[A + 1 ] + (1 ) v: : (b) Show: the derivative of T at v = v 0 < 0 is: = (v 0 ) (1 ) ; [A + 1 ]

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1 Homework #2 Economics 4- Due Wednesday, October 5 Christiano. This question is designed to illustrate Blackwell's Theorem, Theorem 3.3 on page 54 of S-L. That theorem represents a set of conditions that are sucient for a mapping, T; to be a contraction, so that T j w 0 = w as j! for all w 0 belonging to a specied set. The question draws attention to the fact that the conditions of Blackwell's theorem are not necessary. Consider the following functional equation: T (v) = max 0A+ Suppose > and (A + ) <. (a) Show: T (v) = for v > 0, T (0) = ( ) [A + ] + ( ) v: [A+ ]( ) : (b) Show: the derivative of T at v = v 0 < 0 is: where dt (v 0 ) dv = (v 0 ) ( ) ; ( ) [A + ] (v 0 ) = argmax 0A+ + ( ) v 0 : (c) Explain why T does not satisfy the conditions of Theorem 3.3 in S- L, page 54. (Hint: does T : B(X)! B(X), where the `functions' we consider here are actually points in R? Is discounting satised?) (d) What happens to (v) as v!? (e) What does the graph of T (v) versus v for v 0 look like? Does it cross a 45 0 line drawn in the negative orthant? Draw this graph by hand, emphasizing its qualitative features (i.e., you need not compute the graph numerically, using numerical values for the parameters of the function.)

2 (f) Explain, using the graph you just developed, why T j v 0 = v as j! ; for every v 0 < 0; where v is unique. 2. This question asks you to redo Theorem 4.5 in a model that takes into account uncertainty. Suppose that at each date t a random variable, s t ; is realized. It can take on any one of N possible values: s(); s(2); :::; s(n): Call s t the state of nature at date t: Let s t denote the history of states of nature up to time t: s t = (s 0 ; s ; :::; s t ): At date 0, s 0 is known. Thus, as of date 0, there is one possible history, s 0 ; at date there are N possible histories, s ; at date 2, N 2 possible histories, s 2 ;... at date t; N t possible histories s t ; etc. Let the probability of history s t be denoted by (s t ): Then, by the denition of a probability, (s t ) 0; for all s t ; and X s t (s t ) = ; for every t = 0; ; 2; :::; where P s t denotes `the sum over all N t possible values of s t '. Let the N N matrix be dened by: ij = Pr obability[s t+ = s(j)js t = s(i)]: (a) Suppose v(s t ) = v i if s t = s(i); for i = ; :::; N. That is, the value taken on by v(s t ) is a function only of the current state of nature. Let the N N matrix 2 be dened by 2 = : Similarly, dene 3 = 2 ;..., k = k : i. Prove that each row of k is a probability distribution (i.e., all elements of k are non-negative and k satises k = ; where is the N vector = (; ; :::; ) 0 ): Strictly speaking, the notation should be t (s t ): I omit the t subscript on the probability to keep from proliferating notation. 2

3 ii. Suppose s 0 = s(k): Show that: X X t=0 s t (s t ) t v(s t ) = [I ] v; () where v is an N column vector, v = (v ; :::; v N ) 0 ; and is a N row vector with all zeros, except a one in the k th entry. Recall the denition of a double sum: X X q ij [q 00 + q 0 + q 02 + :::] + [q 0 + q + q 2 + :::] + :::; i=0 j=0 where q ij is an arbitrary set of numbers. (Hint: start by writing the expression on the left of the equality in () explicitly for t=0,,2,..., and stare.) iii. Show that: X q(s t+ ) = X X q(s t+ ); (2) s t+ s t s t+ js t where s t+ j s t signies `all possible histories s t+ ; given history s t has occurred' and q(s t ) is an arbitary function of s t. It's enough to establish this for t = and N = 2: (b) Consider the utility function: and resource constraint: X t X (s t )u(c(s t )); (3) t=0 s t c(s t ) + k(s t ) f(k(s t ); s t ): (4) Note that s t shifts the production function. Assume u and f satisfy the same conditions stated above (with the obvious modications to reect the absence of hours worked from the problem!). Suppose c (s t ); k (s t ) > 0 satisfy (4) for all s t ; t = 0; ; 2; :::; with k (s ) = k 0 ; the given initial stock of capital. Suppose also that the `Euler equations' are satised: u c (c (s t )) = X s t+ js t (st+ ) (s t ) u c(c (s t+ ))f k (k (s t ); s t+ ); 3

4 for all s t ; t 0; and the `transversality condition': X lim T (s T )u c (c (s T ))f k (k (s T ); s T )k (s T )! 0: T! s T Prove that fc (s t ); k (s t ); t 0; all s t g yields the highest value of (3) within the set of all sequences that satisfy (4) and the nonnegativity constraints on consumption and the stock of capital. (Hint: imitate the proof strategy of Theorem 4.5 as closely as you can, and make use of (2) when you group terms in the capital stock). The Euler and transversality conditions are sometimes stated using the expectation operator: u c;t = E t u c;t+ f k;t+ and lim E 0 T u c;t f k;t k T = 0; T! where E t denotes the mathematical expectation operator, conditional on information dated t and earlier (to understand the conditional expectation operator in the euler equation, recall that (s t+ ) signies the conditional probability of s t+ ; given s t.) (s t ) 3. Consider the standard neoclassical model (i.e., the one in the previous question, with c = 0). Replace the non-negativity condition, k t+ 0; with the following alternative, i t 0: (a) Show that monotonicity of (k), Assumption 4.6 in S-L, fails so that one of the conditions of Theorem 4.7 which guarantee a strictly increasing value function, v; is not satised. (b) Show that the feasible set for this economy satises the following `quasi-monotonicity property': if ~ k k; then (k) + ( )( ~ k k) ( ~ k): Here, the sum of a set, say X; and a number, say a; is a new set, X + a; where X + a fx + a : x 2 Xg: (c) Show: v is an increasing function in k: (Hint: (i) following the basic strategy of the proof of Theorem 4.7, it's enough to establish that the assumptions of Theorem 4.7 with the monotonicity 4

5 assumption on replaced by quasi-monotonicity guarantee T w is increasing if w is; (ii) make use of the fact that if k 0 2 (k); then ~k 0 = k 0 + ( )( k ~ k) 2 ( k), ~ k ~ 0 > k 0 ; and f( k) ~ + ( ) k ~ k ~ 0 > f(k) + ( )k k 0.) Can you provide intuition for the fact that v is increasing even though fails to satisfy monotonicity? 4. (Boldrin-Montrucchio 986). Consider the policy rule, g : [0; ]! [0; ] : g(x) = 4x( x). Draw this function, along with the 45 degree line, by hand in the unit box. Find an economy, (F; ; ; X); for which the above function is the policy rule, where the economy satises all of our assumptions (i.e., assumptions A4.3-A4.9 in Stokey and Lucas). Here, x is the aggregate stock of capital at the beginning of the period and x 0 is its value at the end of the period. Some hints: Recall, where ; 2 (0; ) and F satisfy g(x) = max F (x; y) + v(y); y2 (x) v(x) = F (x; g(x)) + v(g(x)): Recall that the denition of a function or a correspondence must include a specication of the domain and range. Recall too that a function, say f(x; y); is strictly concave i: f xx (x; y) < 0; f yy (x; y) < 0; f xx (x; y)f yy (x; y) f xy (x; y) 2 > 0; for all (x; y) in the domain of f: Also, it is easy to verify that g(x) = arg max y2 (x) when (x; y) is dened as follows: (x; y); (x; y) = 2 y2 + yg(x) 2 Lx2 + ax; and L; a are known constants. Finally, note that v(x) = (x; g(x)); and use this to back out F: You can think of your task as having to identify values of a and L that ensure assumptions A4.3-A4.9 are satised. How many such values are there? 5