Answer part d:

Hint: For d, when using the Hessian criterion, one of the numbers in the sequence requires the computation of the determinant of a 4 4 matrix. You can just show the matrix without computing its determinant.

' or CobbsDouglas Production Function with Three Inputs [50 points] In lecture 16, we have introduced the (lobbsDouglas function: suppose the output Q can be modeled by the amount of labor L and capital investment X such that our, L) = AKLl-a, where A and o are positive constants with {l f. c c: 1. Now, let's consider the output Q with three inputs: the capital investment m, amount of labor 3;, and materials and supply 2'. The new model becomes Q(w. to Z) = Awayszl, where A and o are positive constants with {l 1'. (2,3,1 c: 1 and o + +v= 1. Suppose that capital investment, labor and materials all have di'erent unit cost: ,0, o' and r respectively. The total amount of cost can be expressed by C(m, y. z} = pm + oy+ are. If we restrict the total amount of cost to be capped at positive constant M, we have Clix, y, z] :1 M. Since our domain is a closed and bounded region, by the extreme value theorem, the output must attain at least one absolute minimum and at least one absolute maximum. How do we nd these points? First, we divide the domain into two regions: the open set U = {[w, y, z] | at + cry + rz r. M} and the boundary set 3U = ame} | pa: + my + T2 = M}. By the rstderivative test1 we can locate the local extrema by locating the critical points of f. (a) (10 points) Locate all critical point(s} of 0(3, y, z} where the domain is the open set U. [b] (10 points} Use the secondderivative test to determine whether these critical point(s] is ,i'are mini mum, maximum, saddle point or degenerate. Now, consider the output Q when restricted to the boundary, that is the level set 3U = ag, 2'} | C (1:, y, z] = M}. Apply the method of Lagrange multiplier, we can locate all the critical points of Qlay. (c) (10 points) Locate all critical point(s} of Q[;r, y, a} restricted to the set 3U. [d] (20 points) Use the Hessian criterion for constrained extrema to determine whether these constrained critical points of Q|au are constrained minimum, constrained maximum, saddle point or degenerate. \fA x y B ZY Page 4 = _ M Putting value of I in (5), we get, Add ly z - SAXY PZY = 0 M 10 M 18 8 x = M M X = M f similarly, using valye of ) in (@ &D, we get, 7 - BM 6 and Z = YM I Thus ( x , 3 , 2 ) = am BM YM I is a critical point of a restricted to the set JU