Consider the minimization problem min rK + wL (K,L)ER? subject to Y = VK +VI, where w, r, Y > 0. 1. Apply the theorem of Lagrange to find necessary conditions for a minimum (ignoring the constraint qualification). Find the candidate solution: K = ab sin (a) OO a ? ab sin (a) OC a ? P , L =2. Find the Lagrange multiplier for the problem (and explain to yourself what you can infer from it economically). 1 = sin (a) OO 3. Through the relevant second-order condition, show that this candidate is indeed a solution to the minization problem. For this part, denote f(K, L) = rK + wL and g(K, L) = Y - VK - VI. 3.a. The space of functions orthogonal to Dg(a* ) is O Z E R2 : 2 - C [ W] , CER } O {ZER : 2 = C [ Y/W] , CER} O {ZERO : 2 - C [ ], CER }3.b. The matrix D2 L* = D2 f(K*, L*) + 1* D2g(K*, L*) is: O positive definite, and therefore z'D2 L*> > 0 for all z E Z(K*, L*) so that (K* , L*) is a local minimiser. It is a global minimiser because f is convex. not positive definite, although z'D2 L* > > 0 for all z E Z (K*, L*) so that (K* , L*) is a local minimiser. It is a global O minimiser because f is convex. O positive definite, and therefore z'D2 L* > > 0 for all z E Z (K*, L*) so that (K*, L* ) is a local minimiser. It is a global minimiser because f is strictly convex. not positive definite, although z'D2 L* > > 0 for all z E Z (K*, L*) so that (K* , L* ) is a local minimiser. It is a global O minimiser because f is strictly convex