Notes on Kehoe Perri

<strong>Notes</strong> on Kehoe PerriJonathan HeathcoteOctober 10, 2002There is nothing in these notes that is not in Kehoe Perri NBER WorkingPaper 7820 or Kehoe and Perri Econometrica 2002. However, I have gatheredall the equations together, and added a few more steps in the computations inplaces.1 ModelPlanner’s objective"X∞ Xmax λ 1 β t π(s t )U ¡ c 1 (s t ),l 1 (s t ) ¢ X∞ X+ λ 2 β t π(s t )U ¡ c 2 (s t ),l 2 (s t ) ¢#t=0 s t t=0 s tResource constraintsX £ci (s t )+k i (s t ) ¤ = X£F (ki (s t−1 ),A i (s t )l i (s t )) + (1 − δ)k i (s t−1 ) ¤ ∀t, s ti=1,2i=1,2Enforcement constraints∞X Xr=ts r β r−t π(s r |s t )U ¡ c i (s r ),l i (s t ) ¢ ≥ V i (k i (s t−1 ),s t ) ∀i, t, s tAutarky problem∞X XV i (k i (s t−1 ),s t )=max β r−t π(s r |s t )U(c i (s r ),l i (s r ))r=t s rsubject toc i (s r )+k i (s r ) ≤ F (k i (s r−1 ),A i (s r )l i (s r )) + (1 − δ)k i (s r−1 )∀r, s r1

Lagrangian for planner’s problem∞Xmax λ X1 β t π(s t )U ¡ c 1 (s t ),l 1 (s t ) ¢ X∞ X+ λ 2 β t π(s t )U ¡ c 2 (s t ),l 2 (s t ) ¢{c i (s t ),l i (s t ),k i (s t )}t=0 s t t=0 s t∞X X+ β t π(s t ) X "X ∞Xµ i (s t ) β r−t π(s r |s t )U ¡ c i (s r ),l i (s t ) ¢ #− V i (k i (s t−1 ),s t )t=0 s ti=1,2 r=t s⎡r ⎤∞X X+ β t π(s t )γ(s t ) ⎣ X £F (ki (s t−1 ),A i (s t )l i (s t )) + (1 − δ)k i (s t−1 ) ¤ − X £ci (s t )+k i (s t ) ¤ ⎦t=0 s t i=1,2i=1,2Now note that π(s r )=π(s r |s t )π(s t ).Note also that there is a ’partial summation formula of Abel’ which pointsout that∞P X∞β t µ t β r−t Pu(c r )= ∞ β t M t u(c t )t=0r=tt=0whereM t = M t−1 + µ t ,M −1 =0Lets apply this to the Lagrangianmax + X∞ X{c i (s t ),l i (s t ),k i (s t )}t=0 s t+ resource constraintsXβ t π(s t ) £ M i (s t )U ¡ c i (s t ),l i (s t ) ¢ − µ i (s t )V i (k i (s t−1 ),s t ) ¤i=1,2whereM i (s t )=M i (s t−1 )+µ i (s t )M i (s −1 )=λ iFor some reason KP instead choose to write choose instead to write∞X X Xmaxβ t π(s t ) £ M i (s t−1 )U ¡ c i (s t ),l i (s t ) ¢ + µ i (s t ) £ U ¡ c i (s t ),l i (s t ) ¢ − V i (k i (s t−1 ),s t ) ¤¤{c i (s t ),l i (s t ),k i (s t )}t=0 s t i=1,2+ resource constraints2 First order conditions1. c i (s t )β t π(s t )M i (s t−1 )U ic (s t )+β t π(s t )µ i (s t )U ic (s t ) − β t π(s t )γ(s t )=0Combining the FOCs for the two countries, we getU 1c (s t ) ¡ M 1 (s t−1 )+µ 1 (s t ) ¢ = U 2c (s t ) ¡ M 2 (s t−1 )+µ 2 (s t ) ¢2

orU 1c (s t )U 2c (s t ) = M 2(s t−1 )+µ 2 (s t )M 1 (s t−1 )+µ 1 (s t )2. l i (s t )β t π(s t )M i (s t )U il (s t )+β t π(s t )µ i (s t )U il (s t ) − β t π(s t )γ(s t )F ilCombining this with the FOC for c i (s t ) gives3. k i (s t )U il (s t )(M i (s t )+µ i (s t ))F il= U ic (s t ) ¡ M i (s t )+µ i (s t ) ¢or U il(s t )F il= U ic (s t )−β t π(s t )γ(s t )= β X t+1 π(s t+1 |s t )γ(s t+1 )[F ik +(1− δ)] − β X t+1 π(s t+1 |s t )µ i (s t+1 )V ik (k i (s t ),s t+1 )s t+1 s t+1U ic (s t ) ¡ M i (s t−1 )+µ i (s t ) ¢= β X π(s t+1 |s t ) £ U ic (s t ) ¡ M i (s t )+µ i (s t+1 ) ¢ [F ik +(1− δ)] − µ i (s t+1 )V ik (k i (s t ),s t+1 ) ¤s t+1U ic (s t )=β X ·π(s t+1 |s t ) U ic (s t ) M i(s t+1 )M i (s t [F ik +(1− δ)] − µ i(s t+1 ¸))M i (s t ) V ik(k i (s t ),s t+1 )s t+1As in the simpler Kocherlakota economy, Kehoe and Perri then define somenew variablesv i (s t )= µ i(s t )M i (s t )z(s t )= M 2(s t )M 1 (s t )ThenM i (s t )= M i(s t−1 )1 − v i (s t )and thusz(s t )= 1 − v 1(s t )1 − v 2 (s t ) z(st−1 )Now the first order conditions for consumption and capital can be rewrittenasU 1c (s t )U 2c (s t ) = M 2(s t )+µ 2 (s t )M 1 (s t−1 )+µ 1 (s t ) = z(st )= 1 − v 1(s t )1 − v 2 (s t ) z(st−1 )3

andU ic (s t )=β X ·π(s t+1 |s t ) U ic (s t 1)1 − v i (s t+1 ) [F ik +(1− δ)] − v i(s t+1 ¸)1 − v i (s t+1 ) V ik(k i (s t ),s t+1 )s t+1Kehoe and Perri focus on Markov shocks, so that π(s t |s t−1 )=π(s t |s t−1 ).Then look for decision rules c i (x t ),l i (x t ),k i (x t ) together with policy rulesfor z(x t ) and v i (x t ) where the state x t = ¡ z(s t−1 ),k 1 (s t−1 ),k 2 (s t−1 ),s t¢. Thusthis is the same state space we used to solve the Kocherlakota model, exceptthat k 1 (s t−1 ) and k 2 (s t−1 ) have been added.3 Computation procedureLet x =(z,k 1 ,k 2 ,s) be the state (note that the notation is a bit confusing, sincethe understanding is that z, k 1 and k 2 are inherited from the previous period,and do not depend on the current state s).Define also value functionsW i (x) =U(c i (x),l i (x)) + β X s 0π(s 0 |s)W i (x)Define a grid X on the state space (i.e a three dimensional grid over thethree continuous variables z, k 1 and k 2 for each possible (discrete) value for s.Guess © c 0 i (x),l0 i (x),k00 i (x),z00 (x),vi 0(x),W0 i (x)ª ∀x ∈ X. One good initialguess is the solution to the planning problem without enforcement constraints.Note that these allocations correspond to the allocations that would emerge in adecentralized economy with a complete set of asset markets and no enforcementproblems.Now procede to update the guess. Suppose, we are on the n th iteration inupdating the vector of unknown functions, and suppose we are at point q in thegrid.First assume neither enforcement constraint binds. Thus immediately v i (q) =0 and z 0 (q) =z.Compute (c i (q),l i (q),ki 0 (q)) (6 numbers) that satisfy1. The consumption leisure FOC for both agents (2 equations)U il (q)F il (q) = U ic(q)2. The expression for z (1 equation)U 1c (q)U 2c (q) = z0 (q)3. The world resource constraint (1 equation)X[c i (q)+ki(q)] 0 = X[F (k i (q),A i (q)l i (q)) + (1 − δ)k i (q)]i=1,2i=1,24

4. The FOCs for capital (2 equations)U c (c i (q),l i (q))= β X ·π(s0|s) U c (c n−1i (x 0 ),lin−1s0(x 0 )) F k(ki 0(q),A(s0 )l n−1i(x 0 )) + (1 − δ)1 − v n−1−i(x 0 )Note that, to evaluate c i ,l i and v i in the next period, we use the oldguesses for the decision rules policy functions (superscript n − 1). The’unknowns’ in this equation are c i (q), l i (q) and ki 0 (q). Note, however, thatalthough we use the old functions for next period variables, we do evaluatethem at the correct point: x 0 =(z 0 (q),k1(q),k 0 2(q),s 0 0 ) .Thus at this point we solve a system of 6 non-linear equations in 6 unknowns.Then we check whether either of the two enforcement constraints is violated.If neither is, we are done with this grid point.Alternatively suppose one of the enforcement constraints is violated (say theone for country 1). Then we solve a different system of equations. Now we lookfor (c i (q),l i (q),k 0 i (q),v 1(q),z 0 (q)) (8 numbers) that satisfy1. The consumption leisure FOC for both agents (2 equations)2. The expression for z (1 equation)U il (q)F il (q) = U ic(q)U 1c (q)U 2c (q) = z0 (q)3. The law of motion for z (1 equation)z0(q) = 1 − v 1(q)z14. The world resource constraint (1 equation)X[c i (q)+ki(q)] 0 = X[F (k i (q),A i (q)l i (q)) + (1 − δ)k i (q)]i=1,2i=1,25. The FOCs for capital (2 equations)U c (c i (q),l i (q))= β X ·π(s0|s) U c (c n−1i (x 0 ),lin−1s0(x 0 )) F k(ki 0(q),A(s0 )li n−1 (x 0 )) + (1 − δ)1 − vi n−1−(x 0 )6. The enforcment constraint for country 1 is an equality (1 equation)vn−1 i(x 0 ¸)1 − v n−1i(x 0 ) V ik(k0(q),s 0 )vn−1 i (x 0 ¸)1 − vi n−1 (x 0 ) V ik(k0(q),s 0 )U c (c i (q),l i (q)) + β X s 0π(s 0 |s)W n−1i (x 0 )=V i (k i ,s)5

Note that in equilibrium, if the planner is doing its job, it will not be thecase the both enforcement constraints bind simultaneously.Once decision rules have been updated at one point in the grid, we move tothe next point, and continue until decision rules have been updated. Now (andonly now) we can update the value functions, for each x ∈ X. For example, forx = qW ni (q) =U(c n i (q),l n i (q)) + β X s 0π(s 0 |s)Win−1 (x 0 (q))If the new value functions are cloes to the old for every x, we are done.Otherwise, we need to repeat the entire process.It is important the the initial guess for the value functions is uniformlygreater than or equal to the true value of the value function.6