Calculator Inputs
Use the evaluated partial derivatives at the optimum. For constrained problems, the page applies the Lagrangian form automatically.
Example Data Table
This sample shows a constrained case where the theorem matches a centered finite-difference estimate.
| Scenario | Mode | θ | x* | V(θ) | fθ | λ* | gθ | h | V(θ+h) | V(θ-h) | dV/dθ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Sample A | Equality constrained | 4.0000 | 2.5000 | 30.0000 | 6.2000 | 1.1000 | -0.4000 | 0.5000 | 32.8800 | 27.1200 | 5.7600 |
| Sample B | Unconstrained | 3.0000 | 1.8000 | 18.5000 | 2.9000 | N/A | N/A | 0.2500 | 19.2250 | 17.7750 | 2.9000 |
Formula Used
1) Unconstrained optimization
Problem: Maximize or minimize f(x, θ) with optimizer x*(θ).
Envelope theorem: dV/dθ = ∂f(x*(θ), θ)/∂θ
The derivative of the value function ignores the indirect effect from dx*/dθ under regularity conditions.
2) Equality-constrained optimization
Problem: Optimize f(x, θ) subject to g(x, θ)=0.
Lagrangian: L(x, λ, θ)=f(x, θ)+λg(x, θ)
Envelope form: dV/dθ = ∂L/∂θ = fθ + λ*gθ
This calculator applies that expression when the constrained mode is selected.
3) Numerical verification
Forward difference: [V(θ+h) - V(θ)] / h
Backward difference: [V(θ) - V(θ-h)] / h
Centered difference: [V(θ+h) - V(θ-h)] / (2h)
4) Elasticity
Elasticity = (dV/dθ) × (θ / V(θ))
How to Use This Calculator
- Select whether your optimization problem is unconstrained or equality constrained.
- Enter the current parameter value, the optimal decision, and the current value function level.
- Provide the direct partial derivative with respect to the parameter, evaluated at the optimum.
- For constrained cases, also enter the multiplier and the parameter derivative of the active constraint.
- Enter a finite-difference step and optional nearby value-function observations for verification.
- Click Calculate to display the result block above the form.
- Review the theorem-based derivative, numerical checks, elasticity, and local approximation graph.
- Use the CSV or PDF buttons to export a quick report.
Useful Notes
Value function sensitivity Lagrangian derivative Constraint shadow effect Finite-difference check Local approximation Optimization analysisFAQs
1) What does this calculator measure?
It estimates how the optimized objective value changes when one parameter changes slightly. The main output is the derivative of the value function with respect to that parameter.
2) Why does the theorem ignore the derivative of the optimizer?
At the optimum, first-order conditions make the indirect effect vanish under standard assumptions. That is why only the direct parameter effect remains in the value derivative.
3) When should I use constrained mode?
Use constrained mode when the problem has an active equality constraint and you know the multiplier and the parameter derivative of that constraint.
4) What does the multiplier contribution mean?
It measures how the constraint changes the value sensitivity. A nonzero multiplier can raise or lower the total derivative depending on the sign of the constraint derivative.
5) Why compare with finite differences?
Finite differences provide a numerical reality check. If the theorem-based derivative and the numerical estimate are close, your local approximation is usually more credible.
6) What if my centered difference is far away?
Check the step size, derivative inputs, active constraints, and whether the optimum changes sharply near the chosen parameter value. Poor local smoothness can also matter.
7) Can this page solve the optimization problem itself?
No. It evaluates sensitivity after you already know the optimal point, the direct partial derivative, and, when needed, the multiplier and constraint derivative.
8) Is the graph a global solution curve?
No. The graph is a local linear approximation around the current parameter value. It is most useful for small parameter changes, not large jumps.