Differentiating Implicit Models

Conceptual map

1. III-01.01Multivariable differentiation
2. III-01.02Implicit equilibrium maps
3. III-01.03Inverse-matrix derivatives
4. III-01.04Low-rank updates
5. III-01.05Regularity and branch failure

1. Setup and notation

An equilibrium calculation begins with equations and a reference solution. Comparative statics require an additional object: a locally selected equilibrium branch whose derivative is well defined. The Jacobian with respect to endogenous variables determines whether the equations supply that branch.

Let \(A\subset\mathbb{R}^{n}\) and \(\Theta\subset\mathbb{R}^{m}\) be open sets. The endogenous vector is \(a\in A\), and the parameter vector is \(\theta=(s,p)\in\Theta\), where \(s\) collects states and \(p\) collects policy variables. A system of equilibrium conditions is a map \(F:A\times\Theta\to\mathbb{R}^{n}\). Its equilibrium correspondence is

A reference pair \((a_{0},\theta_{0})\) satisfies \(F(a_{0},\theta_{0})=0\). Write \(J_{0}\) = \(D_{a}F(a_{0},\theta_{0})\) for the \(n\times n\) equilibrium Jacobian.

The square-system condition describes the coordinates chosen for the local calculation. Systems with redundant equations, free endogenous coordinates, or inequality constraints require a rank reduction, a parameterization of the solution set, or constrained differentiation.

2. Local existence and first-order comparative statics

What the proposition establishes

The theorem supplies one branch through the reference equilibrium and uniqueness among equilibria inside \(U\) for parameters inside \(V\). Other equilibria can exist outside \(U\). A global uniqueness claim needs additional structure such as global univalence, monotonicity, contraction, or a model-specific equilibrium argument.

Equation (3) is the computational form. A numerical routine should factor \(J_{0}\) and solve for the columns of \(Dg\). Forming \(J_{0}^{-1}\) explicitly creates extra floating-point error and discards useful matrix structure.

3. Second-order responses

Curvature determines how quickly a first-order response ceases to approximate a finite policy change. Suppose \(\theta\) is scalar and \(F\) is twice continuously differentiable. Let \(v\) = \(g^{\prime}(\theta_{0})\). The second derivative follows from differentiating the equilibrium identity twice.

For a vector parameter, (4) becomes a bilinear map. Given directions \(h\) and \(k\), compute \(Dg[h]\) and \(Dg[k]\), assemble the four second-derivative terms, and solve one more system with the same Jacobian factorization.

4. Conditioning and low-rank updates

The derivative magnitude depends on the economic right-hand side and the inverse Jacobian. For a subordinate matrix norm, equation (2) implies

A large condition number \(\kappa(J_{0})\) = \(\lVert J_{0}\rVert \lVert J_{0}^{-1}\rVert\) signals sensitivity of the linear solve to perturbations in the equations and their derivatives. Scaling matters. A report should state the norm and the units or normalization of each equation.

Equation (6) reuses the baseline factorization when a policy changes a few rows, columns, or interaction channels. The small \(r\times r\) matrix also detects failure of the update: its singularity coincides with singularity of the updated Jacobian under the stated conditions.

5. Worked nonlinear equilibrium

Consider two endogenous variables, quantity \(q\) and wedge \(w\), with policy \(p\) and state \(s\):

At (\(q_{0},w_{0}\)) = (1/5, 3/10), the compatible parameter values are (\(p_{0},s_{0}\)) = (29/100, 17/50). The equilibrium Jacobian and its determinant are

Holding \(s\) fixed, \(F_{p}\) = \((-1,0)^{\top}\). Equation (3) gives

Holding \(p\) fixed gives \([q_{s}, w_{s}]^{\top}\) = \([-15/19, 25/19]^{\top}\). For the policy derivative, the Hessian term in (4) is \((2w_{p}^{2}, 2q_{p}^{2})^{\top}\). Hence \(q_{pp}=13750/6859\approx2.004665\) and \(w_{pp}=-29250/6859\approx-4.264470\).

Finite-difference verification

For each step \(h\), solve the nonlinear system at \(p_{0}\) \(\pm\) \(h\) while holding \(s\) fixed, then form the central difference. The table reports the Euclidean error relative to (9). The near-singular comparison uses (\(q,w\)) = (0.49,0.49), where \(\kappa _{2}(J)\) = 99.

The benchmark exhibits the expected second-order decay of central-difference error until floating-point cancellation becomes visible. Near singularity, the derivative is roughly twenty times larger and a coarse step crosses substantial curvature. A derivative report without conditioning and a step-size grid conceals this distinction.

6. Failure at a singular equilibrium

The singular Jacobian blocks the implicit-function theorem for a unique two-parameter equilibrium selection. A differentiable symmetric selection still exists along the one-dimensional restriction \(p\) = \(s\). The example separates three claims: existence of a selected branch, regularity of the full equilibrium system, and uniqueness of the equilibrium correspondence.

7. Implementation protocol

The following procedure keeps the theorem, numerical solve, and diagnostic evidence aligned.

input: F, reference (a0, theta0), parameter direction h
1. verify ||F(a0, theta0)|| is below the declared residual tolerance
2. compute J = D_a F(a0, theta0) and B = D_theta F(a0, theta0)[h]
3. estimate reciprocal conditioning and record the matrix norm
4. solve J v = -B with a factorization; retain the residual ||J v + B||
5. solve F(a_plus, theta0 + delta h) = 0 from a0
6. solve F(a_minus, theta0 - delta h) = 0 from a0
7. compare v with (a_plus - a_minus)/(2 delta) over a step-size grid
8. repeat after rescaling equations when units differ materially
output: derivative, solve residual, condition estimate, finite-difference table

• Automatic differentiation evaluates \(J\) and \(B\); it does not supply equilibrium existence or branch uniqueness.
• Continuation methods reduce branch switching by using the previous solution as the next initial value and monitoring Jacobian singular values.
• Equation scaling changes numerical conditioning. Report the scaling whenever the rows of \(F\) use different units.
• A central difference tests the implemented derivative and nonlinear solver jointly. A residual check for each perturbed equilibrium identifies solver failure.

8. Exercises

1. For the benchmark in (7), derive both columns of \(Dg\) with respect to (\(p,s\)). Verify each column by multiplying it by \(J_{0}\).
   Partial solution

   The two columns are \((25/19, -10/19)^{\top}\) and \((-15/19, 25/19)^{\top}\). Their Jacobian products are \((1,0)^{\top}\) and \((0,1)^{\top}\), matching \(-F_{p}\) and \(-F_{s}\).

2. Use Proposition 2 to derive \(q_{pp}\) and \(w_{pp}\). Compare the quadratic approximation with the exact numerical equilibrium at \(p_{0}\) + 0.01.
   Partial solution

   The second derivatives are 13750/6859 and \(-29250/6859\). The approximation is \(a(p_{0}+\Delta)\approx a_{0}+a_{p}\Delta+\frac{1}{2}a_{pp}\Delta ^{2}\).

3. Prove the rank-one form of Proposition 3 directly by multiplying the proposed inverse by \(J_{0}\) + \(uv^{\top}\). State the scalar condition that permits the update.
   Partial solution

   The condition is \(1+v^{\top}J_{0}^{-1}u\ne0\). After expansion, the two rank-one correction terms cancel.

4. At the singular point in Section 6, draw the three equilibrium branches in (\(t,q\)) coordinates. Identify which derivative diverges as the two asymmetric branches approach \(t\) = 3/4.
   Partial solution

   The asymmetric branches satisfy \(q\) = 1/2 \(\pm\) \(\sqrt{t-3/4}\), so \(dq/dt\) = \(\pm[2\sqrt (t-3/4)]^{-1}\). Their slopes diverge at the branch point; the symmetric branch has slope 1/2.

References and source crosswalk

1. Walter Rudin, Principles of Mathematical Analysis, 3rd ed., §§9.26–9.29, pp. 224–227. Source for Proposition 1: finite-dimensional implicit-function theorem and derivative formula. Reference copy.
2. Martin J. Osborne, Mathematical Methods for Economic Theory, §2.3, Proposition 2.3.1. Economic comparative-statics formulation and worked implicit-derivative examples. Online text.
3. James McKernan, MIT 18.022, Lecture 13, “Implicit Functions.” Multivariable equation systems and the nonsingular endogenous-Jacobian condition. MIT OpenCourseWare.
4. William W. Hager (1989), “Updating the Inverse of a Matrix,” SIAM Review 31(2), 221–239, DOI 10.1137/1031049. Source for Proposition 3 and the history and uses of Sherman–Morrison–Woodbury updates. Author-hosted article.
5. E. Anderson et al., LAPACK Users’ Guide, 3rd ed., “Linear Equations” and “Accuracy and Stability.” Computational guidance for factor-and-solve methods, condition estimation, refinement, and error bounds. Netlib documentation.