Multivariate calculus · EC3304
The implicit
function theorem
Economic models rarely hand you a variable as a tidy formula. More often a relationship is written as an equation, \(f(y,x)=c\), and you want to know whether it secretly defines \(y\) as a function of \(x\), and how \(y\) responds when \(x\) moves. The implicit function theorem answers exactly this, and it is the engine behind comparative statics, the marginal rate of technical substitution, and the slope of any indifference or isoquant curve.
Motivation: when does an equation define a function?
Suppose two variables are tied together by a single equation
$$f(y,x)=0.$$
Generally this makes \(y\) and \(x\) implicitly related: for each \(x\) the equation pins down one or more values of \(y\), but without giving us a formula \(y=y(x)\). Three questions naturally follow:
- When can we find a genuine function \(y(x)\) satisfying \(f(y(x),x)=0\) near a point of interest?
- Is that function \(y(x)\) continuous and differentiable in \(x\)?
- If so, what is its derivative \(y'(x)\)?
The implicit function theorem gives conditions under which all three hold, and hands us the derivative for free.
The theorem (single equation)
Let \(f(y,x)\) be a \(C^1\) function on a ball about \((y_0,x_0)\) in \(\mathbb{R}^2\) with
$$f(y_0,x_0)=c \qquad\text{and}\qquad \frac{\partial f(y_0,x_0)}{\partial y}\neq 0.$$
Then there exists a \(C^1\) function \(y=y(x)\), defined on an interval \(I\) about \(x_0\), such that
$$f(y(x),x)=c \text{ for all } x\in I,\qquad y(x_0)=y_0,\qquad y'(x_0)=-\frac{\;\partial f(y_0,x_0)/\partial x\;}{\partial f(y_0,x_0)/\partial y}.$$
A function is \(C^1\) if all of its first-order partial derivatives exist and are continuous. The single decisive condition is that the partial derivative with respect to the variable you want to solve for, here \(\partial f/\partial y\), is non-zero at the point. Geometrically this rules out the degenerate case where the level curve is momentarily vertical, so \(y\) cannot be written as a single-valued function of \(x\). (This result is Theorem 15.3 in Simon & Blume.)
The derivative formula is the workhorse. It comes straight from differentiating the identity \(f(y(x),x)=c\) with respect to \(x\) using the chain rule:
$$\frac{\partial f}{\partial y}\,y'(x)+\frac{\partial f}{\partial x}=0 \quad\Longrightarrow\quad y'(x)=-\frac{\partial f/\partial x}{\partial f/\partial y}.$$
Worked example: the marginal rate of technical substitution
Consider the Cobb–Douglas production function \(f(k,l)=20\,k^{1/4}l^{3/4}\), where \(k\) is capital and \(l\) is labour. Along an isoquant, output is fixed at \(f(k,l)=c\), so capital and labour are implicitly related. The marginal rate of technical substitution is the slope of that isoquant, \(dk/dl\), and by the implicit function theorem it equals
$$\mathrm{MRTS}_{k,l}=\frac{dk}{dl}=-\frac{\partial f/\partial l}{\partial f/\partial k}.$$
Computing the two marginal products,
$$\frac{\partial f}{\partial k}=5\,k^{-3/4}l^{3/4},\qquad \frac{\partial f}{\partial l}=15\,k^{1/4}l^{-1/4},$$
so that
$$\mathrm{MRTS}_{k,l}=-\frac{15\,k^{1/4}l^{-1/4}}{5\,k^{-3/4}l^{3/4}}=-\frac{3k}{l}.$$
The familiar microeconomics result, that the isoquant slope equals minus the ratio of marginal products, is nothing more than the implicit function theorem applied to the production relationship.
The gradient is perpendicular to the level set
A closely related geometric fact underlies the theorem. Let \(f\) be \(C^1\) on a neighbourhood of \((y_0,x_0)\), and suppose \((y_0,x_0)\) is a regular point (the gradient is non-zero there). Then the gradient vector \(\nabla f(y_0,x_0)\) is perpendicular to the level set of \(f\) at that point. Because the gradient points in the direction of steepest ascent, moving along a level curve, where \(f\) does not change, must be at right angles to it. This is why the derivative formula above can be read as "the level curve has a well-defined tangent whenever the gradient is non-zero in the relevant direction".
Systems of equations and the Jacobian
Economic models usually involve several equations and several endogenous variables at once. Write the system as
$$f_1(y_1,\dots,y_m;\,x_1,\dots,x_n)=c_1,\ \dots,\ f_m(y_1,\dots,y_m;\,x_1,\dots,x_n)=c_m,$$
or compactly \(\mathbf f(\mathbf y;\mathbf x)=\mathbf c\). Here the \(\mathbf y\) are endogenous variables (those the model determines) and the \(\mathbf x\) are exogenous variables (parameters or inputs treated as given). The distinction is a modelling choice rather than anything intrinsic to the maths.
A closed-form solution \(\mathbf y=\mathbf y(\mathbf x)\) is usually impossible for non-linear systems. But the system may still implicitly define such functions, and that is often all we need, for instance to know how equilibrium quantities \(\mathbf y\) respond to a change in a policy parameter \(\mathbf x\) at a particular point. The tool is the Jacobian matrix, the matrix of first-order partial derivatives:
$$J^f_{\mathbf y}=\begin{pmatrix}\dfrac{\partial f_1}{\partial y_1}&\cdots&\dfrac{\partial f_1}{\partial y_m}\\[6pt]\vdots&&\vdots\\[4pt]\dfrac{\partial f_m}{\partial y_1}&\cdots&\dfrac{\partial f_m}{\partial y_m}\end{pmatrix},\qquad J^f_{\mathbf x}=\begin{pmatrix}\dfrac{\partial f_1}{\partial x_1}&\cdots&\dfrac{\partial f_1}{\partial x_n}\\[6pt]\vdots&&\vdots\\[4pt]\dfrac{\partial f_m}{\partial x_1}&\cdots&\dfrac{\partial f_m}{\partial x_n}\end{pmatrix}.$$
For the system \(\mathbf f(\mathbf y;\mathbf x)=\mathbf c\), if the Jacobian with respect to \(\mathbf y\), \(J^f_{\mathbf y}\), is invertible at the point, then the endogenous variables are locally \(C^1\) functions of the exogenous variables, and the matrix of comparative-statics derivatives is
$$J_{\mathbf x\mathbf y}=\begin{pmatrix}\partial y_1/\partial x_1&\cdots&\partial y_1/\partial x_n\\ \vdots&&\vdots\\ \partial y_m/\partial x_1&\cdots&\partial y_m/\partial x_n\end{pmatrix}=-\big[J^f_{\mathbf y}\big]^{-1}J^f_{\mathbf x}.$$
Notice this is the exact matrix analogue of the single-equation formula \(y'=-\,(\partial f/\partial x)/(\partial f/\partial y)\): dividing by the derivative becomes multiplying by the inverse Jacobian. As a sanity check, take the linear system \(\mathbf y=A\mathbf x\), i.e. \(\mathbf f(\mathbf y;\mathbf x)=\mathbf y-A\mathbf x=\mathbf 0\). Then \(J^f_{\mathbf y}=I\) and \(J^f_{\mathbf x}=-A\), so the theorem returns \(\partial y_i/\partial x_j=a_{ij}\), exactly as expected.
Worked example: comparative statics from a non-linear system
Find \(dy_1/dx\) and \(dy_2/dx\) from the system (with \(x>0\))
$$x-3y_1^{2}+2y_2=0,\qquad e^{y_2}-2y_1+\ln x=0.$$
The Jacobian with respect to \((y_1,y_2)\) is
$$J^f_{\mathbf y}=\begin{pmatrix}-6y_1 & 2\\ -2 & e^{y_2}\end{pmatrix},\qquad \det J^f_{\mathbf y}=-6y_1e^{y_2}+4.$$
This is invertible, so the theorem applies, precisely when \(-6y_1e^{y_2}+4\neq 0\), i.e. when \(3y_1e^{y_2}\neq 2\). Now differentiate each equation totally in \(x\) (treating \(y_1,y_2\) as functions of \(x\)):
$$dx-6y_1\,dy_1+2\,dy_2=0,\qquad e^{y_2}\,dy_2-2\,dy_1+\tfrac{1}{x}\,dx=0.$$
Writing this as a linear system in \((dy_1,dy_2)\),
$$\begin{pmatrix}-6y_1 & 2\\ -2 & e^{y_2}\end{pmatrix}\begin{pmatrix}dy_1\\ dy_2\end{pmatrix}=\begin{pmatrix}-dx\\ -\tfrac{1}{x}\,dx\end{pmatrix},$$
and solving by Cramer's rule gives
$$\frac{dy_1}{dx}=\frac{xe^{y_2}-2}{2x\,(3y_1e^{y_2}-2)},\qquad \frac{dy_2}{dx}=\frac{x-3y_1}{x\,(3y_1e^{y_2}-2)}.$$
Even though we could never solve the original system for \(y_1\) and \(y_2\) explicitly, we have their exact rates of change with respect to \(x\) at any admissible point.
Why this matters in economics
The implicit function theorem is the mathematical backbone of comparative statics: how do equilibrium prices, quantities or choices shift when a parameter changes? Equilibrium conditions (market clearing, first-order conditions, budget constraints) are typically a system \(\mathbf f(\mathbf y;\mathbf x)=\mathbf c\) that cannot be solved explicitly. The theorem lets us differentiate the equilibrium implicitly and read off signs and magnitudes of responses, exactly what is needed to sign the effect of a tax, a shift in demand, or a change in income. The same machinery gives the slope of indifference curves (marginal rate of substitution) and isoquants (marginal rate of technical substitution), both of which are ratios of partial derivatives delivered by the theorem.
Check your understanding
Use the implicit function theorem to find \(dy/dx\) for \(x^2+y^2=25\) at the point \((3,4)\).
Set \(f(y,x)=x^2+y^2\) with \(c=25\). The partial derivatives are \(\partial f/\partial x=2x\) and \(\partial f/\partial y=2y\). At \((y,x)=(4,3)\) we have \(\partial f/\partial y=8\neq0\), so the theorem applies and
$$\frac{dy}{dx}=-\frac{\partial f/\partial x}{\partial f/\partial y}=-\frac{2x}{2y}=-\frac{x}{y}=-\frac{3}{4}.$$
The condition \(\partial f/\partial y\neq0\) fails at \((\pm5,0)\), the two points where the circle is vertical and \(y\) cannot be written as a single function of \(x\), exactly the cases the theorem excludes.
For a utility function \(u(x_1,x_2)\), show that the slope of an indifference curve is minus the ratio of marginal utilities.
An indifference curve is a level set \(u(x_1,x_2)=\bar u\). Treating \(x_2\) as an implicit function of \(x_1\) along the curve, the implicit function theorem gives
$$\frac{dx_2}{dx_1}=-\frac{\partial u/\partial x_1}{\partial u/\partial x_2}=-\frac{MU_1}{MU_2}.$$
The magnitude \(MU_1/MU_2\) is the marginal rate of substitution: the amount of good 2 the consumer will give up for one more unit of good 1 while staying on the same indifference curve. The requirement \(\partial u/\partial x_2\neq0\) is just local non-satiation in good 2.
When does the general (system) version of the theorem fail to apply, and what does that mean economically?
It fails when the Jacobian with respect to the endogenous variables, \(J^f_{\mathbf y}\), is singular (\(\det J^f_{\mathbf y}=0\)). Economically, at such a point the equilibrium conditions do not locally pin down the endogenous variables uniquely as smooth functions of the parameters: the system may have no local solution, multiple solutions, or a solution that fails to vary smoothly. Comparative statics is then not well defined there, which is often a signal of a bifurcation, a knife-edge case, or an indeterminacy in the model.
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These notes cover the implicit function theorem strand of the multivariate calculus material. The complete slide deck also includes the Hessian, concavity, Taylor series and integration.
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