Suggested solutions to problem set 2
1. Paraboloid symmetrical around origo with min f(x,y)= -10 at (x,y) = (0,0).
Circular level curves.
2.
a) y=(x/2)-(K/2) . Ex: y=(x/2)-1 and so on, straight parallel lines with slope 1/2
b) y=±√(x2-K) , defined for |x|≥√K . As two "cut off in half" horizontal parabolas
that meets on the x-axis at √K and - √K " . If K=9 they meet at 3 and -3 respectively
(becomes "wider and wider" apart when K increases)
c) y=K/x , ex. y=1/x . As (usual) indifference curves symmetrical around origo
3.
a)
fx= 4x3-3x2y+2xy2-y3 , fy= - x3+2x2y-3xy2+4y3
fxx= 12x2- 6xy+2y2 , fyy= 2x2-6xy+12y2 , fxy= fyx=-3x2+4xy -3y2
b) fx =2xexy+x2yexy, fy= x3exy
2 2
2
fxx= (2+4xy+x y ) exy , fyy= x4exy , fxy= fyx=(3x +x3y) exy
c) fx= 4x3 , fxx= 12x2 , fy= -16z , fyy= 0 , fz= -16y , fzz= 0 , fxy= 0 , fxz = 0 , fyz = -16
d) fK =αAKα-1Lβ,fL = βAKαLβ-1, fKK =α(α-1)AKα-2Lβ,
fLL = β(β-1)AKαLβ-2 , fKL =αβAKα-1Lβ-1
4.
a) x=3, y= -4 ,fxx= -2 < 0 , fxx fyy - (fxy)2= 4>0 → max
b) fx=2x+y = 0, fy=2y+x=0 → y=-2x → x=y=0 , fxx = 2>0 , fxx fyy - (fxy)2= 1>0 → min
c) x=y=0 , fxx = 2>0 , fxx fyy - (fxy)2= 3>0 → min
5
2
a) dy/dx = - (fx/fy) = -(3x +2y)/(x2-10) → dy/dx | (2,1) = 8/3
b) ∂z/∂x = - (Fx/Fz) = - 2x/2z . NB! Tangent to a circle in the xz-plane with
radius 1 and midpoint (0,0)
c) ∂z/∂x = - (Fx/Fz) = - (yz+z2-y2z5)/(xy+2xz-5xy2z4 )
6
Hint : calculate (∂w/∂x)(dx/dt) + (∂w/∂y)(dy/dt) !
7. 3(tx)2ty -(ty)3 = 3t3x2y + t3y3 = t3 ( 3x2y-y3) = t3f(x,y) QED
α αβ β
(α+β)
8. At x t y = t
f(x,y) , ( homogenous of degree α+β)
9. max L(x,y,λ) = 4xy- λ(1-x2-y2) → y=x=1/√2 . The rectangle with the greatest
area that fits into a circle with radius 1!
(α-1) (β)
(α) (β-1)
10-11 Hint1 : note that αx
y /βx y
= MRS = αy /βx = px/py
α β
Hint 2 : ln(Ax y ) = lnA + αlnx + βlnx (MRS unchanged!)
12 dY = (h'm'/(1-f'))dM - (f'/(1-f'))dT + (1/(1-f'))dG ! Look at the last term!
To totally differentiate the function means that you follow a generalized plane tangent
to the curved surface instead of following the surface itself (approximation!). If dM
and dT = 0 then we follow the plane in direction of G. Note the use of this method in
macroeconomics! (Multipliers)