Lecture Notes 3: Quantitative Analysis in DSGE Models: New Keynesian Model

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1 Lecure Noes 3: Quaniaive Analysis in DSGE Models: New Keynesian Model Zhiwei Xu, The moneary policy plays lile role in he basic moneary model wihou price sickiness. We now urn o he New Keynesian (NK) model. The NK model has wo basic feaures: (i) monopolisic compeiion; (ii) price is sicky. The former speci caion ensures ha rms make posiive pro, and he laer one implies ha he price is no exible and hus he money injecion may play a non-rivial role. Basic Economic Environmen The economy has one nal good which can be used eiher for consumpion or invesmen. The nal good is produced by he nal good rms. In paricular, he nal good rms use he inermediae goods as inpu. The nal good marke is compeiive. The inermediae goods rms are monopolisic compeiive, hey have monopoly power o se heir prices. However, he prices canno be exibly se. Therefore, he prices presen some exen of sickiness. To make he households have incenive o hold money (does no earn ineres), we inroduce he Cash-in-Advance (CIA) consrain. We rs sar wih households problem. 2 Household Problem Denoe he aggregae price in he economy as : The represenaive household choose money holding M ; bond holding B, consumpion c ; labor n, nex-period capial sock k + o solve he following opimizaion problem: subjec o budge consrain max fm ;B ;c ;n ;k + g E X (log c a n n ) () = c + m + k + ( ) k + b R b w n + r k + b + d + m + x ; (2) where d is he real pro disribued from rm side, m = M is he real money balance, b = B is he real bond holding, x = X is he real money injecion, = P is he in aion, w and r are real wage rae and renal rae. The CIA consrain is given by c m : (3)

2 2 Le and denoe he Lagrangian muliplier for he CIA consrain and budge consrain, respecively. FOCs w.r. fc ; n ; m ; b ; k + g are given by where in aion is de ned as = o (8). =c = + ; (4) a n = w ; (5) = E ( + = + ) + (6) = R b E ( + = + ) ; (7) = E + (r + + ) : (8) P : The household s side are summarized by equaions (2) 3 Final Good Secor Final good marke is compeiive, he rm combines a coninuum inermediae goods y i as inpus o produce nal good y : The producion funcion is assumed o be CES form Z y = y i di : (9) The pro maximizaion problem is Z max y P i y i di; () y i subjec o (9). The opimal y i implies ha he demand funcion of y i is Pi y i = y : () Puing las equaion ino he producion funcion, we ge he price indexaion funcion Z P = P i di: (2) Noe ha as he nal good marke is compeiive, wih he CRS producion funcion, he rm earns zero pro. 4 Inermediae Goods Secor The inermediae goods secor is monopolisic. Firm i produces good i wih Cobb-Douglas echnology y i = A kin i : (3)

3 3 The real pro of rm i is de ned as i = P i y i w n i r k i : (4) where y i = Pi y : As he labor and capial decisions are saic, he above pro funcion can be reduced o (hrough a cos-minimizaion problem) i = Pi Pi y ; (5) where is he marginal cos = A The demand for capial and labor are given by w r : (6) w = ( ) y i n i ; (7) r = y i k i : (8) De ne he aggregae capial, labor and oal oupu in inermediae good secor as Z Z Z k = k i di; n = n i di; ~y = y i di: (9) The facor demand funcions and he Cobb-Douglas producion funcion imply From (), he aggregae oupu y can be expressed as k i n i = k n : (2) y = ~y ; (2) where = R P i di: Therefore, he facor demand funcions can be re-wrien as And he aggregae producion funcion can be wrien as w = ( ) y n ; (22) r = y k : (23) ~y = A k n : (24)

4 4 4. Opimal Price-Seing Decision The opimizaion problem for he rm i is o se price p i o maximize he discouned pro ows. To model he price sickiness, we follow Calvo (982) o assume ha in he period, he rm, wih probabiliy ; can se is price exibly. Wih probabiliy ; he rm canno se price and hus he price remains he same as he previous period ( can adjus is price is ). The Bellman equaion for he rm ha V ; = max fp i g i + E + [( ) V ;+ + V ;+ ] ; (25) where V ; is he value of he rm ha can adjus is price, and V j; is he value of rm ha adjus is price j period ago and sill canno adjus is price in curren period (). For insance, V ;+ is he value of rm ha adjus is price one period before and sill canno adjus in period +. The opimal price P i for an acive rm is se o i + E + According o he pro funcion (5), he rs erm ;+ = : i i P i P i! Pi Pi y : (27) To derive i ; we need o specify he value funcion of inacive rm who adjus he price j periods before. V j; (P i j ) = i (P i j ) + E + [( ) V ;+ + V j+;+ ] : (28) Noe ha here is no max operaor in he above Bellman equaion because he rm is inacive, i jus akes he previous price as oday s price. Taking derivaive w.r.. P i From he above recursive srucure of j j; i (P i j ) j+;+ + E : i i i i j ; we can j; i (P i j ) i+ (P i j ) + E + () 2 i+2 (P i j ) i i i j = X = () i+ (P i j ) i j i j + ::: For i ; we hen ;+ = X () i++ (P i ) E + : i = i

5 5 Plugging las equaion ino (26), we have X E () i+ (P i ) = (3) i+ (P i i = = + P i + + P i + obain he opimal pricing rule: P i = P =! Pi i + Pi + y+ : Wih some algebra, we X E () + P+ y + + = E X = () + P + y + : (32) Noe ha he opimal price P i is idenical across rm index i; ha is, once rms can adjus heir price, hey se an idenical opimal level. As a resul, by law of large number, he price indexaion funcion (2) implies P = P + ( ) (P ) : (33) 4.2 Summary of Producion Side Equaions The equaions in producion side can be summarized as follows. Facor demand funcions: Aggregae producion w = ( ) y n ; (34) r = y k : (35) ~y = A k n : (36) Aggregae oupu where Opimal price P where y = ~y ; (37) Z Pi = di = + ( ) ( ) : (38) is given by P P = : (39) 2 = P y + E + ; (4) 2 = P y + E 2+ : (4)

6 6 Aggregae price indexaion equaion P = P + ( ) (P ) : (42) 5 Moneary Auhoriy We sill need o deermine he money supply. Here, we consider wo ypes of moneary policy rule. The rs one assumes he cenral bank issues he money follow a simple exogenous rule, i.e., he growh rae of money supply g m = AR() process here " m is he money supply shock. M M, has seady-sae value of, and follows an exogenous ln g m = ln g m + " m ; (43) The second ype of rule is Taylor rule, i.e., he nominal ineres rae arges he oupu and in aion: where y and > : R y b y = R b y ss ss exp(" m ); (44) 6 General Equilibrium To close he full sysem, we need o specify he marke clearing condiions. Since we use he same noaions for capial demand and supply, labor demand and supply, so here wo markes are implicily se o achieve he equilibrium. For he money marke, he clearing condiion is given by M = M + X ; or m = g m m = : (45) From he resource consrain, he Walras s law implies ha he bond marke clearing condiion is B = B = : (46) The full dynamic sysem consiss of (2) o (8), (34) o (42), (43) (or (44)), (45) and (46). 7 Seady Sae We now derive he seady sae of he sysem. Since we assume in he seady sae he growh rae of money supply g m = ; herefore (45) implies seady-sae in aion is = : We normalize he

7 7 price level in he seady sae as, so (42) implies P = P = : From he opimal price rule (39), he marginal cos is given by = : We also specify he seady sae labor as n = :33: (6), (7) imply R b = =; (47) = ( ) ; (48) herefore, since he nominal ineres rae is greaer han, he CIA consrain is always binding in he seady sae ( > ). In addiion, (4) implies = The Euler equaion for capial (8) implies r = = + : (2 ) c : (49) The aggregae oupu equaion (37) implies y = ~y due o = : Recall ha he facor demand funcions (34) and (35) imply ha r = y k = k n : (5) We can solve k as k = r n: (5) From producion funcion, we could furher obain y = k n obain seady sae wage rae: : From labor demand (34), we can w = ( ) y n = ( ) k n : (52) The resource consrain implies ha c = y k: (53) Then we can obain = =c and = R b : From labor supply (5), we can pin down he value of parameer a n a n = w: (54) From he CIA consrain, he real money balance is m = c: (55)

8 8 8 Loglinearizaion The loglinearized sysem is given by: ^c = ^m ; ^y = c y ^c + k y ^k + ( ) k y ^k ; ^c = + ^ + + ^ ; = ^ + ^w ; ^ = ^R b + E ^+ E ^ + ; ^ = E ^+ ^ + + ( ) ^ ; r ^ = E ^+ + r + E ^r + ; ^w = ^ + ^y ^n ; ^r = ^ + ^y ^k ; ^y = ^A + ^k + ( ) ^n ; ^ = ^ ^ 2 ; ^ = ( ) ^ + ^ + ^y + ^ + E ^ + ; ^ 2 = ( ) h^ + ( ) ^P i + ^y + E ^ 2+ ; ^ = ^ + ( ) ^P ; ^m = ^g m + ^m ^ ; ^g m = m^g m + " m ; or ^R b = y ^y + ^ ^ + " m : 9 Simulaion We rs calibrae he parameers as follows. We se = :99; = :25; = :4; n ss = :33: The parameer is se o be, implying a markup of %. We se = :75 which implies he rm on average adjuss is price once in four quarers. For he moneary policy parameers, we specify heir values as follows: m = :6; m = :; = :5; y = :: Figures below repor responses of aggregae variables o an expansionary moneary policy shock.

9 9. c.5 i. y n r w pi P phi x 3 Rb m gm Figure. IRF o money growh shock: exogenous money supply rule. 4 x 3 c.2 i. y n r x 3 w x 3 pi x 3 P phi x 3 Rb x 3 m gm Figure 2. IRF o a negaive ineres rae shock: Taylor rule.