University of Delhi 2011 M.A Economics winter semester 701 - population and development (admissions of 1999 and onwards) - Question Paper
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This question paper contains 3 printed pages.
M.A. I Winter Semester A
ECONOMICS Course 701 Population and Development (Admissions of 1999 and onwards)
Time : 2i/2 hours Maximum Marks : 70
(Write your Roll No. on the top immediately on receipt of this question paper.)
AU questions are compulsory. Please be brief in your answers or marks would be deducted for verbosity.
Question 1. Consider a game with two players, 1 and 2. Suppose both players have the same strategy set A. Let Ui : A2 * Sf be player is payoff function. Suppose
(a) A is a nonempty, compact and convex subset of 5ft1, where I Af,
(b) for all a, 6 A, i (a, b) u2(b,a),
(c) for every a 6 A, ui(.,a) : A * Sft is quasi-concave, and
(d) U\ is continuous.
Show that:
(A) U2 is continuous and U2{a,.) : A > Si is quasi-concave for every a A.
(B) diag A2 = {(a, 6) A2 | a 6} is nonempty, convex and compact.
(C) Provide a fixed point argument on diagyl2 to show the existence of a* 6 A such that (a,a*) > Ui(b,a*) and u2(a',a) > u2(a*,6) for every 6 e A.
(D) Provide an alternative simpler proof of (C) using a fixed point argument on A.
(3,5,9,6|)
Question 2. Consider an open set C C 31 and / : C > S. / is said to locally bounded if, for every x C, there exists r > 0 and M 6 3t+, such that \f(y)\ < M for every y Br(x). (A) Suppose C is convex and / is convex. Show that / is locally bounded.
(Hint: Find r > 0 such that Br(x) C C. Consider y 6 Br(x). Use the convexity of / to get an upper bound on }(y) that is independent of y, say M. Noting that there exists
Hanover
z Br(x) such that x y/2 + z/2, use the convexity of / to get a lower bound on f(y) that is independent of y.) |
(B) By (A), there exists r > 0 and M 3?+ such that B2r(x) C C and \f(y)\ < M for every y 6 B2t(x). Consider distinct y,z Br(x) and set w = z -f (r/a){z y), where a ~ \y - z\. Show that w G B2r(x) and that is a convex combination of y and w.
(C) Use the convexity of / to show that |
/(*) - f(y) < -I/M ~ f(y)I < I* - y!
r r
(D) Show that | f(z) f(y)\ < (2M/r)\z j/|, and therefore / is continuous.
(E) Provide two economic applications of (this result.
(5,3,8, 3,4)
Question 3. Consider the pair (X, y, wher h is a binary relation on the set X. Let x ~ y if and only if x y y and y y x. Let x >- y if and only if x y y and -*y y x. Suppose
(a) >r is a complete preordering on X\ i
(b) there exist a, 6 X such that by x ahd xy a for every x X\
(c) X is convex;
(d) for every the sets Ux = {t [o| 1] | tb + (1 t)a y x} and Ux {t c [0, ll | xytb+( 1 t)a} are closed in [0,1]; |
(e) for all s, t (0,1), s > t if and only if $6 + (1 s)a y tb + (1 - t)a\
(f) for all x, y, z e X and t [0,1], if x then tx + (1 - t)z ~ ty + (1 l)z.
Given the above assumptions, show that .
I:
(A) there exists f : X + 9? such that x y y if and only if f(x) > /(y), and f(tx + (1 - t)y) ~ tf(x) + (1 - t)f(y) for all x,y X and t e [0,1].
(Hint: Show that, for every x X, there exists t G [0,1] such that x ~ tb + (1 )a.)
(B) Provide an economic application of (A).
I
Question 4. Suppose there are two dates, 0 and 1. Suppose the world will be in one of p states at date 1, but the true state of the world1 at date 1 is unknown at date 0. Let. there be n financial assets. Let A be a p x n matrix where Oi3 is interpreted as the payment by asset j in state i at date 1; assume that a* = (a*i,..., a) 0 for i = 1,... ,p. Let b 9in .{0} be the vector of financial asset prices at date 0, with b3 as the price of asset j-
(A) Provide an economic interpretation of the condition: for every x ftn,
Ax > 0 => (6, x) > 0 (*)
(B) Show that, if (*) is satisfies, then there exists c 2ft+ {0} such that b -- cA.
(C) Provide economic interpretations of (B).
(2,16$, 5)
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