Partial Jacobians¶

Definition¶

Let $\boldsymbol{F}\colon A\subseteq\mathbb{R}^{p+q}\to\mathbb{R}^m$ be differentiable, and write the variables as $(\boldsymbol{x},\boldsymbol{y})$ with $\boldsymbol{x}\in\mathbb{R}^p$ and $\boldsymbol{y}\in\mathbb{R}^q$${\mathbb{R}}^q$. The partial Jacobians of $\boldsymbol{F}$ are the blocks

Together they form the full Jacobian:

Interpretation¶

A partial Jacobian isolates how an equation system responds to one chosen block of variables while the other block is held fixed. It is the matrix version of a partial derivative.

Each row of the Jacobian belongs to one component $F_i$ and each column to one variable. In the figure $m=3$$3$, $p=2$ and $q=3$$3$: the two blue columns, derivatives with respect to $x_1$ and $x_2$, form $\boldsymbol{D}_x\boldsymbol{F}$ ($3\times$$3\times2$$2$), and the three red columns form $\boldsymbol{D}_y\boldsymbol{F}$ ($3\times$$3\times3$$3$). So the full Jacobian is split by columns, $\boldsymbol{D}\boldsymbol{F}=(\boldsymbol{D}_x\boldsymbol{F}\mid\boldsymbol{D}_y\boldsymbol{F})$${\mathbf{D}}_y\mathbf{F})$, never by rows.

The split matters when the two blocks play different roles. In the implicit function theorem, $\boldsymbol{x}$ is exogenous and $\boldsymbol{y}$ is solved for. If $\boldsymbol{y}=\boldsymbol{g}(\boldsymbol{x})$ satisfies $\boldsymbol{F}(\boldsymbol{x},\boldsymbol{g}(\boldsymbol{x}))=\boldsymbol{0}$$0$, the multivariable chain rule gives $\boldsymbol{D}_x\boldsymbol{F}+\boldsymbol{D}_y\boldsymbol{F}\,\boldsymbol{D}\boldsymbol{g}=\boldsymbol{0}$$0$, so $\boldsymbol{D}\boldsymbol{g}=-(\boldsymbol{D}_y\boldsymbol{F})^{-1}\boldsymbol{D}_x\boldsymbol{F}$ when the block $\boldsymbol{D}_y\boldsymbol{F}$ is square and invertible. Square means $q=m$$m$, as many unknowns as equations, as in the figure. For the circle $F(x,y)=x^2+y^2-1$ from the Examples this gives $g'(x)=-x/y$ where $y\neq0$$0$; at $(1,0)$ the block $D_yF=0$$0$, and indeed near that point $y$ is not a function of $x$.

A common slip is to reverse the product. $(\boldsymbol{D}_y\boldsymbol{F})^{-1}$ is $q\times q$ and $\boldsymbol{D}_x\boldsymbol{F}$ is $q\times p$$p$, so $\boldsymbol{D}\boldsymbol{g}$ is $q\times p$$p$: one row for each endogenous variable and one column for each exogenous one. In the other order the sizes do not match unless $p=q$$q$.

Key Results¶

• Partial Jacobians are the matrix version of holding some variables constant in partial differentiation.
• The implicit function theorem uses the partial Jacobian with respect to the variables being solved for.
• Block notation makes comparative statics formulas easier to write and remember.

Examples¶

• For $F(x,y)=x^2+y^2-1$$1$, $D_xF=2x$ and $D_yF=2y$$2y$.
• For $\boldsymbol{F}(x_1,x_2,y)=(x_1+y,x_2y)$$y,x_{2}y)$, $\boldsymbol{D}_x\boldsymbol{F}=\begin{pmatrix}1&0\\0&y\end{pmatrix}$ and $\boldsymbol{D}_y\boldsymbol{F}=\begin{pmatrix}1\\x_2\end{pmatrix}$$\left(\begin{array}{c}1\\ x_2\end{array}\right)$.

Prerequisites¶

• Jacobian Matrix
• Matrix

Related Concepts¶

• Implicit Function Theorem
• Exogenous and Endogenous Variables
• Total Derivative as a Linear Map

Sources¶

• Math IV: Multivariate Calculus · Week 2, Lecture 2

Aliases: partial jacobian Keywords: Jacobian with respect to variables, block Jacobian Symbols: element of, real numbers, subset

Last updated: 22 October 2026 19:04