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Theory Macroeconomic II Homework 4 - Solution Professor Gianluca Violante, Teacher Assistant: Nikita Roketskiy New York University Spring 2008 1 1.1 Problem 1 A stationary recursive competitive equilibrium De ne Q ((a; ") ; A E) as the probability that an individual with current state (a; ") transits to the set A E next period, formally Q : S B ! [0; 1], and Q ((a; ") ; A E) = X I fa0 (a; ") 2 Ag ("0 ; ") (1) "0 2E where I is the indicator function, and a0 (a; ") is the optimal saving policy. The household problem in recursive form is: 8 < v (a; "; ) = max u (c) + c;a0 : s:t: c + a0 a 0 "0 2f"h ;"l g X 9 = v (a0 ; "0 ; ) ("0 ; ") ; (2) = (1 + r ( )) a + " b A stationary recursive competitive equilibrium is a value function v : S ! <; This document is based on the previous version by Ofer Setty. All errors are my own. 1 policy functions for the household a0 : S ! <, and c : S ! <+ ; price r; and a stationary measure 2 such that: given the price r, the policy functions c and a0 solve the household probs lem (2) and v is the associated value function, R the goods market clears: A E c (a; ") d (a; ") = Y satis es Z for all (A; E) 2 B, the invariant probability measure (A; E) = Q ((a; ") ; A E) d (a; ") ; AE where Q is the transition function de ned in (1). Existence of the recursive competitive equilibrium The major di erence between this exchange economy and the production economy discussed in class is that assets need to be in zero net supply. This is equivalent to de ning the demand of assets as 0 8r. The supply of capital behaves in a similar way to the discussion in class. To prove the existence of a stationary equilibrium we need to verify that there exists a unique invariant distribution . To satisfy theorem 12.13 in SLP for the existence of the equilibrium we need the existence and uniqueness of the invariant distribution, which follows from the following assumptions1 : For compactness of the state space we need to assume DARA preference. Then compactness is guaranteed for every r that satis es (1 + r) < 1. The Feller property is immediate and does not require additional assumptions. For the monotonicity of the transition function we need to assume that hh > ll : MMC is satis ed if the Markov chain is ergodic. Uniqueness of the recursive competitive equilibrium As in the production economy we only know that A (r) is continuous but we do not know if it is increasing. However, for the exchange economy this is su cient because the demand for assets is perfectly inelastic. Unfortunately, this is not enough for existence as there could be more than one interest rates that correspond to zero assets, so uniqueness is still not guaranteed. 1 Please, see some notes at the end of the document. 2 1.2 The No-borrowing constraint case In this case, agents cannot borrow against future endowment. Although some agents (those with high endowments would like to lend some of their endowment for consumption smoothing, the strict limit on borrowing there is no one who could borrow. This means that in equilibrium there are is no trade across agents (and since it is an endowment economy there is no insurance across periods) and the equilibrium is autarky. The agents with low endowment are constrained by the no-borrowing constraint. For the equilibrium, we need that the interest rate will be such that the agents with the high endowment would not want to lend. In fact we look for the interest rate that would cause the one with the high endowment to want to borrow (but he will be constrained by the no-borrowing constraint). The Euler equation uc (ct ) (1 + r) Eu0 (ct+1 ) can be written in explicit form for the agent with the high endowment as follows: uc ("h ) (1 + r) f hh uc ("h ) + (1 hh ) uc ("l )g and using the functional form of the utility function: "h (1 + r) hh "h + (1 hh ) "l Therefore the highest interest rate that supports equilibrium is: (1 + r) = Assuming > 0, " we get that the term 1 hh "h "h + (1 hh ) "l hh h is a decreasing function and then by assuming hh < 1 "h is lower than 1, where I have used the " +(1 )" hh l trivial assumption that "h > "l . Therefore (1 + r) < Comparative statics: 1 : As "h ="l increases (1 + r) ! 0. The intuition is that when the rich agents have a high endowment, they realize that future endowment might be very bad and their incentive to save increases. Note that they would like to save even if r < 0. However, when (1 + r) = 0 there is no point in saving so there would always be a small enough value for (1 + r) to hold the equilibrium. As the persistence parameter hh increases the term hh "h "h +(1 hh )"l is 3 closer to one and (1 + r) increases towards 1 . The intuition here is that when the persistence increases, rich agents have a lower incentive to save because it is them more likely that they would not need the savings. h As the risk aversion coe cient increases, the term hh "h +(1 hh )"l gets smaller. The intuition is that risk averse agents have a stronger incentive to smooth consumption and thus they are more concerned about possible future low levels of endowments and their incentive to save is stronger. In our model to o set that incentive to save the interest rate decreases. " 4 2 2.1 Problem 2 The household problem The state of the agent is the assets level ait , last period undeclared taxes xi;t s (note the di erence in notation) and the current shock it . Let V (a; x; ; 1) be the value for an agent who is monitored and let V (a; x; ; 0) be the value for an agent who is nor monitored. For brevity of notation, I only use x and not use . Finally, let V (a; x; ) = V (a; x; ; 1) + (1 ) V (a; x; ; 0) The household problem in recursive form is: V (a; x; ; n) = c;a0 ;h;x0 max u (c) + e v(1 h) + EV a0 ; x0 ; 0 ; 1 + (1 ) EV a0 ; x0 ; 0 ;0 (3) s:t: 0 a 0 x0 ra )) + x0 c+b nz (x) = hw + a (1 + r (1 0 a0 where n 2 f0:1g, E is the expected value w:r:t equivalent to 2 [0; 1]. The FOC w.r.t. x0 is: uc (c) = EV2 a0 ; x0 ; 0 0 , and the rst constraint is 2.2 The tax evasion choice From the envelope condition w.r.t. x we get: V2 (a; x; ; n) = uc (c) nz 0 (x) Now use the de nition for V (a; x; ) and get: 5 V (a; x; ) = V2 (a; x; ) = = V (a; x; ; 1) + (1 V2 (a; x; ; 1) + (1 u (c) z (x) 0 0 ) V (a; x; ; 0) ) V2 (a; x; ; 0) and iterate forward to get: V2 a0 ; x0 ; 0 = u0 (c0 ) z 0 (x0 ) (4) Finally, use this in the FOC above to get: uc (c) = z 0 (x0 ) Eu0 (c0 ) (5) The aggregate state variables are: r; w 2.3 A stationary recursive competitive equilibrium A stationary recursive competitive equilibrium is a value function V (a; x; ; n) : S ! <; policy functions for the household a0 : S ! <, h : S ! [0; 1], and c; x0 : S ! <+ ; policies for the rm H and K; prices r and w; government policies ; b; and a stationary measure 2 such that: given prices r; w; and government policy ; ; b, the policy functions solve the household problem (3) with V as the associated value function, given r; w, the rm optimally chooses H and K, i.e. r + = FK (K; H) and w = FH (K; H) R the labor market clears: H = A X h (a; x; ) d (a; x; ) R the goods market clears: A X c (a; x; ) d (a; x; ) + K = F (K; H) R the asset market clears: K = A X a0 (a; x; ) d (a; x; ) R the government budget is balanced: A X ( (a; x; ) ra + z (x)) d (a; x; ) = b for all (A; X ; ) 2 B, the invariant probability measure 6 satis es (A; X ; ) = Z AX Q ((a; x; ) ; A; X ; ) d (a; x; ) ; where the transition function is: Q (0 (a; x; ) ; A X )= X 0 I fa0 (a; x; ) 2 A \ x0 (a; x; ) 2 X g 0 ; 2.4 Optimal monitoring probability maxV a; x s:t: Z 1 ; d (a; x; ) b = ( (a; x; ) ra + (x)) z d (a; x; ) m( ) AX where for each , all the variables may be di erent (policies, value function, distributions). The trade o are as follows: Monitoring bene ts are redistribution and insurance. Monitoring losses are distortion and the administrative cost 7 3 3.1 Problem 3 Aggregate production function Since the capital stock at any given period is xed ,the only decision the rm needs to make is how much labor to hire. The rst order condition for labor inputs is: ii Fn zt ; kt ; ni = w t (6) where w is xed in the stationary equilibrium. 1 2 1 2 Consider two rms with kt = kt and zt 6= zt . Then, in general it will be that: 2 1 nt 6= nt and therefore the capital-labor ratio will be di erent. The intuition is that now the productivity factor shifts the decisions of the rm. A rm with a low capital level that gets a high productivity shock decides to hire (relatively to the aggregate factor) more labor for each unit of capital. Obviously, the cost of the absent market for physical capital is ine cient production. Still, with a continuum of agents and a stationary economy, by using the law of large numbers, it is possible to aggregate the production technology. 3.2 Natural borrowing constraint Assuming that (zmin jz) > 0 8z 2 Z we get that the worst possible realization zmin = 0 is possible in the next period (and all future periods) and therefore the natural borrowing constraint is b = 0. Note that since labor supply is endogenous, future labor income cannot be used as a collateral as well. 3.3 The household problem i The individual state variables are: the capital level kt the level of assets ai ; and t i the productivity shock zt ,: The problem of the household is: v (k; a; z; ) = c; h;k0 ;a0 max ( u (c; 1 h) + s:t: c + k 0 + a0 a 0 z 0 2Z X v (k ; a ; z ; ) (z ; z) 0 0 0 0 ) (7) = F (z; k; n) + (1 0 8 ) k + (1 + r ( )) a + (h n) w where n is determined by the FOC of the rm (6) 3.4 A stationary recursive competitive equilibrium To simplify the equilibrium, I follow the assumption in section (b), and conclude that in equilibrium in this economy there is no trade in the asset, simply because no household can lend. One interest rate that supports this equilibrium is r = 1. At this interest rate agents will want to borrow up to their borrowing constraint which is 0. Now we can go back to the household problem and set a0 = 0 and de ne the recursive competitive equilibrium as follows. The state of the agent is (k; z). A stationary recursive competitive equilibrium is a value function v : S ! <; policy functions for the household h : S ! [0; 1], and k0; c : S ! <+ ; price w; and a stationary measure 2 such that: given the price w, the policy functions c and k 0 and h solve the household s problem (7) and v is the associated value function, R R the labor market clears: K Z n (k; z) d (k; z) = K Z h (k; z) d (k; z) R R the goods market clears: K Z c (k; z) d (k; z)+ K Z k (k; z) d (k; z) = R F (z; k; n (k; z)) d (k; z) KZ for all (K; Z) 2 B, the invariant probability measure (K; Z) = Z Q ((k; z) ; K; Z) d satis es (k; z) KZ where the transition function is: Q ((k; z) ; K Z) = X I fk 0 (k; z) 2 Kg (z 0 ; z) z 0 2Z 3.5 Existence and Uniqueness The existence of the stationary invariant distribution is similar to the discussion in question (1) above. However, unlike the model discussed in class where capital supply is unbounded, here labor supply is bounded by 1 and therefore it is possible (although not likely) that there will be no equilibrium as the labor supply and labor demand would not cross. 9 As for uniqueness, consider the labor market and its response to a change in the price of labor w. As in the capital market, demand for labor n is strictly decreasing in w. However, as r increases we have a similar problem to the one we had with capital supply: the actual e ect is ambiguous because of the income a ect (less leisure) and the substitution e ect (more leisure). Hours increase because wages are higher and hours decrease because the worker needs to work less to generate the same income. Therefore uniqueness cannot be guaranteed. 3.6 Additional market for physical capital In the presence of open markets for both capital and labor, all rms use the same capital-labor ratio. Take a look at the rst problem set if you are not sure why. Therefore, the new economy can be represented by a representative agent. 4 Notes on FP, Monotonicity and MMC Conditions 4.1 Feller Property For this property we need operator T to preserve continuity and boudedness. Boundedness is straightforward from compactness of A E. For continuity argument we need to provide some details of the proof. Lets take an arbitrary 1 sequence f(an ; n )gn=1 such that limn!1 (an ; n ) = (a; ) : We know that Q ((an ; i for any f 2 C n!1 2 n ) ; s) !weakly Q ((a; ) ; s) as n ! 1 lim Z f (s) Q ((an ; n ) ; ds) = Z f (s) Q ((a; ) ; ds) By Aleksandrov Theorem Q ((an ; n ) ; s) !weakly Q ((a; ) ; s) as n ! 1 that are associated with probability measures 2 Here convergence is in terms of r.v. Q ((an ; n ) ; ) and Q ((a; ) ; ) 10 i Q ((an ; n ) ; S) ! Q ((a; ) ; S) for all S such that Q ((a; ) ; @S) = 0 Note that since a0 is continuos then limn!1 a0 (an ; n ) = a0 (a; ) : Also note that Q ((an ; n ) ; (A E)) ! Q ((a; ) ; (A E)) is violated only if a0 (a; ) 2 @A. Hence if a0 (a; ) 62 @A then Q ((an ; n ) ; S) ! Q ((a; ) ; S) : One can show that Q ((a; ) ; @ (A E)) = 0 i (a) or (b) hold: a a0 (a; ) 62 @A P b ( 0; ) = 0 0 2E In case (a) we just have shown that Q ((an ; n ) ; (A E)) ! Q ((a; ) ; (A E)) : In case of (b) Q ((an ; n ) ; S) ! Q ((a; ) ; S) obviously holds since both Q ((a; ) ; S) and Q ((an ; n ) ; S) are zero starting from some n: Combining this altogether we get that T preserves continuity. 4.2 Monotonicity Monotonicity has two components: monotonicity in a and in : T preserves monotonicity i for any f that is increasing in both arguments T f is also increasing. First lets prove monotonicity in a: We need to show that for all a; a : a a ^^ Z f (s) Q ((^; ) ; ds) a Z f (s) Q ((a; ) ; ds) for all increasing f We know that this is equivalent to Q ((^; ) ; ds) <F SD Q ((a; ) ; ds) : We also a know that there is a criterion for FSD relation which is g <F SD h i G (x) H (x) for all x where G ( ) is cdf for g and H ( ) is cdf for h: In our case cdf is Q ((^; ) ; S ( ; e)) a where S ( ; e) = f(a; ) j (a; ) 2 (A E) and (a; ) ( ; e)g Note that since a0 (a; ) is monotonic then Q ((^; ) ; S ( ; e)) looks like a step a and threshold for Q ((^; ) ; S ( ; e)) is higher then for Q ((a; ) ; S ( ; e)) hence a 11 Q ((^; ) ; S ( ; e)) a Q ((a; ) ; S ( ; e)) :From this we can conclude that Q ((^; ) ; S ( ; e)) <F SD Q ((a; ) ; S ( ; e)) a and hence Z Z f (s) Q ((^; ) ; ds) a Z Z f (s) Q ((a; ) ; ds) for all increasing f Now lets prove monotonicity in :We need to show that for all ; ^ : ^ f (s) Q ((a; ^) ; ds) f (s) Q ((a; ) ; ds) for all increasing f We will show it for the case of E = f H ; L g : The argument that we need to use is also based on FSD relation. However we need to impose more conditions. Since Q ((a; ) ; A ( ; e) E ( ; e)) = I fa0 (a; ) 2 A ( ; e)g X ( 0; ) ;e) 0 2E( then I fa0 (a; ^) 2 A ( ; e)g and X ( 0 ; ^) ;e) I fa0 (a; ^) 2 A ( ; e)g for all ( ; e) X ( 0 ; ) for all ( ; e) ;e) (8) (9) 0 2E( 0 2E( implies that Q ((a; ^) ; A ( ; e) E ( ; e)) Q ((a; ) ; A ( ; e) E ( ; e)) for all ( ; e) ( H; H) It is easy to check that increasing a0 is su cient for 8 to hold and ( L ; L ) implies 9. QED 4.3 MMC Note that necessary condition for MMC to hold is ergodicity of Markov chain. 12

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