Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 23 pdf

Chapter 23 Two topics in international trade 23.1. Two dynamic contracting problems This chapter studies two models in which recursive contracts are used to over- come incentive problems commonly thought to occur in international trade. The first is Andrew Atkeson’s model of lending in the context of a dynamic setting that contains both a moral hazard problem due to asymmetric information and an enforcement problem due to borrowers’ option to disregard the contract. It is a considerable technical achievement that Atkeson managed to include both of these elements in his contract design problem. But this substantial technical accomplishment is not just showing off. As we shall see, both the moral hazard and the self-enforcement requirement for the contract are required in order to explain the feature of observed repayments that Atkeson was after: that the occurrence of especially low output realizations prompt the contract to call for net repayments from the borrower to the lender, exactly the occasions when an unhampered insurance scheme would have lenders extend credit to borrowers. The second model is Bond and Park’s recursive contract that induces moves to free trade starting from a Pareto non-comparable initial condition. The new policy is accomplished by a gradual relaxation of tariffs accompanied by trade concessions. Bond and Park’s model of gradualism is all about the dynamics of promised values that are used optimally to manage participation constraints. – 817 – 818 Two topics in international trade 23.2. Lending with moral hazard and difficult enforce- ment Andrew Atkeson (1991) designed a model to explain how, in defiance of the pattern predicted by complete markets models, low output realizations in various countries in the mid-1980s prompted international lenders to ask those countries for net repayments. A complete markets model would have net flows to a borrower during periods of bad endowment shocks. Atkeson’s idea was that information and enforcement problems could produce the observed out- come. Thus, Atkeson’s model combines two features of the models we have seen in chapter 19: incentive problems from private information and participation constraints coming from enforcement problems. Atkeson showed that the optimal contract the remarkable feature that the job of handling enforcement and information problems is done completely by a repayment schedule without any direct manipulation of continuation values. Continuation values respond only by updating a single state variable – a mea- sure of resources available to the borrower – that appears in the optimum value function, which in turn is affected only through the repayment schedule. Once this state variable is taken into account, promised values do not appear as in- dependently manipulated state variables. 1 Atkeson’s model brings together several features. He studies a “borrower” who by himself is situated like a planner in a stochastic growth model, with the only vehicle for saving being a stochastic investment technology. Atkeson adds the possibility that the planner can also borrow subject to both participation and information constraints. A borrower lives for t =0, 1, 2, He begins life with Q 0 units of a single good. At each date t ≥ 0, the borrower has access to an investment technology. If I t ≥ 0 units of the good are invested at t, Y t+1 = f (I t ,ε t+1 ) units of time t+1 goods are available, where ε t+1 is an i.i.d. random variable. Let g(Y t+1 ,I t )be the probability density of Y t+1 conditioned on I t .Itisassumedthatincreased investment shifts the distribution of returns toward higher returns. The borrower has preferences over consumption streams ordered by U =(1−δ)E 0 ∞  t=0 δ t u(c t )(23.2.1) 1 To understand how Atkeson achieves this outcome, the reader should also digest the approach described in the following chapter. Lending with moral hazard and difficult enforcement 819 where δ ∈ (0, 1) and u(·) is increasing, strictly concave, and twice continuously differentiable. Atkeson used various technical conditions to render his model tractable. He assumed that for each investment I , g(Y, I) has finite support (Y 1 , ,Y n )with Y n >Y n−1 > > Y 1 . He assumed that g(Y i ,I) > 0 for all values of I and all states Y i , making it impossible precisely to infer I from Y .Hefurther assumed that the distribution g (Y,I) is given by the convex combination of two underlying distributions g 0 (Y )andg 1 (Y ) as follows: g(Y,I)=λ(I)g 0 (Y )+[1− λ(I)]g 1 (Y ), (23.2.2) where g 0 (Y i )/g 1 (Y i ) is monotone and increasing in i,0≤ λ(I) ≤ 1, λ  (I) > 0, and λ  (I) ≤ 0 for all I .Notethat g I (Y,I)=λ  (I)[g 0 (Y ) − g 1 (Y )], where g I denotes the derivative with respect to I . Moreover, the assumption that increased investment shifts the distribution of returns toward higher returns implies  i Y i [g 0 (Y i ) −g 1 (Y i )] > 0. (23.2.3) We shall consider the borrower’s choices in three environments: (1) autarky, (2) lending from risk-neutral lenders under complete observability of the bor- rower’s choices and complete enforcement, and (3) lending under incomplete observability and limited enforcement. Environment 3 is Atkeson’s. We can use environments 1 and 2 to construct bounds on the value function for performing computations described in an appendix. 820 Two topics in international trade 23.2.1. Autarky Suppose that there are no lenders. Thus the “borrower” is just an isolated house- hold endowed with the technology. The household chooses (c t ,I t ) to maximize expression (23.2.1) subject to c t + I t ≤ Q t Q t+1 = Y t+1 . The optimal value function U(Q) for this problem satisfies the Bellman equation U(Q)= max Q≥I≥0  (1 −δ) u(Q −I)+δ  Q  U  Q   g(Q  ,I)  . (23.2.4) The first-order condition for I is −(1 −δ)u  (c)+δ  Q  U(Q  )g I (Q  ,I)=0 (23.2.5) for 0 <I<Q. This first-order condition implicitly defines a rule for accumu- lating capital under autarky. 23.3. Investment with full insurance We now consider an environment in which in addition to investing I in the technology, the borrower can issue Arrow securities at a vector of prices q(Y  ,I),wherewelet  denote next period’s values, and d(Y  )thequantity of one-period Arrow securities issued by the borrower; d(Y  )isthenumberof units of next period’s consumption good that the borrower promises to deliver. Lender observe the level of investment I and so the pricing kernel q(Y  ,I) depends explicitly on I . Thus, for a promise to pay one unit of output next period contingent on next-period output realization Y  ,foreachlevelofI ,the borrower faces a different price. (As we shall soon see, in Atkeson’s model, the lender cannot observe I , making it impossible to condition the price on I .) We shall assume that the Arrow securities are priced by risk-neutral investors who also have one-period discount factor δ. As in chapter 8, we formulate the borrower’s budget constraints recursively as c −  Y  q(Y  ,I)d(Y  )+I ≤ Q (23.3.1a) Q  = Y  − d(Y  ). (23.3.1b) Investment with full insurance 821 Let W(Q) be the optimal value for a borrower with goods Q. The borrower’s Bellman equation is W (Q)= max c,I,d(Y  )  (1 −δ)u(c)+δ  Y  W [Y  − d(Y  )]g(Y  ,I) + λ[Q −c +  Y  q(Y  ,I)d(Y  ) −I]  , (23.3.2) where λ is a Lagrange multiplier on expression (23.3.1a). First-order conditions with respect to c, I, d(Y  ), respectively, are c:(1− δ)u  (c) −λ =0, (23.3.3a) I: δ  Y  W [Y  − d(Y  )]g I (Y  ,I)+ λq I (Y  ,I)d(Y  ) −λ =0, (23.3.3b) d(Y  ): − δW  [Y  − d(Y  )]g(Y  ,I)+λq(Y  ,I)=0. (23.3.3c) Letting risk-neutral lenders determine the price of Arrow securities implies that q(Y  ,I)=δg(Y  ,I), (23.3.4) which in turn implies that the gross one-period risk-free interest rate is δ −1 . At these prices for Arrow securities, it is profitable to invest in the stochastic technology until the expected rate of return on the marginal unit of investment is driven down to δ −1 :  Y  [Y  − d(Y  )]g I (Y  ,I)=δ −1 , (23.3.5) and after invoking equation (23.2.2) λ  (I)  Y  [Y  − d(Y  )] (g 0 (Y  ) − g 1 (Y  )) = δ −1 . This condition uniquely determines the investment level I , since the left side is decreasing in I and must eventually approach zero because of the upper bound on λ(I). (The investment level is strictly positive as long as the left-hand side exceeds δ −1 when I =0.) The first-order condition (23.3.3c) and the Benveniste-Scheinkman condi- tion, W  (Q  )=(1− δ)u  (c  ), imply the consumption-smoothing result c  = c. 822 Two topics in international trade This in turn implies, via the status of Q as the state variable in the Bellman equation, that Q  = Q. Thus, the solution has I constant over time at a level determined by equation (23.3.5), and c and the functions d(Y  ) satisfying c + I = Q +  Y  q(Y  ,I)d(Y  )(23.3.6a) d(Y  )=Y  − Q (23.3.6b) The borrower borrows a constant  Y  q(Y  ,I)d(Y  ) each period, invests the same I each period, and makes high repayments when Y  is high and low repayments when Y  is low. This is the standard full-insurance solution. We now turn to Atkeson’s setting where the borrower does better than under autarky but worse than with the loan contract under perfect enforcement and observable investment. Atkeson found a contract with value V (Q)forwhich U(Q) ≤ V (Q) ≤ W (Q). We shall want to compute W(Q)andU(Q)inorder to compute the value of the borrower under the more restricted contract. 23.4. Limited commitment and unobserved investment Atkeson designed an optimal recursive contract that copes with two impedi- ments to risk sharing: (1) moral hazard, that is, hidden action: the lender cannot observe the borrower’s action I t that affects the probability distribution of returns Y t+1 ; and (2) one-sided limited commitment: the borrower is free to default on the contract and can choose to revert to autarky at any state. Each period, the borrower confronts a two-period-lived, risk-neutral lender who is endowed with M>0 in each period of his life. Each lender can lend or borrow at a risk-free gross interest rate of δ −1 andmustearnanexpectedreturn of at least δ −1 if he is to lend to the borrower. The lender is also willing to borrow at this same expected rate of return. The lender can lend up to M units of consumption to the borrower in the first period of his life, and could repay (if the borrower lends) up to M units of consumption in the second period of his life. The lender lends b t ≤ M units to the borrower and gets a state-contingent repayment d(Y t+1 ), where −M ≤ d(Y t+1 ), in the second period of his life. That the repayment is state-contingent lets the lender insure the borrower. A lender is willing to make a one-period loan to the borrower, but only if the loan contract assures repayment. The borrower will fulfill the contract only if Limited commitment and unobserved investment 823 he wants. The lender observes Q, but observes neither C nor I . Next period, the lender can observe Y t+1 . He bases the repayment on that observation. Where c t + I t −b t = Q t , Atkeson’s optimal recursive contract takes the form d t+1 = d (Y t+1 ,Q t )(23.4.1a) Q t+1 = Y t+1 − d t+1 (23.4.1b) b t = b(Q t ). (23.4.1c) The repayment schedule d(Y t+1 ,Q t ) depends only on observables and is de- signed to recognize the limited-commitment and moral-hazard problems. Notice how Q t is the only state variable in the contract. Atkeson uses the apparatus of Abreu, Pearce, and Stacchetti (1990), to be discussed in chapter 22, to show that the state can be taken to be Q t , and that it is not necessary to keep track of the history of past Q’s. Atkeson obtains the following Bellman equation. Let V (Q) be the optimum value of a borrower in state Q under the optimal contract. Let A =(c, I, b, d(Y  )), all to be chosen as functions of Q. The Bellman equation is V (Q)=max A  (1 −δ) u ( c )+δ  Y  V [Y  − d (Y  ,Q)] g (Y  ,I)  (23.4.2a) subject to c + I − b ≤ Q, b ≤ M, −d(Y  ,Q) ≤ M, c ≥ 0,I≥ 0(23.4.2b) b ≤ δ  Y  d(Y  ) g (Y  ,I)(23.4.2c) V [Y  − d (Y  )] ≥ U (Y  )(23.4.2d) I =argmax ˜ I[0,Q+b]   1 − δ  u  Q + b − ˜ I  + δ  Y  V  Y  − d (Y  ,Q)  g (Y  , ˜ I)  (23.4.2e) Condition (23.4.2b) is feasibility. Condition (23.4.2c) is a rationality con- straint for lenders: it requires that the gross return from lending to the borrower be at least as great as the alternative yield available to lenders, namely, the risk-free gross interest rate δ −1 . Condition (23.4.2d)saysthatineverystate tomorrow, the borrower must want to comply with the contract; thus the value of affirming the contract (the left side) must be at least as great as the value of autarky. Condition (23.4.2e) states that the borrower chooses I to maximize his expected utility under the contract. 824 Two topics in international trade Therearemanyvaluefunctions V (Q) and associated contracts b(Q),d(Y  ,Q) that satisfy conditions (23.4.2). Because we want the optimal contract, we want the V (Q) that is the largest (hopefully, pointwise). The usual strategy of it- erating on the Bellman equation, starting from an arbitrary guess V 0 (Q), say, 0, will not work in this case because high candidate continuation values V (Q  ) are needed to support good current-period outcomes. But a modified version of the usual iterative strategy does work, which is to make sure that we start with a large enough initial guess at the continuation value function V 0 (Q  ). Atkeson (1988, 1991) verified that the optimal contract can be constructed by iterating to convergence on conditions (23.4.2), provided that the iterations be- gin from a large enough initial value function V 0 (Q). (See the appendix for a computational exercise using Akkeson’s iterative strategy.) He adapted ideas from Abreu, Pearce, and Stacchetti (1990) to show this result. 2 In the next subsection, we shall form a Lagrangian in which the role of continuation values is explicitly accounted for. 23.4.1. Binding participation constraint Atkeson motivated his work as an effort to explain why countries often expe- rience capital outflows in the very-low-income periods in which they would be borrowing more in a complete markets setting. The optimal contract associated with conditions (23.4.2) has the feature that Atkeson sought: the borrower makes net repayments d t >b t in states with low output realizations. Atkeson establishes this property using the following argument. First, to permit him to capture the borrower’s best response with a first order condition, he assumes the following conditions about the outcomes: 3 2 See chapter 22 for some work with the Abreu, Pearce, and Stacchetti struc- ture, and for how, with history dependence, dynamic programming principles direct attention to sets of continuation value functions. The need to handle a set of continuation values appropriately is why Atkeson must initiate his iterations from a sufficiently high initial value function. 3 The first assumption makes the lender prefer that the borrower would make larger rather than smaller investments. See Rogerson (1985b) for conditions needed to validate the first-order approach to incentive problems. Limited commitment and unobserved investment 825 Assumptions: For the optimum contract  i d i  g 0 (Y i ) − g 1 (Y i )  ≥ 0. (23.4.3) This makes the value of repayments increasing in investment. In addition, as- sume that the borrower’s constrained optimal investment level is interior. Atkeson assumes conditions (23.4.3) and (23.2.2) to justify using the first- order condition for the right side of equation (23.4.2e) to characterize the in- vestment decision. The first-order condition for investment is −(1 − δ)u  (Q + b − I)+δ  i V (Y i − d i )g I (Y i ,I)=0. 23.4.2. Optimal capital outflows under distress To deduce a key property of the repayment schedule, we will follow Atkeson by introducing a continuation value ˜ V as an additional choice variable in a programming problem that represents a form of the contract design problem. Atkeson shows how (23.4.2) can be viewed as the outcome of a more elementary programming problem in which the contract designer chooses the continuation value function from a set of permissible values. 4 Following Atkeson, let U d (Y i ) ≡ ˜ V (Y i −d(Y i )) where ˜ V (Y i −d(Y i )) is a continuation value function to be chosen by the author of the contract. Atkeson shows that we can regard the contract author as choosing a continuation value function along with the elements of A, but that in the end it will be optimal for him to choose the continuation values to satisfy the Bellman equation (23.4.2). We follow Atkeson and regard the U d (Y i )’s as choice variables. They must satisfy U d (Y i ) ≤ V (Y i − d i ), where V (Y i − d i ) satisfies the Bellman equation 4 See Atkeson (1991) and chapter 22. 826 Two topics in international trade (23.4.2). Form the Lagrangian J(A, U d ,µ)=(1−δ)u(c)+δ  i U d (Y i )g(Y i ,I) + µ 1 (Q + b − c −I) + µ 2  δ  i d i g(Y i ,I) −b  + δ  i µ 3 (Y i )g(Y i ,I)  U d (Y i ) − U(Y i )  + µ 4  −(1 −δ)u  (Q + b −I)+δ  i U d (Y i )g I (Y i ,I)  + δ  i µ 5 (Y i )g(Y i ,I)  V (Y i − d i ) − U d (Y i )  , (23.4.4) where the µ j ’s are nonnegative Lagrange multipliers. To investigate the conse- quences of a binding participation constraint, rearrange the first-order condition with respect to U d (Y i )toget 1+µ 4 g I (Y i ,I) g(Y i ,I) = µ 5 (Y i ) −µ 3 (Y i ), (23.4.5) where g I /g = λ  (I)  g 0 (Y i )−g 1 (Y i ) g(Y,I)  , which is negative for low Y i and positive for high Y i . All the multipliers are nonnegative. Then evidently when the left side of equation (23.4.5) is negative,wemusthaveµ 3 (Y i ) > 0, so that condition (23.4.2d) is binding and U d (Y i )=U(Y i ). Therefore, V (Y i − d i )=U(Y i )for states with µ 3 (Y i ) > 0. Atkeson uses this finding to show that in states Y i where µ 3 (Y i ) > 0, new loans b  cannot exceed repayments d i = d(Y i ). This conclusion follows from the following argument. The optimality condition (23.4.2e) implies that V (Q) will satisfy V (Q)= max I∈[0,Q+b] u(Q + b −I)+δ  Y  V (Y  − d(Y  ))g(Y  ; I). (23.4.6) Using the participation constraint (23.4.2d)ontherightsideof(23.4.6) implies V (Q) ≥ max I∈[0,Q+b]  u(Q + b −I)+δ  Y  U(Y  i )g(Y  ; I)  ≡ U(Q + b)(23.4.7) where U is the value function for the autarky problem (23.2.4). In states in which µ 3 > 0, we know that, first, V (Q)=U(Y ), and, second, that by (23.4.7) [...]... for P (y) from (23. 14.5e ) Next solve for P (vL ) from (23. 14.5a) Get eS from (23. 14.5b ) Finally, equations (23. 14.5a), (23. 14.5b ), and (23. 14.5d) imply that equation (23. 14.5e ) holds, which establishes the reduced rank of the system of equations We can use (23. 14.3 ) to compute eS , the transfer from the period 1 onward In particular, eS satisfies y = uL (0) + eS + βy Subtracting (23. 14.5b ) from... cannot be sustained by a time-invariant transfer ∗∗ scheme Values vL > vL also fail to be sustainable by a time-invariant transfer ∗ ∗∗ because the required eS is too high For initial values vL < vL or vL > vL , Bond and Park construct time-varying tariff and transfer schemes that sustain continuation value vL They proceed by designing a recursive contract similar to ones constructed by Thomas and Worrall... either leisure or work, 1 = + n1 + n2 , (23. 6.2) where nj is the labor input in the production of intermediate good xj , for j = 1, 2 The two intermediate goods can be combined to produce additional units of the final consumption good The technology is as follows x1 = n1 , x2 = γ n2 , (23. 6.3a) γ ∈ [0, 1], (23. 6.3b) y = 2 min{x1 , x2 }, (23. 6.3c) c = y + y, ¯ (23. 6.3d) 6 Bond and Park say that in practice... with our study of Thomas and Worrall’s and Kocherlakota’s model, we place non-negative multipliers θ on (23. 13.2a) and µL , µS on (23. 13.2b ) and (23. 13.2c), respectively, form a Lagrangian, and obtain the following first-order necessary conditions for a saddlepoint: tL : uS (tL ) + (θ + µL )uL (tL ) ≤ 0, = 0 if tL > 0 (23. 13.3a) Base line policies 839 y : P (y)(1 + µS ) + (θ + µL ) = 0 (23. 13.3b) eS :... indicated We can solve for tL , eS for period zero as follows For a given θ > 1 , solve the following equations for y, P (y), tL , eS , P (vL ): uS (tL ) + θuL (tL ) = 0 (23. 14.4a) vL = uL (tL ) + eS + βy (23. 14.4b) N − eS + βP (y) = βvS (23. 14.4c) P (vL ) = uS (tL ) − eS + βP (y) uW (0) y + P (y) = 1−β (23. 14.4d) (23. 14.4e) To find the maximized value P (vL ), we must search over solutions of (23. 14.4... − 0.5 2 ˜ ˜ d 2γ Substituting for x(γ) from (23. 6.4 ) gives 1+γ = 1 − which can be rearranged ˜ to become 1−γ = L(γ) = (23. 6.5) 1+γ It follows that per-capita the equilibrium quantity of each intermediate good is given by 2 γ2 ˜ x = χ(γ) = χ(γ)[1 − L(γ)] = ˜ (23. 6.6) (1 + γ)2 23. 6.1 Two countries under autarky Suppose that there are two countries named L and S (denoting large and small) Country... country L immediately to reduce its tariff to zero by paying transfers that rise between period 0 and period 1 and that thereafter remain constant That the initial tariff is zero means that we are immediately on the unconstrained Pareto frontier It just takes time-varying transfers to put us there P(v ) L N vL * vL v ** L Figure 23. 14.1: The constrained Pareto frontier vS = P (vL ) in the Bond-Park model... Multiplicity of payoffs and continuation values 845 Equation (23. 15.1a) determines the transfer and continuation value needed to deliver the promised value vL to country L under free trade eS + βy = vL − uL (0) The incentive-compatibility condition for country S requires that inequality (23. 15.1c) is satis ed which can be rewritten as N eS + βy ≤ β(W − vS ) (23. 15.2) Since we are postulating that we are in region... once again given by (23. 15.3 ) The lower bound of vL is pinned down by the incentive-compatibility condition (23. 15.1b ) for country L and this implies an immediate transition out of region III into region I N For the lowest possible promised value vL = vL , the range of continuation values in (23. 15.3 ) becomes degenerate with only one admissible value of y = ∗ N ∗ vL From equation (23. 15.1a), we can... X(d) = (23. A.1) i The first-order condition for the borrower’s problem (23. 4.2e ) is −(1 − δ)u (Q + b − I) + δλ (I)X ≥ 0, = 0 if I > 0 Given a candidate continuation value function V j , a value Q , and b, d1 , , dn , solve the borrower’s first-order condition for a function I = f (b, d1 , , dn ; Q) 11 We recommend the Schumaker shape-preserving spline mentioned in chapter 4 and described by Judd . borrower under the more restricted contract. 23. 4. Limited commitment and unobserved investment Atkeson designed an optimal recursive contract that copes with two impedi- ments to risk sharing: (1). long as the left-hand side exceeds δ −1 when I =0.) The first-order condition (23. 3.3c) and the Benveniste-Scheinkman condi- tion, W  (Q  )=(1− δ)u  (c  ), imply the consumption-smoothing result. Chapter 23 Two topics in international trade 23. 1. Two dynamic contracting problems This chapter studies two models in which recursive contracts are used to over- come incentive