ECON009: Advanced Mathematical Methods for Economics
Problem Set 3 (Lagrange Method: When it could fail)
Instructor: Mr. Rohit
Question 1: Violation of the 3rd condition of the Lagrange theorem
Consider the following constrained optimisation problem.
max − y subject to y3 = x2
or
min y subject to y3 = x2
(a) Find all points where the constraint qualification does not satisfy.
(b) Use the Lagrange method and check whether it gives any stationary points.
(c) Does any solution to the problem exist? If it does, write the solution.
(d) Explain the solution graphically by drawing level curves and the constraint equation.
Help: What do we learn from this question?
1. If the optimal point exists at a point where the constraint qualification fails, the Lagrange
method won’t fetch that point as its stationary point. Therefore, we always check the value of
the objective function at the points where the constraint qualification fails on the constraint set.
Question 2: Violation of the 3rd condition of the Lagrange theorem
Consider the following constrained optimisation problem.
max 2x 3 − 3x 2 subject to (3 − x)3 − y 2 = 0
(a) Find all points where the constraint qualification does not satisfy.
(b) Use the Lagrange method and check whether it gives any stationary points. Use a bordered
Hessian to determine local maximum and local minimum points.
(c) Does any global max point exist? If it does, write it.
(d) Explain the solution graphically by drawing level curves and the constraint equation.
Help: What do we learn from this question?
1. It is possible that while local optima exist for the problem but none of them is the global
optimum. This could be because, although a global optimum exists, constraint qualification is
violated at that point.
Question 3: Violation of the 2nd condition of the Lagrange theorem
Consider the following constrained optimisation problem.
max x 2 − y 2 subject to x +y =1
(a) Is there any point where the constraint qualification fails?
(b) Use the Lagrange method and check whether it gives any stationary points.
ECON009: Advanced Mathematical Methods for Economics
Problem Set 3 (Lagrange Method: When it could fail)
(c) Draw a graph of the level curves and the constraint equation.
(d) Does any solution to the problem exist? If it does, write the solution.
Help: What do we learn from this question?
1. Even if the constraint qualification holds everywhere on the constraint set, the Lagrange method
may fail to identify the optimal point because of the reason that an optimal point may not exist.
(i.e. violation of the 2nd condition of the Lagrange theorem)
Question 4: All three conditions are satisfied
Consider the following constrained optimisation problem.
max(min) x 2 − y 2 subject to x2 + y2 = 1
(a) Is there any point where the constraint qualification fails? If there is, is it on the constraint set?
Do you need to bother about this point?
(b) Use the Lagrange method and check whether it gives any stationary points. Use bordered
Hessian to determine local max and local min points.
(c) Determine if any of the local optimal points is a global optimal point. Use Hessain of the
Lagrange function.
(d) Explain the solution graphically by drawing level curves and the constraint equation.
Question 5: Violation of the 3rd condition of the Lagrange theorem
Consider the following constrained optimisation problem.
max y − 2x 2 + x subject to (x + y)2 = 0
(a) Find all points where the constraint qualification does not satisfy.
(b) Use the Lagrange method and check whether it gives any stationary points.
(c) Does any global max point exist? If it does, write it.
(d) Explain the solution graphically by drawing level curves and the constraint equation.
Help: What do we learn from this question?
1. The Lagrangean method could fail to give a solution because, although an optimum exists, the
constraint qualification is not satisfied at that point.
2. (Taken from question 3) The Lagrange method may fail to give the optimal point because an
optimal point may not exist.
Question 6: All three conditions are satisfied
Consider the following constrained optimisation problem.
max y − 2x 2 + x subject to x +y =0
ECON009: Advanced Mathematical Methods for Economics
Problem Set 3 (Lagrange Method: When it could fail)
(a) Is there any point where the constraint qualification fails?
(b) Use the Lagrange method and check whether it gives any stationary points. Use Bordered
Hessain to determine whether the stationary point is a local optimum.
(c) Use the Hessian of the Lagrangean to determine whether the stationary point is a global
optimum.
(d) Draw a graph of the level curves and the constraint equation.
Question 7: Violation of 1st condition of the Lagrange theorem.
max(min) x + y subject to x+ y=4
(a) Is there any point where the constraint qualification fails?
(b) Use the Lagrange method and check whether it gives any stationary points. Use bordered
Hessian to determine local max and local min points.
(c) Determine if any of the local optimal points is a global optimal point. Use Hessain of the
Lagrange function.
(d) Does any global max point exist? If it does, write it.
(e) Explain the solution graphically by drawing level curves and the constraint equation.
Help: What do we learn from this question?
1. The Lagrangean method could fail to give a solution because, although an optimum exists,
either the objective function or the constraint function is not differentiable at the optimal point
Note 1: This question is done in class.
Note 2: Try this question: max(min) x+ y subject to x +y = 4
Question 8: Wrong Understanding: The Lagrange method transforms a constrained
optimisation problem into one of finding an unconstrained optimum of the Lagrangean.
Consider the following constrained optimisation problem.
max (x − 1)2 − (y − 1)
2
subject to 2x + y = 3
(a) Is there any point where the constraint qualification fails?
(b) Use the Lagrange method and check whether it gives any stationary points. Use bordered
Hessian to determine local optimal points.
(c) Does the stationary point found in part (b) maximise the Lagrange function?
(d) Draw a graph of the level curves and the constraint equation.
(e) Does a global optimal point exist? If it does, write the solution.
Help: What do we learn from this question?
ECON009: Advanced Mathematical Methods for Economics
Problem Set 3 (Lagrange Method: When it could fail)
1. Second-order global sufficient conditions are sufficient, but not necessary. It means there might
be a stationary point which does not satisfy the second-order global sufficient conditions and is
still a global optimal point.
2. The stationary point which maximises the constraint optimisation problem may not maximise
the Lagrange function.
Note: Ex 18.2, question 4 is another example of such a wrong understanding.
Question 9: Local optima exist, but global optima don’t
Consider the following constrained optimisation problem.
1 3
max x 3 − y 2 + 2x subject to x −y =0
3 2
(a) Use the Lagrange method and check whether it gives any stationary points. Use bordered
Hessian to determine local optimal points.
(b) Use the Hessian of the Lagrangean to determine whether the stationary point is a global
optimum.
(c) Draw a graph of the level curves and the constraint equation.
(d) Does a global optimal point exist?
Help: What do we learn from this question?
1. It is possible that while local optima exist for the problem but none of them is the global
optimum. This could be because, although a global optimum exists, constraint qualification is
violated at that point or because a global optimum does not exist. Thus, even the existence of
local optima to the problem doesn’t enable us to make inferences about the existence of global
optima.
Question 10: Bordered Hessian determinant is 0
Consider the following constrained optimisation problem.
max(min) y − x 3 − 3x 2 subject to 3x + y = 1
(a) Solve the problem using the Lagrange method. Use the bordered Hessian to determine the local
optimum.
(b) Use the Hessian of the Lagrangean to determine whether the stationary point is a global
optimum.
(c) Draw a graph of the level curves and the constraint equation.
Help: What do we learn from this question?
1. If the Bordered Hessian determinant is 0, then the stationary point is ________________.
ECON009: Advanced Mathematical Methods for Economics
Problem Set 3 (Lagrange Method: When it could fail)
2. If the Hessian determinant of the Lagrangean is 0, then the stationary point is ________________
3. Bonus (2 marks):- Can you find a scope of improvement in the Lagrange theorem with the
help of this question? If yes, then mention the improvement. (Deadline: 18 August, 5 pm)
