Optimal Monetary Policy with Staggered Wage and Price Contracts

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 640 July 1999 OPTIMAL MONETARY POLICY WITH STAGGERED WAGE AND PRICE CONTRACTS Christopher J. Erceg, Dale W. Henderson, and Andrew T. Levin NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.bog.frb.fed.us.

OPTIMAL MONETARY POLICY WITH STAGGERED WAGE AND PRICE CONTRACTS Christopher J. Erceg, Dale W. Henderson, and Andrew T. Levin* Abstract: We formulate an optimizing-agent model in which both labor and product markets exhibit monopolistic competition and staggered nominal contracts. The unconditional expectation of average household utility can be expressed in terms of the unconditional variances of the output gap, price inflation, and wage inflation. Monetary policy cannot replicate the Pareto-optimal equilibrium that would occur under completely flexible wages and prices; that is, the model exhibits a tradeoff between stabilizing the output gap, price inflation, and wage inflation. The Pareto optimum is only attainable if either wages or prices are completely flexible. For reasonable calibrations of the model, we characterize the optimal policy rule. Furthermore, strict price inflation targeting is clearly suboptimal, whereas rules that also respond to either the output gap or wage inflation are nearly optimal. Keywords: monetary policy, inflation targeting, sticky wages, sticky prices, staggered contracts. *All three authors are economists in the International Finance Division of the Federal Reserve Board. Earlier versions of the paper were presented at the Federal Reserve Board, the Swedish National Bank - Institute for International Economic Studies (IIES) Conference on Monetary Policy Rules, the Meetings of the Western Economic Association, the Federal Reserve Bank of Chicago, and the University of Michigan. The authors would like to thank Stefan Gerlach, their discussant at the Conference, for helpful comments; John Taylor and Michael Woodford for useful discussions; and Carolina Marquez for valuable research assistance. We have benefitted from suggestions made by an anonymous referee and by Robert King. We particularly appreciate the recent contributions of Julio Rotemberg and Michael Woodford, who not only have performed path breaking welfare analysis of monetary policy, but also have developed intuitively appealing explanations of their results and have made substantial efforts to provide the details of their derivations. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. The email addresses of the authors are ercegc@frb.gov, hendersd@frb.gov, and levina@frb.gov, respectively.

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Wrlqwurgxfhqrplqdosulfhvwlfnlqhvvlqwrwkhprgho/zhdvvxphwkdwsurgxfhuvvhw sulfhv lq Fdoyr0vw|oh vwdjjhuhg frqwudfwv1 Wkh gxudwlrq ri hdfk frqwudfw lv udqgrpo| ghwhuplqhg= lq dq| jlyhq shulrg/ wkh (cid:31)up lv doorzhg wr uhvhw lwv sulfh frqwudfw zlwk suredelolw| +(cid:129)(cid:3)1 ,1 Zkhqhyhu wkh (cid:31)up lv qrw doorzhg wr uhvhw lwv frqwudfw/ wkh (cid:31)up*v R sulfh lv dxwrpdwlfdoo| lqfuhdvhg dw wkh xqfrqglwlrqdo phdq udwh ri jurvv lq dwlrq/ (cid:31)1 Wkxv/ li (cid:31)up s kdv qrw dgmxvwhg lwv frqwudfw sulfh vlqfh shulrg |/ wkhq lwv sulfh lq shulrg | n (cid:26) lv (cid:27) Es(cid:28) ’ (cid:27) Es(cid:28) (cid:31)(cid:26)1 Qrwh wkdw wkh suredelolw| wkdw d (cid:31)up zloo eh |n(cid:26) | doorzhgwruhvhwlwv frqwudfwsulfhlqdq|jlyhqshulrggrhv qrwghshqgrqkrzorqj lwv h{lvwlqj frqwudfw kdv ehhq lq h(cid:30)hfw/ dqg wklv suredelolw| lv lqyduldqw wr wkh djjuhjdwh vwdwh yhfwru1 Wkxv/ d frqvwdqw iudfwlrq +(cid:129) (cid:3) 1 , ri (cid:31)upv duh doorzhg wr uhvhw wkhlu R frqwudfw sulfhv hdfk shulrg1 Zkhq d (cid:31)up lv doorzhg wr uhvhw lwv sulfh lq shulrg |/ wkh (cid:31)up pd{lpl}hv wkh iroorzlqj sur(cid:31)w ixqfwlrqdo zlwk uhvshfw wr lwv frqwudfw sulfh/ (cid:27) Es(cid:28)= | [" H 1 (cid:26)(cid:6) EE(cid:129)n(cid:12) (cid:28)(cid:31)(cid:26)(cid:27) Es(cid:28)t Es(cid:28)(cid:3)(cid:127)(cid:28) t Es(cid:28)(cid:28) +9, | R |c|n(cid:26) R | |n(cid:26) |n(cid:26) |n(cid:26) (cid:26)’f Wkh rshudwru H uhsuhvhqwv wkh frqglwlrqdo h{shfwdwlrq edvhg rq lqirupdwlrq wkurxjk | shulrg | dqg wdnhq ryhu vwdwhv ri qdwxuh lq zklfk wkh (cid:31)up lv qrw doorzhg wr uhvhw lwv sulfh1 Wkh (cid:31)up*v rxwsxw lv vxevlgl}hg dw d (cid:31){hg udwh (cid:12) 1 Wkh (cid:31)up glvfrxqwv sur(cid:31)wv R uhfhlyhg dw gdwh |n(cid:26) e| wkh suredelolw| wkdw wkh (cid:31)up zloo qrw kdyh ehhq doorzhg wr uhvhw lwv sulfh +1 (cid:26), dqg e| wkh vwdwh0frqwlqjhqw glvfrxqw idfwru (cid:6) 18 R |c|n(cid:26) Wkh (cid:31)uvw0rughu frqglwlrq iru d sulfh0vhwwlqj (cid:31)up lv (cid:21) (cid:22) [" (cid:31)(cid:26)(cid:27) Es(cid:28) (cid:127)(cid:28) H 1(cid:26)(cid:6) E(cid:129)n(cid:12) (cid:28) | (cid:3)E(cid:129)nw (cid:28) |n(cid:26) t Es(cid:28) ’ f(cid:23) +:, | R |c|n(cid:26) R (cid:27) R (cid:27) |n(cid:26) |n(cid:26) |n(cid:26) (cid:26)’f Wkxv/ wkh (cid:31)up vhwv lwv sulfh vr wkdw wkh vxp ri lwv h{shfwhg glvfrxqwhg uhdo uhyhqxh +lqfoxvlyh ri vxevlglhv, lv htxdo wr wkh sulfh pdunxs idfwru E(cid:129)nw (cid:28) pxowlsolhg e| wkh R vxp ri glvfrxqwhg uhdo frvwv1 Lq wkh olplwlqj fdvh lq zklfk doo (cid:31)upv duh doorzhg wr vhw wkhlu sulfhv hyhu| shulrg +1 $ f,/ htxdwlrq +:, uhgxfhv wr wkh frqglwlrq +idploldu R iurp vwdqgdug lpshuihfw frpshwlwlrq dqdo|vlv, wkdw wkh pdujlqdo surgxfw ri oderu +(cid:127)(cid:27)u (cid:28) pxowlsolhg e| wkh vxevlg| idfwru E(cid:129)n(cid:12) (cid:28) htxdov wkh uhdo zdjh pxowlsolhg e| | R wkh sulfh pdunxs idfwru= E(cid:129)n(cid:12) (cid:28)(cid:127)(cid:27)u ’ E(cid:129)nw (cid:28) ‘| +;, R | R (cid:27)| (cid:127)(cid:27)u ’ E(cid:129)(cid:3)k(cid:28)g7 ku3kf +<, | | | Zh dvvxph wkdw surgxfwlrq lv vxevlgl}hg wr holplqdwh wkh prqrsrolvwlf glvwruwlrq dvvrfldwhg zlwk d srvlwlyh pdunxs1 Wklv uhtxluhv vhwwlqj (cid:12) ’ w 1 Wkxv/ wkh vwhdg| R R vwdwh ri wkh prgho vdwlv(cid:31)hv wkh h!flhqf| frqglwlrq wkdw wkh pdujlqdo surgxfw ri oderu htxdov wkh uhdo zdjh/ dv lq d shuihfwo| frpshwlwlyh hfrqrp|1 8Wkh glvfrxqw idfwru # lv dq lq(cid:31)qlwh0glphqvlrqdo yhfwru zlwk hohphqwv fruuhvsrqglqj wr w>w.m glvwlqfw vwdwhv ri qdwxuh lq shulrg w.m1 Hdfk hohphqw ri # lqglfdwhv wkh sulfh lq shulrg w ri d w>w.m fodlpwkdwsd|vrqhgrooduliwkhfruuhvsrqglqjvwdwhrffxuvlqshulrgw.m>glylghge|wkhsuredelolw| wkdw vwdwh zloo rffxu1 7

515 Krxvhkrogv dqg Zdjh Vhwwlqj Zh dvvxph wkdw d frqwlqxxp ri prqrsrolvwlfdoo| frpshwlwlyh krxvhkrogv +lqgh{hg rq wkh xqlw lqwhuydo, vxsso| gl(cid:30)huhqwldwhg oderu vhuylfhv wr wkh surgxfwlrq vhfwru> wkdw lv/ jrrgv0surgxflqj (cid:31)upv uhjdug hdfk krxvhkrog*v oderu vhuylfhv / E(cid:5)(cid:28)/ (cid:5) 5 dfc(cid:129)o/ dv | dq lpshuihfw vxevwlwxwh iru wkh oderu vhuylfhv ri rwkhu krxvhkrogv1 Lw lv frqyhqlhqw wr dvvxph wkdw d uhsuhvhqwdwlyh oderu djjuhjdwru +ru (cid:19)hpsor|phqw djhqf|(cid:20), frpelqhv krxvhkrogv* oderu krxuv lq wkh vdph sursruwlrqv dv (cid:31)upv zrxog fkrrvh1 Wkxv/ wkh djjuhjdwru*vghpdqgiruhdfkkrxvhkrog*v oderulvhtxdowrwkh vxpri(cid:31)upv*ghpdqgv1 Wkh oderu lqgh{ u kdv wkh Gl{lw0Vwljolw} irup= | (cid:23)] (cid:24) (cid:129) (cid:129)nw(cid:25) u ’ / E(cid:5)(cid:28) (cid:129) _(cid:5) +43, | | (cid:129)nw(cid:25) f zkhuh w : f1 Wkh djjuhjdwru plqlpl}hv wkh frvw ri surgxflqj d jlyhq dprxqw ri (cid:25) wkh djjuhjdwh oderu lqgh{/ wdnlqj hdfk krxvhkrog*v zdjh udwh ‘ E(cid:5)(cid:28) dv jlyhq/ dqg | wkhq vhoov xqlwv ri wkh oderu lqgh{ wr wkh surgxfwlrq vhfwru dw wkhlu xqlw frvw ‘ = | (cid:23)] (cid:24) ‘ ’ (cid:129) ‘ E(cid:5)(cid:28) 3(cid:129) _(cid:5) 3w(cid:25) +44, | | w(cid:25) f Lwlvqdwxudowrlqwhusuhw‘ dvwkhdjjuhjdwhzdjhlqgh{1 Wkhdjjuhjdwru*vghpdqgiru | wkhoderukrxuvrikrxvhkrog(cid:5) (cid:24)ruhtxlydohqwo|/ wkhwrwdoghpdqgiruwklvkrxvhkrog*v oderu e| doo jrrgv0surgxflqj (cid:31)upv (cid:24) lv jlyhq e| (cid:23) (cid:24) ‘ E(cid:5)(cid:28) 3(cid:129)nw(cid:25) / E(cid:5)(cid:28) ’ | w(cid:25) u +45, | ‘ | | Wkh shulrg xwlolw| ixqfwlrq ri krxvhkrog (cid:5) lv vhsdudeoh lq wkuhh dujxphqwv= qhw frqvxpswlrq/ qhw ohlvxuh/ dqg uhdo prqh| edodqfhv1 Qhw frqvxpswlrq lv gh(cid:31)qhg e| vxewudfwlqjwkhfrqvxpswlrqvkrfnL iurpwkhkrxvhkrog*vfrqvxpswlrqlqgh{(cid:28) E(cid:5)(cid:28)1 | | Qhwohlvxuhlv gh(cid:31)qhge|vxewudfwlqjkrxuv zrunhg/ E(cid:5)(cid:28) dqgwkh ohlvxuh vkrfn~ iurp | | wkhkrxvhkrog*vwlphhqgrzphqw+qrupdol}hgwrxqlw|,1 Wkhfrqvxpswlrqdqgohlvxuh vkrfnv duh frpprq wr doo krxvhkrogv1 Uhdo prqh| edodqfhv duh qrplqdo prqh| kroglqjv (cid:127) E(cid:5)(cid:28) gh dwhg e| wkh djjuhjdwh sulfh lqgh{ (cid:27) 1 Wkxv/ wkh krxvhkrog*v | | shulrg xwlolw| ixqfwlrq lv= XE(cid:28) E(cid:5)(cid:28)cL (cid:28)nYE/ E(cid:5)(cid:28)c~ (cid:28)n> PE(cid:127)|E(cid:5)(cid:28)(cid:28) | | | | f (cid:27)| +46, (cid:19) (cid:20) ’E(cid:28)|E(cid:5)(cid:28)3L|(cid:28)(cid:129)3jnE(cid:129)3w |E(cid:5)(cid:28)3~|(cid:28)(cid:129)3(cid:29) n >f (cid:127)|E(cid:5)(cid:28) (cid:129)3> E(cid:129)3j(cid:28) (cid:129)3(cid:29) E(cid:129)3>(cid:28) (cid:27)| Krxvhkrog (cid:5)*v exgjhw frqvwudlqw lq shulrg | vwdwhv wkdw frqvxpswlrq h{shqglwxuhv soxv dvvhw dffxpxodwlrq pxvw htxdo glvsrvdeoh lqfrph= (cid:27) (cid:28) E(cid:5)(cid:28)n(cid:127) E(cid:5)(cid:28)(cid:3)(cid:127) E(cid:5)(cid:28)nB (cid:18) E(cid:5)(cid:28)(cid:3)(cid:18) E(cid:5)(cid:28) | | | |3(cid:129) |n(cid:129)c| | |3(cid:129) +47, ’ E(cid:129)n(cid:12) (cid:28)‘ E(cid:5)(cid:28)u E(cid:5)(cid:28)nK E(cid:5)(cid:28)nA E(cid:5)(cid:28) (cid:25) | | | | 8

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U U (cid:129)YE/E(cid:5)(cid:28)c~(cid:28)_(cid:5) * Yn (cid:129)Y /y_wE(cid:5)(cid:28)_(cid:5)nY ~7 _~ f f u w7 ~ ~7 (cid:21) (cid:19) (cid:20) (cid:22) +E19, U U (cid:3) (cid:4) n(cid:129) (cid:129)Y /y 2 _wE(cid:5)(cid:28) 2 _(cid:5)n (cid:129)Y /y~7_wE(cid:5)(cid:28)_~_(cid:5)nY ~7 2 _~ 2 2 f uu w7 f u~ w7 ~7 ~~ ~7 Pdnlqj xvh ri wkh uhvxow lq htxdwlrq +E14, |lhogv U (cid:3) (cid:4) (cid:3) (cid:4) (cid:129)YE/E(cid:5)(cid:28)c~(cid:28)_(cid:5) * YnY /y H OE(cid:5)(cid:28)n (cid:129)H OE(cid:5)(cid:28)2 nY ~7 5 n (cid:129)52 f u (cid:5) 2 (cid:5) ~ 2 +E1:, (cid:3) (cid:4) n(cid:129) Y /y 2H OE(cid:5)(cid:28)2 n2Y /y~75H OE(cid:5)(cid:28)nY ~7 252 2 ww (cid:5) w~ (cid:5) ~~ k l U Wkh lqgh{ u djjuhjdwhv oderu krxuv dfurvv krxvhkrogv= u @ (cid:129)/E(cid:5)(cid:28) (cid:129) (cid:129)nw ‘1 (cid:129)nw(cid:25) f Wkxv/ k l (cid:19) (cid:20) U , ’ *? f (cid:129)/E(cid:5)(cid:28) (cid:129)n (cid:129) w(cid:25) (cid:129)nw(cid:25) (cid:3)*?u 7 * H (cid:5) OE(cid:5)(cid:28)n (cid:129) 2 (cid:129)n (cid:129) w(cid:25) (cid:15)@o (cid:5) OE(cid:5)(cid:28) +E1;, U Wkh lqgh{ u lv dovr wkh vxp ri oderu vhuylfhv xvhg e| (cid:31)upv= u ’ (cid:129)uEs(cid:28)_s1 Wkxv/ f U , ’ *? (cid:129)uEs(cid:28)_s (cid:3)*?u 7 * H ,Es(cid:28)n (cid:129)(cid:15)@o ,Es(cid:28) +E1<, f (cid:19) s (cid:20) 2 s (cid:3) (cid:4) Doo (cid:31)upv fkrrvh lghqwlfdo fdslwdo oderu udwlrv gEs(cid:28) htxdo wr wkh djjuhjdwh udwlr g uEs(cid:28) u ehfdxvh wkh| idfh wkh vdph idfwru sulfhv/ vr (cid:19) (cid:20) (cid:3) (cid:4) k t Es(cid:28) ’ gEs(cid:28) uEs(cid:28)f ’ g kuEs(cid:28)f +E143, uEs(cid:28) u Vlqfh wkh wrwdo dprxqw ri fdslwdo lv (cid:31){hg/ htxdwlrq +E143, lq wxuq lpsolhv +Es(cid:28) ’ %(cid:3)k,n,Es(cid:28)c H +Es(cid:28) ’ %(cid:3)k,nH ,Es(cid:28)c (cid:15)@o +Es(cid:28) ’ (cid:15)@o ,Es(cid:28) +E144, s s s s Vxevwlwxwlqj wkh uhodwlrqvklsv lq htxdwlrq+E144, lqwrhtxdwlrq+E1<,/ dqgxvlqj htxd0 wlrq +E15, wr vxevwlwxwh iru H +E (cid:28)/ zh rewdlqhtxdwlrq +59, lq wkh wh{w/ uhshdwhg khuh s s iru frqyhqlhqfh= (cid:19) (cid:20) (cid:3) (cid:4) , * (cid:129) E+(cid:3)%(cid:28)n (cid:129) (cid:129) wR (cid:15)@o +Es(cid:28) +E145, (cid:129)3k 2 (cid:129)3k (cid:129)nwR s Vroylqj htxdwlrq +E1;, iru H OE(cid:5)(cid:28) dqg holplqdwlqj , xvlqj htxdwlrq +E145, |lhogv (cid:5) (cid:19) (cid:20) (cid:19) (cid:20) (cid:3) (cid:4) H OE(cid:5)(cid:28) * (cid:129) E+(cid:3)%(cid:28)n (cid:129) (cid:129) wR (cid:15)@o +Es(cid:28)(cid:3) (cid:129) (cid:129) (cid:15)@o OE(cid:5)(cid:28) +E146, (cid:5) (cid:129)3k 2 (cid:129)3k (cid:129)nwR s 2 (cid:129)nw(cid:25) (cid:5) Xvlqj htxdwlrq +E15,/ wkh whup (cid:15)@o OE(cid:5)(cid:28)n dH OE(cid:5)(cid:28)o2 fdq eh vxevwlwxwhg iru hdfk (cid:5) (cid:5) rffxuuhqfh ri H OE(cid:5)(cid:28)2 lq htxdwlrq +E1:,1 Wkhq vxevwlwxwlqj wkh uljkw0kdqg vlgh ri (cid:5) htxdwlrq +E146, iru hdfk rffxuuhqfh ri H OE(cid:5)(cid:28)/ zh rewdlq= (cid:5) 5:

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Figure 1: Welfare Costs of Output Gap Deviations Real wage mrs t g slope(mrs ) t B * (cid:133) t g slope(mpl) t g mpl * (cid:131) A t t-1 0 g Output gap t

80 79 78 0 1 2 3 4 −8 Output Gap Variance (x 10 ) 8− ) 01 x( ecnairaV noitalfnI ecirP Figure 2 Three Views of the Monetary Policy Frontier 82 80 78 76 74 72 8.5 9 9.5 10 10.5 11 11.5 12 −8 Wage Inflation Variance (x 10 ) 8− ) 01 x( ecnairaV noitalfnI ecirP 9.7 9.6 9.5 9.4 9.3 1 1.5 2 2.5 −8 Output Gap Variance (x 10 ) 8− ) 01 x( ecnairaV noitalfnI egaW Monetary Policy Frontier Indifference Curve at Optimum

−8 A. Price Inflation Variance (x10 ) 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Mean Wage Contract Duration noitaruD tcartnoC ecirP naeM 5 1236 0 Figure 3 Optimal Monetary Policy for Alternative Contract Durations −8 B. Wage Inflation Variance (x10 ) 10 9 8 7 5 2 6 50 5 4 100 3 250 2 1 1 2 3 4 5 6 7 8 9 10 Mean Wage Contract Duration noitaruD tcartnoC ecirP naeM 0 3 0 3 1 0 2 1 1 1 0 −8 C. Output Gap Variance (x10 ) 10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10 Mean Wage Contract Duration noitaruD tcartnoC ecirP naeM −5 D. Welfare Loss (x10 ) 10 9 5 0. 6 8 7 6 6 3 3 5 6 4 6 3 0 3 2 0.5 0 1 1 2 3 4 5 6 7 8 9 10 Mean Wage Contract Duration noitaruD tcartnoC ecirP naeM 0 3 .5 2 3 1 3 2 2 1 0 0

Figure 4: Strict Price Inflation Targeting Real wage mrs t B = 0 * (cid:133) t t D (cid:131) t (cid:131) C mpl t * t-1 (cid:131)A 0 g Output gap t ‘