Fundamental equations in economics

Question

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?

Answer 1

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".

Answer 2

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 $$V^\prime(y)=f_y(x,y)$$
• "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

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

• Karush-Kuhn-Tucker 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 $$\theta_{i}^\star = \arg \max_{\theta_i} u_i(\theta_i ,\theta_{-i}^\star)$$
• Revelation principle: which to be fair isn't so much an equation as a theorem...
• Bellman equation $$V(x)=\max_{c\in\Omega(x)} U(x,z)+\beta\left[V(x^\prime)\right]$$

Answer 3

Most of intro econ is intersecting lines. Specifically,

$$MB = MC$$ * Equilibrium is achieved when Marginal Benefit is equal to Marginal Cost*

$$\dfrac{MU_x}{p_x}=\dfrac{MU_y}{p_y}.$$ Marginal Utility per unit cost should always be equal

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.

Answer 4

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.") 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). 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.) Some more resources about how his work in this regard can be found here and here.

Linear duality can be used to describe the no arbitrage condition. More generally, this theorem is typically proved with the 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.

Answer 5

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. 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 $R_t^{-1} = p_t = E \left[ \frac{u'(c_{t+1})}{u'(c_t)} \right]$

Answer 6

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.

Answer 7

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.

Answer 8

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

Answer 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.

Answer 10

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!

$p⋅x \leq w$

Answer 11

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).

Answer 12

The foundation of intertemporal economics is the net present value equation. 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.

Answer 13

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.}