`econ-model`¶

Pack: 100xOS shared skills

Category: drafting

Field: economics

License: private (curator-owned)

Updated: 2026-05-20

Stages: paper-drafting

Curator-private skill — copy text from 100xOS/shared/skills/latex/econ-model.md.

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LaTeX Conventions for Economic Models¶

Guidelines for typesetting economic theory in LaTeX. Follow these conventions to produce clean, readable, publication-quality mathematical economics.

Preamble essentials¶

TeX

\usepackage{amsmath}      % Core math environments
\usepackage{amssymb}      % Additional math symbols
\usepackage{amsthm}       % Theorem environments
\usepackage{mathtools}    % Extensions to amsmath (dcases, coloneqq, etc.)
\usepackage{bbm}          % Blackboard bold for indicator functions: \mathbbm{1}
\usepackage{bm}           % Bold math symbols (for vectors if needed)

Equation environments¶

Single equations¶

Use equation for numbered, equation* for unnumbered. Never use $$...$$.

TeX

% Numbered equation (use when you will reference it)
\begin{equation}\label{eq:utility}
  U_i = \sum_{t=0}^{\infty} \beta^t u(c_{it})
\end{equation}

% Unnumbered (use sparingly; most equations in a theory paper should be numbered)
\begin{equation*}
  \pi_f = p \cdot q - C(q)
\end{equation*}

Multi-line aligned equations¶

Use align for related equations that should be aligned (typically at = or inequality signs). Use aligned inside equation when you want a single equation number for the group.

TeX

% Multiple related equations, each numbered
\begin{align}
  \max_{c_t, k_{t+1}} \quad & \sum_{t=0}^{\infty} \beta^t u(c_t) \label{eq:objective} \\
  \text{s.t.} \quad & c_t + k_{t+1} = f(k_t) + (1-\delta) k_t \label{eq:budget} \\
                     & c_t \geq 0, \quad k_0 \text{ given} \label{eq:constraints}
\end{align}

% Single equation number for a system
\begin{equation}\label{eq:foc-system}
  \begin{aligned}
    u'(c_t) &= \beta u'(c_{t+1}) \bigl[ f'(k_{t+1}) + 1 - \delta \bigr] \\
    c_t + k_{t+1} &= f(k_t) + (1 - \delta) k_t
  \end{aligned}
\end{equation}

Cases (piecewise functions)¶

Use cases from amsmath or dcases from mathtools (the latter provides displaystyle inside cases, which looks better for fractions).

TeX

\begin{equation}\label{eq:tax-schedule}
  T(y) =
  \begin{dcases}
    0                          & \text{if } y \leq \bar{y} \\
    \tau (y - \bar{y})         & \text{if } y > \bar{y}
  \end{dcases}
\end{equation}

Standard notation conventions¶

Agents, goods, time¶

Symbol	Convention	Example
`i, h`	Individual agents, households	$i \in {1, \ldots, N}$
`f, j`	Firms	$f \in {1, \ldots, F}$
`j, k`	Goods, sectors	$j \in {1, \ldots, J}$
`t`	Time period	$t = 0, 1, 2, \ldots$
`s`	State of the world	$s \in S$

Core economic objects¶

TeX

% Preferences
u(\cdot)              % Utility function (lowercase)
U_i                   % Lifetime/indirect utility
\beta                 % Discount factor (0 < \beta < 1)
\delta                % Depreciation rate

% Prices and quantities
p_j                   % Price of good j
w                     % Wage
r                     % Interest rate / rental rate of capital
c_{it}                % Consumption of agent i at time t
k_t                   % Capital stock at time t
\ell_i                % Labor supply (use \ell, not l, to avoid confusion with 1)
y_i                   % Income of agent i
Y                     % Aggregate output (uppercase for aggregates)

% Production
f(k, \ell)            % Production function (lowercase)
F(K, L)               % Aggregate production function (uppercase)
A_t                   % Total factor productivity

% Probability and expectations
\Pr(\cdot)            % Probability
\mathbb{E}[\cdot]     % Expectation operator
\mathbb{E}_t[\cdot]   % Conditional expectation given info at t
\mathbb{V}[\cdot]     % Variance (use \operatorname{Var} as alternative)

% Sets
\mathbb{R}            % Real numbers
\mathbb{R}_+          % Non-negative reals
\mathbb{R}^n          % n-dimensional Euclidean space
\Delta^{n-1}          % Simplex (for mixed strategies, distributions)

Operators and functions¶

Define custom operators in the preamble to get proper spacing:

TeX

\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\Cov}{Cov}
\DeclareMathOperator{\Corr}{Corr}
\DeclareMathOperator{\plim}{plim}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\sgn}{sgn}

Usage:

TeX

x^* = \argmax_{x \in X} f(x)

Indicator functions¶

TeX

\mathbbm{1}\{x > 0\}        % Preferred: blackboard bold 1
\mathbf{1}\{x > 0\}         % Alternative: bold 1
\mathbbm{1}_{x > 0}         % Subscript notation (also common)

Optimization problems¶

Economics papers are full of optimization problems. Typeset them consistently:

TeX

% Consumer's problem
\begin{equation}\label{eq:consumer}
  \max_{\{c_t, \ell_t\}_{t=0}^{\infty}} \sum_{t=0}^{\infty}
    \beta^t u(c_t, 1 - \ell_t)
  \quad \text{s.t.} \quad
  c_t + a_{t+1} = (1+r_t) a_t + w_t \ell_t
\end{equation}

% Firm's problem (static)
\begin{equation}\label{eq:firm}
  \max_{K, L \geq 0} \; F(K, L) - rK - wL
\end{equation}

% Planner's problem with Lagrangian
\begin{equation}\label{eq:lagrangian}
  \mathcal{L} = \sum_{t=0}^{\infty} \beta^t
    \Bigl[ u(c_t) + \lambda_t \bigl( f(k_t) + (1-\delta)k_t - c_t - k_{t+1} \bigr) \Bigr]
\end{equation}

For multi-line constraints, use the align approach shown earlier.

Equilibrium definitions¶

Use the definition environment from amsthm. Define it in the preamble:

TeX

\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem{assumption}{Assumption}
\newtheorem{proposition}{Proposition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
\newtheorem{remark}{Remark}
\newtheorem{example}{Example}

Competitive equilibrium¶

TeX

\begin{definition}[Competitive Equilibrium]\label{def:ce}
  A competitive equilibrium is a collection of allocations
  $\{c_t^*, k_{t+1}^*\}_{t=0}^{\infty}$ and prices
  $\{r_t^*, w_t^*\}_{t=0}^{\infty}$ such that:
  \begin{enumerate}
    \item \textbf{Household optimality:} Given prices $\{r_t^*, w_t^*\}$,
      the allocation $\{c_t^*, k_{t+1}^*\}$ solves the household's problem
      \eqref{eq:consumer}.
    \item \textbf{Firm optimality:} In each period $t$, factor prices satisfy
      $r_t^* = F_K(K_t, L_t)$ and $w_t^* = F_L(K_t, L_t)$.
    \item \textbf{Market clearing:} For all $t$,
      \begin{align*}
        c_t^* + k_{t+1}^* &= F(k_t^*, 1) + (1 - \delta) k_t^*
          && \text{(goods market)} \\
        L_t &= 1 && \text{(labor market, inelastic supply)}
      \end{align*}
  \end{enumerate}
\end{definition}

Nash equilibrium (game theory)¶

TeX

\begin{definition}[Nash Equilibrium]\label{def:nash}
  A strategy profile $\sigma^* = (\sigma_1^*, \ldots, \sigma_N^*)$ is a Nash
  equilibrium if for every player $i \in \{1, \ldots, N\}$ and every
  alternative strategy $\sigma_i' \in \Sigma_i$:
  \begin{equation*}
    u_i(\sigma_i^*, \sigma_{-i}^*) \geq u_i(\sigma_i', \sigma_{-i}^*)
  \end{equation*}
\end{definition}

Proofs¶

Use the proof environment from amsthm. It automatically adds a QED symbol.

TeX

\begin{proposition}\label{prop:euler}
  In any interior solution, the Euler equation holds:
  \begin{equation*}
    u'(c_t) = \beta (1 + r_{t+1}) u'(c_{t+1})
  \end{equation*}
\end{proposition}

\begin{proof}
  The first-order conditions of \eqref{eq:lagrangian} with respect to $c_t$
  and $k_{t+1}$ are:
  \begin{align}
    \frac{\partial \mathcal{L}}{\partial c_t}:     \quad & u'(c_t) = \lambda_t \label{eq:foc-c} \\
    \frac{\partial \mathcal{L}}{\partial k_{t+1}}: \quad & \lambda_t = \beta \lambda_{t+1} \bigl[ f'(k_{t+1}) + 1 - \delta \bigr] \label{eq:foc-k}
  \end{align}
  Substituting \eqref{eq:foc-c} into \eqref{eq:foc-k} and using
  $r_{t+1} = f'(k_{t+1}) - \delta$ yields the result.
\end{proof}

Proof sketches and appendix proofs¶

TeX

% In the main text
\begin{proof}[Proof sketch]
  The argument proceeds in three steps. First, we show existence by
  Brouwer's fixed point theorem. Second, we establish uniqueness via
  contraction mapping. The full proof is in Appendix~\ref{app:proofs}.
\end{proof}

Common patterns in economics papers¶

First-order conditions (FOCs)¶

TeX

% Vertically aligned FOCs
\begin{align}
  [c_t]: \quad & u'(c_t) = \lambda_t \label{eq:foc1} \\
  [k_{t+1}]: \quad & \lambda_t = \beta \lambda_{t+1} [f'(k_{t+1}) + 1 - \delta] \label{eq:foc2} \\
  [\lambda_t]: \quad & c_t + k_{t+1} = f(k_t) + (1-\delta)k_t \label{eq:foc3}
\end{align}

Value functions (dynamic programming)¶

TeX

\begin{equation}\label{eq:bellman}
  V(k) = \max_{k' \in [0, f(k) + (1-\delta)k]}
    \left\{ u\bigl(f(k) + (1-\delta)k - k'\bigr) + \beta V(k') \right\}
\end{equation}

Welfare and surplus¶

TeX

% Compensating variation
\begin{equation}\label{eq:cv}
  V(p^1, y - CV) = V(p^0, y)
\end{equation}

% Social welfare function
\begin{equation}\label{eq:swf}
  W = \sum_{i=1}^{N} \omega_i \, u_i(x_i)
  \quad \text{where } \sum_{i} \omega_i = 1, \; \omega_i \geq 0
\end{equation}

Elasticities¶

TeX

\varepsilon_{x,p} = \frac{\partial x}{\partial p} \cdot \frac{p}{x}
  = \frac{\partial \ln x}{\partial \ln p}

Cross-referencing¶

Use \eqref for equations (adds parentheses automatically) and \ref for theorems, propositions, and definitions. Use the cleveref package for automatic formatting:

TeX

\usepackage[capitalise,noabbrev]{cleveref}

% Then in text:
As shown in \cref{eq:euler}, the Euler equation implies...
By \cref{prop:euler,prop:tvc}, the solution is characterized by...

Style tips¶

Use \left( ... \right) sparingly. Prefer \bigl( ... \bigr) or manual sizing for better control. \left/\right can produce ugly spacing.
Use \text{} inside math for short verbal descriptions (e.g., "s.t.", "if", "for all"). Use \quad for spacing around these.
Use \cdot for multiplication when needed (not \times unless for cross products or dimensions).
Use \ldots for comma-separated lists ($x_1, \ldots, x_n$) and \cdots for operator-separated lists ($x_1 + \cdots + x_n$).
Subscripts for identity, superscripts for indices when both are needed: $c_i^t$ (consumption of agent $i$ at time $t$), though $c_{it}$ is also common and acceptable.
Bold lowercase for vectors ($\bm{x}$), bold uppercase for matrices ($\bm{A}$), but many economics papers avoid bold entirely and just state "let $x$ be a vector."
Calligraphic letters for sets, information sets, or Lagrangians: $\mathcal{I}$, $\mathcal{F}$, $\mathcal{L}$.

econ-model¶