Small enu Coss an Large Business Ccles: An Exension of ankiw oel * Hirana K Nah Deparmen of Economics an Inl. Business Sam Houson Sae Universi an ober Srecher Deparmen of General Business an Finance Sam Houson Sae Universi November 23 Absrac: Using a muli-perio general equilibrium moel, his paper exens he resuls of ankiw (99) b showing ha monopolisicall compeiive firms ma require relaivel large menu coss o issuae hem from changing prices in response o an aggregae eman shock ha is perceive o be permanen. Thus, small menu coss ma be insufficien o conribue o large business ccles. JEL Classificaion: E3 Kewors: enu Coss; Nominal igiiies; General Equilibrium * We hank Don Freeman for useful commens. Corresponing auhor. Deparmen of Economics an Inernaional Business, Sam Houson Sae Universi, Hunsville TX 7734-28; Phone: 936-294-476; Fax: 936-294-3488; E-mail: eco_hkn@shsu.eu
. Inroucion I is b now a commonl accepe view among economiss ha nominal rigiiiies are he mos ap characerizaion of he shor run behavior of he econom. However, he heories ha have been propose o explain sluggish ajusmens of prices an wages are varie an numerous. One of he heories ha gaine populari among a secion of economiss in recen ears suggess ha firms are require o incur some coss o change prices. These coss are ofen associae wih prining menus, an herefore referre o as menu coss. Accoring o his menu coss heor, since changing prices is cosl, man firms o no respon immeiael o a shock b changing prices, an as a resul, real variables such as oupu have o bear he brun. Some economiss, however, cas oubs abou his explanaion because hese menu coss are evienl small. Using parial as well as general equilibrium moels, ankiw (99) shows ha hese small menu coss are in fac capable of proucing large business ccles. Consiering monopolisicall compeiive firms ha se prices, he shows ha hough menu coss ma be small, he incremenal profis ha resul from price changes ma be even smaller an, herefore, firms are beer off b no changing prices in response o a eman shock. In ankiw s moel he ecision of he firm epens on a comparison beween one-ime menu coss an he change in single-perio profi. This paper argues ha if he firms consier changes in heir fuure sream of profis ha woul resul from he ecision o change price hen small menu coss ma no be able o issuae hem from changing prices. I esseniall exens he resuls of Srecher (22), which presens a parial equilibrium analsis of non-marke clearing firm o show ha inroucion of he For a comprehensive surve of hese compeing heories, see Bliner e al (998) an Talor (998)
opporuni cos of capial o iscoun fuure incremenal profis will reuce he abili of small menu coss o generae large business ccles. In his paper, we buil a general equilibrium moel which iffers from he one in ankiw (99) in wo was: firs, he represenaive consumer maximizes life-ime uili ha involves iner-emporal ransfer of resources. Secon an more imporanl, he monopolisicall compeiive firm bases is ecision o change price on a comparison of he menu coss eiher wih he change in single-perio profi, or wih he iscoune presen value of he changes in all fuure profis, epening upon wheher i perceives he aggregae eman shock o be emporar or permanen. The res of he paper is organize as follows. Secion 2 presens a general equilibrium moel, wih maximizing rules for consumers an firms. In secion 3, we inrouce menu coss an iscuss how he affec firms price seing behavior. This secion also inclues he main proposiions of his paper. Secion 4 inclues a few concluing remarks. 2. A General Equilibrium oel wih onopolisicall Compeiive Firms The econom consiss of a coninuum of monopolisicall compeiive firms, isribue along he uni inerval. 2.. Consumers an Preferences We assume ha he econom is populae b a large number of ienical infiniel-live consumers. The represenaive consumer has ime-separable preferences summarize b he following uili funcion: U β ( ) i, i θ log P L () 2
where <β< is he iscoun facor, i, is he quani of goo i she consumes in perio, is he reciprocal of he elasici of subsiuion beween ifferen goos prouce b he firms an <<, is her mone eman in perio, P is he general price level, L is he labor suppl 2, an θ is he mone eman parameer (θ > ). The general price level P is he geomeric average of all P i, s, where P i, is he nominal price of he goo prouce b firm i in perio, an is given as follows: P exp log Pi, i (2) The consumer earns wage income b suppling labor, an ineres income from lening in he previous perio. She also receives mone suppl. In aiion o spening on consumpion, he consumer lens. Thus he buge consrain for he represenaive consumer is given b Pi, i, i B W L B Π (3) where W is he nominal wage 3 in perio, B is he amoun len in perio, is he ineres rae in perio, is he mone suppl an Π is he oal profis of he firms. Noe ha Walras s Law requires ha he profis of he firms go o he iniviual. The iniviual, however, consiers profis as fixe in her uili maximizaion problem. 2.2. Firms an proucion Each firm prouces is oupu using labor onl, an he echnolog is given b he proucion funcion: 2 We ma spli his labor suppl, b making he consumer ecie he amoun of labor she is willing o suppl o each firm. Bu since labor is perfecl mobile across firms his wis in he moel is inconsequenial. Also, he marke clearing in he labor marke requires ha his labor suppl is exacl equal o he oal eman for labor b he firms in he econom. 3 Since labor is mobile across firms, nominal wage rae is he same in all firms. 3
i, L i, (4) where L i, is he labor inpu use b firm i in perio. Thus he cos funcion of he firm is given b: C i, W L i, W i, (5) The firm faces a eman funcion implie b he uili maximizaion an he firm chooses i, an P i, in each perio such ha is profi is maximize. 2.3. Uili an Profi aximizaion The represenaive consumer maximizes her life-ime uili given b equaion () subjec o her buge consrain given b equaion (3). The firs-orer coniions are given below: β i, λ P i, (6) β θ λ (7) P P β λ (8) W λ E λ (9) Pi, i, i B W L B Π () Noe ha λ is he Lagrange muliplier for he buge consrain (3) in he consumer s uili maximizaion problem. earranging equaion (8), we have β λ () W Subsiuing ino equaions (6), (7) an (9), an rearranging we obain 4
i, W (2) P i, W (3) θ W E β W (4) eman. Thus, Equilibrium in he mone marke implies ha mone suppl equals mone (5) Subsiuing (5) ino (3), we obain: W (6) θ Then subsiuing (6) ino (2) an (4), i, θp (7) i, an E β (8) earranging equaion (7) P i, (9) θ i, This is he inverse eman funcion face b firm i in perio. Also, subsiuing for W from (6) ino he cos funcion (5), we obain C i, θ i, (2) The implie profi funcion can be wrien as: 5
π i, ( i, i, ) θ (2) Firm i chooses i, in such a wa ha π i, is maximize. The firs-orer coniion of profi maximizaion iels: This implies ( ) ) ( i, * ( i, ) (22) where * i, is he profi maximizing oupu of firm i in perio. Subsiuing for i, ino equaion (9) we obain he following profi-maximizing price for firm i in perio : * Pi, (23) θ ( ) As we can see from equaions (22) an (23), a change in mone suppl oes no affec he profi-maximizing choice of oupu of firm i. I affecs price onl. Uner ceeris paribus, a one percen increase in mone suppl will increase he price of firm b one percen. Thus, if all firms full ajus prices in response o a monear shock, hen he general price level will ake he enire brun of he shock leaving oupu unalere. 3. enu Coss an he Firm s Decision o Change Price Suppose he firm is require o incur a cos o change price. Following ankiw (99), we assume ha changing price involves a small labor inpu g. Thus, le he menu cos of firm i be z i, g (i) W g (i) θ (24) 6
The firm s ecision o change price epens on a comparison of hese coss wih poenial gains from such a change. To sar wih, suppose he mone suppl is in each perio an each firm chooses quani an price accoring o equaions (22) an (23), ha maximize is profis. Le an P be he profi-maximizing quani an price in each perio corresponing o his mone suppl. Suppose ha suenl he mone suppl is change o in perio. If he firm ecies o change is price, hen he new price will be given b (23). Oherwise, i remains a P. The nominal wage, however, changes from θ ( ) W θ ow. Through prouc eman (equaion (7)), oupu changes from θ o. The firm s ecision o change price is base on wheher he incremenal profi ha resuls from he change in price ouweighs he menu cos. However, i is imporan o consier wheher he firm perceives he shock o be ransior or permanen. 3.. When he monear shock is perceive o be ransior If he firm perceives he change in mone suppl o be ransior, i will compare he menu cos wih he incremen in profi in perio onl. Because if he shock is emporar hen he mone suppl in he nex perios will be, an an P will sill be he profi-maximizing quani an price. In ha case, he marginal firm I ha is inifferen over changing price woul be I g π W i, g ( ) ( )) (25) 7
8 If i<i, hen he firm fins i profiable o change price even hough i has o incur he menu cos. If i>i, on he oher han, he firm leaves is price unalere a P an prouces. Thus, POPOSITION : Following a monear shock ha is perceive o be ransior, if z i > ( ) ( ) ( ) W, hen he firm oes no change is price o P. 4 3.2. When he monear shock is perceive o be permanen If he firm perceives he change in mone suppl o be permanen, on he oher han, i will compare he menu cos wih he iscoune presen value of all fuure incremens in profi in perio onwars. Because if he shock is permanen hen he mone suppl in all subsequen perios will remain a. If he firm oes no change price hen will be he oupu in perio an in all subsequen perios. In ha case, he marginal firm I ha is inifferen over changing price woul be ( ) ( ) [ ] ( ) [ ] ( )... g W g I k k k l l π (26) From equaion (8), β l for all l,, 2, 3.. (27) Thus, (26) becomes ( ) ( ) [ ][ ] ( ) ( ) ( ) [ ] ( ) β β β π g... g W g I - 2 k k k l l (28) 4 If he shock is, in fac, emporar an he firm respons o he shock b changing is price o P hen in he nex perio i will have o change he price back o P. In ha case, he firm will incur he menu coss wice an herefore will compare 2z i wih he incremenal profi in orer o make a ecision abou price change. I reinforces ankiw s (99) resul.
If i<i, hen he firm changes price; oherwise, i leaves is price unchange a P. Thus, POPOSITION 2: Following a monear shock ha is perceive o be permanen, if z i > [ ( ) ( )] W ( ) β, hen he firm oes no change is price o P. I is no ifficul o show ha [ ( ) ( )] ( ) ( ) ( )) > β. 5 Thus for given menu coss, he number of firms changing prices in he laer case will be larger han in he former. In oher wors, if he firms perceive he monear shock o be permanen he will require relaivel larger menu coss o issuae hem from changing prices. In boh cases, oal oupu is Y i, i I ( I ) The general price level is P exp log Pi, i exp ( I log P ( I )log P ) When a monear shock is perceive o be ransior, for given z i s (even if i is small), I will be closer o, an mos firms will no change price. We will hus observe a relaivel larger effec of he monear shock on oupu. On he oher han, if he monear shock is perceive o be permanen, I will be closer o an mos of he shock will be absorbe b changes in prices. In ha case, small menu coss ma no be a likel cause of large business ccles. 5 For example, for a value β.95, he firs erm of his inequali is 2 imes higher han he secon erm. 9
4. Concluing emarks Using a simple general equilibrium framework, his paper shows ha if he firms perceive he aggregae eman shock o be permanen he ma require no small bu relaivel large menu coss o issuae hem from changing prices. In ha case, heir ecision o change prices will epen on a comparison beween one-ime menu coss an iscoune presen value of all fuure incremenal profis ha woul resul from such price changes.
eferences: Bliner, Alan S, Elie E.D. Canei, Davi E. Lebow an Jerem B. u, 998, Asking Abou Prices: A New Approach o Unersaning Price Sickiness. New York: ussell Sage Founaion ankiw, N. Gregor, 99, Small enu Coss an Large Business ccles: A acroeconomic oel of onopol, in N. Gregor ankiw an Davi omer (es): New Kenesian Economics, vol, Cambrige, A: The IT Press. Srecher, ober, 22, Discouning Price igiiies, Journal of Economics an Economic Eucaion esearch, vol. 3(2): 93-3. Talor, John B. 998, Saggere Price an Wage Seing in acroeconomics, NBE Working Paper # 6754, Cambrige, A.