Consider the following Lagrange problem: Maximise f(u,v) = u + 3v subject to g(u,v) = u² + av² = 10. Use the envelope theorem… [Full question continued]

Part a) Lagrange Problem and Envelope Theorem

Given:

• Maximize: f(u, v) = u + 3v
• Subject to: g(u, v) = u² + av² = 10
• We are told to apply the Envelope Theorem and estimate f* when a = 1.01

Step 1: Lagrangian Function

L(u, v, λ) = u + 3v + λ(10 − u² − av²)

Step 2: First-Order Conditions

• ∂L/∂u = 1 − 2λu = 0 → λ = 1/(2u)
• ∂L/∂v = 3 − 2λav = 0 → λ = 3/(2av)
• Equating both expressions for λ:

1/(2u) = 3/(2av) → v = 3au

Step 3: Plug into Constraint

u² + a(3au)² = 10

u² + 9a²u² = 10 → u²(1 + 9a²) = 10

u² = 10 / (1 + 9a²) → u = √[10 / (1 + 9a²)]

Step 4: Calculate f*(a)

v = 3au → f(u,v) = u + 3v = u + 3(3au) = u(1 + 9a)

So, f*(a) = √[10 / (1 + 9a²)] * (1 + 9a)

Step 5: Estimate f*(1.01)

• a = 1.01 → a² = 1.0201
• Denominator = 1 + 9(1.0201) = 10.1809
• Numerator = 1 + 9(1.01) = 10.09
• f*(1.01) ≈ √(10 / 10.1809) * 10.09
• √(0.9822) ≈ 0.99106
• Estimated f* ≈ 0.99106 × 10.09 ≈ 10.00

Envelope Theorem Check:

Envelope Theorem tells us:

df*/da = −λ * ∂g/∂a = −λ * v²

Since v = 3au and u = √[10 / (1 + 9a²)], this can be verified by substitution if exact change is required.

Part b) Maximize f(u;a) = −u² + 2au + 4a

Given:

• f(u; a) = −u² + 2au + 4a
• We need to maximize over u, where a > 0

Step 1: First Order Condition

∂f/∂u = −2u + 2a = 0 → u = a

Step 2: Plug back into f(u; a)

f(a; a) = −a² + 2a² + 4a = a² + 4a

Thus, optimal value of function:

f* = a² + 4a

Implications:

• The function increases with increasing a
• As a → ∞, f* → ∞
• So maximum increases with a for a > 0

Conclusion

We solved two Lagrange problems — one using Envelope Theorem to estimate the optimal value with a slight change in parameter a, and the second by directly optimizing a function with respect to u. These techniques are common in economic optimization and sensitivity analysis. Envelope Theorem helps simplify comparative statics when a constraint changes slightly, while direct optimization gives closed-form solutions under known conditions.