Real Business Cycles. Jesus Fernandez-Villaverde University of Pennsylvania
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1 Real Buine Cycle Jeu Fernandez-Villaverde Univerity of Pennylvania 1
2 Buine Cycle U.S. economy uctuate over time. How can we build model to think about it? Do we need dierent model than before to do o? Traditionally the anwer wa ye. Nowaday the anwer i no. We will focu on equilibrium model of the cycle. 2
3 Buine Cycle and Economic Growth How dierent are long-run growth and the buine cycle? Change in Output per Worker Secular Growth Buine Cycle Due to change in capital 1/3 0 Due to change in labor 0 2/3 Due to change in productivity 2/3 1/3 We want to ue the ame model with a lightly dierent focu. 3
4 Stochatic Neoclaical Growth Model Ca (1965) and Koopman (1965). Brock and Mirman (1972). Kydland and Precott (1982). Hanen (1985). King, Ploer, and Rebelo (1988a,b). 4
5 Reference King, Ploer, and Rebelo (1988a,b). Chapter by Cooley and Precott in Cooley' Frontier of Buine Cycle Reearch (in fact, you want to read the whole book). Chapter by King and Rebelo (Reurrection Real Buine Cycle Model) in Handbook of Macroeconomic. Chapter 12 in Ljungqvit and Sargent. 5
6 Preference Preference: E 0 1 X t=0 t (1 + n) t u c t t ; l t t where n i population growth. for c t t 0; l t t 2 (0; 1) Standard technical aumption (continuity, dierentiability, Inada condition, etc...). However, thoe till leave many degree of freedom. Retriction impoed by economic theory and empirical obervation. 6
7 Retriction on Preference Three obervation: 1. Rik premium relatively contant)crra utility function. 2. Conumption grow at a roughly contant rate. 3. Stationary hour after the SWW)Marginal rate of ubtitution between labor and conumption mut be linear in conumption. u c = w t t ) u l c t t f l t t = w t t ) t c 0 f l t t = t w 0 Explanation: income and ubtitution eect cancel out. 7
8 Parametric Family Only parametric that atify condition (King, Ploer, and Rebelo, 1988a,b): Retriction on v (l) : 1. v 2 C 2 (cv (l)) 1 1 if 1 > 0, 6= 1 log c + log v (l) if = 1 2. Depending on : (a) If = 1, log v (l) mut be increaing and concave. (b) If < 1, v 1 (c) If > 1, v 1 mut be increaing and concave. mut be decreaing and convex. 3. v (l) v 00 (l) > (1 2) v 0 (l) 2 to enure overall concavity of u: 8
9 1. CRRA-Cobb Dougla: Three Ueful Example E 0 1 X t=0 t (1 + n) t c t t 1 l t t Log-log (limit a! 1): E 0 1 X t=0 t (1 + n) t n log c t t + log 1 l t t o 3. Log-CRRA E 0 1 X t=0 8 >< t (1 + n) t >: log c t t l t t >= >; 9
10 Houehold Problem Let me pick log-log for implicity: E 0 1 X t=0 t (1 + n) t n log c t t + log 1 l t t o Budget contraint: c t t + x t t = w t t l t t + r t t k t t 1, 8 t > 0 Complete market and Arrow ecuritie. We can price any ecurity. 10
11 Problem of the Firm I Neoclaical production function in per capita term: y t t = e z t k t t 1 (1 + ) t l t t 1 Note: labor-augmenting technological change (Phelp, 1966). We are etting up a model where the rm rent the capital from the houehold. However, we could alo have a model where rm own the capital and the houehold own hare of the rm. Both environment are equivalent with complete market. 11
12 Problem of the Firm II By prot maximization: e z t k t t 1 1 (1 + ) t l t t 1 = rt t (1 ) e z t k t t 1 (1 + ) t l t t = wt t Invetment x t induce a law of motion for capital: (1 + n) k t+1 t = (1 ) k t t 1 + x t t 12
13 Evolution of the technology t = z t z t change over time. It follow the AR(1) proce: z t = z t 1 + " t " t N (0; 1) Interpretation of and : 13
14 Arrow-Debreu Equilibrium A Arrow-Debreu equilibrium are price. f bp t ( t ); bw t ( t ); br t ( t )g 1 t=0; t 2S t and allocation f^c t ( t ); l b t ( t ); k b t ( t )g 1 t=0; t 2S t uch that: 1. Given f^p t ( t )g 1 t=0; t 2S t ; f^c t ( t ); b l t ( t ); b k t ( t )g 1 t=0; t 2S t olve max f^c t ( t );b lt ( t );b kt ( t )g 1 t=0; t 2S t E 0 1X t=0.t. X 1X t=0 t 2S t bp t ( t ) 1X t=0 t (1 + n) t n log c t t + log 1 X ^p t ( t ) ct ( t ) + (1 + n) k t+1 t t 2S t bw t ( t )l t ( t ) + br t ( t ) + 1 c t ( t ) 0 for all t k t+1 t l t t o 14
15 2. Firm pick f b l t ( t ); b k t ( t )g 1 t=0; t 2S t to minimize cot: 3. Market clear: e z tb k t t 1 1 (1 + ) t b l t t 1 = br t t (1 ) e z tb k t t 1 (1 + ) t b l t t = bw t t bc t ( t ) + (1 + n) b k t+1 t = e z tb k t t 1 (1 + ) t b l t t 1 + (1 ) b k t t 1 for all t; all t 2 S t 15
16 Sequential Market Equilibrium I. We introduce Arrow ecuritie.. Houehold problem: olve c t ( t ); l t ( t ); k t ( t ); n a t+1 ( t ; t+1 ) o 1 t+1 2S t=0; t 2S t X 1 max E 0 t (1 + n) t n log c t t + log 1 t=0.t. c i t( t ) + (1 + n) k t+1 t + X l t t o t+1 j t b Q t ( t ; t+1 )a t+1 ( t ; t+1 ) bw t ( t )l t ( t ) + br t ( t ) + 1 k t+1 t + a t ( t ) c t ( t ) 0 for all t; t 2 S t a t+1 ( t ; t+1 ) A t+1 ( t+1 ) for all t; t 2 S t Role of A t+1 ( t+1 ): 16
17 Sequential Market Equilibrium II A SM equilibrium i price for Arrow ecuritie f b Q t ( t ; t+1 )g 1 t=0; t 2S t ; t+1 2S, allocation ^c i t (t ); l b t ( t ); k b t ( t ); n^a t+1. ( t ; t+1 ) o t+1 2S and in- t=0; t 2St put price f bw t ( t ); br t ( t )g 1 t=0; t 2S t ; uch that: 1 1. Given f b Q t ( t ; t+1 )g 1 t=0; t 2S t ; t+1 2S and f bw t ( t ); br t ( t )g 1 t=0; t 2S t ^c t ( t ); l b t ( t ); k b t ( t ); n^a t+1 ( t ; t+1 ) o 1 t+1 2S the prob- t=0; t 2Stolve lem of the houehold. 2. Firm pick f b l t ( t ); b k t ( t )g 1 t=0; t 2S t to minimize cot: e z tb k t t 1 1 (1 + ) t b l t t 1 = br t t (1 ) e z tb k t t 1 (1 + ) t b l t t = bw t t 3. Market clear for all t; all t 2 S t bc t ( t )+(1 + n) b k t+1 t = e z tb k t t 1 (1 + ) t b l t t 1 +(1 ) b k t t 1 17
18 Recurive Competitive Equilibrium Often, it i convenient to ue a. third alternative competitive equilibrium concept: Recurive Competitive Equilibrium (RCE). Developed by Mehra and Precott (1980). RCE emphaize the idea of dening an equilibrium a a et of function that depend on the tate of the model. Two interpretation for tate: 1. Pay-o relevant tate: capital, productivity, Other tate: promied utility, reputation,... Recurive notation: x and x 0. 18
19 Value Function for the Houehold Individual tate: k:. Aggregate tate: K and z: Recurive problem: n v (k; K; z) = max log c + log (1 l) + (1 + n) Ev k 0 ; K 0 ; z 0 jz o c;x;l :t: c + x = r (K; z) k + w (K; z) l (1 + n) k 0 = (1 ) k + x (1 + n) K 0 = (1 ) K + X (K; z) z 0 = z + " 0 19
20 Denition of Recurive Competitive Equilibrium A RCE for our economy i a value function v (k; K; z), houehold policy function, c (k; K; z) ; x (k; K; z) ; and. l (k; K; z), aggregate policy function C (K; z) ; X (K; z), and L (K; z), and price function r (K; z) and w (K; z) uch that thoe function atify: 1. Recurive problem of the houehold. 2. Firm maximize: e z K 1 ((1 + ) L (K; z)) 1 = r (K; z) (1 ) e z K ((1 + ) L (K; z)) = w (K; z) 3. Conitency of individual and aggregate policy function, c (k; K; z) = C (K; z), x (k; K; z) = X (K; z), l (k; K; z) = L (K; z) ; 8 (K; z) : 4. Aggregate reource contraint: C (K; z) + X (K; z) = e z K ((1 + ) L (K; z)) 1 ; 8 (K; z) 20
21 Equilibrium Condition 1 c t ( t ) = E 1 t c t+1 t+1 rt+1 t c t t 1 l t ( t ) = w t t r t t = e z t k t t 1 1 (1 + ) t l t t 1 w t t = (1 ) e z t k t t 1 (1 + ) (1 )t lt t c t t + (1 + n) k t+1 t = e z t k t t 1 (1 + ) t l t t 1 + (1 ) kt t 1 z t = z t 1 + " t 21
22 Scaling the Economy I Economy ha long-run growth rate equal to (n + ). Per capita term, the economy grow at a rate. Hence, the model i non-tationary and we need to recale it. General condition: tranform every non-tationary variable into a tationary one by dividing it by (1 + ) t t ex t t = x t (1 + ) t 22
23 Scaling the Economy II We can rewrite the preference (and adding a uitable contant): E 0 1 X t=0 X 1 = E 0 t (1 + n) t t=0 E 0 1 X t=0 ct t v l t t 1 t (1 + n) t 1 (1 + ) t c et t v l t t 1 t (1 + n) t t(1 ) (1 + ) 1 1 ec t t v l t t ) 1 23
24 Scaling the Economy III We can rewrite the preference (and adding a uitable contant):and E 0 1 X t=0 t (1 + n) t n log c t t + log v l t t o = E 0 1 X t=0 t (1 + n) t n log (1 + ) t ec t t + log v l t t o ) E 0 1 X t=0 t (1 + n) t n log ec t t + log v l t t o 24
25 Scaling the Economy IV The reource contraint, diving both ide by (1 + ) t ec t t + (1 + n) (1 + ) e k t+1 t = e z te k t t 1 lt t 1 + (1 ) e k t t 1 Input price: r t t = e z te k t t 1 1 lt t 1 ew t t = (1 ) e z te k t t 1 lt t 25
26 A New Competitive Equilibrium We can dene a competitive equilibrium in the recaled economy. Equilibrium condition (log cae): (1 + ) ec t ( t ) = E 1 t ec t+1 t+1 ec t t 1 l t ( t ) = ew t rt+1 t t r t t = e z te k t t 1 1 lt t 1 ew t t = (1 ) e z te k t t 1 lt t ec t t + (1 + n) (1 + ) e k t+1 t = e z te k t t 1 lt t 1 + (1 ) e k t t z t = z t 1 + " t 26
27 Exitence and Welfare Theorem There i a unique equilibrium in thi economy once we impoe the right tranverality condition. Both welfare theorem hold. We can move back and forth between the market equilibrium and the ocial planner' problem. 27
28 Behavior of the Model We want to characterize behavior of the model. Three type of dynamic: 1. Balanced growth path. 2. Tranitional dynamic (Ca, 1965, and Koopman, 1965). 3. Ergodic behavior. 28
29 Stochatic Behavior We have an initial hock: productivity change. We have a tranmiion mechanim: intertemporal ubtitution and capital accumulation. We can look at a imulation from thi economy. Why only a imulation? To imulate the model we need: 1. To elect parameter value. 2. To compute the olution of the model. 29
30 Selecting Parameter Value How do we determine the parameter value? Two main approache: 1. Calibration. 2. Statitical method: Method of Moment, ML, Bayeian. Advantage and diadvantage. 30
31 Calibration a an Empirical Methodology Emphaized by Luca (1980) and Kydland and Precott (1982). Two ource of information: 1. Well accepted microeconomic etimate. 2. Matching long-run propertie of the economy. Problem of 1. and 2. Reference: 1. Browning, Hanen and Heckman (1999) chapter in Handbook of Macroeconomic. 2. Debate in Journal of Economic Perpective, Winter 1996: Kydland and Precott, Hanen and Heckman, Sim. 31
32 Calibration of the Standard Model Parameter:,,,,, n,, : n: population growth in the data. : per capita long run growth. : capital income. Proprietor' income? 32
33 Balanced Growth Path Equilibrium condition in the BGP: 1 + = 1 (r + 1 ) ec ec ec 1 l = ew r = k e 1 l 1 ew = (1 ) e k l ec + (1 + n) (1 + ) e k = e k l 1 + (1 ) e k A ytem of 5 equation on 5 unknown. 33
34 Three Condition of the Balanced Growth Path Firt: r = e y ek = Alo: (1 + n) (1 + ) k e = (1 ) k e + ex ) x = e + 1 (1 + n) (1 + ) ek Finally, ec 1 l = (1 ) y e l ) c e ey = 1 1 l l 34
35 Uing the Three Condition to Calibrate the Model Firt, we ue data on hour of work to nd y = (1 ) e 1 l ec l Second, give data and we determine. = e x ek + 1 (1 + n) (1 + ) Finally, we get : = (1 + ) e y ek
36 Frich Elaticity I Dene the Frich Elaticity a: d log l d log w c contant For our parametric family: 1. c (1 l) : 1 l l : 2. log c + log (1 l): 1 l l : 3. log c l1+ 1+ : 1=: 36
37 Frich Elaticity II Empirical evidence i that l 1=3 (Ghez and Becker, 1975). Then, our Frich Elaticity i 2. Empirical evidence: 1. Traditional view: MaCurdy (1981), Altonji (1986), Browning, Deaton and Irih (1985) between 0 and Reviionit view: between 0.5 and 1.6 (Browning, Hanen, and Heckman, 1999). Some etimate (Imai and Keane, 2004) even higher (3.8). 37
38 Equivalence between Utility Function With log c t + log (1 l t ), the tatic FOC i: c t 1 l t = w t while with log c l1+ t t 1+, the tatic FOC i c t l t = w t Loglinearize both expreion: c 1 l b c t + bc t + cl (1 l) 2b l t = w bw t ) l bl t = bw t 1 l 38
39 cl bc t + b l t = w bw t ) bc t + b l t = bw t If we calibrate the model to l 1=3: bc t + 1 2b l t = bw t and hence, both utility function are equivalent if we make = In the cae l 1=3, = 1=2: l 1 l. 39
40 Solow Reidual Lat tep i to calibrate z t = z t 1 + " t Obtain the Solow reidual after a time trend ha been removed. Etimate and by OLS. Problem of etimate. 40
41 Solution Method Value function iteration. Projection. Perturbation: 1. Generalization of linearization. 2. Dynare. 41
42 General Structure of Linearized Sytem There are many linear olver. Fundamental equivalence. \A Toolkit for Analyzing Nonlinear Dynamic Stochatic Model Eaily" by Harald Uhlig. Given m tate x t ; n control y t ; and k exogenou tochatic procee z t+1, we have: Ax t + Bx t 1 + Cy t + Dz t = 0 E t (F x t+1 + Gx t + Hx t 1 + Jy t+1 + Ky t + Lz t+1 + Mz t ) = 0 E t z t+1 = Nz t where C i of ize l n; l n and of rank n; that F i of ize (m + n l) n; and that N ha only table eigenvalue. 42
43 Policy Function I We gue policy function of the form: x t = P x t y t = Rx t 1 + Qz t 1 + Sz t where P; Q; R; and S are matrice uch that the computed equilibrium i table. 43
44 Policy Function I For implicity, uppoe l = n: See Uhlig for general cae (I have never be in the ituation where l = n did not hold). Then: 1. P atie the matrix quadratic equation: F JC 1 A P 2 JC 1 B G + KC 1 A P KC 1 B + H = 0 The equilibrium i table i max (ab (eig (P ))) < R i given by: R = C 1 (AP + B) 44
45 3. Q atie: N 0 F JC 1 A + I k JR + F P + G KC 1 A vec (Q) = vec JC 1 D L N + KC 1 D M 4. S atie: S = C 1 (AQ + D) 45
46 How to Solve Quadratic Equation To olve P 2 P = 0 for the m m matrix P : 1. Dene the 2m 2m matrice: = " I m 0 m # ; and = " 0 m 0 m I m # 2. Let be the generalized eigenvector and be the correponding generalized eigenvalue of with repect to : Then we can write 0 = x 0 ; x 0 for ome x 2 < m : 46
47 3. If there are m generalized eigenvalue 1 ; 2 ; :::; m together with generalized eigenvector 1 ; :::; m of with repect to ; written a 0 = h x 0 i ; x0 ii for ome xi 2 < m and if (x 1 ; :::; x m ) i linearly independent, then: P = 1 i a olution to the matrix quadratic equation where = [x 1 ; :::; x m ] and = [ 1 ; :::; m ]. The olution of P i table if max j i j < 1. Converely, any diagonalizable olution P can be written in thi way. 47
48 Comparion with US economy Simulated Economy output uctuation are around 70% a big a oberved uctuation. Conumption i le volatile than output. Invetment i much more volatile. Behavior of hour. 48
49 Aement of the Baic Real Buine Model It account for a ubtantial amount of the oberved uctuation. It account for the covariance among a number of variable. It ha ome problem accounting for the behavior of the hour worked. More important quetion: where do productivity hock come from? 49
50 Negative Productivity Shock The model implie that half of the. quarter we have negative technology hock. I thi plauible? What i a negative productivity hock? Role of trend: negative hock alo include growth of technology below the trend..d. of hock i Mean quarter productivity growth i (to give u a 1.9% growth per year). A a conequence, we would only oberve negative technological hock when " t < 0:0047. Thi happen in the model around 25% of time. Comparion with the data. 50
51 Some Policy Implication The baic model i Pareto-ecient. Fluctuation are the optimal repone to a changing environment. Fluctuation are not a ucient condition for ineciencie or for government intervention. In fact in thi model the government can only woren the allocation. Receion have a \cleaning" eect. 51
52 Aet Market Implication I We will have the fundamental aet pricing equation: Q t ( t ; t+1 ) = t+1 j t u 0 ct+1 t+1 ; l t+1 t+1 u 0 (c t ( t ) ; l t ( t )) If utility i eparable and log in conumption: Q t ( t ; t+1 ) = t+1 j t c t t c t+1 t+1 Now, c t t i the equilibrium conumption. Since c t t i mooth in the model, Q t ( t ; t+1 ) will alo be mooth. Hence, we will have the tandard equity premium puzzle. 52
53 Aet Market Implication II Return to invet in an uncontingent. bond old at face value 1: c t t E t c t+1 t+1 Rt b t Return to invet in capital: c t t E t c t+1 t+1 rt+1 t By non-arbitrage: c t E t c t+1 t+1 Rt b t c t t = E t c t+1 t+1 rt+1 t Preence of capital tie down return. 53
54 Further Extenion We can extend our model in everal. direction. Two objective: 1. Fix empirical problem. 2. Addre additional quetion. Example: 1. Indiviible labor upply. 2. Capacity utilization. 3. Invetment Specic technological change. 4. Monopolitic Competition. 54
55 Lotterie Our rt extenion i to introduce. lotterie: Hanen (1985). Rogeron (1988) and General procedure to deal with non-convexitie. For example, an agent can either work 0 hour or l hour. Why? Extenive veru intenive margin. Then, expected utility: pu (c 1 ; l ) + (1 p) u (c 2 ; 0) Reource contrain in the economy (law of large number): pc 1 + (1 p) c 2 = c 55
56 Aggregation Firt order condition: u c (c 1 ; l ). = u c (c 2 ; 0). For our log-log utility function log c + log (1 l) ; we have c = c 1 = c 2 Alo, In the aggregate, we have that l = pl. Then, expected utility i log c + p log (1 l ) + (1 p) log 1 ) log c + Al where A = log(1 l ) l : Note that thi utility function belong to the cla log c l1+ 1+ with = 0, i.e., with innite Frich elaticity. 56
57 Capacity Utilization In benchmark model, the hort run elaticity of capital i zero while in the long run i innite. Empirical evidence of ue of machinery, number of hift, or electricity conumption. Modied production function: y t t = e z t ut t k t t 1 (1 + ) t l t t 1 where u t i the utilization rate. Depreciation: (1 + n) k t+1 t = 1 u t t k t t 1 + x t t 57
58 Combining Both Extenion We can generate 70 percent of aggregate uctuation with a.d. of How do we look at the Solow reidual in thi model? Thi implie negative technological growth in around 5 percent of quarter, roughly obervation in the data. 58
59 Invetment-Specic Technological Change Greenwood, Herkowitz, and Kruell (1997 and 2000): importance of technological change pecic to new invetment good for undertanding potwar U.S. growth and aggregate uctuation. Obervation: fall in the relative price of capital. Implication for NIPA. A imple way to model it: (1 + n) k t+1 t = (1 ) k t t 1 + v t x t t where v t i the invere of the relative price of capital. Two dierent technological hock with dierent implication. 59
60 Monopolitic Competition Final good producer with competitive behavior. Continuum of intermediate good producer with market power. Alternative formulation: continuum of good in the utility function. Otherwie, the model i the ame a the tandard RBC model. 60
61 The Final Good Producer Production function: y t ( t ) = Z 1 (y it ( t )) " 1 " di 0 where " control the elaticity of ubtitution.! " " 1 Final good producer i perfectly competitive and maximize prot, taking a given all intermediate good price p ti ( t ) and the nal good price p t ( t ). 61
62 Maximization Problem Thu, it maximization problem i: max y it ( t ) p t ( t ) y t ( t ) Z 1 0 p it ( t ) y it ( t ) di Firt order condition are for 8i: p t " " 1 Z 1 (y it ( t )) " 1 " di 0! " " 1 1 " 1 " (y it ( t )) " 1 " 1 p it ( t ) = 0 62
63 Working with the Firt Order Condition Dividing the rt order condition for two intermediate good i and j,. we get: or: p it ( t ) p jt ( t ) = y! 1 it ( t ) " y jt ( t ) p jt ( t ) = y!1 it ( t ) " pit ( t ) y jt ( t ) Hence: p jt ( t ) y jt ( t ) = p it ( t ) y it ( t ) 1 " yjt ( t ) " 1 " Integrating out: Z 1 0 p jt ( t ) y jt ( t ) dj = p it ( t ) y it ( t ) 1 " Z 1 " 1 y " 0 jt dj = p it ( t ) y it ( t ) 1 " yjt ( t ) " 1 " 63
64 Input Demand Function By zero prot (p t ( t ) y t ( t ) = R 1 0 p jt ( t ) y jt ( t ) dj), we get: p t ( t ) y t ( t ) = p it ( t ) y it ( t ) 1 " yjt ( t ) " 1 " ) p t ( t ) = p it ( t ) y it ( t ) 1 " y t ( t ) 1 " Conequently, the input demand function aociated with thi problem are: y it ( t ) = p! " it ( t ) y t ( t ) 8i p t ( t ) Interpretation. 64
65 Price Level By the zero prot condition p t ( t ) y t ( t ) = R 1 0 p it ( t ) y it ( t ) di and plug-in the input demand function: Z! 1 " p t ( t ) y t ( t ) = p p it ( t ) it ( t ) y t ( t ) di 0 p t ( t ) ) p t ( t ) 1 " = Z 1 0 p it ( t ) 1 " di Thu: p t ( t ) = Z 1 0 p it ( t ) 1 " di! 1 1 " 65
66 Intermediate Good Producer Continuum of intermediate good producer. No entry/exit. Each intermediate good producer i ha a production function y it ( t ) = A t k it ( t ) l it ( t ) 1 A t follow the AR(1) proce: log A t = log A t 1 + z t z t N (0; z ) 66
67 Maximization Problem I Intermediate good producer olve a two-tage problem. Firt, given w t and r t, they rent l it and k it in perfectly competitive factor market in order to minimize real cot: min l it ( t );k it ( t ) fw t ( t ) l it ( t ) + r t ( t ) k it ( t )g ubject to their upply curve: y it = A t k it ( t ) l it ( t ) 1 67
68 Firt Order Condition The rt order condition for thi problem are: w t ( t ) = % (1 ) A t k it ( t ) l it ( t ) r t ( t ) = %A t k it ( t ) 1 l it ( t ) 1 where % i the Lagrangian multiplier or: k it ( t ) = 1 w t ( t ) r t ( t ) l it ( t ) Note that ratio capital-labor only i the ame for all rm i: 68
69 Real Cot The real cot of optimally uing l it i: w t ( t ) l it ( t ) + 1 w t ( t ) l it ( t ) Simplifying: 1 1 w t ( t ) l it ( t ) 69
70 Marginal Cot I The rm ha contant return to cale. Then, we can nd the real marginal cot mc t ( t ) by etting the level of labor and capital equal to the requirement of producing one unit of good A t k it ( t ) l it ( t ) 1 = 1 Thu:! A t k it ( t ) l it ( t ) 1 w t ( t ) = A t 1 r t ( t ) l it ( t ) l it ( t )! w t ( t ) = A t l it ( t ) = 1 1 r t ( t ) 70
71 Marginal Cot II Then: mc t ( t ) = = ! w t ( t ) 1 w t ( t ) A t 1 r t ( t ) 1 w t ( t ) 1 r t ( t ) A t 1 1 Note that the marginal cot doe not depend on i. Alo, from the optimality condition of input demand, input price mut atify: k t ( t ) = 1 w t ( t ) r t ( t ) l t ( t ) 71
72 Maximization Problem II The econd part of the problem i to chooe price that maximize dicounted real prot, i.e., ubject to max p it ( t ) ( p it ( t ) p t ( t ) mc t ( t )! y it ( t ) y it ( t ) = p! " it ( t ) y t ( t ) ; p t ( t ) ) Firt order condition:! " p it ( t ) yt ( t ) " p it ( t ) p t ( t ) p t ( t ) p t ( t ) mc t ( t )! p it ( t ) p t ( t )! " 1 yt ( t ) p t ( t ) = 0 72
73 Mark-Up Condition From the t order condition: 1 " p!! 1 it ( t ) p mc t ( t ) it ( t ) = 0 ) p t ( t ) p t ( t ) p it ( t ) = " (p it ( t ) mc t ( t ) p t ( t )) ) p it ( t ) = " " 1 mc t ( t ) p t ( t ) Mark-up condition. Reaonable value for ": 73
74 Aggregation I To derive an expreion for aggregate output, remember that: k it ( t ) l it ( t ) = 1 w t ( t ) r t ( t ) Since thi ratio i equivalent for all intermediate rm, it mut alo be the cae that: k it ( t ) l it ( t ) = k t ( t ) l t ( t ) = 1 w t ( t ) r t ( t ) If we ubtitute thi condition in the production function of the intermediate good rm A t k it ( t ) l it ( t ) 1 we derive:!! k y it = A it ( t ) k t ( t ) t l it ( t ) = A t l it ( t ) l it ( t ) l t ( t ) 74
75 Aggregation II The demand function for the rm. i: y it ( t ) = p! it ( t ) p t ( t ) " y t ( t ) 8i; Thu, we nd the equality:! "! p it ( t ) k t ( t ) y t ( t ) = A t l it ( t ) p t ( t ) l t ( t ) If we integrate in both ide of thi equation: y t ( t ) Z 1 0! " p it ( t ) k t ( t ) di = A t p t ( t ) l t ( t )! Z 1 0 l it ( t ) di = A t k t ( t ) l t ( t ) 1 75
76 Aggregation III Then: where. y t ( t ) = A t v t ( t ) k t ( t ) l t ( t ) 1 v t ( t ) = Z 1 0! " p it ( t ) di = j t ( t ) " p t ( t ) p t ( t ) " But note that: p t ( t ) =! 1 p it ( t ) 1 " 1 " di = pit ( t ) 0 Z 1 ince all intermediate good producer charge the ame price. Then: v t ( t ) = R 1 pit ( t ) 0 p t ( t ) " di = 1 and: y t = A t k t ( t ) l t ( t ) 1 76
77 Behavior of the Model Preence of monopolitic competition i, by itelf, pretty irrelevant. Why? Contant mark-up. Similar to a tax. Solution: 1. Shock to mark-up (maybe endogenou change). 2. Price rigiditie. 77
78 Further Extenion We can extend our model in many other direction. Example we are not going to cover: 1. Fical Policy hock (McGrattan, 1994). 2. Agent with Finite Live (Ro-Rull, 1996). 3. Home Production (Benhabib, Rogeron, and Wright, 1991). 78
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