1) (10 points) 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) (10 points) 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) (10 points) 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 2 3g. 4) (10 points) 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); ::::; x n(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. 1 Exercise 3: Gradient and Directional Derivative (20 points) Let f(x; y; ) = x3ey=x where e is the exponential function. 1) (8 points) 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 2 . Estimate the corresponding change in the value of f using the tangent plane of f at z. 2) (8 points) 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) (4 points) 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 (35 points) 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) (10 points) 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) (10 points) How does the solution (x; z) and the welfare level change when changes? 3) (15 points) 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. 2