SMALL MENU COSTS AND LARGE BUSINESS CYCLES: AN EXTENSION OF THE MANKIW MODEL

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1 SMALL MENU COSTS AND LARGE BUSINESS CYCLES: AN EXTENSION OF THE MANKIW MODEL 2 Hiranya K. Nah, Sam Houson Sae Universiy Rober Srecher, Sam Houson Sae Universiy ABSTRACT Using a muli-period general equilibrium model, his paper can be used o enhance classroom presenaion of new Keynesian heory by exending he resuls of Mankiw (99) by showing ha monopolisically compeiive firms may require 'relaively large' menu coss o dissuade hem from changing prices in response o an aggregae demand shock ha is perceived o be permanen. Thus, "small" menu coss may be insufficien o conribue o large business cycles. INTRODUCTION I is by now a commonly acceped view among economiss ha nominal rigidiies are he mos ap characerizaion of he shor run behavior of he economy. However, he heories ha have been proposed o explain sluggish adjusmens of prices and wages are varied and numerous. One of he heories ha gained populariy among a secion of economiss in recen years suggess ha firms are required o incur some coss o change prices. These coss are ofen associaed wih prining menus, and herefore referred o as 'menu coss'. According o his menu coss heory, since changing prices is cosly, many firms do no respond immediaely o a shock by changing prices, and as a resul, real variables such as oupu have o bear he brun. Some economiss, however, cas doubs abou his explanaion because hese menu coss are evidenly small. Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

2 22 Using parial as well as general equilibrium models, Mankiw (99) shows ha hese small menu coss are in fac capable of producing large business cycles. Considering monopolisically compeiive firms ha se prices, he shows ha hough menu coss may be small, he incremenal profis ha resul from price changes may be even smaller and, herefore, firms are beer off by no changing prices in response o a demand shock. In Mankiw's model he decision of he firm depends on a comparison beween one-ime menu coss and he change in single-period profi. This paper argues ha if he firms consider changes in heir fuure sream of profis ha would resul from he decision o change price hen 'small menu coss' may no be able o dissuade hem from changing prices. I essenially exends he resuls of Srecher (22), which presens a parial equilibrium analysis of non-marke clearing firm o show ha inroducion of he opporuniy cos of capial o discoun fuure incremenal profis will reduce he abiliy of 'small menu coss' o generae large business cycles. In his paper, we build a general equilibrium model which differs from he one in Mankiw (99) in wo ways: firs, he represenaive consumer maximizes life-ime uiliy ha involves iner-emporal ransfer of resources. Second and more imporanly, he monopolisically compeiive firm bases is decision o change price on a comparison of he menu coss eiher wih he change in single-period profi, or wih he discouned presen value of he changes in all fuure profis, depending upon wheher i perceives he aggregae demand shock o be emporary or permanen. The res of he paper is organized as follows. Secion 2 presens a general equilibrium model, wih maximizing rules for consumers and firms. In secion 3, we inroduce menu coss and discuss how hey affec firms' price seing behavior. This secion also includes he main proposiions of his paper. Secion 4 includes a few concluding remarks. A GENERAL EQUILIBRIUM MODEL WITH MONOPOLISTICALLY COMPETITIVE FIRMS The economy consiss of a coninuum of monopolisically compeiive firms, disribued along he uni inerval. Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

3 Consumers and Preferences 23 We assume ha he economy is populaed by a large number of idenical infiniely-lived consumers. The represenaive consumer has ime-separable preferences summarized by he following uiliy funcion: U β ( ) y i, d M di +θlog L P () where <$< is he discoun facor, y i, is he quaniy of good i consumed in period, N is he reciprocal of he elasiciy of subsiuion beween differen goods produced by he firms and <N<, M d is he individual's money demand in period, P is he general price level, L is he labor supply 2, and 2 is he money demand parameer (2 >). The general price level P is he geomeric average of all P i, s, where P i, is he nominal price of he good produced by firm i in period, and is given as follows: P exp logp i,di (2) The consumer earns wage income by supplying labor, and ineres income from lending in he previous period. She also receives money supply. In addiion o spending on consumpion, he consumer lends. Thus he budge consrain for he represenaive consumer is given by Pi,y i, di d + B + M WL + R B + M + Π (3) where W is he nominal wage 3 in period, B is he amoun len in period, R is he ineres rae in period, M is he money supply and A is he oal profis of he firms. Noe ha Walras's Law requires ha he profis of he firms go o he individual. The individual, however, considers profis as fixed in her uiliy maximizaion problem. Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

4 24 Firms and Producion Each firm produces is oupu using labor only, and he echnology is given by he producion funcion: y i, L i, (4) where L i, is he labor inpu used by firm i in period. Thus he cos funcion of he firm is given by: C i, W L i, W y i, (5) The firm faces a demand funcion implied by he uiliy maximizaion and he firm chooses y i, and P i, in each period such ha is profi is maximized. Uiliy and Profi Maximizaion The represenaive consumer maximizes her life-ime uiliy given by equaion () subjec o her budge consrain given by equaion (3). The firs-order condiions are given below: βy λ P i, i, (6) β θ M P d P λ (7) β +λ W (8) λ + E λ R (9) Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

5 d Pi,yi, di + B + M WL R B M Π () 25 Noe ha 8 is he Lagrange muliplier for he budge consrain (3) in he consumer's uiliy maximizaion problem. Rearranging equaion (8), we have λ β W () Subsiuing ino equaions (6), (7) and (9), and rearranging we obain y i, W Pi, φ (2) W d M θ (3) W R E β W + (4) Equilibrium in he money marke implies ha money supply equals money demand. Thus, d M M (5) Subsiuing (5) ino (3), we obain: W M θ (6) Then subsiuing (6) ino (2) and (4), Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

6 26 y i, M θp i, φ (7) and M R E β M + (8) Rearranging equaion (7) P i, M θy φ i, (9) This is he inverse demand funcion faced by firm i in period. Also, subsiuing for W from (6) ino he cos funcion (5), we obain C i, M y θ i, (2) The implied profi funcion can be wrien as: π i, φ M ( yi, yi, ) θ (2) Firm i chooses y i, in such a way ha B i, is maximized. The firs-order condiion of profi maximizaion yields: ( )y ) ( i, This implies Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

7 y i *, ( )φ (22) 27 where * y i, is he profi maximizing oupu of firm i in period. Subsiuing for y i, ino equaion (9) we obain he following profi-maximizing price for firm i in period : P * i, M θ ( ) (23) As we can see from equaions (22) and (23), a change in money supply does no affec he profi-maximizing choice of oupu of firm i. I affecs price only. Under ceeris paribus, a one percen increase in money supply will increase he price of firm by one percen. Thus, if all firms fully adjus prices in response o a moneary shock, hen he general price level will ake he enire brun of he shock leaving oupu unalered. Menu Coss and he Firm's Decision o Change Price Suppose he firm is required o incur a cos o change price. Following Mankiw (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) M (24) θ The firm's decision o change price depends on a comparison of hese coss wih poenial gains from such a change. To sar wih, suppose he money supply is M in each period and each firm chooses quaniy and price according o equaions (22) and (23), ha maximize is profis. Le y and P be he profi-maximizing quaniy and price in each period corresponding o his money supply. Suppose ha suddenly he money supply is changed o M in period. If he firm decides Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

8 28 o change is price, hen he new price will be given by (23). Oherwise, i remains a M P θ ( ). The nominal wage, however, changes from M W θ o M. W θ Through produc demand (equaion (7)), oupu changes from y o. M φ y y M The firm's decision o change price is based on wheher he incremenal profi ha resuls from he change in price ouweighs he menu cos. However, i is imporan o consider wheher he firm perceives he shock o be ransiory or permanen. When he Moneary Shock is Perceived o be Transiory If he firm perceives he change in money supply o be ransiory, i will compare he menu cos wih he incremen in profi in period only. Because if he shock is emporary hen he money supply in he nex periods will be M, and y and P will sill be he profi-maximizing quaniy and price. In ha case, he marginal firm I ha is indifferen over changing price would be π i, I g g W ( y y ) ( y y )) (25) Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

9 If i<i, hen he firm finds i profiable o change price even hough i has o incur he menu cos. If i>i, on he oher hand, he firm leaves is price unalered a P and produces y. Thus: 4 PROPOSITION : Following a moneary shock ha is perceived o be ransiory, if zi > y y y y, hen he firm does no change is price o P. ( ) ( ) ) W 29 When he Moneary Shock is Perceived o be Permanen If he firm perceives he change in money supply o be permanen, on he oher hand, i will compare he menu cos wih he discouned presen value of all fuure incremens in profi in period onwards. Because if he shock is permanen hen he money supply in all subsequen periods will remain a M. If he firm does no change price hen y will be he oupu in period and in all subsequen periods. In ha case, he marginal firm I ha is indifferen over changing price would be k I g R k l + l π + k g W [ ] ([( y y ) ( y y )] + R + ( RR ) +...) + (26) From equaion (8), R + l β for all l,, 2, 3.. (27) Thus, (26) becomes I g k R k l + l π + W k g - g 2 ([( y y ) ( y y )][ +β+β +...]) [( y y ) ( y y )] ( ) β (28) Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

10 3 If i<i, hen he firm changes price; oherwise, i leaves is price unchanged a P. Thus, PROPOSITION 2: Following a moneary shock ha is perceived o be permanen, if z i >, hen he firm does no change [ ( y ) ( ) ] y y y W ( ) β is price o P. I is no difficul o show ha 5 > y y y ( y ) ( ) y y y β [( ) ( )] ( ) y ( ) Thus for given menu coss, he number of firms changing prices in he laer case will be larger han in he former. In oher words, if he firms perceive he moneary shock o be permanen hey will require relaively larger menu coss o dissuade hem from changing prices. In boh cases, oal oupu is Y yi,di Iy + ( I) y The general price level is P exp logp i,di exp IlogP ( + ( I)log P ) When a moneary shock is perceived o be ransiory, for given zis (even if i is small), I will be closer o, and mos firms will no change price. We will hus observe a relaively larger effec of he moneary shock on oupu. On he oher hand, if he moneary shock is perceived o be permanen, I will be closer o and mos of he shock will be absorbed by changes in prices. In ha case, small menu coss may no be a likely cause of large business cycles. Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

11 CONCLUDING REMARKS 3 Using a simple general equilibrium framework, his paper shows ha if he firms perceive he aggregae demand shock o be permanen hey may require 'no small' bu 'relaively large' menu coss o dissuade hem from changing prices. In ha case, heir decision o change prices will depend on a comparison beween one-ime menu coss and discouned presen value of all fuure incremenal profis ha would resul from such price changes. This enhances he radiional presenaion of he Mankiw price rigidiy model o include discouning of fuure profis when comparing o menu coss. This is especially useful when consisency (concerning a posiive opporuniy cos of capial) beween macro resuls and microfoundaional models is desired. ENDNOTES. For a comprehensive survey of hese compeing heories, see Blinder e al (998) and Taylor (998) 2. We may spli his labor supply, by making he consumer decide he amoun of labor she is willing o supply o each firm. Bu since labor is perfecly mobile across firms his 'wis' in he model is inconsequenial. Also, he marke clearing in he labor marke requires ha his labor supply is exacly equal o he oal demand for labor by he firms in he economy. 3. Since labor is mobile across firms, nominal wage rae is he same in all firms. 4. If he shock is, in fac, emporary and he firm responds o he shock by changing is price o P hen in he nex period i will have o change he price back o P. In ha case, he firm will incur he menu coss wice and herefore will compare 2z i wih he incremenal profi in order o make a decision abou price change. I reinforces Mankiw's (99) resul. 5. For example, for a value $.95, he firs erm of his inequaliy is 2 imes higher han he second erm. Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25

12 32 REFERENCES Blinder, A. S, Elie E.D. Canei, D. E. Lebow & J. B. Rudd. (998). Asking Abou Prices: A New Approach o Undersanding Price Sickiness. New York: Russell Sage Foundaion. Mankiw, N. G. (99). Small Menu Coss and Large Business Cycles: A Macroeconomic Model of Monopoly. In N. Gregory Mankiw & David Romer (eds): New Keynesian Economics, vol, Cambridge, MA: The MIT Press. Srecher, R. (22), Discouning Price Rigidiies, Journal of Economics and Economic Educaion Research, 3(2), Taylor, J. B. (998). Saggered Price and Wage Seing in Macroeconomics, NBER Working Paper # 6754, Cambridge, MA. Journal of Economics and Economic Educaion Research, Volume 6, Number 2, 25