1)  Consider the maximization problem of f(x; y) subject to the constraint g(x; y) = c. Assume that (x; y) is a global maximum. Then rf(x; y) = rg(x; y) where  is the lagrange multiplier.

2) Consider function f(x; y) and a given point (x0; y0). Assume that @f @y (x0; y0) 6=0. Then the gradient of f at x0; y0 is perpendicular to the level of curve of f going through(x0; y0).

3) Let f(x; y) = x2 y2 2xy x3. Then the greatest open set over which f is a concave function is X = f(x; y) 2 R2 : y  23g.

4) Consider the following function f(x; a), where x 2 Rn and a 2 R is a parameter. The solution to the unconstrained maximization problem of f is x(a) =(x1(a); ::::; xn(a)). The value function associated to the problem is f(a)  f(x(a); a).

Then, by the envelope theorem,
df(x(a); a)
da
=
Xn
i=1
@f(x(a); a)
@xi
(a)
dxi
(a)
da
+
@f(x(a); a)
@a
Exercise 2: Extrema I (20 points)
Let f(x; y; z) = x4 + x2y + y2 + z2 + xz + 1. Find the critical points of f and characterize them using the second order conditions.

Exercise 3: Gradient and Directional Derivative
Let f(x; y; ) = x3ey=x where e is the exponential function.
1) Compute the gradient of f at z = (2; 0). Compute the tangent plane of f at z. Next, suppose that, starting from z = (2; 0), x goes up by 1 and y goes up by 1 Estimate the corresponding change in the value of f using the tangent plane of f at z.

2) Compute the tangent plane to the level curve f(x; y) = 8 at the point (2; 0). Show that rf(2; 0) is perpendicular to this level curve. Next, if x goes up by 2, estimate the corresponding change in y along the level curve f(x; y) = 8 using the tangent plane at
(2; 0).

3) Compute the directional derivative of f at point z in the direction of the vector v = ( 1
3 ; 1

4 ). What is the direction of maximal increase of f at z? What is the maximal value of the directional derivative of f at z?
Exercise 4: Social Welfare

Society's welfare is given by u(x; z) = ln(1 + x)  ln(1 + z), where x 2 R2
+ is production,
z 2 R2
+ is pollution. Notice that @u
@z 6= 0 for any z  0. Finally, let z = h(x) = x2 + 1
where  > 0 is a parameter. Society's goal is to maximize social welfare.

1)  Find the optimal level of production x and deduce the optimal level of pollution z {Since  > 0, you can easily discard one of the two solutions you will nd.

2) How does the solution (x; z) and the welfare level change when  changes?

3) Find an expression for the slopes of the level curves of u(x; z), then compute

dz
dx for any (x; z) in the (x; z) plane as well as d2z
dx2 . Next, describe in the (x; z) plane the
level curves of u(x; z) as well as h(x). Explain the maximization problem and its solution
graphically.