I made in my answer. As an aphorism, the world tries to be Walrasian and in attempting so , it ends us being Schumpeterian. And then it tries again to become Walrasian... ad infinitum
Alecos Papadopoulos
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2015-05-28T14:21:11.073Z
That is exactly the problem with partial equilibrium reasoning. I am of course familiar with the stories one tells econ 101 students about excess demand leads to higher prices and excess supply to lower prices, so that "markets tend to equilibrium". What the story conveniently hides is that in the process, other markets might be upset. And of course Walrasian equilibrium theory is highly idealized- but partial equilibrium models are even more so.
Michael Greinecker
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2014-11-19T04:28:54.327Z
2016-01-23T03:35:55.833Z
2016-01-23T03:35:55.833Z
Alecos Papadopoulos
<p>For the other sciences it´s easy to point to the most important equations that ground the discipline. If I want to explain Economics to a physicist say, what are considered to be the most important equations that underly the subject which I should introduce and attempt to explain?</p>

CC BY-SA 3.0
2016-01-24T02:39:21.653Z
[What are the foundational equations, inequalities, propositions or concepts of Finance?](http://economics.stackexchange.com/questions/10361)
BCLC
CC BY-SA 3.0
2014-11-20T14:47:29.163Z
I beg to differ. I think this is an important question for people who would like to get an overview of a field, which can certainly be answered in all the other sciences - and indeed several excellent answers have been posted below. It could be broken up into macro/micro etc., but I think that would miss the point.
Lumi
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2014-11-20T15:49:31.977Z
I find this question broad but nevertheless interesting and worth discussing. Proof of that are the very interesting answers.
user157623
CC BY-SA 3.0
2014-11-20T17:53:21.073Z
I disagree with the "on hold" decision. By characterizing this question as "too broad", we essentially state that the "foundational equations" of Economics are too many and too diverse. Are they really?
Alecos Papadopoulos
CC BY-SA 3.0
2014-11-21T02:03:36.023Z
@MartinVanderLinden This is a very good question.  But, I would suggest making in more narrow.  What part of economics are these equations coming from? Interest rates? GDP? 
Even topics such as "finance" and "international economics" are very broad.
Mathematician
<macroeconomics><microeconomics><mathematical-economics><soft-question>
Fundamental equations in economics
Lumi
# Fundamental equations in economics

(Asked by: username_0 on 2014-11-19)

For the other sciences it´s easy to point to the most important equations that ground the discipline. If I want to explain Economics to a physicist say, what are considered to be the most important equations that underly the subject which I should introduce and attempt to explain?

**Tags:** macroeconomics, microeconomics, mathematical-economics, soft-question
> **Comments:**
> **username_0**: I beg to differ. I think this is an important question for people who would like to get an overview of a field, which can certainly be answered in all the other sciences - and indeed several excellent answers have been posted below. It could be broken up into macro/micro etc., but I think that would miss the point.  
> **username_2**: I find this question broad but nevertheless interesting and worth discussing. Proof of that are the very interesting answers.  
> **username_3**: I disagree with the "on hold" decision. By characterizing this question as "too broad", we essentially state that the "foundational equations" of Economics are too many and too diverse. Are they really?  
> **username_4**: @MartinVanderLinden This is a very good question.  But, I would suggest making in more narrow.  What part of economics are these equations coming from? Interest rates? GDP?  Even topics such as "finance" and "international economics" are very broad.  
> **username_1**: [What are the foundational equations, inequalities, propositions or concepts of Finance?](http://economics.stackexchange.com/questions/10361)  

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## 13 answers

### Answer by username_3 on 2014-11-19. Score: 55 | Accepted answer

Instead of proposing specific equations, I will point to two concepts that *lead* to specific equations for specific theoretical set ups:

**A) Equilibrium**  
The most fundamental and the most misunderstood concept in Economics. People look around and see constant movement -how more irrelevant can a concept be, than "equilibrium"? So the job here is to convey that Economics models the observation that things most of the time *tend* to "settle down" -so by characterizing this "fixed point", it gives us an anchor to understand the movements outside and around this equilibrium (which may be changing of course).

It is *not* the case that "**quantity supplied equals quantity demanded**" (here is a foundational equation)

$$Q\_d = Q\_s$$

but it *is* the case that supply *tends to equal* demand (of *anything*) for reasons that any economist should be able to convincingly present to anyone interested in listening (and deep down they all have to do with finite resources).

Also, by determining the conditions for equilibrium, we can understand, when we observe divergence, which conditions were violated.

**B) Marginal optimization under constraints**  
In a **static environment**, it leads to the equation of marginal quantities/first derivatives of functions.  
Goods market: **marginal revenue equals marginal cost**.  
Inputs market: **marginal revenue product equals marginal reward** (rent, wage).  
Etc. (I left "utility maximization" out of the picture on purpose, because, here first one would have to present what this "utility index" is all about, and how crazy we are (*not*), by trying to model human "enjoyment" through the concept of utility).

Perhaps you could cover it all under the umbrella "marginal benefit equal marginal cost" as other questions suggested:

$$MB = MC$$

Economists live in marginal optimization and most consider it self-evident. But if you try to explain it to an outsider, there is a respectable probability that he will object or remain unconvinced, instead usually proposing "average optimization" as "more realistic", since "people do not calculate derivatives" (we don't argue that they do, only that their thought processes can be modeled *as if* they were). So one has to get his story straight about marginal optimization, with convincing examples, and a discussion about "why not average optimization".

In an **intertemporal setting**, it leads to the discounted trade-off between "the present and the future", again "at the margin" -starting with the **"Euler equation in consumption"**, which in its discrete deterministic version reads

$$u'(c\_{t})=\beta(1+r\_{t+1})u'(c\_{t+1})$$

...and one cannot avoid the theme of utility, after all: $u'()$ is marginal utility from consumption, $0<\beta<1$ is a discount rate and $r\_{t+1}$ is the interest rate

(*don't* consult wikipedia article on Euler's equation in consumption, the concept behind it is much more generally applicable and foundational than the specific application that the wikipedia article discusses).

Interestingly, although dynamic economics are more technically demanding, I find this more intuitively appealing since people seem to understand way better "what you save today will determine what you will consume tomorrow", than "your wage rate will be the marginal revenue product of all labor employed".

> **Comments:**
> **username_22**: -1 "for reasons that any economist should be able to convincingly present to anyone interested in listening" Unless of course those economists who actually tried to explain how this dynamics are supposed to work. See for example [this sobering survey](http://economics.mit.edu/files/6988) by Franklin M. Fisher, arguably the leading authority on the subject.  
> **username_3**: @MichaelGreinecker I am one of "those economists" and I never have trouble explaining it. Thanks for the link by the way, although this link refers to the "competitive general equilibrium" concept - which is a Platonian ideal that has not much  to do with the concept of "equilibrium" as I understand it... CONTD  
> **username_3**: @MichaelGreinecker CONTD...-and I understand it as a tendency and not as a situation in which we usually find ourselves. Because if we found ourselves in equilibrium, things wouldn't move -which is the opposite of what we observe... And this is exactly the distinction  I made in my answer. As an aphorism, the world tries to be Walrasian and in attempting so , it ends us being Schumpeterian. And then it tries again to become Walrasian... ad infinitum  
> **username_22**: That is exactly the problem with partial equilibrium reasoning. I am of course familiar with the stories one tells econ 101 students about excess demand leads to higher prices and excess supply to lower prices, so that "markets tend to equilibrium". What the story conveniently hides is that in the process, other markets might be upset. And of course Walrasian equilibrium theory is highly idealized- but partial equilibrium models are even more so.  

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### Answer by username_5 on 2014-11-19. Score: 36

As has already been said, the MOST fundamental equation is surely:
$$\text{MB}=\text{MC}$$

EDIT: This equation is fundamental in terms of the way economists think. As pointed out in the comments below, in terms of fundamental equations of economic models, the most fundamental equations describe equivalences between the uses and supplies of items (money, goods, etc.). These provide the tension of the marginal cost side of this equation.

I would add equations related to comparative statics:

* **[Envelope theorem](http://en.wikipedia.org/wiki/Envelope_theorem)**
  $$V^\prime(y)=f\_y(x,y)$$
* **"Delta" analysis**, as described in Samuelson's Foundations of Economic Analysis:
  $$\Delta p\Delta y-\Delta w\Delta x\geq0$$
  (this examines responses of price-taking producers in terms of vectors of production $y$ and uses of inputs $x$, to their prices $p$ and $w$, essentially revealed preference for producers)
* **[Revealed preference](http://en.wikipedia.org/wiki/Revealed_preference)**

If we can claim game theorists or mathematicians whose equations we use constantly:

* **[Karush-Kuhn-Tucker conditions](http://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions)**, especially complementary slackness. There's no single equation for linear programming, but I think econ has a claim to Kantorovich too. \
  Stationarity:
  $$\nabla f(x^\*) = \sum\_{i=1}^m \mu\_i \nabla g\_i(x^\*) + \sum\_{j=1}^l \lambda\_j \nabla h\_j(x^\*)$$
  Primal feasibility:
  $$g\_i(x^\*) \le 0, \mbox{ for all } i = 1, \ldots, m$$
  $$h\_j(x^\*) = 0, \mbox{ for all } j = 1, \ldots, l \,\!$$
  Dual feasibility:
  $$\mu\_i \ge 0, \mbox{ for all } i = 1, \ldots, m$$
  Complementary slackness:
  $$\mu\_i g\_i (x^\*) = 0, \mbox{for all}\; i = 1,\ldots,m.$$
* **[Nash equilibrium](http://en.wikipedia.org/wiki/Nash_equilibrium)**
  $$\theta\_{i}^\star = \arg \max\_{\theta\_i} u\_i(\theta\_i ,\theta\_{-i}^\star)$$
* **[Revelation principle](http://en.wikipedia.org/wiki/Revelation_principle)**: which to be fair isn't so much an equation as a theorem...
* **[Bellman equation](http://en.wikipedia.org/wiki/Bellman_equation)**
  $$V(x)=\max\_{c\in\Omega(x)} U(x,z)+\beta\left[V(x^\prime)\right]$$

> **Comments:**
> **username_6**: I would suggest that there are some inequalities which are even more fundamental than the first equation above. Unlike equations which represent approximations, some of the inequalities represent absolute.  For example, the total quantity of something people will be able to afford cannot exceed the total quantity that will exist.  If the number of people who would like to have something exceeds the quantity that exists, unless more of the thing are produced or some people stop wanting it, not everyone who wants one will get one, period, no matter what else is done.  
> **username_5**: That's fair. I suppose budget constraints are also "more fundamental" in that sense.  
> **username_6**: If someone proposes a policy which, if successful, would violate one of the normal equations associated with economics, such a person should be called upon to justify the expectation that the equation would not hold in that case, but since most equations don't hold 100% of the time it would be plausible that the policy might work despite the equation suggesting otherwise.  On the other hand, a policy which couldn't achieve its stated aims without violating some fundamental inequalities cannot reasonably be expected to achieve those aims; no wise person could plausibly expect otherwise.  
> **username_5**: Does my edit above get at what you're trying to express? I see this as a difference in framing of the term "fundamental". You seem to mean that physical constraints are the most fundamental element of any given *economic model*, with which I agree. But I see $\text{MB}=\text{MC}$ as the most fundamental element in an *economists toolkit* because it combines these constraints with notions of efficient use. I'm especially fond of it because it is a general equation, whereas physical constraints tend to be stated differently for different situations.  
> **username_6**: If one imagines the state of an economic system as being a marble rolling over a hilly surface, the equations define grooves in which the marble will tend to roll, but the limiting inequalities define boundaries.  Merely knowing the boundaries in which the marble is constrained without knowing how it will behave within them isn't very useful, but likewise a prediction of the marble's behavior which ignores the existence of a boundary between its present position and expected future position is apt to be very wrong.  In a sense, though, I think the constraints are somewhat more foundational...  
> **username_6**: ...since in their absence the natural "solution" would simply be to have everyone get everything they could possibly desire.  It is only the existence of resource limits that makes all the other compromises necessary.  Also, BTW, a concept I've not seen mentioned much is that it's *much* easier to turn \$1 into \$0.95 than \$1.05; if it seems like everyone in some particular field is turning \$1 into \$1.05 with little effort, odds are what's really happening is they're turning \$1 into \$0.95 (or less) but making it *look* like \$1.05.  The fact that the market price of something is \$1.05...  
> **username_6**: ...doesn't mean that it will ever be possible to convert a million units of that commodity into \$1,050,000 or even \$1,000,000.  If people too-easily convert \$1,000,000 into 1,000,000 units of something, odds are good that the total liquidation value will end up being noticeably less than \$1,000,000.  

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### Answer by username_7 on 2014-11-19. Score: 24

Most of intro econ is intersecting lines. Specifically,

$$MB = MC$$
[\* Equilibrium is achieved when Marginal Benefit is equal to Marginal Cost\*](https://ctaar.rutgers.edu/gag/NOTES/micnotes5.html)

$$\dfrac{MU\_x}{p\_x}=\dfrac{MU\_y}{p\_y}.$$
[*Marginal Utility per unit cost should always be equal*](http://dev.business.und.edu/Goenner/teaching/econ201/Resources/Book%20supplemental/07sg.pdf)

Economics is about the logic of human behavior, how we make decisions in a world of scarcity. These equations describe constrained optimization under some usual assumptions like continuity, convex preferences, and no corner solutions. I'd also give prominence to consumer theory over producer. Most of undergrad producer theory can be understood with the same tools used in consumer theory.

> **Comments:**
> **username_7**: It makes sense to assume that consumers are rational optimizers. That is a toothless statement (complete and transitive preferences). It's just much harder to know what a human's objective is. I think of producer theory as often being a special kind of consumer. They are risk neutral consumers who get utility from dollars.  

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### Answer by username_8 on 2014-11-19. Score: 23

I once heard Roger Myerson talk about why he thought Economics has, as a Social Science, been so successful in applying (or has so readily incorporated) mathematics. He suggested that perhaps it was due to some of the fundamental linearities within the world. Two examples would be the flow-balance constraints of scarce goods (commodity constraints) and
no-arbitrage conditions. These are fundamentally linear constraints.

* It's important to emphasize the importance of these because we can get a surprising amount out of the two. For example, a lot of people think that the law of demand is a consequence of assuming rationality (specifically, preferences that exhibits a diminishing marginal rate of substitution). A result due to Gary Becker shows that the law of demand (albeit just a slightly weaker version) can be derived *from the budget constraint alone*. (See Becker 1962, "[Irrational Behavior and Economic Theory](https://www.journals.uchicago.edu/doi/pdf/10.1086/258584).") That is, this fundamental economic result can be derived from the reality of scarce resources alone---without assuming rationality.
* The no-arbitrage condition is an application of the linear duality theorem ([Farkas' lemma](http://en.wikipedia.org/wiki/Farkas%27_lemma)). A lot of economics and finance (asset pricing) can be done just by the assumption that in economic equilibrium there is no arbitrage.

Extra Notes:

Gary Becker made a lot of advances in the field by studying the way constraints affect human behavior. One famous quote, taken from his Nobel prize lecture, is the remark that "different constraints are decisive for different situations, but the most fundamental constraint is limited time." (Some discussion [here](http://conversableeconomist.blogspot.com/2014/05/gary-becker-and-time-constraint.html).) Some more resources about how his work in this regard can be found [here](https://cehd.uchicago.edu/sites/heckman2013.uchicago.edu/files/uploads/Becker-slides_2014-10-31.pdf) and [here](http://bfi.uchicago.edu/video/impact-work-gary-s-becker).

Linear duality can be used to describe the no arbitrage condition. More generally, this theorem is typically proved with the [Hyperplane Separation Theorem](http://en.wikipedia.org/wiki/Hyperplane_separation_theorem), which is mathematical tool that shows up a lot in economics textbooks.

Also, keep in mind that it's enough just to assume that in economic equilibrium, there
is approximately no arbitrage.


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### Answer by username_9 on 2014-11-22. Score: 21

I think one of the most important equations (at least within macroeconomics) is:

$$E\left[ m R \right] = 1$$

This equation has been used to derive many foundational results. This equation motivated the [Hansen–Jagannathan bound](http://en.wikipedia.org/wiki/Hansen%E2%80%93Jagannathan_bound). It is fundamental for asset pricing as well.

Also, something interesting I saw from once from Tom Sargent. If you use the stochastic discount factor for a standard model $m = \beta E\_t \left[ \frac{u'(c\_{t+1})}{u'(c\_t)} \right]$ then depending on which piece of the equation you allow to be exogenous you can get some fundamental results of macro:

* Permanent Income Hypothesis: Let $\beta R = 1$ then we get $c\_t = E [c\_{t+1}]$
* Lucas Asset Pricing Model: Let the process for consumption be a given. Then the price of an asset can be described by $R\_t^{-1} = p\_t = E \left[ \frac{u'(c\_{t+1})}{u'(c\_t)} \right]$


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### Answer by username_10 on 2014-11-19. Score: 19

Although I agree with Jyotirmoy Bhattacharya that the most interesting ideas in economics are not always best expressed through equations, I still want to mention the Slutsky or compensated law of demand from consumer theory

$$ (p' -p) \Big[ x\big(p', p' x(p,w)\big) – x\big(p,w\big)\Big]^T \leq 0,$$

where $p',p \in \mathbb{R}\_{++}^n$ are any two price vectors, $w \in \mathbb{R}\_+$ is any level of income, and $x(\cdot,\cdot) \in \mathbb{R}^n$ is the demand function.

The underlying relation is a couple of orders of certitude away from fundamental equations in other fields. Also, it does not ground the discipline, in the sense that it is not used all that often.

However, I tend to view it as fundamental because

* It is an *absolutely non-trivial* consequence of three simple and fundamental assumptions in consumer theory, namely,
  + That the demand function $x(\cdot,\cdot)$ is homogenous of degree zero (no money illusion)
  + Walras' law (people do not burn money)
  + The weak axiom of revealed preferences (if you chose A when B is available “today”, you will not chose B “tomorrow” if A remains available)
* Therefore testing the inequality is equivalent to testing these three assumptions jointly.
* The three assumptions are used in the vast majority (maybe more than 90%?) of the models including consumers in economic theory.
* Their validity (at least as approximations) is therefore crucial to the validity of most models in economic theory (at least as approximations).
* Although it is not always obvious how to relate the notions of prices, goods and income to observables, all the element in the equation are observable *in principle* (as opposed to utility levels for instance) and the validity of the inequality can therefore be tested empirically.

> **Comments:**
> **username_11**: I'd add that it gets even better: there are _three_ laws of demand, which are equivalent (and boil down to Slutsky negative semidefiniteness) in the infinitesimal case but are distinct in general. After the price change from $p$ to $p'$, you can either (1) adjust wealth such that it can purchase the old bundle, (2) adjust wealth such that utility is constant, or (3) adjust wealth such that the newly chosen bundle could have been purchased yesterday - in all cases you get a law of demand. (These are arguably the laws of overcompensated, compensated, and undercompensated demand, respectively.)  

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### Answer by username_17 on 2014-11-19. Score: 13

I don't think there are any economics equations with the same status as, say, Maxwell's equations in physics. In its place we have concepts like the equimarginal principle, competitive equilibrium or Nash equilibrium which are at the core of the "economist's approach". But I think that the real worth of economics is not even in these ideas themselves but in what we know about concrete problems in specific areas of applications: for example what we know about business cycles in macro. In this economics may be more like medicine than physics.

> **Comments:**
> **username_18**: The acknowledgment that the totality of activities has a scale limit is understandably slow because economic developments are evaluated in the conceptual and quantitative terms of a system that repudiates the existence of such limits; tough, Maxwell can marginally be introduced to the core of the "economist's approach": [Entropy, limits to growth, and the prospects for weak sustainability](http://www.embl-hamburg.de/aboutus/science_society/discussion/discussion_2006/ref1-13July06.pdf) and the axiomatic base: [Ten Proofs of the Generalized Second Law](http://arxiv.org/pdf/0901.3865v2.pdf)  

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### Answer by username_19 on 2017-03-11. Score: 10

A bit late to the game, but I'm surprised no one has named the equation to calculate OLS estimates:
$$
\hat\beta=(X'X)^{-1}X'y
$$

> **Comments:**
> **username_20**: I don't know if that can be called economics. Or if it can, then what about other results from statistics and mathematics that are widely used in economics?  

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### Answer by username_12 on 2014-11-19. Score: 9

Whilst not as foundational as, for example, the Slutsky equation, the condition on the Lerner index that a profit maximising firm with price $p$, cost $c$, and price elasticity of demand $\eta$ has
$$\frac{p-c}{p}=-\frac{1}{\eta}$$
is an important equation in industrial organisation.

This is not only an elegant formulation of the solution of the firm's problem, but it is also practically useful:

* A firm that estimates its $\eta$ and knows its $c$ can use this formula to calculate the profit-maximising price.
* A regulator that observes a $p$ and estimates $\eta$ can use the formula to calculate $c$—important in many forms of regulation.


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### Answer by username_13 on 2014-11-19. Score: 9

For me, one of the most important ones is the budget constraint. It might seem too obvious but a lot of laypersons (though maybe not physicist) don't get it!

$p⋅x \leq w$

> **Comments:**
> **username_14**: It's not that foundational if you remember about borrowing.  

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### Answer by username_15 on 2015-04-24. Score: 7

It is already written but Euler equation in continous time yields

$$\frac{\dot{C}}{C}=\sigma(r-\rho)$$

where $\sigma$ is intertemporal elasticity of substitution, $r$ interest rate and $\rho$ is the discount rate (impatience level).


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### Answer by username_16 on 2014-11-22. Score: 6

The foundation of intertemporal economics is the [net present value equation](http://en.wikipedia.org/wiki/Net_present_value). That is, the net present value of a future income stream is the yearly incomes divided by an appropriate discount factor, based on the prevailing interest rate, r, taken to the nth power, where n is the number of years.

> **Comments:**
> **username_8**: NPV as described in the linked Wikipedia article doesn't seem as general and central to economics as does $E[mR] = 1$, for example.  
> **username_16**: @jmbejara: It's the foundation of finance, as it relates to the value of bonds, the mortgage on you house, etc.  
> **username_8**: I know. What I mean to point out is that, if we think of $E[mR] = 1$ more generally (e.g., drop the equilibrium interpretation), it can encompass NPV as you described it. But it can also do much more. If you write it as $E[m X] = P$ and you treat $X$ as a stream of future cash flows and $m$ as the appropriate discount factor, you can recover your definition of NPV.  
> **username_8**: http://en.wikipedia.org/wiki/Stochastic_discount_factor  

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### Answer by username_21 on 2017-11-10. Score: 4

Well for microeconomics there are several, however they all follow the same pattern.

> ***Here I'll attempt to teach an entire intermediate microeconomics course in one post.***

## Most microeconomics problems follow this format:

Though leaving out some minor details, if you do enough microeconomics practice sets the problems end up looking the same after a while. This is what I got to share.

## Production/Utility functions

There are three main types of utility/production functions you will be exposed to in an intermediate microeconomics course1. They are:

1. Cobb Douglas  
   $$f(x\_1,x\_2)=x\_1^ax\_2^b$$
2. Leontif/ Perfect Complement  
   $$f(x\_1,x\_2)=\min\{x\_1,x\_2\}$$
3. Perfect substitutes
   $$f(x\_1,x\_2)=x\_1+x\_2$$

## Budget lines and Cost functions

In consumer theory, you have a budget line represented by the formula:

$$m=p\_1x\_1+p\_2x\_2$$

In producer theory we call it a cost function.
$$C(x\_1,x\_2)=w\_1x\_1+w\_2x\_2$$

we either want to maximize consumption given a budget/cost function or minimize costs holding your utility/output level constant. To do this we use another equation:

## The Lagrangian Multiplier:

Though not exclusive to economics tool per say, its the primary tool of all intermediate microeconomics students.

$$\mathcal{L}=f(x\_1,x\_2)\pm\lambda(H-g(x\_1,x\_2))$$

where $H-g(x\_1,x\_2)$ is either a budget line/cost function or Utility/Production function when its equal to zero.

We use this for calculating utility/profit maximizing consumption bundles/inputs or Minimize Costs holding profit/utility constant.

And thats a wrap!\*

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\*Though there is what to say on marshallian and hicksian demands I'll leave that for others to fill in.


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