NBER WORKING PAPER SERIES

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1 NBER WORKING PAPER SERIES A DUAL METHOD OF EMPIRICALLY EVALUATING DYNAMIC COMPETITIVE EQUILIBRIUM MODELS WITH MARKET DISTORTIONS, APPLIED TO THE GREAT DEPRESSION AND WORLD WAR II Casey B. Mullgan Workng Paper NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambrdge, MA February 2002 I apprecate the comments of John Cochrane, Jess Gaspar, Lars Hansen, Bob Lucas, Jeff Russell, conference partcpants at the Mnneapols Federal Reserve Bank s October 2000 Great Depressons conference, GSB macro lunch eaters, and Economcs 332 students, the research assstance of Faban Lange, and the fnancal support of the Smth-Rchardson Foundaton and the Unversty of Chcago s George Stgler Center for the Study of the Economy and the State. The vews expressed heren are those of the author and not necessarly those of the Natonal Bureau of Economc Research by Casey B. Mullgan. All rghts reserved. Short sectons of text, not to exceed two paragraphs, may be quoted wthout explct permsson provded that full credt, ncludng notce, s gven to the source.

2 A Dual Method of Emprcally Evaluatng Dynamc Compettve Equlbrum Models wth Market Dstortons, Appled to the Great Depresson and World War II Casey B. Mullgan NBER Workng Paper No February 2002 JEL No. H30, C68, J22 ABSTRACT I prove some theorems for compettve equlbra n the presence of market dstortons, and use those theorems to motvate an algorthm for (smply and exactly) computng and emprcally evaluatng compettve equlbra for dynamc economes. Although a compettve equlbrum models nteractons between all sectors, all consumer types, and all tme perods, I show how my algorthm permts separate emprcal evaluaton of these peces of the model and hence s practcal even when very lttle data s avalable. I then compute a neoclasscal growth model wth dstortonary taxes that fts aggregate U.S. tme seres for the perod and conclude that, f t s to explan aggregate behavor durng the perod, government polcy must have heavly taxed labor ncome durng the Great Depresson and lghtly taxed t durng the war. In other words, the challenge for the compettve equlbrum approach s not so much why output mght change over tme, but why the margnal product of labor and the margnal value of lesure dverged so much and why that wedge perssted so long. In ths sense, explanng aggregate behavor durng the perod has been reduced to a publc fnance queston were actual government polces dstortng behavor n the same drecton and magntude as government polces n the model? Casey B. Mullgan Department of Economcs Unversty of Chcago 1126 East 59th Street Chcago, IL and NBER c-mullgan@uchcago.edu

3 Table of Contents I. Introducton... 1 II. Compettve Equlbra wth Dstortonary Taxes...7 Setup of the Model...8 Consumers...8 Frms and Dstortonary Taxes...9 The Government Resource Balance Constrants Defnton of a Compettve Equlbrum Problems wth the Prmal Approach III. The Dual Procedure for Computng and Evaluatng the Model Demand and Supply Prces Consumer Problem Frms Tax Wedges IV. Applcaton to the Great Depresson and WWII A Neoclasscal growth Model wth Labor and Captal Income Taxes as a Specal Case Smulated Polces Understandng the Great Depresson...26 Productvty Shocks Cannot Explan , or Income and Sales Taxes are not an Important Part of the Labor-Lesure Dstorton Transfer Programs Have Lttle or No Effect How Much Can Internatonal Trade Explan? Labor Market Regulatons Can Monopoly Unons be Part of the Story? What about Monetary Shocks? Intertemporal Dstortons Durng the Perod V. Conclusons Understandng the Amercan Economy Lessons for Other Applcatons...42 The Relaton Between the Dual Method and GMM Forecastng vs. Emprcal Evaluaton...44 Tastes Shfts vs. Market Dstortons...44 VI. Appendx: Data and Calculatons for the perod VII. References...47

4 I. Introducton Explanng aggregate measures of behavor, such as employment, output, consumpton, and nvestment, has for decades been one of the prme nterests of macroeconomsts, and others. Almost as old s the queston of how much aggregate behavor mght be explaned by prvate sector mpulses (n modern parlance: tastes, technology, market structure, and demographc shocks) rather than publc sector mpulses such as government regulatons, taxes, and subsdes. Somewhat more recent are attempts to quanttatvely model prvate sector behavor as a dynamc compettve equlbrum. Kydland and Prescott (1982) s a poneerng, and rather successful, attempt. Ths paper reconsders the nteracton between the tme seres data, constructon of compettve models of prvate behavor, and constructon of models of government polcy, n order to: (a) further mprove and apply the computable dynamc general equlbrum methods that have been used by the many mportant papers followng Kydland and Prescott, and (b) emphasze and explot some of the economc smlartes between computable general equlbrum models and mcroeconometrc models of market supply and demand. One obstacle n the use of quanttatve dynamc general equlbrum models has been ther analytc tractablty, and the nature of the data requred to evaluate them. As they become more realstc, especally as regards to modelng market dstortons, quanttatve general equlbrum models become more complex to solve or smulate, and ths complexty has tempted many takng the compettve equlbrum approach (eg., Braun and McGratten 1993 or Ramey and Shapro 1998) to gnore, for example, the dstortonary effects of taxes, and nearly all studes gnore the dstortonary effects of busness, labor, and product regulatons. A soluton to a dynamc general equlbrum model also depends on the behavor of tme sequences of the exogenous varables nto the dstant future, so emprcally evaluatng a model and ts soluton requres measurement of, or guesses about, the nature of those sequences n the future. I argue that these dffcultes can be avoded, wthout the cost of addtonal approxmaton and wth the beneft of added economc understandng, by changng the computaton procedure. I motvate ths change by borrowng prce theory s concepts of supply

5 Equlbra wth Dstortons - 2 prce and demand prce and usng those concepts to generalze and renterpret a calculaton that has been made by some n the macroeconomc lterature on unmeasured tme-varyng preferences. Although tme-varyng preferences are perfectly consstent wth the computable general equlbrum methodology, n practce only a very small fracton of quanttatve general equlbrum models nclude them. 1 Two of those studes are Parkn (1988) and Hall (1997) and, although ther models do not allow for market dstortons, both authors make an nterestng calculaton. They compare the postwar tme seres behavor of the consumpton-lesure rato, whch they nterpret as one of two determnants of a representatve consumer s margnal rate of consumpton-lesure substtuton, to the tme seres behavor of the average product of labor, whch they nterpret as the one determnant of the margnal product of labor. When these two seres dverge, they suggest, ths s evdence of a preference shft. 2 Regardless of whether preferences should be modeled as stable or not, the purpose of my paper s to suggest that the consumpton-lesure rato mght be nterpreted as (proportonal to) the supply prce of labor, and the average product of labor as (proportonal to) the demand prce. Hence, the Parkn-Hall calculaton mght nstead be nterpreted as generatng measures of gaps between supply and demand prces labor-lesure dstortons and be generalzed n order to emprcally evaluate stable preference compettve equlbrum models from a publc fnance perspectve. My evaluaton method s computatonally economcal, easy to adjust for questons of data qualty, and economcally consstent wth a varety of partal and general equlbrum models of market 1 Whether a utlty functon vares over tme, or t s a stable functon wth some unmeasured and tme-varyng arguments s not relevant for Hall and Parkn s analyss (see Becker 1996 for further dscusson of ths pont). In any case, I use the more common practce of modelng utlty as a stable functon of measured varables. Ingram, Kocherlakota, and Savn (1997) have calculatons smlar Hall and Parkn s, although not dentcal because Ingram et al have two consumpton goods, and nterpret devatons between measured consumpton and work hours as changes n home producton tme rather than preferences shfts. 2 One dfference between Hall and Parkn s that Parkn also consders a tme-varyng labor elastcty n the producton functon. Hence, the gap between the consumpton-lesure rato and the average product of labor s a measure of the preference shft n Parkn s model only when we correct the average product for any change n the labor elastcty.

6 Equlbra wth Dstortons - 3 dstortons. The computable general equlbrum lterature usually, and understandably, constructs compettve equlbrum explanatons of aggregate behavor proceeds as follows: () wrte down a model for government polcy (eg., a set of taxes, transfers, and regulatons) () wrte down a model for prvate sector behavor, ncludng responses to the modeled government polces () choose functonal forms and numercal parameters for the model of the prvate sector (eg., rate of tme preference, elastctes of substtuton n preferences, elastctes of substtuton n producton) (v) choose numercal values for the government polcy parameters (a) based on some observatons of government polcy and (b) so that the model government budget constrant balances n step (v) (v) compute a compettve equlbrum (eg., tme seres for employment, consumpton, nterest rates, etc.) (v) compare the equlbrum quanttes (and perhaps prces) to observed quanttes (and perhaps prces) Steps () - (v) mght be done once, n whch case the procedure s called smulaton, and the success of the model mght be judged on step (v) s metrc of the proxmty of smulated and observed quanttes. 3 Ths s the approach, for example, of Burnsde et al (2000) 4 and, essentally, Mullgan (1998), who conclude that a neoclasscal model cannot explan some tme seres comovements of employment and government expendture, and Cole and Ohanan (1999) who suggest that fscal polcy cannot explan the Great Depresson. Steps () - (v) mght be done many tmes, perhaps wth the objectve of choosng numercal values for the prvate sector parameters n order to maxmze step (v) s metrc of the proxmty of smulated and observed 3 A varety of metrcs have been proposed n the lterature; Watson (1993) s one study wth a detaled proposal, and a revew of prevous approaches. 4 One of the methodologcal contrbutons of the Burnsde et al study s to propose a specfc test statstc for evaluatng the ablty of a fscal polcy model, based on ts smulated quanttes and prces, to explan observed quanttes and prces.

7 Equlbra wth Dstortons - 4 quanttes, n whch case the procedure s called estmaton. Ths s the approach, for example, of Hansen and Sargent s (1991, Chapter 7) study of recursve lnear compettve equlbrum models. In ether procedure, step (v) computng quanttes and prces that maxmze utlty, maxmze profts, and balance the government budget gven numercal values for government polcy s not an easy one, especally when government polcy s dstortonary. Indeed, ths step can be so dffcult that many takng the compettve equlbrum approach are tempted to gnore market dstortons. Even when dstortonary taxes are ncluded n the model, t s dffcult wthout the assstance of a computer to understand exactly how changes n prvate sector parameters or government polces affect equlbrum quanttes and prces. My approach s not to advocate smulaton versus estmaton, but rather to change steps (v)-(v) n order to smplfy computaton and data requrements by orders of magntude, and to hghlght the publc fnance dmenson of the problem. Here are my proposed steps: () wrte down a model for government polcy (eg., a set of taxes, transfers, and regulatons) () wrte down a model for prvate sector behavor, ncludng responses to the modeled government polces () choose functonal forms and numercal parameters for the model of the prvate sector (eg., rate of tme preference, elastctes of substtuton n preferences, elastctes of substtuton n producton) (v)n use observed quanttes to compute margnal rates of substtuton and transformaton (v)n use the compettve equlbrum condtons, and the results from (v)n, to compute numercal values for the government polcy parameters, and perhaps prces (v)n compare the equlbrum polces (and perhaps prces) to observed polces (and perhaps prces) Notce how I have left ()-() ntact, changng only (v)-(v). In partcular, (v)n calculates supply and demand prces, and (v)n nterprets the gap as ndcators of market dstortons. I propose to feed observed quanttes nto the model to nfer polces, rather than feedng polces nto the model to nfer quanttes. For ths reason, I refer to ()-(v) as the prmal or polcy-quantty approach and my approach ()-(v)n as the dual or quantty-polcy

8 Equlbra wth Dstortons - 5 approach, and hghlght ther dfferences n Fgure 1. Fgure 1 Overvew of the Prmal and Dual Approaches We see n the Fgure s left panel that the prmal approach uses a numercal model of prvate behavor and observed polces to smulate quanttes and prces, and the red ovals emphasze some of the practcal dffcultes wth the approach. As shown n the rght panel, the dual approach uses a numercal model of prvate behavor and observed quanttes to smulate polces and prces.

9 Equlbra wth Dstortons - 6 As n the prmal approach, the dual approach has both smulaton and estmaton versons. Steps ()-(v)n could be done once (aka, smulaton ) or steps () - (v)n mght be done many tmes, perhaps wth the objectve of choosng numercal values for the prvate sector parameters n order to maxmze step (v)n s metrc of the proxmty of smulated and observed polces (aka, estmaton ). In ether procedure, step (v)n computng polces that satsfy equlbrum condtons gven observed quanttes s a trval one. Indeed, estmaton s much more economcal wth my dual approach than wth the prmal approach because performng step (v)n many tmes s much easer than performng step (v) many tmes. Consder frst a very smple statc economy wth many dentcal frms producng one unt of the consumpton good c for every unt of labor tme L t hres. The labor tme s suppled by dentcal consumers who earn w per unt tme, pay a prce 1 for the consumpton good, and have utlty functon ln c + 2 ln (1-L). Frms also pay a tax at flat rate J on ts payroll wl, and the government uses the revenue to fnance lump sum taxes n the amount v to the consumers. Followng the usual conventons, let s defne a compettve equlbrum to be, for a gven tax rate J, four scalars {c,l,w,v} such that: () c and L maxmze utlty ln c + 2 ln (1-L) subject to c = wl + v and gven w and v, () the resource constrant c = L bnds, () c and L maxmze profts [c - (1+J)wc] gven w and J, and (v) the government budget constrant v = JwL balances. The prmal approach of emprcally evaluatng ths model s pckng a postve value of the utlty parameter 2, and usng a measure of the tax rate J to smulate equlbrum consumpton and labor. In ths example, the smulaton nvolves just a smple formula, namely c * = L * = [1+(1+J)2] -1, 5 and we can compare the smulated values c * and L * wth observed values. The dual approach uses a parameter value 2 and data on c and L to calculate margnal rates of substtuton (2c/(1-L)) and transformaton (1), to smulate the tax rate J * based on the gap (namely, J * = (1-L)/(2c)-1), and to compare the smulated tax rate wth the observed tax rate. Both methods would lead to analogous conclusons. For example, f the prmal smulaton over-predcted labor ( the model says people should work less than they do ), then the dual smulaton would under-predct the tax rate ( the model says people are actng lke the tax rate s lower than t s ). Both the prmal and dual methods smulatons nvolve a sngle formula n ths example, but one mportant 5 The formula s just the soluton to the three (non-redundant) compettve equlbrum condtons: c=l, w(1+j)=1, and 2c=w(1-L).

10 Equlbra wth Dstortons - 7 attrbute of the dual method s that, as we complcate the model, the prmal smulatons get a lot more complcated than do the dual smulatons. Secton II exposes the generalty and computatonal smplcty of the dual approach by provng two propostons for a dstortonary tax dynamc general equlbrum model wth many agents, consumpton goods, and nvestment goods. The frst shows that, gven a model of prvate sector behavor and observed quanttes, a polcy consstent wth compettve equlbrum can be computed two frst order condtons at a tme, and n any order. The second proposton shows that there s one and only one set of government polces that s consstent wth a compettve equlbrum. It s well known that analogous results cannot be proven for the prmal approach because there can be zero, or multple, equlbrum responses to a gven polcy (the Laffer curve characterzes some of the well known examples of nonunque or nonexstent compettve equlbra), and the equlbrum quantty n any one sector at any one date depends on polces and technologes n all sectors at all dates. Hence, the dual approach does not present equlbrum choce or nonexstence problems, and does not requre much accurate data. Secton III then uses my procedure to explan the perod wth a neoclasscal growth model. The calculatons are so smple that they are reported n a self-contaned appendx for easy verfcaton by the nterested reader. I show that, n order for the neoclasscal growth model to explan aggregate behavor durng the perod, margnal labor ncome tax rates must have been qute hgh durng the Depresson and qute low durng the war. Snce t appears that margnal labor ncome tax rates had a dfferent hstory, I conclude that the neoclasscal growth model cannot explan why there was so lttle employment durng the Depresson and so much durng the War. Perhaps another defensble concluson s that margnal labor ncome tax rates dd have a hstory lke that generated by the model, and that the usual measures of margnal tax rates are not capturng all of the dstortons ntroduced by government regulatons, taxes, and subsdes durng the perod. Under ths nterpretaton of my results, explanng the perod s reduced to the publc fnance problem of dentfyng and quantfyng the varous government polces drvng a wedge between labor supply and labor demand, and showng how actual margnal tax rates had a hstory lke that generated by the model. II. Compettve Equlbra wth Dstortonary Taxes

11 Equlbra wth Dstortons - 8 II.A. Setup of the Model There are a contnuum of nfntely lved consumers and frms, each takng prces and polcy parameters as gven. Consumers are parttoned nto h=1,...,h (equally populated) types accordng to the productvty of ther labor, ther preferences and ther treatment by the government. There are M captal goods, whch are used together wth labor to produce more captal goods or to produce the N consumpton goods. Any frm produces only one of these M+N goods; frms are ndexed by ther sector j=1,...(m+n) wth the frst N sectors producng consumpton goods, and the rest producng captal goods. Snce the economy s assumed to be compettve the ownershp of captal does not affect the allocaton n the economy. For convenence I assume that all captal s owned by the frm producng t and rented out to the other frms for producton purposes, and that there s no uncertanty. 6 Vectors are denoted by underlned letter: x, and are column vectors. Matrces are denoted by capped letters: $x. I use * $ to denote multplcaton element-by-element. Let x -1 stand for the vector of recprocals of x. Tme s dscrete and ndexed by t = 0,...,. Consumpton good prces, gross of taxes, are gven by: pt () [ p(), t p(),... t p ()]' t = 1 2 N II.A.1 Consumers The consumpton of the N consumpton goods by ndvdual h s gven by the vector h h h h c () t [ c (), t c (),... t c ()]' t. = 1 2 N Aggregate consumpton of the economy s gven by h ct () = c() t. Total labor h M+ N = = 1 h h suppled by the household s L () t L () t and L h () t denotes the labor by h to sector. The vector of ownershp shares by h of the N+M frms s gven by nterest rate s gven by q(t) h h h α α α α = 1 2 N+ M h [,,..., ]'. The 6 Mullgan (2001b) studes the applcaton of the dual method to stochastc models.

12 Equlbra wth Dstortons - 9 Preferences are governed by: t = 1 h t s h h h U = e u ( c ( t), L ( t)) t= 0 π( ) (1) where B(t) s the consumer s date (t-1) one perod forward rate of tme preference. The budget constrant s gven by: t h 1 h h h h qs w tl t pt c t v t z t= ( + ( )) [ ( ) ( ) ( )' ( ) + ( )] + α ' = 0 s= (2) where z s the value at date zero of the frms and v h () t denotes the lump sum transfers at date t. The resource constrant of the ndvdual can also be expressed usng a seres of constrants as follows: a h h h h h 1 h w () t L () t + v () t + a () t = p()' t c () t + [ 1+ q( t + 1)] a ( t + 1) t = 0,..., h h () 0 = α ' z where w h (t) and a h (t) are scalars denotng household h s date t wage rate and asset holdngs. (3) II.A.2 Frms and Dstortonary Taxes The producton functons are gven by f ( K ( t), L( t), t) where N + N M N K ( t) = [ K 1 + ( t), K 2 + ( t),..., K ( t)]' H and L () t = [ L 1 (), t L 2 (),..., t L ()]' t are the vectors of captal and labor nputs used by frm at date t. K () t wll denote the date t aggregate amount of type captal. The rental rate vectors of nputs at date t are gven by N+ N M N rt () = [ r 1 + (), t r 2 + (),... t r ()]' t and 1 2 H wt ( ) = [ w( t), w( t),... w ( t)]' Taxes are leved on labor at the rates 1 2 τ( t) = [ τ ( t), τ ( t),..., τ H ( t)]', and on

13 Equlbra wth Dstortons - 10 captal nputs at rates are then () t [ N (), t N + 1 (),..., t N + = M ()]' t. The nput prces faced by frms γ γ γ γ ~ r() t = r() t *[ $ + ( t)] and. 1 γ w ~ () t = w() t *[ $ 1 + τ ( t)] M H The objectve n the consumpton good sector s: z () t = max[ p() t f ( K (), t L(),) t t ~ r ()' t K () t w ()' t L ()] t (4) = 1,..., N and t = 0,..., The problem n the producton goods sector can be expressed recursvely as V( K ( t)) = max { K ( t), L ( t)} r () t K () t ~ r ()' t K () t w ()' t L () t 1 + ( 1+ qt ( )) V( K( t+ 1)) (5) s.t. K ( t + 1) = f ( K (), t L(),) t t + ( 1 δ ) K () t (6) K ( 0) = K = N + 1,... N + M 0 The value of the frms at date t s gven by zt ( ) [ z( t), z( t),..., z ( t)] = 1 2 N+ M In ths set up the nvestment by frms s reversble. The producton of captal and consumpton goods however s restrcted to be non-negatve,.e. f ( K ( t), L( t), t) 0 t,. An nteror soluton s assumed to hold, although see Houthakker (1995) or Mullgan (2000, 2001a), for some dscrete-choce nterpretatons of the nteror condtons. II.A.3. The Government The government budget constrant s gven by: N+ M h gt ()' pt () = [ γ() t * $ K( t)]' rt ( ) + [ τ ( t) * $ L()]' t wt () v() t = 1 H h= 1 (7) where gt ( ) [ g( t),..., g ( t)]' s the vector of government consumpton. Ths constrant = 1 N smply says that government spendng (consumpton and net transfers) equals the sum of labor

14 Equlbra wth Dstortons - 11 and captal ncome taxes. II.A.4 Resource Balance Constrants Markets for consumer goods, captal goods, labor, and assets clear at each date. In other words, government and prvate purchases equal output n each of the N consumpton good sectors, captal demanded by frms equal supply (captal type-by-type), labor demanded by frms equal supply (labor type-by-type), and net household asset holdngs equal the value of the frm sector. Algebracally, market clearng mples (8)-(11). gt () + ct () = y () t = [ y1(),..., t y ()]' t wth y () t = f ( K (), t L(),) t t t (8) N N K () t =Σ K () t, = N + 1,... N + M, t (9) j j N+ M 1 2 [ L( t), L ( t),..., L H ( t)]' = L ( t), t (10) Σ N = = 1 + M h 1 z() t = Σha () t, t (11) II.A.4 Defnton of a Compettve Equlbrum Gven a polcy sequence { gt (), (),{ t v h H N+ M γ ()} t,{()} τ t } h= 1 = 1 t= 0, ntal captal stocks K 0 and ntal ownershp shares { } α h H h=1 sequences {{ L h (), t a h (), t c h ()} t H=1, { ( )} N+ M, K t h, a compettve equlbrum s gven by quantty = N+ { pt ( ), wt ( ), rt ( ), qt ( )} and a sequence of frm values {()} t= 0 N+ M 1 { K ( t), L( t)} } = 1 t= 0 zt t = 0 such that:, prce sequences () {{ L h ( t), a h ( t), c h H ( t)} } h= 1 t= 0 maxmze (1) subject to (2) (or (3)) for all h N () {{ K ( t), L( t)} } maxmze (4) for =1,...,N and all t = 1 t= 0

15 Equlbra wth Dstortons - 12 N+ M {{ K ( t), L( t)} } = N+ 1 t= 0 maxmze (5) for =N+1,...,N+M () (v) z t ( )= 0 for all =1,...,N and all t t 1 s z() t = { 1+ q()} s ( 1 δ) r () t K () t for =N+1,...,N+M and for all t t= 0 (7)-(11) hold at all t s= 0 () requres that households wllngly consume the equlbrum consumpton bundle, wllngly hold equlbrum assets, and wllngly supply equlbrum labor. () requres each type of frm to wllngly demand the equlbrum nputs. () s a free-entry condton, and requres that frms are only valued at the value of ther assets. (v) says the government budget constrant must hold and all markets clear. II.B. Problems wth the Prmal Approach Proposton 1a Gven a polcy sequence { gt ( ),{ v h H N+ M ( t)},{ γ( t), τ( t)} } h= 1 = 1 t= 0, ntal α h H captal stocks K 0 and ntal ownershp shares { }, a compettve equlbrum may not exst. h=1 Proof (by example) Consder a 1 household, 1 good economy wthout captal and wth a polcy havng no government consumpton, no captal taxes: = τ() t = τ * gt () = γ () t = 0 and vt () v*, as well as,. Ths mples from the gov t BC that: Lt () = * * v τ wt (). (12) Takng v* and J* as gven the household problem yelds a labor supply functon: * * Lt () = Lwt ( (); τ, v) (13) Unless the relaton (12) and (13) ntersect n the postve quadrant ths economy does not admt

16 Equlbra wth Dstortons - 13 an equlbrum. The exstence problem s more severe than suggested by proposton 1a: even f part of the polcy sequence s treated as a free parameter, there are stuatons where no compettve equlbrum exsts. Ths s outlned n proposton 1b: Proposton 1b Gven a polcy sequence { gt ( ),{ v h H N+ M ( t)},{ γ ( t)} } α h K 0 and ntal ownershp shares { } that admts a compettve equlbrum. h= 1 = 1 t= 0, ntal captal stocks N+ M =1, there mght not exst a sequence {()} } H h τ t = 1 t= 0 Proof (by example) Consder a 1 household, 1 good economy wthout captal wth the polcy g(t) * = ((t) = 0 and vt () = v. Also consder preferences that admt a (monotone, contnuous) nverse labor supply functon s.t. w S (L=0,t)=0 and a producton functon s.t. the nverse (monotone, contnuous) labor demand functon has w D (L=0,t)< 4. Then n equlbrum the tax revenue s gven by the area J(t)w*(t)*L*(t). Contnuty mples that ths area s bounded by the area A between the nverse supply and demand functon and s thus fnte. Thus for any v* > are no tax rates compatble wth a compettve equlbrum. A, there Here the labor tax rate s taken as a free parameter, and only government spendng, transfers and captal taxes are taken as gven. Stll t s easy to construct an example n whch for the gven polcy sequences there does not exst a equlbrum-compatble labor tax rate. Smlar examples can be constructed for cases n whch other subsets of the polcy sequence are labeled free parameters and gvens. Note the proof gven for proposton 1b s a smple case of an economy wth a contnuous, bounded Laffer curve. Any polcy allocaton requrng revenues greater than the bound on the Laffer curve can not possbly be supported. A smlar dea can be used to prove an analogous dea for the multple equlbrum case, as n Proposton 2.

17 Equlbra wth Dstortons - 14 Proposton 2 There may be multple compettve equlbra consstent wth a gven polcy sequence { gt ( ),{ v h H N+ M ( t)},{ ( t), ( t)} } = 1 γ λ = 1 = 0 ntal captal stocks K 0 and ntal ownershp shares α h { } H h=1. h t Proof (by example) Consder the same economy as n the proof for proposton 1b. Equatons (12) and (13) can have multple ntersectons ( backward bendng supply curve ). In other words, for a set of requred tax revenues there are two or more possble labor tax rates that rase the requred revenue n equlbrum. Proposton 3 If there s any mssng data for the polcy sequence N+ M γ τ ntal captal stocks K 0 and ntal ownershp { gt ( ),{ v h H ( t)}, ( t),{ ( t)} } α h shares { } H h=1 h= 1 = 1 t= 0, then compettve equlbra and prces are not computable. Proof Immedate Proposton 3 emphaszes how nfnte polcy sequences are requred nputs for the prmal approach. In practce, ths dffculty s handled by extrapolatng future polces from past polces, and often by truncatng the horzon. The next secton shows how nether of these approxmatons are requred by the dual procedure. III. The Dual Procedure for Computng and Evaluatng the Model The dual procedure smply uses the frst order condtons ()-() mpled by the defnton of compettve equlbrum to calculate tax rates.

18 Equlbra wth Dstortons - 15 III.A. Demand and Supply Prces III.A.1 Consumer Problem Let h h h h h h δu ( c ( t), L ( t), t)/ δc ( t) mrs( c ( t), L ( s)) = log h h h h δu ( c ( s), L ( s), s)/ δl ( s). Wth a known utlty functon, and known date t quanttes, mrs can readly be calculated. Item () of the defnton of compettve equlbrum requres that consumers wllngly demand the equlbrum quanttes. If these quanttes are postve, then () mples the frst order condton equatng margnal rates of substtuton to the relatve after-tax prce of goods (note normalzaton of p1()= t 1 t = 0,..., ): h h mrs( c ( t), L ( t)) = log( p ( t)) log( w ( t)) = 1,..., N, h = 1..., H t mrs( c ( t), c ( s)) = log( 1 + q( k)) log( 1 + q( k)) 1 1 k= 0 k= 0 s [C.1] [C.2] Equatons [C.1] are the wthn-perod frst order condtons and [C.2] the between-perod condtons. These condtons are related to the wthn- and between- perod condtons for frms, as shown below. III.A.2 Frms Item () of the defnton of compettve equlbrum requres that frms wllngly demand the equlbrum quanttes. If these quanttes are postve, then () mples the frst order condton equatng margnal products to the net-of-tax nput rental rates: δf( K1( t), L( t), t) h h log h = log( w ( t)) + log( 1+ τ ( t)) log( p( t)) δl () t [F.1] = 1,...,N, h = 1,...,H

19 Equlbra wth Dstortons - 16 δf( K1( t), L( t), t) h h log h = log( w ( t)) + log( 1+ τ ( t)) log( λ ( t)) δl () t [F.2] = N+1,...,N+M, h = 1,...,H δf( K1( t), L( t), t) j log j = log( rj( t)) + log( 1+ γ ( t)) log( p( t)) δk () t [F.3] = 1,...,N, j = N+1,...,N+M δf( K1( t), L( t), t) j log j = log( rj( t)) + log( 1+ γ ( t)) log( λ ( t)) δk () t [F.4] = N+1,...,N+M, j = N+1,...,N+M where s 1 λ () t = ( 1+ q( t + m)) r( t + s)( 1 δ ) s= 1 m= 0 s represent the PDV of a unt of captal of type 1. III.B. Tax Wedges The dual procedure as suggested n Secton I allows us to evaluate the model wthout explctly calculatng the (prmal) soluton to the maxmzaton problem. Even wth lmted data we can derve model mplcatons wth mnmal computatonal effort. The maxmzaton problems of the consumer and the frms mply a set of FOCs as gven above. Gven numercal functons for technologes and preferences, 7 and observatons of the quantty data, mutual 7 In other words, the quantty data s suffcent to calculate any margnal product or margnal rate of substtuton. For example, wth Cobb-Douglas producton and preferences, ths means that the numercal values of the share parameters and multplcatve productvty

20 Equlbra wth Dstortons - 17 consstency of the FOCs allows the dervaton of a set of tax wedges. These tax wedges n turn can be used to deduce the polcy sequences consstent wth the model. Proposton 4 establshes that n the present set-up for mnmal data t s possble to deduce the labor tax rate at date t for households h. Proposton 4 Gven a sample contanng no more data than labor supply and consumpton by one household h { L h h ( t), c ( t)} and data on the producton nputs for one of the consumpton frms{ K ( t), L( t)} at date t t s possble to obtan the labor ncome tax rate τ h () t. Proof: solve C.1 for log w h ( t) log p( t) and nsert nto F.1. To do ths s not even necessary to observe all quantty data at date t nor do we need any observatons from other tme perods. Proposton 5 shows how usng quantty data from 2 adjacent tme perods t s possble to deduce the complete set of prces and polces for the frst perod. These 2 propostons contrast wth the result from proposton 3 that the compettve equlbrum n the prmal problem s only computable f all polces are observed. Thus t s possble here to evaluate the model wthout access to the complete set of data and wthout the computatonal effort mpled by the prmal problem. Proposton 5 Gven observatons on quanttes {{ L h h H N+ M N+ M (), t c ()} t,{ K ()} t,{ K (), t L()} t } T h= 1 = N+ 1 = 1 t= 0 and t s possble to compute the prce sequences { pt ( ), w h ( t)} T t=0, and the polcy sequences for each date have been chosen. As mentoned above, wth some restrctons these parameters could be estmated by teratng on steps ()-(v)n. Or the parameters could be separately estmated other data sets, as s done for Solow resduals n much of the macroeconomcs lterature.

21 Equlbra wth Dstortons - 18 { γ( t), τ( t), v( t), g( t)} t T =0 for t=0,..., T. Proof Step 1. Begnnng wth =1 and p 1 (t)=1, C.1 yelds wt () for t=0,...,t. Gven wt (), p (t) for > 1 can be calculated from C.1's other condtons, t=0,...,t. Step 2. Use proposton 4 to get τ () t for t=0,...,t. Step 3. Use the condton F.2 to get λ( t) for t=0,...,t. Step 4. From C.2 for t and t-1 get qt () for t=0,...,t λ () t Step 5. From the defnton of : 1 λ( t 1) = ( 1+ q( t)) [ r( t)( 1 δ ) + ( 1 δ ) λ( t)] λ(), t λ( t 1), q() t Usng (from Step 3 and 4) and (specfed n the set-up) ths solves for r t ( ) Step 6. From F.3 obtan γ () t δ Step 7. Use the RBC (8) and the budget constrant (2) to obtan vt () and gt () Thus proposton 5 shows how to use quantty data and the consstency requrements to dentfy prce and polcy sequences consstent wth the compettve equlbrum assumpton. It s possble to obtan polcy data for only a subset of data. To llustrate proposton 5 consder how t s appled to the canoncal example of an economy wth one good and lesure, where the good serves both as the consumpton and captal good. There exsts only one type of labor. Step 1. Gven there s only one good p(t)=1 and from C.1 obtan w(t) for all t. Ths step smply explots the margn between lesure and consumpton for the consumer to deduce the prce of lesure faced by the consumer. Step 2. From F.1 obtan J(t). Gven the prce of lesure faced by the consumer and the margnal product of labor one can obtan the wedge between the wage pad by the

22 Equlbra wth Dstortons - 19 frm and the wage receved by workers Step 3. Snce the good doubles as a consumpton and captal good we mmedately have p(t)=1=8(t). Step 4. C.2 gves the nterest factor q(t) (where q(0)=0). Here the ntertemporal margn s exploted. Step 5. Here (1+q(t))=(1-*)(1+r(t)). The return on a captal unt produced equals the nomnal nterest rate. Ths gves the rental prce for a captal unt net of taxes. Step 6. From F.3 one obtans ((t). The rental prce gross of taxes has to equal the margnal product of captal. Ths allows deducng the captal tax rate. Step 7. Fnally as before use the RBC and budget constrant to obtan vt () and gt (). IV. Applcaton to the Great Depresson and WWII To see the usefulness of these methods, consder the queston How can aggregate U.S. behavor be explaned for the perod ? A frst step n answerng ths s to pck a model of the economy, say, the neoclasscal growth model wth dstortonary taxes and changng productvty. Second, I use the dual approach to generate the margnal tax rates renderng the observed quanttes to be exactly a compettve equlbrum of the model. I show how the requred margnal labor ncome tax rates change sgnfcantly over tme, suggestng that a model wthout dstortonary taxes, or wth tme-nvarant taxes, cannot ft the quantty data. I then look at some of the evdence on taxes and regulaton durng the perod, and suggest that t s mplausble for those polces to have generated the large margnal tax rate changes that are requred to replcate observed behavor n the model. IV.A. A Neoclasscal growth Model wth Labor and Captal Income Taxes as a Specal Case Here we lmt our attenton to the specal case of the model wth one type of household (H=1), one captal good (M=1), and one consumer good (N=1) that s perfectly substtutable for nvestment goods. The model government only consumes, lump sum transfers, taxes labor ncome, and taxes captal nputs. Gven a polcy sequence {g(t),v(t)} 4 0, and an ntal captal stock K 0, a compettve equlbrum wth labor ncome taxes s smply a constant z and sequences {c(t),l(t),k(t+1),w(t),q(t),((t),j(t)} 4 0 such that:

23 Equlbra wth Dstortons - 20 () gven z and {(1-J(t))w(t),q(t+1),v(t)} 0 4, {c(t),l(t)} 0 4 solve: 4 j t'0 e 't s'0 B(s) u(c(t),l(t)) s.t. 4 j t'0 Q(t)[(1& J(t))w(t)L(t) % v(t) & c(t)] % z ' 0 ln Q(t) / & j t s'1 ln [1 % q(t)] where B(t) s the consumer s date (t-1) one perod forward rate of tme preference. () The resource constrant bnds at each date t: f(l(t),k(t),t) - *K(t) = c(t) + (K(t+1)-K(t)) + g(t) () gven {w(t),q(t+1),((t)} 0 4 and K(0), z and {L(t),K(t+1)} 0 4 solve: z ' max 4 j {L(t),K(t%1)} t'0 Q(t)[f(L(t),K(t),t) & (K(t%1) & (1&*)K(t)) & w(t)l(t) & ((t)q(t)k(t)] (v) {g(t),v(t),j(t),((t),w(t),l(t)} 0 4 balances the government budget constrant at each date: g(t) + v(t) = J(t)w(t)L(t) + ((t)q(t)k(t) Gven data (L(t),K(t+1),K(t)) on quanttes for any perod t, and numercal utlty and producton functons, t s straghtforward to compute the polcy varables (J ( (t),0 ( (t),v ( (t),g ( (t)) that are consstent wth a compettve equlbrum:

24 Equlbra wth Dstortons - 21 J ( (t) ' 1 % u L (c(t),l(t)) u c (c(t),l(t))f L (L(t),K(t),t), (( (t) ' f (K(t),L(t),t) & * K e B(t) u c (c(t&1),l(t&1)) u c (c(t),l(t)) v ( (t) ' c(t) % [K(t%1) & (1&*)K(t)] & f(l(t),k(t),t) & J ( (t)l(t)f L (L(t),K(t),t) & 1 & 1 (14) g ( (t) ' f(l(t),k(t),t) & c(t) & [K(t%1) & (1&*)K(t)] where the term n square brackets s smply gross nvestment. The smulated tax rates are just the gap between the smulated demand and supply prces. I use producton and utlty functons famlar from the real busness cycle lterature (eg., Kng, Plosser, and Rebelo 1988): 8 u(c,l) / ln c % 2 ln (1&L) f(l,k,t) / A(t)L $ K 1&$ where L s measured as manhours as a rato of the annual tme endowment (2500 hours per person) for the populaton aged 15 and over, and all other quanttes are measured per person aged 15+. Appendx Table 1 reports {L(t),c(t)/Y(t)} for t = (where Y(t) s date t output). 9 Four adjustments are made durng wartme ( ) 10 to reflect the msmeasurement of output and the nvoluntary nature of wartme mltary labor supply (not captured n the model above). Frst, output s measured for the cvlan sector only, under the assumpton that cvlan and 8 Mullgan (2000) studes two other functonal forms as well, fndng very smlar results for the Great Depresson and somewhat dfferent results for WWII and other tme perods. 9 Data sources, and the wartme adjustments below, are explaned n Mullgan (2000). 10 Results are qute nsenstve to small changes n the defnton of war years because these adjustments are trval when the mltary s small, or there s a volunteer force.

25 Equlbra wth Dstortons - 22 mltary personnel produce measured output n proporton to ther measured labor ncome. To be consstent wth ths adjustment, the second adjustment s to measure labor nput as cvlan manhours only. Most wartme solders were drafted, so t s questonable whether ther consumpton and lesure s as voluntary as modeled above. My thrd adjustment s therefore to calculate consumpton as cvlan consumpton expendture per cvlan aged Ths adjustment slghtly ncreases measured wartme consumpton. IV.B. Smulated Polces Gven the numercal utlty and producton functons, the formulas for the polcy varable J ( (t) consstent wth the model s compettve equlbrum are: J ( (t) ' 1 & L(t) (1&L(t)) 2 $ c(t) f(l(t),k(t),t) (14)N Recall that J ( (t) s just the gap between the smulated demand and supply prces of labor whch, gven the assumed Cobb-Douglas functonal forms, are proportonal to the consumpton-lesure and output-labor ratos, respectvely. The last column of Appendx Table 1 calculates {J ( (t)} for t = , usng parameters $ = and 2 = The dual approach does not have mplcatons for transfers and government consumpton that can be tested wth natonal accounts data because the natonal accounts calculate these to ft the model (at least f we nterpret purchases and sales of government debt as lump sum transfers and taxes), so (14)N neglects the equatons smulatng transfers and government consumpton. 11 Cvlan consumpton s measured as the dfference between aggregate personal consumpton expendtures and one half of mltary wages (assumng that half of mltary wages are saved, pad n taxes, or pad to cvlan famly members). 12 These are bascally those used n the lterature, wth small dfferences due to the dfferent tme perod studed, and my explct modelng of dstortonary taxes.

26 Equlbra wth Dstortons - 23 Fgure 2 Smulated and Measured Margnal Labor Income Tax Rates Compared, Fgure 2 compares the margnal labor ncome tax rates {J * (t)} consstent wth the model s compettve equlbrum wth the margnal labor ncome tax rates calculated by Barro and Sahasakul from IRS data (1986). We see that the model predcts Depresson tax rates that are much hgher, and Wartme tax rates that are lower, than measured drectly from government tax records. It s easy to study the economc and statstcal reasons for the fluctuatons n the smulated margnal labor ncome tax rate {J * (t)}. To understand the statstcal reasons, recall from (14)N that J * (t), up to the rato 2/$ of constants, s one mnus the product of the laborlesure and consumpton-output ratos. Fgure 3 dsplays the measured tme seres for those ratos, and we see how the consumpton-output rato s pretty steady except durng the war when t s a bt lower. So most of the varaton n J * (t) comes from the labor-lesure rato whch s low

27 Equlbra wth Dstortons - 24 n the depresson and hgh n the war, so that smulated margnal tax rates are hgh durng the war and low durng the Depresson. The basc patterns n the data are hardly controversal see, for example, Fredman (1957, p. 117f) on low-to-medum frequency constancy of the consumpton-output rato and Lucas and Rappng (1969) labor fluctuatons. Fgure 3 Components of Smulated Labor Tax Rates: By Data Source, I have not removed trends from the data, but we see from Fgure 3 that trends are not partcularly notceable n the data I use to smulate margnal tax rates. Perhaps ths s one advantage of the dual approach there s less reason to remove trends of from the basc data (because there s not much trend!) and we mght worry less about the senstvty of results to

28 Equlbra wth Dstortons - 25 trend estmaton. Fgure 4 dsplays the economc components of the smulated tax rate, namely the margnal product of labor and the margnal rate of substtuton (see equaton (14)). The margnal product of labor, computed as tmes the average product of labor, s dsplayed as a sold lne. It follows a pretty steady trend over tme, except a bump durng the war and no growth Fgure 4 Components of Smulated Labor Tax Rates: By Economc Margn, For the most part, the smulated margnal rate of substtuton (MRS), or margnal value of lesure tme, s less than the margnal product of labor (MPL). Perhaps surprsng s the dramatc dvergence of MRS from MPL durng the perod (30 or 40 percentage ponts!), a wedge whch perssts untl the war. As I dscuss n the next subsecton, the rapd emergence of ths

29 Equlbra wth Dstortons - 26 wedge, and ts persstence, are crucal for understandng the Great Depresson. IV.C. Understandng the Great Depresson Fgure 2 and 4 make an mportant pont f an aggregatve compettve equlbrum model s to explan the Great Depresson, at least wth Cobb-Douglas producton and utlty functons, t must explan why MRS and MPL dverged so dramatcally and why the wedge perssted. Ths pont has mplcatons for many theores explored n the lterature: IV.C.1. Productvty Shocks Cannot Explan , or Cole and Ohanan (1999, p. 3) suggest that, f t could be argued that productvty shocks ({A(t)} n my notaton) were large and persstent enough, then a neoclasscal growth model could ft the 1930's data pretty well. They reject ths explanaton because they see no reason why productvty would have been low after 1933, but my analyss rejects t for a very dfferent reason: there s no productvty seres {A(t)} that can be fed nto the neoclasscal growth model (wthout some of the dstortons mentoned below) to ft the Depresson data because that model equates MRS and MPL for any realzaton of the productvty seres. In other words, whle the productvty parameter A affects the relaton between nputs and outputs, t does not affect ether the relaton between MRS and the consumpton-lesure rato or the relaton between the average and margnal products of labor. Hence, accordng to the model, technology shocks should not affect the gap between observed and smulated labor ncome tax rates, because these smulated rates are calculated from the consumpton-lesure and the output-labor ratos. Smlarly, Cole and Ohanan (1999, p. 3) and Prescott (1999, p. 26) suggest that the perod s not puzzlng for the real busness cycle approach, because there are lots of canddates for productvty shocks durng that perod. Perhaps there are good canddates, but productvty shocks do not cause MRS and MPL to dverge n the neoclasscal growth model and my Fgures 2 and 4 shows that such dvergence s what happened In summary, n addton to (or nstead of?) the rght tme seres for productvty shocks, the neoclasscal growth model needs to be amended to explan why MRS and MPL dverged and why that wedge perssted. 13 To put t another way, an adverse productvty shock decreases the MRS and MPL together n the neoclasscal growth model.

30 Equlbra wth Dstortons - 27 IV.C.2. Income and Sales Taxes are not an Important Part of the Labor-Lesure Dstorton Cole and Ohanan (1999, p. 6) suggest that government purchases, or taxes on factor ncomes, mght help explan some of the Depresson economy. However, my analyss suggests that government purchases, and taxes on captal, cannot explan why MRS and MPL would be dfferent, let along why and how that wedge would persst over tme. Of course, taxes on labor ncome create such a wedge, but Barro and Sahasakul s study suggests that federal taxes on payroll and ndvdual ncome were trval, and unchangng, durng the perod. Indeed, IRS records (IRS, varous ssues) show that the vast majorty of the populaton dd not fle ndvdual ncome tax returns durng the 1930's, so that any IRS-nduced tax wedge affected very few people (not to menton small for the few affected). 14 Taxes on consumpton expendture are also expected to drve a wedge between MRS and MPL (n the absence of other dstortons, consumers equate ther MRS to MPL/(1+F), where F s the margnal sales tax rate). The federal government dd not have a general sales tax, although t does have (and has had) excse taxes on goods such as cgarettes, gasolne, and mports. More general sales taxes have been collected by states and localtes. However, the revenues from these taxes are too few, and not changng enough over tme, to drve much a of wedge. Furthermore, gven the assumed logarthmc functonal forms and the fact that my measure of consumpton s nclusve of sales taxes, sales taxes do not drve a wedge between measured MRS and MPL. IV.C.3. Transfer Programs Have Lttle or No Effect Government transfer payments, such as those used by Socal Securty, welfare, and unemployment systems are also expected to affect the gap between MRS and MPL. Unfortunately (for the analyst), there are many transfer programs at the federal, state, and local levels that mght be expected to drve a wedge, and the ncentve effects of even one of those programs are complcated, heterogeneous, and changng over tme. Indeed, a entre paper or lterature mght be devoted to the wedge created by one enttlement program n one year, for one 14 State and local ncome taxes are not ncluded n my calculatons. However, snce these taxes tend to be flat (e, relatvely few tax brackets and a relatvely broad tax base), and revenues from these taxes were essentally zero durng the 1930's (Census Bureau 1975 seres Y-658), t seems that these taxes had practcally no effect on the wedge between MRS and MPL.

31 Equlbra wth Dstortons - 28 subset of the populaton (eg., Feldsten and Samwck 1992 on 1990 Socal Securty beneft formulas and the workng-aged populaton, Blnder, Gordon and Wse 1980 on 1977 Socal Securty beneft formulas and the populaton aged 62-69, or Fraker, Mofftt, and Wolf 1985 on 1981 AFDC). My approach s therefore to calculate an upper bound on the potental aggregate ncentve effects to see f transfer programs mght credbly explan the large tax wedge changes smulated from aggregate behavor. Fgure 5 dsplays as a sold red lne government transfers (ncludng those pad by federal, state, and local governments) as a fracton of labor ncome for the years Transfers ncreased slghtly n nomnal terms durng the 1930's whle nomnal labor ncome declned, so Fgure 5 shows an ncrease n the transfer-labor ncome rato. The transfer-labor ncome rato was relatvely hgh between WWII and the Korean War.

32 Equlbra wth Dstortons - 29 Fgure 5 Spendng on Transfer Programs vs Smulated Tax Rates, Whle calculatng the average margnal tax rate mplct n the portfolo of federal, state, and local transfer programs s very dffcult, the transfer-labor ncome rato shown n Fgure 5 s probably an upper bound on a more thorough and more accurate calculaton of that rate (see Mullgan 2000 for more dscusson of ths pont). Fgure 5 dsplays as a dashed lne the smulated tax rate mnus the measured ncome tax rate, whch I nterpret as that part of the smulated tax wedge that s unexplaned by ncome tax polcy. Wth ts sold red lne as an upper bound on the composte margnal rate from transfer programs, Fgure 5 suggests that, durng the 1930's, smulated tax rates ncreased by an order of magntude more than dd the rates from transfer programs, so that transfer programs cannot be an mportant part of an explanaton of Depresson