Appendix A: Mathematical Formulas and Tables

A.1 Differentiation Rules¶

Basic rules. For differentiable functions $f, g$ and constant $c$:

Common functions:

Implicit differentiation. If $F(x, y) = 0$ defines $y$ as a function of $x$: $dy/dx = -F_x/F_y$.

Logarithmic differentiation. For $y = f(x)^{g(x)}$: $\ln y = g(x)\ln f(x)$, then differentiate both sides.

A.2 Integration Formulas¶

Integration by parts: $\int u\,dv = uv - \int v\,du$.

Gaussian integral: $\int_{-\infty}^\infty e^{-ax^2}dx = \sqrt{\pi/a}$ for $a > 0$.

Present value integral: $\int_0^\infty e^{-\rho t}f(t)dt = \mathcal{L}\{f\}(\rho)$ (Laplace transform at $\rho$).

A.3 Taylor Series Expansions¶

Standard expansions around 0:

Key approximations (small $x$): $\ln(1+x) \approx x$, $(1+x)^\alpha \approx 1 + \alpha x$, $e^x \approx 1 + x + x^2/2$.

A.4 Matrix Identities¶

Woodbury identity: $(A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}$.

Sherman–Morrison: $(A + \mathbf{u}\mathbf{v}')^{-1} = A^{-1} - \frac{A^{-1}\mathbf{u}\mathbf{v}'A^{-1}}{1+\mathbf{v}'A^{-1}\mathbf{u}}$.

Matrix determinant lemma: $\det(A + \mathbf{u}\mathbf{v}') = (1+\mathbf{v}'A^{-1}\mathbf{u})\det(A)$.

Trace and determinant: $\text{tr}(AB) = \text{tr}(BA)$. $\det(AB) = \det(A)\det(B)$. $\det(A^{-1}) = 1/\det(A)$.

Kronecker product: $(A\otimes B)(C\otimes D) = (AC)\otimes(BD)$. $\text{vec}(AXB) = (B'\otimes A)\text{vec}(X)$.

Differentiation: $\partial(A\mathbf{x})/\partial\mathbf{x} = A$. $\partial(\mathbf{x}'A\mathbf{x})/\partial\mathbf{x} = (A+A')\mathbf{x}$. $\partial\ln\det(A)/\partial A = (A^{-1})'$.

A.5 Special Functions in Macroeconomics¶

Log-normal: If $\ln X \sim \mathcal{N}(\mu,\sigma^2)$, then $X \sim \text{LogNormal}(\mu,\sigma^2)$ with $\mathbb{E}[X] = e^{\mu+\sigma^2/2}$, $\text{Var}[X] = (e^{\sigma^2}-1)e^{2\mu+\sigma^2}$.

Gamma function: $\Gamma(n) = (n-1)!$ for integer $n$; $\Gamma(1/2) = \sqrt{\pi}$; $\Gamma(n+1) = n\Gamma(n)$.

Normal CDF: $\Phi(z) = P(Z\leq z)$ for $Z\sim\mathcal{N}(0,1)$; $\Phi(-z) = 1-\Phi(z)$.

Appendix B: Linear Algebra Review¶

B.1 Vector Spaces¶

Definition. A vector space over $\mathbb{R}$ is a set $V$ with addition and scalar multiplication satisfying 8 axioms (closure, associativity, commutativity, identity, inverses, distributivity).

Basis and dimension. A set $\{\mathbf{v}_1,\ldots,\mathbf{v}_n\}$ is a basis if it is linearly independent and spans $V$. The dimension $\dim V$ is the cardinality of any basis.

Standard basis of $\mathbb{R}^n$: $\mathbf{e}_i$ has 1 in position $i$ and 0 elsewhere.

B.2 Matrix Operations¶

Rank. $\text{rank}(A)$ = number of linearly independent rows (= columns). $\text{rank}(A) \leq \min(m,n)$ for $A\in\mathbb{R}^{m\times n}$. Full column rank: $\text{rank}(A) = n$ (columns independent). Full row rank: $\text{rank}(A) = m$.

Null space. $\mathcal{N}(A) = \{\mathbf{x}: A\mathbf{x}=\mathbf{0}\}$. Dimension: $n - \text{rank}(A)$ (rank-nullity theorem).

Norms. $\|\mathbf{x}\|_1 = \sum|x_i|$; $\|\mathbf{x}\|_2 = \sqrt{\sum x_i^2}$; $\|\mathbf{x}\|_\infty = \max|x_i|$. Matrix norm: $\|A\|_2 = \sigma_{\max}(A)$ (largest singular value); $\|A\|_F = \sqrt{\text{tr}(A'A)}$ (Frobenius).

B.3 Determinants and Cramer’s Rule¶

For $A\in\mathbb{R}^{2\times2}$: $\det(A) = a_{11}a_{22} - a_{12}a_{21}$.

Cramer’s rule. For $A\mathbf{x} = \mathbf{b}$ with $\det(A)\neq0$: $x_i = \det(A_i)/\det(A)$, where $A_i$ is $A$ with column $i$ replaced by $\mathbf{b}$.

Properties. $\det(AB) = \det(A)\det(B)$. $\det(A') = \det(A)$. $\det(\alpha A) = \alpha^n\det(A)$ (for $n\times n$). $\det(A) = \prod_i\lambda_i$ (product of eigenvalues).

B.4 Eigenvalues and Eigenvectors¶

$A\mathbf{v} = \lambda\mathbf{v}$, $\mathbf{v}\neq\mathbf{0}$. Eigenvalues are roots of $\det(A-\lambda I) = 0$ (characteristic polynomial).

Spectral decomposition (symmetric $A=A'$): $A = Q\Lambda Q'$ where $Q$ is orthonormal and $\Lambda = \text{diag}(\lambda_1,\ldots,\lambda_n)$.

Stability: Discrete: $A^t \to 0$ iff $|\lambda_i| < 1$ for all $i$. Continuous: $e^{At} \to 0$ iff $\text{Re}(\lambda_i) < 0$ for all $i$.

Generalized eigenvalues. $A\mathbf{v} = \lambda B\mathbf{v}$ has generalized eigenvalues $\lambda_i = T_{ii}/S_{ii}$ from the QZ decomposition $QAZ = S$, $QBZ = T$.

B.5 The Perron–Frobenius Theorem¶

Theorem (Perron–Frobenius). Let $A$ be a square matrix with all positive entries. Then:

1. $A$ has a unique largest real eigenvalue $\lambda_1 > 0$ (the Perron root).

2. The corresponding eigenvector $\mathbf{v}_1$ has all positive entries (Perron vector).

3. $|\lambda_i| < \lambda_1$ for all other eigenvalues.

Application in input-output analysis (Chapter 2). For the Leontief technical coefficient matrix $A$ (with $A_{ij} \geq 0$), the Perron root $\lambda_1(A) < 1$ guarantees $(I-A)^{-1}$ exists and is non-negative — the Leontief inverse has all non-negative entries (backward linkage multipliers are non-negative).

Application in Markov chains. For a stochastic matrix $P$ (row sums = 1, all entries $\geq 0$), Perron–Frobenius guarantees a unique stationary distribution $\bm\pi$ with $\bm\pi'P = \bm\pi'$ and $\pi_i > 0$ for all $i$ (ergodic chains).

B.6 QR and LU Decompositions¶

LU decomposition. $PA = LU$: $P$ permutation, $L$ unit lower triangular, $U$ upper triangular. Solves $A\mathbf{x}=\mathbf{b}$ in $O(n^3/3)$ flops (Chapter 25).

QR decomposition. $A = QR$: $Q$ orthonormal ($Q'Q=I$), $R$ upper triangular. Used for numerically stable OLS (Chapter 25). $\kappa(R) = \kappa(A)$ vs. $\kappa(A'A) = \kappa(A)^2$ for normal equations.

Cholesky. For positive definite $A$: $A = LL'$ ($L$ lower triangular). Fastest symmetric system solver. Used for sampling from multivariate normal (Chapter 26).

SVD. $A = U\Sigma V'$: $U, V$ orthonormal, $\Sigma = \text{diag}(\sigma_1,\ldots,\sigma_r,0,\ldots)$. Rank = number of positive singular values. Used for numerical rank determination (Chapter 41).

Appendix C: Calculus Review¶

C.1 Limits and Continuity¶

Limit. $\lim_{x\to a}f(x) = L$: for every $\varepsilon>0$ there exists $\delta>0$ such that $|x-a|<\delta \Rightarrow |f(x)-L|<\varepsilon$.

L’Hôpital’s rule. If $\lim f = \lim g = 0$ (or $\pm\infty$): $\lim f/g = \lim f'/g'$ (when the right-hand limit exists).

Useful limits: $\lim_{x\to0}(1+x)^{1/x} = e$. $\lim_{n\to\infty}(1+r/n)^n = e^r$. $\lim_{x\to0}\frac{\sin x}{x} = 1$.

C.2 Multivariate Differentiation¶

Partial derivative. $\partial f/\partial x_i$ = derivative of $f$ treating all $x_j$ ($j\neq i$) as constants.

Gradient. $\nabla f = (\partial f/\partial x_1, \ldots, \partial f/\partial x_n)' \in \mathbb{R}^n$.

Hessian. $H_f = [\partial^2f/\partial x_i\partial x_j]$ — symmetric $n\times n$ matrix of second partial derivatives. $H_f \succ 0$ (positive definite) iff $f$ is strictly convex.

Jacobian. For $F:\mathbb{R}^n\to\mathbb{R}^m$: $J_F = [\partial F_i/\partial x_j]$ — the $m\times n$ matrix of first partial derivatives.

Chain rule (vector form). For $h = f\circ g$ ($h:\mathbb{R}^m\to\mathbb{R}^p$, $g:\mathbb{R}^n\to\mathbb{R}^m$, $f:\mathbb{R}^m\to\mathbb{R}^p$): $J_h = J_f(g(\mathbf{x}))\cdot J_g(\mathbf{x})$.

C.3 Optimization Conditions¶

Unconstrained. FOC: $\nabla f(\mathbf{x}^*) = \mathbf{0}$. SOC (minimum): $H_f(\mathbf{x}^*) \succ 0$.

Equality constraints (Lagrange). Maximize $f(\mathbf{x})$ s.t. $g(\mathbf{x}) = 0$: Lagrangian $\mathcal{L} = f - \lambda g$; FOCs: $\nabla f = \lambda\nabla g$.

Inequality constraints (KKT). Maximize $f$ s.t. $g_j(\mathbf{x}) \leq 0$: KKT conditions: $\nabla f = \sum_j\mu_j\nabla g_j$; $\mu_j \geq 0$; $\mu_jg_j = 0$ (complementary slackness).

Envelope theorem. For $V(\alpha) = \max_x f(x,\alpha)$ s.t. $g(x,\alpha)=0$: $dV/d\alpha = \partial\mathcal{L}/\partial\alpha|_{x=x^*(\alpha)}$.

C.4 Integration¶

Fundamental theorem of calculus. $\frac{d}{dx}\int_a^x f(t)dt = f(x)$. $\int_a^b f'(x)dx = f(b) - f(a)$.

Fubini’s theorem. For integrable $f$: $\int\!\int f(x,y)dxdy = \int\!\left[\int f(x,y)dx\right]dy$.

Change of variables. $\int_{\phi(a)}^{\phi(b)}f(x)dx = \int_a^b f(\phi(t))\phi'(t)dt$.

Leibniz rule. $\frac{d}{d\alpha}\int_{a(\alpha)}^{b(\alpha)}f(x,\alpha)dx = f(b,\alpha)b'(\alpha) - f(a,\alpha)a'(\alpha) + \int_a^b\frac{\partial f}{\partial\alpha}dx$.

Dominated convergence theorem. If $f_n \to f$ pointwise and $|f_n| \leq g$ (integrable), then $\int f_n \to \int f$.

Appendix D: Probability Distributions and Statistical Tables¶

D.1 Core Distributions¶

Normal Distribution: $\mathcal{N}(\mu, \sigma^2)$¶

PDF: $f(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp\!\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]$.

Mean: $\mu$. Variance: $\sigma^2$. MGF: $M(t) = e^{\mu t + \sigma^2t^2/2}$.

Standard normal $\mathcal{N}(0,1)$: CDF $\Phi(z)$. Key quantiles: $\Phi^{-1}(0.025) = -1.960$, $\Phi^{-1}(0.05) = -1.645$, $\Phi^{-1}(0.10) = -1.282$.

Log-Normal Distribution: $\text{LogN}(\mu, \sigma^2)$¶

$X \sim \text{LogN}(\mu,\sigma^2)$ iff $\ln X \sim \mathcal{N}(\mu,\sigma^2)$.

Mean: $e^{\mu+\sigma^2/2}$. Variance: $(e^{\sigma^2}-1)e^{2\mu+\sigma^2}$. Median: $e^\mu$.

Role in macroeconomics: Income and productivity shocks ($A_t = e^{z_t}$, $z_t \sim \mathcal{N}$); asset prices; firm sizes (Chapter 33).

Gamma Distribution: $\text{Gamma}(\alpha, \beta)$¶

PDF: $f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$, $x > 0$.

Mean: $\alpha/\beta$. Variance: $\alpha/\beta^2$.

Role: Prior for positive parameters (CRRA $\sigma$, shock persistence); the chi-squared is $\chi^2(k) = \text{Gamma}(k/2, 1/2)$.

Beta Distribution: $\text{Beta}(\alpha, \beta)$¶

PDF: $f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$, $x\in(0,1)$.

Mean: $\alpha/(\alpha+\beta)$. Variance: $\alpha\beta/[(\alpha+\beta)^2(\alpha+\beta+1)]$.

Role: Prior for parameters in $[0,1]$ — Calvo probability $\theta$, AR persistence $\rho$, capital share $\alpha$.

Inverse-Gamma Distribution: $\text{IG}(\alpha, \beta)$¶

$X \sim \text{IG}(\alpha,\beta)$ iff $1/X \sim \text{Gamma}(\alpha,\beta)$.

Mean: $\beta/(\alpha-1)$ ($\alpha>1$). Variance: $\beta^2/[(\alpha-1)^2(\alpha-2)]$ ($\alpha>2$).

Role: Prior for variance parameters ($\sigma^2_\varepsilon$) — conjugate prior for the normal variance.

D.2 Key Statistical Tables¶

Normal Distribution Quantiles¶

ADF Critical Values (asymptotic, constant + trend)¶

Johansen Trace Statistic Critical Values (5%)¶

D.3 Moment-Generating Functions¶

Key property: $\mathbb{E}[X^k] = M^{(k)}(0)$ (the $k$-th derivative of MGF at 0).