as each term is positive.

(d) JX2) = A[(S(iJf|.)-'" + (1 -¿)(.vA-2rrrl/r

= Sf(X\ ■ Xi) and so /' is homogeneous of degree I.

sax; ' + t i -«)%'] + (i -¿»/txr^r'H-d -A'W] = A [¿A-,-' + ( I - + < I -

which, since / is homogeneous of degree one. corresponds to Euler's theorem

Chapter 12

12.1 Exercises

(h) d/S. 0.-5/2) (i) (0.0) and d. I) (j) (0,0), (i/2, —x/2) and (—\/2, sfl)

3. />, = 60 : q\ = It). <fl = 20, />: = 80

5. Cournot: </' = 33.33, qj = 30.83. p' = 3.58. Joint-profit maximization: q\ = 48.75, q' = 0, p' = 5.13. Profits increase.

12.2 Exercises

(d) maximum

(c) saddle point

(f) maximum

(g) maximum

(h) maximum

(i) (0,0): neither an extremum nor saddle poini (I. I) : minimum

(j) (ft, 0) ; neither extremum nor saddle point ('1/2, — ^2) : maximum (-V2. J2): maximum

3. The Hessian matrices

are negative delinile. 5. K' = L" = 40%. The Hessian matrix

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