University of Delhi 2011 M.A Economics winter semester 106- topics in economic theory (admissions of 1999 & onwards) - Question Paper

This question paper contains 3 printed pages.

Your Roll No.

M.A. / Winter Semester    

ECONOMICS Course 106 Topics in Economic Theory (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.)

Answer any three of the 4 questions given below. Each question carries a total of 23m marks. Marks for each part of a question are indicated in parentheses.

(1).    Consider a Markov Process on a finite state space S (with \S\ = k), with transition probability matrix M. Use the norm ||y||_L = |y| on 3i^fc.

Suppose there is a state and e > 0 s.t. for all states i S, > e.

Let 5jj₀ be the indicator variable that equals 1 if j = j?₀, and equals 0 if the state j jo.

(A).    Let y 6 s.t. y* = 0. Show that for all states j,

l(yM)yl <     - eS_jijo) '

tS

(B).    Infer from the above that

lls/MHi < (l-e)||y||,

(C).    Notice that if <p and ip are probability vectors, then (B) holds with y replaced by <j> t/>, as the coordinates of this add up to 0. Now let p. be a probability vector, and write p_n pMⁿ. By repeatedly iterating the result in (B), show that, with n> m,

IK - Mm||l < (1 -    - Jilll < C( 1 - e)^m

for some C > 0.

(D).    Hence the sequence (fai)%Lo is Cauchy, and converges to a probability vector 7r. Show that tt = nM, i.e., that 7r is stationary.

(6,6,6,5|)

(2).    (A). Let {S,p) be a complete metric space and let the function / :

S' S satisfy p(f(x)_y f(y)) < p(x>y) for all distinct x,y e S. Let f^m{x) =

Turn over

/(/( (/(*)))) be the function obtained by applying / m- times. Fix x e S. Show that then the sequence of distances W/^m+l(^:c)>/^m(^j:)))m-i ^Ls a convergent sequence.

(2).    (B). Let (S,p) be a complete metric space, and let    .] be a sequence of uniformly strict contractions with modulus A,0 < A < 1, from S into S. Let (x_m) be the corresponding unique fixed points of (3>_m) (due to Banach's Theorem). Suppose there exists a function $ : S - + S such that

sup{/9($_m(x), $(x))|x 6 5} -* i as m > oo

Show that then $ is a uniformly strict contraction with unique fixed point x* ~ lhnx_m. Hint: Estimate the distance p($(x), $(y)) by breaking it up into distances about which you have information regarding the sup assumption above, or about contractions. \

. (10,13|)

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(3).    Let S be a state space and T> be the set of all prospects on it (all real valued functions on S taking on a finite number of values).

(A).    Suppose a decisionmakers (DMs) preference relation >r_ on V is a weak order and satisfies monotonicity. Suppose alrfo that for every prospect x\ there exists a certainty equivalent CE(x). Show that then CE represents >.

(B).    Suppose in addition (to the assumptions![in (A)) that >. satisfies additivity. Show that then for every pair of prospects x, y, CE(x f y) -CE(x) + CE(y).

(C).    Suppose a coin is tossed, giving H or T. Suppose > is a weak order, and that all outcomes a, 0, we have an/3 ~ /3h&> Assume risk aversion in the sense that there exist outcomes 7,/?, with y > p s.t. CE(yf{P) < (P + 7)/2. Show that the preference contains a Dutch Book.

(7,7,9|)

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(4).    Consider the following infinite-horizon md#el of the market for a commodity Time is discrete ( = 0,1,2,...). Harvests (W)o are i.i.d. according to the density <j> on 5 [a, 00), a > 0. Final Consumers demand is D(p), if the market price is p in any period, and th inverse demand function P is strictly decreasing and continuous. I_t units purchased by speculators at time t yields al_t units at time t + 1, (a 6 (0, l)). Risk-neutral speculators expected profits are EtPt+i&hpth, where p*,pi+i are market prices at times , t 4* 1, and E_t refers to expectation conditional on information available at time t.

The supply of the commodity at time 0 is given, and equals X₀ 6 S. Note that supply at time t, X_t alt-1 + Wt and derfaand - D(p_t) -f- It

An equilibrium is a sequence    X_t)_t>o of random variables such that

there is a function p* : S * (0,00) with p_t ~ *(X_t),Vt_y and the following conditions are satisfied:    ;

(i) (110 arbitrage):    - p_t < 0,V. 1

(ii)    Profit Maximization by Speculators: If aE_tp_t+i ~Pt <0, then I_t 0.

(iii)    Market clearing: X_t = alt-i + W_t = D(p_t) + I_t.

(2.1).    Show that there is a unique function p* : S > (0, oo) that solves

p*(x) = max jc* J p*{al(x) + z)<f>(z)dz,P(z) ,Vx e S

(Hint: Use Blackwells Theorem and Banachs Contraction Mapping Theorem, taking their conclusions as given).

(2.2).    Show that the p* defined above can serve as the price functional required in the definition of an equilibrium.

(18,5f)

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