Assignment
Due at 5:30pm of October 27 (Friday)
This assignment is related to Chapter 1-5. You can use any formula taught in class without proof but cannot use formulas from other courses.
1. (15 points) This problem is related to point-set topology.
(i) (5 points) Let a set E ≤ R deÖned as {1; 
; 
; . . . ; 
; . . . }. What are E , Eo , @E, the limit points and the isolated points?
(ii) (5 points) For two sets A and B, suppose A U B is open. Is A necessarily open? Is Ac ∩ Bc necessarily closed? Justify your answer.
(iii) (5 points) Is the set {0; 1; 
; 
; . . . ; 
; . . . } compact as a set in R?
2. (20 points) This problem is related to calculus.
(i) (5 points) Let f : [0; 1] U (2; 3] → [0; 2] deÖned as
What is f-1 (y), y ∈ [0; 2]? Is f-1 a function or not? Is f continuous? Is f-1 continuous? Justify your answer.
(ii) (5 points) Let f : R → R be given by
Show that f(x) is di§erentiable but is not continuously di§erentiable on R+ 三 [0; ∞). (iii) (5 points) Suppose the cdf of a r.v., F (x) : R → (0; 1), is a strictly increasing C1
function, so we can deÖne its inverse function, say Q(T), T ∈ (0; 1). Find the derivative of Q(T) at any T ∈ (0; 1).
(iv) (5 points) Let f : R2 → R be given by f(0; 0) = 0, and for (x; y) 
0,
Show that f is partially di§erentiable at (0; 0) (i.e., 
(0; 0) and 
(0; 0) exist) but not continuous at (0; 0).
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3. (20 points) This problem is related to nonlinear programming.
An agent allocates the H hours of time available to her between labor (l) and leisure (H — l). Her only source of income is from the wages she obtains by working. She earn w per hour of labor; thus, if she works l ∈ [0; H] hours, her total income is wl. She spends her income on food (f) and entertainment (e), which cost p and q per unit respectively. Her utility function is given by u(f; e; l) and is increasing in f and e, and is decreasing in l.
(i) (5 points) Describe the consumerís utility maximization problem.
(ii) (5 points) Describe the equations that deÖne the critical points of the Lagrangian.
(iii) (10 points) Assuming H = 16; w = 3; and p = q = 1, Önd the utility-maximizing consumption bundle if u(f; e; l) = f1=3 e1=3 — l2 .
4. (15 points) This problem is related to convex sets and convex functions.
(i) (2 points) Give an example of an open set in R that is not convex.
(ii) (3 points) Give an example of concave function that is not continuous.
(iii) (5 points) Is f (x) = x+ xx+ x —3×1 —8×2 a concave or convex function or neither?
Justify your answer.
(iv) (5 points) Show that a function f : Rn → R is convex if and only if for each x; x0 ∈ Rn , the function ‘ : [0; 1] → R deÖned by
‘(t) = f (tx + (1 — t)x0 )
is convex on [0; 1]. Is the above statement still true if replace covex with concave?
5. (10 points) This problem is related to the implicit function theorem.
A competitive Örm chooses the quantity of labor L to be hired in order to maximize proÖts, taking as given the salary w and the value of a productivity parameter θ . That is, the Örm solves
max (θf (L) — wL) .
L
Assume that the production function f (.) is twice continuously di§erentiable, strictly increas- ing, and strictly concave (i.e., f0 > 0, f00 < 0).
(i) (5 points) Write out the Örst order condition for the Örmís problem. Does the second order su¢ cient condition for a maximum hold?
(ii) (5 points) Interpret the Örst order condition as an equation that implicitly deÖnes a labor demand function of the form L* = L (w; θ). Determine the signs of 
and 
.
6. (20 points) This problem is related to probability theory.
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(i) (8 points) What are the mean, variance, skewness and kurtosis of a standard uniform r.v. X? Is the standard uniform distribution skewed? Is it heavy-tailed?
Suppose that the random variables Y and X only take the values 0 and 1, and have the following joint probability distribution
| X = 0 X = 1 | |
| Y = 0
Y = 1 |
.1 .2
.4 .3 |
(ii) (4 points) What are E [X], E [Y], Var (X), Var (Y)?
(iii) (4 points) What is Corr (X; Y)? Are X and Y independent?
(iv) (4 points) What are E[YjX = x] and Var(YjX = x) for x = 0 and x = 1?
Bonus (10 points) (i) (5 points) Exercise 1 in the lecture note of Chapter 4.
(ii) (5 points) Show that two continuous r.v.ís X and Y are independent if and only if there exist functions g(x) and h(y) such that the joint pdf f (x; y) = g(x)h(y).
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