Let the production function x = f(v1,v2) = be given. a) What is the exchange ratio dv2/dv1 of the factors of production if the production quantity x* = 1,000 is produced with v1* = v2* = 100 units? Give a brief justification for the plausibility of this value! (6 points)

Isoquant v2 = g(v1) = Exchange ratio dv2/dv1 = Justification: ___________________________________________________________________ (b) Output is to be maximized for a given cost budget. What is the optimal factor input ratio v2*/v1* for factor prices q1 = 20 and q2 = 5? Give a short justification for this! With which factor input quantities (v1*, v2*) is production carried out at a cost budget of 4,000? (6 points)

Optimal factor input ratio v2* / v1* = Justification:

Extreme value candidate (v1*, v2*) =. _________________________________________________________________ c) For the extreme value candidate from part b) the following Hessian matrix results:

Show that here the determinant criterion (DK) for testing the sufficient condition fails! What values must the second total differential d2L take for this extreme value candidate to actually be a maximum? Which direction dv2 = a . dv1 results from the constraint (NB)? (5 points)

DK fails because Necessary values of d2L for a maximum: NB: Direction according to NB: