C&M − QM − TH 3
FINAL SOLUTIONS
Jérémie Gross
10/07/16
1
An account which pays 0.9% monthly corresponds to an Annual Percentage
Rate of 11.35% → I 1 better than I 2
2
A. 1149.32 ( if continuously = 1134.93) / B. 2980.91 (2959.3) / C. 45216.47
(44448.98)
3
Yes, NPV=50000
4
1. The slope at (x,y) of the level curve through (x,y) is
−fx ’(x,y)/fy ’(x,y)=−(4x+y)/(x+2y). Thus the slope at the tangent (2,0) to the level curve through (2,0) is −4. The equation of
the straight line through an arbitrary point (x0 ,y0 ) with slope m is
y−y0 =m(x−x0 ) so the equation of the tangent at (2,0) to the level curve
that passes through this point is y=−4x+8
2. The slope of a level curve is 0 if and only if 4x+y=0. The point (x,y)
is on the level curve for the value 8 if 2x2 +xy+y2 =8. The 2 equations
imply that 2x2 −4x2 +16x2 =8, or 7x2 =4, or x=(4/7)1/2 . Thus, the points
at which the slope of the level curve for the value 8 is 0 are (a,−4a) and
(−a,4a), where a=(4/7)1/2
5
p1 =97.6 / p2 =101.8
1
6
Hessian matrix at (x1 ,x2 ) is :
−5/3 1/2
(−8/3)x1 x2
−2/3 −1/2
2x1 x2
−2/3 −1/2
2x1 x2
1/3 −3/2
−3x1 x2
!
The leading principal minors are (-8/3)x1 −5/3 x2 1/2 < 0 and 8x1 −4/3 x2 −1 4x1 −4/3 x2 −1 =4x1 −4/3 x2 −1 > 0. Hence the Hessian is negative definite, so that f
is concave. Thus, given that the firm’s cost p1 x1 +p2 x2 is linear, the firm profit,
qf (x1 ,x2 )-p1 x1 -p2 x2 is concave.
7
1. Hessian matrix at (x1 ,x2 ) is :
−7/4 1/4
−(3/4)x1 x2
−3/4 −3/4
(1/4)x1 x2
−3/4
(1/4)x1 x2 −3/4
1/4 −7/4
−(3/4)x1 x2
!
The leading principal minors are -(3/4)x1 −7/4 x2 1/4 < 0 and
(9/16)x1 −3/2 x2 −3/2 -(1/16)x1 −3/2 x2 −3/2 =(1/2)x1 −3/2 x2 −3/2 >0. Hence
the Hessian is negative definite, so that f is concave. Thus, given that
the firm’s cost p1 x1 +p2 x2 is linear, the firm profit, qf (x1 ,x2 )-p1 x1 -p2 x2
is concave.
2. The FOC for a maximum of profit are
−3/4 1/4
x2
− p1 = 0
1/4 −3/4
− p2 = 0
qf10 (x1 , x2 ) − p1 = qx1
qf20 (x1 , x2 ) − p2 = qx1 x2
These equations have a unique solutions :
x∗1 = q 2 /(p31 p2 )1/2
x∗2 = q 2 /(p1 p32 )1/2
The objective function is concave, so this input combination globally
maximizes the firm’s profit.
2