Assignment:
1. Find and categorize the extreme values of the following functions as local maximum, local minimum, or, inflection point. Then, use the concepts of concavity and convexity to discuss the global uniqueness of the extreme values.
(a) f(x) = 5x2 + x
(b) f(x) = 13x3 - x2 + x + 10
(c) f(x, y) = 3x2 - 4xy + 7y2
(d) f(x, y, z) = -x3 + 3xz + 2y - y2 - 3z2
(optional)
You can determine if (c) and (d) are concave using the Hessian. For (c) you can also check the signs of fxx, fyy, and fxxfyy - (fxy)2 as described in the text. See the chapter extensions for checking the Hessian.
2. Sketch a plot of the upper contour set of the following functions. Using the graph, discuss the relation between the convexity of the upper contour set (defined by the level curves) of the function and the concept of quasi-concavity.
(a) f(x, y) = √xy
(b) f(x, y) = x2 + y2
(c) f(x, y) = -(x + y)
(d) f(x, y) = ax + by a, b > 0
(e) f(x, y) = min(ax, by) a, b > 0
(f) f(x, y) = max(ax, by) a, b > 0
3. Use Lagrange's method to solve the following optimization problems with equality constraints. Optional: Use the second order conditions to verify that the solution is actually a maximum/minimum (this step requires checking the bordered Hessian - see chapter extensions.)
(a) Max x,y∈R2 √ xy
subject to x + 2y = 10
(b) Max x,y∈R2 x + 5 ln(y)
subject to x + y = k k ≥ 0
(c) Min x,y∈R2 ax + by
subject to xy = 100 a, b ≥ 0
(d) Min x,y∈R2 x + y
subject to ax + by = M a, b, M ≥ 01
4. Consider a farmer that wants to maximize the area of a rectangular fenced surfaced. The farmer has a limited budget for fencing (M ≥ 0) and each foot of fence cost the farmer p > 0 dollars.
(a) Assume the constraint binds. Write this as an optimization problem with an equality constraint, and write the corresponding Lagrangian.
(b) Find the first order conditions.
(c) Find the extreme value of this problem as a function of the budget M and the price p.
(d) Use the envelope theorem to show that the effect of an increase in (i) the budget M, and (ii) the price p, on the area of the fenced surface.
5. An agent lives on an island where she produces two goods, x and y. Her production possibility frontier is x2 + y2 ≤ 200, and she consumes all goods herself. Her utility function is xy3. The consumer also faces an environmental constraint on her total output of both goods. The environmental constraint is given by x + y ≤ 20.
(a) Write the Kuhn-Tucker first order conditions. (Hint: There are two choice variables each with an associated first order conditions, and two constraints each with an associated first-order condition, and two complementary slackness conditions.)
(b) Find the optimal x and y. Determine which restrictions are binding. (Hint: The easiest way to determine which constraints are binding is two solve each of the four cases. (i) Both constraints bind - or a = b = 0, (ii) only the first constraint binds - or a = 0, (iii) only the second constraint binds - or b = 0, and (iv) neither constraint binds - so λ1 = λ2 = 0.)
(c) Use the envelope theorem to show by how much the utility function increases if the restriction of x2 + y2 ≤ 200 is relaxed and changed to x2 + y2 ≤ 300.
6. Prove the following statements. (Hint: Start by reminding yourself of the relevant formal definitions. What is a concave function? What is a quasi-concave function? What does homogeneous of degree α mean? What is a concave function of two variables? What is Euler's theorem?)
(a) If a function f(·) is concave, then the function is also quasi-concave.
(b) For the function f(x) = xα , where α ∈ [0, 1], (i) show that f(x) is concave, (ii) show that the function g(x1, x2, ...xn) = Σnif(xi) is concave, and, (iii) show that g(x1, x2, ...xn) = Σni f(xi) is homogenenous of degree α.
(c) Given the Cobb-Douglas function, f(x, y) = xα yβ , where α, β ∈ (0, 1), then (i) if α + β < 1 then the function is concave, and (ii) if α + β > 1 then the function is not concave.
(d) The function f(x1, x2, ...xn) =IIni xiαi is homogeneous of degree Σni αi.
(e) The function f(x, y) = xα y1-α , where α ∈ [0, 1], is homogeneous of degree one, satisfies Euler's theorem (i.e., xfx(x, y) + yfy(x, y) or x · ∇f(x, y) = f(x, y) where x denotes the vector (x, y)).