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University of California, Berkeley Economics 204–First Midterm Test Tuesday August 25, 2003; 6-9pm Each question is worth 20% of the total Please use separate bluebooks for Parts I and II
Part I
- Prove that (^) n ∑
k=
k =
n(n + 1) 2
- Consider the function
f (x, y) = ex
(^2) − 6 xy+y 2
Recall that (^) dzd ez^ = ez^.
(a) Compute the first order conditions for a local maximum or minimum of f. Find the unique (x∗, y∗) at which the first order conditions are satisfied. (b) Find the second order Taylor series expansion of f at the point (x∗, y∗) determined in part (a); your answer should in- volve a symmetric matrix A representing the quadratic terms in the expansion. (c) Diagonalize the matrix A you found in part (b). Find an or- thonormal basis {v 1 , v 2 } of R^2 such that the quadratic terms of the Taylor expansion can be written as
g((x∗, y∗) + γ 1 v 1 + γ 2 v 2 ) = λ 1 (γ 1 )^2 + λ 2 (γ 2 )^2
Use this information to determine whether f has a local max- imum, a local minimum, or neither, at (x∗, y∗) and to describe the level sets of f near (x∗, y∗).
- Give a proof that does not involve sequential compactness of the following Theorem: Let (X, d) and (Y, ρ) be metric spaces, and f : X → Y a continuous function. If C is a compact set in (X, d), then f (C) is compact in (Y, ρ).
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Part II
- Let C([0, 1]) denote the set of real-valued, continuous functions from [0, 1] to R, and consider the metric
d(f, g) = sup{|f (t) − g(t)| : t ∈ [0, 1]}
Let X = {f ∈ C([0, 1]) : sup{|f (t)| : t ∈ [0, 1]} ≤ 1 } Show that X is not compact.
- Consider the parametrized utility function
u : R^2 ++ × R^2 ++ → R, u(x, ω) = ω 1 x 1 + ω 2 x 2 + x 1 x 2
Here, R^2 + = {x ∈ R^2 : x 1 ≥ 0 , x 2 ≥ 0 }, R^2 ++ = {x ∈ R^2 : x 1 > 0 , x 2 > 0 }, x = (x 1 , x 2 ) denotes a consumption vector in R^2 ++, ω = (ω 1 , ω 2 ) denotes a parameter vector in R^2 ++, and I ∈ R++ denotes income.
(a) Define demand Z : R^2 ++ × R++ × R^2 ++ → R^2 + by Z(p, I, ω) maximizes u(x, ω) subject to x ∈ R^2 +, p · x = I. You may assume without proof that Z(p, I, ω) is uniquely defined. As- suming that Z(p, I, ω) ∈ R^2 ++, write down the first order con- ditions for the maximization problem that defines Z(p, I, ω). (b) Find a function
F : R^2 ++ × R^2 ++ × R++ × R^2 ++ → R^2
such that if Z(p, I, ω) ∈ R^2 ++, Z(p, I, ω) satisfies F (x, p, I, ω) = 0, i.e. F (Z(p, I, ω), p, I, ω) = 0. Hint: The first component of F should encode the Lagrange multiplier condition, and the second component should encode the budget constraint, that you found in part (a). (c) Use the Implicit Function Theorem to show that if Z(p∗, I∗, ω∗) ∈ R^2 ++, then there is an open set U containing (p∗, I∗, ω∗) such that Z is a C^1 function on U. Compute the Jacobian matrix DZ.
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