Sem#10: brief hints & a clarification
- Compendium 2-01:
Use that strictly increasing transformations preserve quasiconcavity/quasiconvexity. - For problem 3 on the 2011 exam
- Problem 3a: exp(quadratic form). exp is an increasing transformation and preserves quasiconcavity/quasiconvexity.
- Problem 3c, clarification: The constraint qualification has not been given on any other ordinary exam than this, and a sufficient answer would be
"the constraint qualification must fail at the optimum"
- that is, given the question formulation you would not have to verify that by computing gradients. The question basically does not ask you to apply the CQ, just asks for familiarity with the concept. (You might however note if you sketch the the admissible set, that it ends in a tip where the two curves are tangent - the CQ always fails at such a point.)
- Elasticity of substitution >0 everywhere:
The form isF′KF′LKL⋅KF′K+LF′LB where B is the bordered Hessian determinant. For this to be positive, B must be positive (this because we are assuming F nondecreasing in each variable!) and since we have two variables, B>0 means (strict) quasiconcavity.
- The CES functions:
F(x)=A⋅(n∑i=1aixρi)q/ρ (incl. Cobb-Douglas)
The point is that a quasiconcave positive function that is homogeneous of order k in (0,1], is concave. Here k = q.