= 1980 and let time run out to infinity. , and working age population,, T. L, and capital stocks, , labor,

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1 Solving he Model The base case model feaures a represenaive household ha chooses pahs of consumpion, leisure, and invesmen o maximize uiliy. The pahs of TFP and populaion are exogenously given (in daabase.x), and he agen has perfec foresigh over heir values. We sar he model a dae T 98 and le ime run ou o infiniy. Definiion. Given sequences of produciviy, A, and working age populaion, T, T +,..., and he iniial capial sock, w, ineres raes, r, consumpion, N, for, an equilibrium is sequences of wages, T, labor, L, and capial socks,, such ha. given he wages and ineres raes, he represenaive household chooses consumpion, labor, and capial o maximize he uiliy funcion () subjec o he budge consrains (2), appropriae nonnegaiviy consrains (3), and he consrain on (4); () max β ( log + ( )log( hn L) ) s.. clk,, T (2) + wl + ( δ + r) + (3),, L, L hn (4) T given T 2. he wages and ineres raes, ogeher wih he firms choices of labor and capial, saisfy he cos minimizaion and zero profi condiions; (5) w ( α) A α L α (6) r α A L α α 3. consumpion, labor, and capial saisfy he feasibiliy condiion. (7) α α ( δ ) A L We urn hese equilibrium condiions ino a sysem of equaions ha can be solved o find he equilibrium of he model. We begin by aking he firs-order condiions of he household s problem o obain

2 (8) w( hn L) + (9) β( δ + r + ) ombining he household s opimaliy condiions (8) and (9), he firm opimaliy condiions (5) and (6), and he feasibiliy condiion (7), we can specify a sysem of equaions ha can be solved o find he equilibrium of he model. Plugging (5) and (6) ino (8) and (9), and using he feasibiliy condiion (7), we obain he sysem of equaions α α () ( α) A L ( hn L) () + β( δ + αa L ) α α (2) A L α α ( δ ) Solving for an equilibrium involves choosing sequences of consumpion, capial socks, and hours worked such ha hese equaions are saisfied, given he iniial condiion T and final condiion, he ransversaliy condiion, (3) lim β + In principle, he sysem of equaions ha characerize he equilibrium, ()-(2), involves an infinie number of equaions and unknowns. To make he compuaion of an equilibrium racable, we assume ha he economy converges o he balanced-growh pah a some dae T, which allows us o runcae he sysem of equaions. Using he feasibiliy condiion (2) o solve for, we can wrie hese equaions as (4) α α α α ( α) A L ( hn L) ( A L + ( δ) + ), T, T +,..., T (5) where A L + ( δ ) α α α α A L + ( δ ) + + gη. T T β( δ + αa L ), T, T +,..., T α α We choose T so ha T T is large, say 6, so ha we are solving he model over he period We hen consruc he exogenous variables. The exogenous variables A, N for are as hey are in he daa. For 26 24, we assume ha TFP grows a a consan rae equal o he average growh rae of TFP over he period 98

3 25 and ha he working-age populaion grows a he same rae as in These are he growh raes g α and η in he specificaion of he balanced-growh pah. Solving he model now consiss of choosing T +, T + 2,..., T, and L T, L T+,..., LT o solve he sysem of equaions (4) and (5). This sysem of 2( T T + ) nonlinear equaions in 2( T T + ) unknowns can be solved relaively quickly using numerical mehods.

4 Running he Program Program Inpus The user mus provide wo files o he program. The firs file should be named parambase.x and consis of a single column vecor of he parameers β,, δ, α, g, η, and. The second file should be named daabase.x and conain a ( T T) 6 T marix of values: levels of TFP, A, working-age populaion, N, available hours, hn, consumpion ax raes, labor ax raes, and capial ax raes. These files mus be in a form ha can be inerpreed by MATLAB. One mehod is o ener he daa ino a Microsof Excel spreadshee and save he file as a ab delimied file. Program Oupu Upon successful compleion, he program will save a ( T T) 6 marix of values o he file oupu.xls which is a ab delimied file. This file can be opened in Excel for inspecion or o creae plos. The daa can also be direcly manipulaed in MATLAB. The variables in he file are Y /N, X /Y, L /(hn ), /Y, /Y and r -δ. Soluion Mehod hoosing T +, T + 2,..., T and L T, L T+,..., LT o saisfy (4) for T, T +,..., T, and (5) for T, T +,..., T requires solving 2( T T + ) equaions in 2( T T + ) unknowns. The accompanying MATLAB program uses Newon s mehod o solve he sysem of equaions. Define he sacked vecor of variables x [,,...,, L, L,..., L ] x [ and T+ T+ 2 T T T+ T arrange he sysem of equaions so ha hey are of he form f( x ) v, where v is a 2( T T + ) vecor of zeros. The algorihm involves making an iniial guess a he i+ i i i variables, x, and updaing he guess by x x Df( x ) f( x ), where Df ( x i ) is he marix of parial derivaives of f ( x ) evaluaed a i x. The sysem of equaions does no have closed-form expressions for he parial derivaives needed o compue Df ( x i ), and so he derivaives have o be evaluaed numerically. A soluion is obained when he funcion, evaluaed a he new ierae of x, has a maximum error less han some value ε, where ε is a small number. Alhough his mehod of solving a sysem of nonlinear equaions can converge o a soluion quickly, his mehod is no globally convergen and can become suck away form a zero of f ( x ) or may no converge a all. The iniial guess, x, is imporan. Furher deails on he implemenaion of Newon s mehod can be found in Press, Flannery, Teukolsky, and Veerling (22). To increase he probabiliy of he algorihm converging o he correc answer, we solve a sequence of models, beginning wih a simple version of he model, which we know how

5 o solve, and progressing o he model ha we would like o solve. The firs model we solve is he one in which TFP, populaion, and available hours are consan and equal o heir average values from 98 o 25, and he ax raes are all zero. The soluion o his problem is relaively easy o find. The nex model akes TFP, populaion, available hours, and ax raes o be convex combinaions of he consan values used in he iniial model and he acual values of TFP, populaion, available hours, and ax raes from he daa. Le λ be he weigh on he consan values, so ha ( λ) is he weigh on he values from he daa. The algorihm requires repeaedly decremening λ and solving he resuling model, each ime using he soluion o he model before i as he iniial guess. The algorihm proceeds unil i solves he case in which λ, which corresponds o he model whose soluion we desire.