Part I Modelling Money in General Equilibrium: a Primer Lecture 3 Welfare Cost of Inflation in the Basic MIU model

Transcription

1 Part I Modellng Money n General Equlbrum: a Prmer Lecture 3 Welfare Cost of Inflaton n the Basc MIU model Leopold von Thadden European Central Bank and Unversty of Manz Monetary and Fscal Polcy Issues n General Equlbrum Summer / 43

2 I Motvaton Some features of fat money regmes: prvate opportunty cost of holdng real balances (m) depends on the level of the short-term nomnal nterest rate () socal opportunty cost of provdng money s essentally zero, e the central bank can make the amount of money n crculaton arbtrarly large (and thereby depress ) the wedge between the prvate and socal cost at postve nterest rates creates an neff cency moreover, n equlbrum, the nomnal nterest rate, the nflaton rate and the quantty of money n crculaton are lnked to each other These features gve rse to two types of (closely lnked) polcy questons whch have been addressed n monetary economcs for centures: 1) From a postve perspectve: How large s the welfare cost of nflaton? 2) From a normatve perspectve: What s the optmal rate of nflaton? 2 / 43

3 I Motvaton The MIU-model offers a transparent and tractable general equlbrum perspectve, e t s a wdely used startng pont to address these questons, qualtatvely and quanttatvely (n partcular, see Lucas, 2000 ): Why s the MIU model a good model to analyze these questons? Because: The costs of nflaton (va the nterest rate) are clearly captured by the expresson u m (c t, m t ) = t u c (c 1 + t, m t ) (1) t Steady-state superneutralty mples that we don t have to worry about the real sde of the economy when comparng steady states wth dfferent nflaton and nterest rates The flow objectve u(c, m) offers a smple and drect metrc for welfare comparsons 3 / 43

4 I Motvaton Queston 2: What s the optmal rate of nflaton? In the basc MIU model the answer to ths queston s rather straghtforward, snce monetary polcy nstruments ( or θ) do not affect the steady-state value of c, only of m Money growth rule (va θ) : Optmalty requres u(c,m) θ = u(c,m) m m θ = 0, or u(c,m) m = 0, mplyng n eqn (1) = 0 Ths s the Fredman rule whch mples (usng r + π) π r,.e. the (long-run) optmal nflaton rate s a rate of deflaton approxmately equal to the return on captal and government bonds (Fredman, 1969) 4 / 43

5 I Motvaton Fredman rule: prelmnary comments Comment 1: Sataton vs. approxmate valdty of Fredman rule Dependng on the assumed utlty functon the strct equalty u(c,m) m = 0 wth = 0 obtans f there exsts a fnte sataton level (as n (A 1)) In other cases (lke the dscussed case of log-utlty) m should be made arbtrarly large such that 0 (approxmate valdty of the Fredman rule) Comment 2: Implementaton The result s equally vald f monetary polcy s drectly mplemented va an nterest rate rule (rather than a money growth rule) 5 / 43

6 I Motvaton Fredman rule: an mportant qualfcaton Ths reasonng s not yet a bass for polcy advce, snce the nterest rate (and ndrectly π) s the only dstortonary polcy nstrument Recall from the model set-up: government has access to lump-sum transfers/taxes τ, makng dstortonary segnorage revenues costly The dscusson about the optmalty of the Fredman rule becomes only meanngful when we ntroduce dstortonary taxes Then, as argued by Phelps (1973), the Fredman rule may well break down We wll return to Queston 2 n detal n Part II whch covers the Fredman vs. Phelps debate, assumng dstortonary taxaton 6 / 43

7 I Motvaton Let us return to Queston 1: How large s the welfare cost of nflaton? Lucas (2000) uses the MIU model to provde quanttatve gudance for the lkely range of welfare costs of nflaton The methodology proposed by Lucas has been nfluental snce t explots the general equlbrum dmenson of the MIU model, as opposed to partal equlbrum estmates that were tradtonally used (Baley 1956) To reproduce the fndngs from Lucas, we proceed n 2 steps: Step I): Two types of money demand specfcatons and dscusson of the emprcal evdence Step II): Partal vs. general equlbrum measures of welfare costs of nflaton 7 / 43

8 II Money demand specfcatons and emprcal evdence Any welfare measure of the costs of nflaton depends crtcally on the form of the money demand equaton and the senstvty of money demand to the opportunty cost of holdng real balances In the basc MIU model these ssues are drectly related to the specfcaton of the utlty functon whch drves the shape of u m (c t, m t ) = u c (c 1 + t, m t ) t The emprcal lterature commonly dstngushes between log-log specfcatons as opposed to sem-log specfcatons of money demand t 8 / 43

9 II Money demand specfcatons and emprcal evdence CES utlty functon between c and m : Consder the followng CES utlty functon whch dsplays a constant elastcty of substtuton between consumpton and real balances: u(c t, m t ) = [ac 1 b t Ths functon satsfes + (1 a)m 1 b t ] 1 1 b wth: 0 < a < 1, b > 0 (and b = 1) (2) u m (c t, m t ) u c (c t, m t ) = ( 1 a )( c t ) b a m t and, when combned wth the frst-order optmalty condton u m (c t, m t ) = u c (c 1 + t, m t ), t t gves rse to the money-demand functon t m t = ( 1 a ) 1 b t ( ) 1 b ct (3) a 1 + t 9 / 43

10 II Money demand specfcatons and emprcal evdence Log-log specfcaton: In the emprcal lterature t s common to rewrte equatons of type (3), e m t = ( 1 a ) 1 b t ( ) 1 b ct a 1 + t by takng logs, leadng to log(m) = 1 b log( 1 a ) + log(c) 1 a b log( 1 + ) (4) Features of eqn (4): Demand for m depends postvely on c and negatvely on Elastcty of money demand w.r.t. consumpton (η m,c = dm m / dc c ) s 1 often c s replaced by y, yeldng an ncome elastcty of 1 Elastcty of money demand w.r.t. the opportunty cost varable 1+ (e: η m, = dm 1+ m / d 1+ ) s b 1 1+ for smplcty η m, /1+ s often referred to as the nterest elastcty of money demand (and, n any case, for small : 1+ ) snce on the RHS of eqn (4) we consder the log of 1+ (and not the level of 1+ ) t s called a log-log specfcaton of money demand 10 / 43

11 II Money demand specfcatons and emprcal evdence Log-log specfcaton: Specal case of b 1 n eqn (2): Features of (5): u(c t, m t ) = c a t m1 a t (5) The money demand equaton (3) becomes m t = ( 1 a a t ) ( ) 1 c t 1 + t The consumpton and the nterest elastcty of money demand are both equal to 1 11 / 43

12 II Money demand specfcatons and emprcal evdence Sem-log specfcaton: In the emprcal lterature commonly studed alternatves to (4) are sem-log specfcatons of money demand Sem-log money demand equatons regress the log of m t on the level of t (or some alternatve measure of the opportunty cost of real balances lke t 1+ t ) Illustraton: Wth a sem-log specfcaton, eqn (4), ceters parbus, turns nto log(m) = ã + log(c) ξ 1 +, (6) The coeff cent ξ n front of 1+ denotes the semelastcty of money demand w.r.t. 1+, e ξ = dm m /d 1+ Hence, the elastcty and the sem-elastcty are lnked va the relatonshp: ξ = dm m 1+ d η m, 1+ = ξ / 43

13 II Money demand specfcatons and emprcal evdence Emprcal evdence: Lucas (2000): How to account for US annual tme seres of short-term nomnal nterest rates and the money-ncome rato (e the rato of M1 to nomnal GDP: M Py ) over the 95 year perod ? Can we explan the relatonshp between the two seres by log-log or sem-log specfcatons of money demand? Fgures 1-4 n Lucas (2000) ental 3 stylzed emprcal fndngs: 13 / 43

14 II Money demand specfcatons and emprcal evdence Fndng 1 (see Fgure 1 from Lucas, 2000): Over the 95 year perod, US real GDP grew at an average rate of 3%, M1 at 5.6% and the GDP deflator at 3.2% Ths makes the money-ncome rato ( Py M ) essentally trendless over the entre perod (although there has been a sgnfcant declne snce World War II) A value of the ncome elastcty of money demand larger than unty (e η m, y > 1) would have produced an upward trend Statonarty of the money-ncome rato not to be rejected Assumed money demand functons m d (, y ) are of type m d = m() y, satsfyng a unt ncome elastcty of money demand (η m, y = 1) 14 / 43

15 II Money demand specfcatons and emprcal evdence Fndng 2 (see Fgures 2 and 3 from Lucas, 2000): Fgures 2 and 3 plot observatons for the money-ncome rato m/y and the short-term nomnal nterest rate To account for ths relatonshp compare predctons from the log-log specfcaton (usng the η values 0.3, 0.5, 0.7) m y = A η log( m ) = log(a) η log() y and the sem-log specfcaton (usng the ξ values 5, 7, 9) m y = Ã e ξ log( m ) = log(ã) ξ, y where A and Ã, respectvely, are ftted such that the curves pass through the geometrc means of the data pars Log-log curves gve a better ft than sem-log curves Wthn the class of log-log curves η = 0.5 gves the best ft 15 / 43

16 II Money demand specfcatons and emprcal evdence Fndng 3 (see Fgures 1 and 4 from Lucas, 2000): Fgure 4 plots observed levels of real balances (e m) aganst the real balances predcted by the estmated log-log demand curve (wth η = 0.5) success: but: m t = A 0.5 t Ftted values successfully track secular ncrease n m/y pror to World War II (n a perod characterzed by declnng nomnal nterest rates) Ftted values track well the declne n m/y after World War II untl 1980s (n a perod characterzed by a secular rse n ) Ftted values become poor snce the md 1980s (a perod of low and sgnfcant fnancal deregulaton) Elastctes needed to ft long-run trends do not permt a good ft on a year-to-year or even quarterly bass y t 16 / 43

17 II Money demand specfcatons and emprcal evdence Related emprcal lterature: Lucas preferred estmate from a log-log specfcaton of η = 0.5, mplyng a value b = 2 n eqn (4), s broadly n lne wth related studes: Char, Kehoe and McGrattan (2000) report a US value of b = 2.6 Hoffmann, Rasche and Teslau (1995) report cross-country evdence and fnd smlar estmates for the US and Canada, a somewhat hgher value for the UK, and lower values for Germany and Japan The falure to obtan a good short-run ft va estmated long-run elastctes fttng secular trends has led to a vast research agenda on money demand specfcatons, wth the am to reconcle evdence at dfferent frequences For example, see the dstrbuted lag specfcatons summarzed by Walsh (Table 2.1), allowng, by constructon, for dfferences between short-run and long-run elastctes Estmatons of more flexble money demand systems, allowng for addtonal varables (varous assets and nterest rates etc.) 17 / 43

18 II Money demand specfcatons and emprcal evdence Related emprcal lterature: The mentoned problems to account for stable money demand relatonshps snce the 1980s have been explaned through varous channels, lke Changes n fnancal regulaton (whch have ncreased the range of money substtutes avalable for transactons at low cost) and assocated portfolo shfts Non-lneartes n money demand equatons (suggestng that n envronments of low and π nterest elastctes of money demand tend to be lower) Ireland (2009) argues that for post-1980 US-data the ft of sem-log specfcatons mproves and outperforms log-log specfcatons: Fndngs of ths type have mplcatons for estmates of the welfare cost of nflaton, as to be dscussed next 18 / 43

19 III Welfare cost of nflaton Partal equlbrum estmates of welfare costs of nflaton: Baley (1956): nflaton taxes real balances focus: fully antcpated nflaton welfare effects of nflaton to be assessed smlar to the effects of any other tax when comparng welfare mplcatons of two dfferent levels of, the natural welfare measure s the area under the (nverse) money demand curve, e the consumer surplus ceters parbus, ths amounts to a partal equlbrum perspectve Lucas (2000) reproduces ths approach, but suggests to consder nstead of the demand for real balances (e m = M P ) the money ncome rato (e m y m) dea: express area under the money demand curve as a fracton of ncome snce hs estmates use m d = m() y ths transformaton s legtmate n terms of dmenson, ths s a more satsfactory measure of how to compensate people to be ndfferent between dfferent steady states 19 / 43

20 III Welfare cost of nflaton Partal equlbrum estmates of welfare costs of nflaton: Welfare costs of nflaton n steady states comparson Consder two pars of observatons s.t. 0 < 0 < 1 wth m 0 > m 1 Here: Fgure I (Welfare costs of nflaton from a partal equlbrum perspectve) Intal steady state: 0, m 0 Area A 0 C : consumer surplus Area A m : surplus extracted by gov t (snce t saves 0 m 0 resources relatve to rasng them va ssuance of gov t bonds at nomnal rate 0 ) New steady state: 1, m 1 Area B 1 C : consumer surplus Area B m : surplus extracted by gov t (snce t saves 1 m 1 resources) Comparson between ntal and new steady state: Area A m 0 m 1 B : Welfare loss (e the dfference between the total surpluses) s the shaded area under the nverted money demand curve (n terms of m), e the combned net loss of surpluses extracted by the gov t and consumers when movng from 0 up to 1 20 / 43

21 III Welfare cost of nflaton Partal equlbrum estmates of welfare costs of nflaton: Fndng 4 (see Fgures 5 and 6 from Lucas, 2000): Lucas (2000) quantfes the partal equlbrum welfare gans from reducng the nomnal nterest rate from 1 > 0 to 0 = 0 by evaluatng the consumer surplus expresson 1 0 m(x)dx 1 m( 1 ) for hs money demand estmates For the benchmark log-log specfcaton wth η = 0.5 a permanent reducton of short-term nomnal nterest rates from 10% to 0% leads to a welfare gan of 1.6% of annual US real GDP Assumng that for the US over the 95 year horzon a nomnal nterest rate of 3% on average leads approxmately to prce stablty, a permanent reducton of short-term nomnal nterest rates from 3% to 0% stll leads to a szeable welfare gan of 0.9% of annual US real GDP 21 / 43

22 III Welfare cost of nflaton Partal equlbrum estmates of welfare costs of nflaton: Comments: For the log-log specfcaton lm 0 m() Ths makes welfare gans at very low levels of relatvely large Gans lke 1.6% or 0.9% of US real GDP may seem small, but they would be avalable every year, e the dscounted present value of them s substantal However, these gans need to be compared wth costs of dsnflaton (see Ball, 1993). Such costs typcally arse because of (short-run) nomnal rgdtes and unexpected dsnflaton surprses Such features are not addressed n the flex-prce ratonal expectatons approach consdered so far 22 / 43

23 III Welfare cost of nflaton Partal equlbrum estmates of welfare costs of nflaton: Comments: For the sem-log specfcaton, m( = 0) s a fnte number (e Ã) Ths makes welfare gans at very low levels of relatvely small What s meant by large vs. small? The numbers can be computed from the expressons (usng A = 0.05, Ã = 0.35): Log-log specfcaton : Sem-log specfcaton : 1 0 m(x)dx η 1 m( 1 ) = A 1 η 1 η m(x)dx 1 m( 1 ) = Ã ξ [1 e ξ 1 (1 + ξ 1 )] Upshot: Under the sem log specfcaton vrtually all gans are realsed f one stops at = 3%, whle under the log-log specfcaton there reman szeable gans f one goes all the way to = 0% 23 / 43

24 III Welfare cost of nflaton Partal equlbrum estmates of welfare costs of nflaton: Lucas (2000): Crtcsm Estmated money demand specfcatons of type (4) or (6) can well be satsfactory n solaton, but they lack a general equlbrum perspectve To use such estmates for welfare comparsons, n general, can be msleadng snce polcy changes affect all equlbrum relatonshps (whle the partal equlbrum approach gong back to Baley reles on the ceters parbus assumpton) 24 / 43

25 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: Lucas (2000): To address ths conceptual challenge, consder as an alternatve the followng general equlbrum strategy: Start out from the estmated money demand relatonshp m = m y = A 0.5 wth: A = 0.05 (7) Fnd an approprately adjusted verson of the basc MIU-model such that eqn (7) can be recovered from a frst-order optmalty condton Solve for the steady state of ths model Identfy the (permanent) welfare cost assocated wth > 0 as the percentage ncrease n annual steady-state consumpton that would be needed to make the representatve household ndfferent between a steady state wth = 0 and > 0 [ For consstency we use from now on the exposton n Walsh, Secton 2.3] 25 / 43

26 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: Smplfcatons and adjustments: 1) Snce the basc MIU model dsplays steady-state superneutralty, c and y are ndependent of, π Smplfy the set-up and consder an economy wth an exogenous and constant endowment (perod by perod) y, yeldng the resource constrant y = c 2) For consstency wth the general frst-order condton of the MIU-model u m = u c 1 + (8) replace n eqn (7) by the alternatve opportunty cost measure / 43

27 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: As a pont of departure for fndng a utlty functon u(c, m) whch dsplays a frst-order condton consstent wth eqn (7) consder the general specfcaton wth ϕ( m c ) to be determned below u(c, m) = 1 1 σ {[c ϕ( m c )]1 σ 1}, (9) Defne x m c For eqn (9) the general frst-order condton of the MIU-model (8) becomes u m u c = ϕ (x) ϕ(x) x ϕ (x) = 1 + (10) 27 / 43

28 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: Choose the functon ϕ(x) such that eqn (10) wll be consstent wth the estmated money-demand functon (7) Accordngly, let mplyng ϕ(x) = Combnng eqns (8) and (12) yelds A 2 x 1 = A 2 cm, (11) ϕ (x) ϕ(x) x ϕ (x) = A2 ( m (12) c )2 m = c A ( 1 + ) 0.5, (13) whch confrms eqn (7), snce we assumed y = c and replaced by / 43

29 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: Comment: Usng (9) and (11), the analyss wll be based on u(c, m) = 1 1 σ {[c ϕ( m c )]1 σ 1}, wth: ϕ( m c ) = A 2 cm and not on the utlty functon (2), e u(c, m) = [ac 1 b + (1 a)m 1 b ] 1 1 b Does ths matter? no (snce the utlty functons are monotonc transformatons of each other, leavng all welfare results establshed below unaffected) 29 / 43

30 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: Intal steady state (wth 0): Normalze the ntal steady-state consumpton s.t. c = 1 Let m denote the level of m yeldng the hghest possble level of utlty f 0 From eqn (13), e m = c A ( 1+ ) 0.5, t s clear that Ths mples m ϕ( m 1 ) = 1 and u(1, m ) = 0 from the defnton of ϕ n eqn (11) and the general utlty specfcaton (9), respectvely 30 / 43

31 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: New steady state (wth > 0): Defne the welfare cost w ( 1+ ( u 1 + w ( ) assocated wth > 0 mplctly va 1 + ), m( ) 1 + ) = u(1, m ) = 0 (14) whch, usng the defnton of m va (13), s equvalent to ( ) u 1 + w, (1 + w ) A ( 1 + ) 0.5 = u(1, m ) = 0 Use eqn (9), e u(c, m) = 1 σ 1 {[c ϕ( m c )]1 σ 1}, to see that ths mples m (1 + w ) ϕ( 1 + w ) = 1 31 / 43

32 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: New steady state (wth > 0): From the defnton of ϕ the last step, e (1 + w ) ϕ( m equvalent to (1 + w ) A 2 1+w (1+w ) A ( 1+ ) 0.5 = 1 1+w ) = 1, s whch can be rearranged to obtan the desred welfare measure w ( 1+ ) of the cost of nflaton w ( 1 + ) = A (15) / 43

33 III Welfare cost of nflaton General equlbrum estmates of welfare costs of nflaton: Fndng 5 (Lucas, 2000): Use the estmated value of A = 0.05 to calculate the welfare gans from a permanent reducton of short-term nomnal nterest rates from 10% to 0% from the expresson w ( 1 + ) = A The welfare gan s % of annual steady-state consumpton (whch s dentcal to output because of the assumpton c = y) 33 / 43

34 IV Comments Quanttatvely, the derved welfare costs of nflaton from a partal and general equlbrum perspectve are very smlar. Ths reflects the steady-state superneutralty of money n the basc MIU model. Conceptually, Lucas pont that one needs to dstngush between partal and general equlbrum welfare measures, of course, s not affected by ths fndng To the contrary, the smple MIU model had been chosen to show that under specal and well understood assumptons the general and the partal equlbrum welfare measures can approxmately concde In larger models wth rcher nteractons such concdence s lkely to dsappear (but researchers are nvted to motvate ther fndngs from transparent benchmark models, to make sure that we understand the drvng forces behnd ther fndngs) 34 / 43

35 IV Comments Recall from above that Ireland (2009) fnds for post-1980 US-data a much mproved ft of the sem-log specfcaton, callng for some cauton when usng welfare costs from a log-log specfcaton See Fgures 1 and 2 from Ireland (2009) whch extend the Fgures from Lucas untl 2006 Moreover, for about the same perod FED polcy was arguably mplemented va an nterest rate rule (and not, for example, a monetary aggregates polcy). Ths mproves the ft of estmated money demand equatons lke (4) or (6) whch use the nterest rate as an nstrument. Ths renforces the cauton expressed by Ireland 35 / 43

36 Appendx: Fgures Source: Lucas, R., Inflaton and welfare, Econometrca, 68, 2, , / 43

37 Appendx: Fgures Source: Lucas, R., Inflaton and welfare, Econometrca, 68, 2, , / 43

38 Appendx: Fgures Source: Lucas, R., Inflaton and welfare, Econometrca, 68, 2, , / 43

39 Appendx: Fgures Source: Lucas, R., Inflaton and welfare, Econometrca, 68, 2, , / 43

40 Appendx: Fgures Source: Lucas, R., Inflaton and welfare, Econometrca, 68, 2, , / 43

41 Appendx: Fgures Source: Lucas, R., Inflaton and welfare, Econometrca, 68, 2, , / 43

42 Appendx: Fgures Source: Ireland, P. On the welfare cost of nflaton and the recent behavor of money, Amercan Economc Revew, 99, 3, , / 43

43 Appendx: Fgures Source: Ireland, P. On the welfare cost of nflaton and the recent behavor of money, Amercan Economc Revew, 99, 3, , / 43