Lecture 2 Real Business Cycle Models

Transcription

1 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 1 Lecture 2 Real Business Cycle Models Version 1.2 5/12/211 Changes from version 1. are in red Changes from version 1. are in purple These are the slides I am using in class. They are not self-contained, do not always constitute original material and do contain some cut and paste pieces from various sources that I am not always explicitly referring to (not on purpose but because it takes time). Therefore, they are not intended to be used outside of the course or to be distributed. Thank you for signalling me typos or mistakes at fportier@cict.fr.

2 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 2 1 Introduction We now know more about business cycle facts (my first lecture). Let us consider a simple stochastic version of an optimal growth model. Let s see how simple versions of this model compare with the business cycle facts. Note that the goal here is to produce quantitative predictions.

3 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 3 2 A First Analytical Model This model is a version of Brock and Mirman (1972, JET) Brock and Mirman provided the first optimizing growth model with stochastic shocks.

4 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Setup Social planner solves subject to max E t n= β n log C(s t+n ) Y t (s t ) = Z t (s t ) ( K t (s t 1 ) ) α K 1+1 (s t ) = Y t (s t ) C t (s t ) with < α < 1, Z t follows some stochastic markovian process and s t = {Z t, K t }.

5 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 5 Note 3 important assumptions: 1. No labor supply decision 2. log utility 3. full depreciation (K t+1 = (1 δ)k t + I t with δ = 1). Let s rewrite the problem in terms of Bellman s equation: V (K t, Z t ) = max log C t + βe t V (K t+1, Z t+1 ) subject to K t+1 = Z t K α t C t

6 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models First Order Conditions Replace in the Bellman s equation K t+1 by its value in the resource constraint and derive to obtain: V (K t, Z t ) K t 1 V (K = t+1, Z t+1 ) βe t (1) C t K t+1 = 1 C t αz t K α 1 t (2) Substituting: [ ] 1 1 = βe t αz C t C t+1 Kt+1 α 1 t+1

7 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 7 [ ] Ct 1 = βe t αz C t+1 Kt+1 α 1 t+1 Note that this look like the standard consumption Euler equation of the permanent income model with R t+1 = αz t+1 K α 1 t+1. 1 = βe t [ ] C t R t+1 C t+1 (3)

8 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Solution Let us guess a solution of the form C t = γy t (constant saving rate) Let s verify that the Euler equation (3) is then satisfied (for a specific value of γ that we need to find). (3) writes [ ] γyt 1 = βe t αz γy t+1 Kt+1 α 1 t+1 Note that Z t+1 K α 1 t+1 = Y t+1 K t+1 so that [ ] Yt 1 = βe t α Y t+1 Y t+1 K t+1

9 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 9 [ ] 1 = βe t α Y t K [ t+1 = βe t α Y t Y t C t ] Y t = βe t [α (1 γ)y t = αβ 1 γ which implies that γ = 1 αβ Then K t+1 = (1 γ)y t = αβz t K α t. In logs : k t+1 = log αβ + z t + αk t. ]

10 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 1 k t+1 = log αβ + z t + αk t This last equation shows that there are two sources of dynamics in the model: 1. The dynamics of the exogenous shock z 2. The own dynamics of k, which is related to consumption smoothing: the social planner will never choose to consume 1% of a shock on impact, but will aim at smoothing consumption using savings (capital).

11 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 11 Using the equation K t+1 = αβy t, one gets the dynamics of output: y t+1 = αy t + z t+1 + α log αβ

12 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Impulse Response to a Permanent Shock Assume z follows a random walk z t+1 = z t + ε t+1 Assume the economy is before t = 1 at its non stochastic steadystate y (defined as the steady state when ε t = t). y = α 1 α log αβ. The process of y then writes y t+1 = αy t + z t+1 + (1 α)y Assume ε 1 = 1 and ε t = t > 1.

13 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 13 Then y 1 = y + 1 y 2 = y α y 2 = y α + α 2... y t = y α + α α t...

14 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 14 1 Figure 1: Response to a permanent shock y time

15 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Impulse Response to a Transitory Shock Assume now that z is a white noise z t = ε t+1 Assume the economy is before t = 1 at its non stochastic steadystate y (defined as the steady state when ε t = t). Assume ε 1 = 1 and ε t = t > 1. Then y 1 = y + 1 y 2 = y + α

16 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 16 y 2 = y + α 2... y t = y + α t...

17 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Figure 2: Response to a temporary shock 1 y time

18 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 18 3 The Standard Real Business Cycle (RBC) Model Perfectly competitive economy Optimal growth model + Labor decisions 2 types of agents: Households and Firms Shocks to productivity Pareto optimal economy We will solve for the competitive equilibrium

19 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models The Household Mass of agents = 1 (no population growth) Identical agents + All face the same aggregate shocks (no idiosyncratic uncertainty) One representative household

20 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 2 Infinitely lived rational agent with intertemporal utility E t β (, 1): discount factor, s= β s U t+s Preferences over a consumption bundle leisure

21 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 21 U t = U(C t, l t ) with U(, ) class C 2, strictly increasing, concave and satisfy Inada conditions compatible with balanced growth [more below]: C t 1 σ 1 σ v(l t) for σ R + \{1} U(C t, l t ) = or log(c t ) + v(l t )

22 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 22 Preferences are therefore given by E t [ s= β s U(C t+s, l t+s ) Household faces three constraints Time constraint h t+s + l t+s T = 1 (for convenience T=1) ]

23 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 23 Budget constraint B t+s }{{} Bond purchases (1 + r t+s 1 )B t+s 1 }{{} Bond revenus Capital Accumulation + C t+s + I t+s }{{} Good purchases + W t+s h t+s }{{} W ages + z t+s K t+s }{{} Capital revenus δ (, 1): depreciation rate K t+s+1 = I t+s + (1 δ)k t+s

24 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 24 The household decides on consumption, labor, leisure, investment, bond holdings and capital formation maximizing utility constraint, taking the constraints into account max {C t+s,h t+s,l t+s,i t+s,k t+s+1,b t+s } s= E t [ s= β s U(C t+s, l t+s ) subject to the sequence of constraints h t+s + l t+s 1 B t+s + C t+s + I t+s (1 + r t+s 1 )B t+s 1 + W t+s h t+s + z t+s K t+s K t+s+1 = I t+s + (1 δ)k t+s K t, B t 1 given ]

25 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 25 subject to max {C t+s,h t+s,k t+s+1,b t+s } s= E t [ s= β s U(C t+s, 1 h t+s ) ] B t+s +C t+s +K t+s+1 (1+r t+s 1 )B t+s 1 +W t+s h t+s +(z t+s +1 δ)k t+s Write the Lagrangian L t = E t s= βs [ U(C t+s, 1 h t+s ) + Λ t+s ((1 + r t+s 1 )B t+s 1 + +W t+s h t+s + (z t+s + 1 δ)k t+s C t+s B t+s K t+s+1 ) ]

26 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 26 First order conditions ( s ) C t+s : E t U c (C t+s, 1 h t+s ) = E t Λ t+s h t+s : E t U l (C t+s, 1 h t+s ) = E t (Λ t+s W t+s ) B t+s : E t Λ t+s = βe t ((1 + r t+s )Λ t+s+1 ) K t+s+1 : E t Λ t+s = βe t (Λ t+s+1 (z t+s δ)) and the transversality condition lim s + βs E t Λ t+s (B t+s + K t+s+1 ) =

27 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 27 Remark : - It is convenient to write and interpret FOC for s = : h t : U l (C t, 1 h t ) = U c (C t, 1 h t )W t B t : U c (C t, 1 h t ) = β((1 + r t )E t U c (C t+1, 1 h t+1 )) K t+1 : U c (C t, 1 h t ) = βe t (U c (C t+1, 1 h t+1 )(z t δ)) and the transversality condition lim s + E tβ s U c (C t+s, 1 h t+s ) (K t+s+1 + B t+s ) =

28 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 28 Simple example : Assume U(C t, l t ) = log(c t ) + θ log(1 h t ) h t+s : E 1 W t 1 h t+s = E t+s t C t+s B t+s : E 1 t C t+s = βe t (1 + r t+s ) 1 C t+s+1 K t+s+1 : E 1 t C t+s = βe 1 t C (z t+s+1 t+s δ) and the transversality condition lim s + E tβ sk t+s+1 + B t+s C t+s =

29 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models The Firm Mass of firms = 1 Identical firms + All face the same aggregate shocks (no idiosyncratic uncertainty) Representative firm Produce an homogenous good that is consumed or invested by means of capital and labor

30 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 3 Constant returns to scale technology (important) Y t = A t F(K t, Γ t h t ) Γ t = γγ t 1 Harrod neutral technological progress (γ 1), A t stationary (does not explain growth)

31 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 31 Remark: one could introduce long run technical progress in three different ways: Y t = Γ t F( Γ t K t, Γ t h t ) Γ t is Hicks Neutral, Γ t is Solow neutral, Γ t is Harrod neutral

32 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 32 Harrod neutral technical progress and the preferences specified above are needed for the existence of a Balanced Growth Path that replicates Kaldor Stylized Facts: 1. The shares of national income received by labor and capital are roughly constant over long periods of time 2. The rate of growth of the capital stock is roughly constant over long periods of time 3. The rate of growth of output per worker is roughly constant over long periods of time 4. The capital/output ratio is roughly constant over long periods of time 5. The rate of return on investment is roughly constant over long periods of time 6. The real wage grows over time For preferences, what is needed is wealth and substitution effect to cancel out in labor supply

33 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 33 Y t = A t F(K t, Γ t h t ) Γ t = γγ t 1 Harrod neutral technological progress (γ 1), A t stationary (does not explain growth) A t are shocks to technology. AR(1) exogenous process with ε t N (, σ 2 ). log(a t ) = ρ log(a t 1 ) + (1 ρ) log(a) + ε t

34 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 34 The firm decides on production plan maximizing profits First order conditions: max {K t,h t } A tf(k t, Γ t h t ) W t h t z t K t K t : A t F K (K t, Γ t h t ) = z t h t : A t F h (K t, Γ t h t ) = W t Simple Example: Cobb Douglas production function First order conditions Y t = A t K α t (Γ t h t ) 1 α K t : αy t /K t = z t h t : (1 α)y t /h t = W t

35 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Equilibrium The (RBC) Model Equilibrium is given by the following equations ( t ): 1. Exogenous Processes : log(a t ) = ρ log(a t 1 )+(1 ρ) log(a)+ε t and Γ t = γγ t 1 2. Law of motion of Capital : K t+1 = I t + (1 δ)k t 3. Bond market equilibrium : B t = 4. Good Markets equilibrium : Y t = C t + I t

36 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Labor market equilibrium : U l(c t,1 h t ) U c (C t,l t ) = A t F h (K t, Γ t h t ) 6. Consumption/saving decision + Capital market equilibrium : U c (C t, 1 h t ) = βe t [U c (C t+1, 1 h t+1 )(A t+1 F K (K t+1, Γ t+1 h t+1 ) + 1 δ)] 7. Financial markets : (1 + r t ) = U c (C t, 1 h t ) βe t U c (C t+1, 1 h t+1 )

37 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models An Analytical Example U(C t, l t ) = log(c t ) + θ log(l t ), Y t = A t K α t (Γ th t ) 1 α Equilibrium θc t = (1 α) Y t 1 h t [ ( h t )] 1 1 = βe t α Y t δ C t C t+1 K t+1 K t+1 = Y t C t + (1 δ)k t Y t = A t Kt α (Γ t h t ) 1 α [ ] Kt+1+s lim s βs E t = C t+s

38 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Stationarization We want a stationary equilibrium (technical reasons) Deflate the model for the growth component Γ t On the example: x t = X t /Γ t

39 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 39 Deflated Equilibrium θc t = (1 α) y t 1 h t [ ( h t )] 1 = β 1 c t γ E t α y t δ c t+1 k t+1 γk t+1 = y t c t + (1 δ)k t y t = A t k α t h1 α t lim s βsγk t+1+s c t+s =

40 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Solving the Model In General Non linear system of stochastic finite difference equations under rational expectations Very complicated In general no analytical solution, need to rely on numerical approximation methods

41 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models The Nice Analytical Case U(c, l) = log(c) + θ log(l), γ = 1 (not needed) and δ = 1 This is the case we solved in section 2 of this lecture Numerical solution Unfortunately: No closed form solution in general Have to adopt a numerical approach Log linearize the equilibrium around the steady state (limits?)

42 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 42 Solve the linearized model using standard techniques The solution to the log linearize version of the model takes the form X t+1 = M x X t + M z Z t (4) Z t+1 = ρz t + ε t+1 (5) Y t = P x X t + P z Z t (6) where X t, Z t and Y t collect, respectively, the state variables, the shocks and the variables of interest Resembles a VAR model

43 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 43 In the basic RBC model X t = {k t } (7) Z t = {a t } (8) Y t = {y t, c t, i t, h t } (9) Let s evaluate the quantitative ability of the model to account for and explain the cycle.

44 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Quantitative Evaluation Calibration Need to assign numerical values to the parameters This is the calibration step What does calibration mean? Make explicit use of the model to set the parameters A lot of discipline, but no systematic recipe Have to set (α, β, θ, δ, γ, ρ, σ, A) Use data: ( log Y, k/y, i/k, h, wh/y)

45 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 45 Compute ( log Y, k/y, i/k, h, wh/y) in the model to set (α, β, θ, δ, γ) Then compute A t on the data and estimate ρ and σ.

46 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 46 In the data y =.9% per quarter γ = 1.9. In the data i/k =.76 on annual data. Use capital accumulation to get annual depreciation. i/k(model) = γ (1 δ) δ =.1 In the data wh/y =.6, in the model wh/y = 1 α α =.4. In the data k/y = 3.32 on annual data. equation which gives β =.98 β = γ αy/k + (1 δ) Using the Euler

47 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 47 h =.31, such that θ = (1 α)(1 h) hc/y We set A = 1 without loss of generality. We have log(a t ) = log(y t ) α log(k t ) (1 α) log(h t ) Estimate log(a t ) = ρ log(a t 1 ) + ε t We obtain ρ =.95 and σ = Can the Business Cycle be Driven by Capital Dynamics? Want to see whether capital dynamics can account for BC.

48 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 48 Perfect foresights dynamics We study a case with fixed labor (h = h) and a variable labor case (U(c, 1 h))

49 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 49 Capital shock - Fixed hours Output Consumption Investment Capital.4 Hours worked Labor productivity (Wages)

50 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 5 Capital shock - Variable hours Output Consumption Investment Capital.4 Hours worked Labor productivity (Wages)

51 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 51 Capital dynamics cannot be the story for the business cycle. More (shocks?) is needed to understand the BC That s why the RBC literature proposes technological shocks (Brock & Mirman, 1972)

52 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Technological Shocks We first stick in the model the estimated ε t and compare model and data.

53 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 53 A good fit with estimated shocks.1.5 Output Data Model.4.2 Hours Worked Consumption Investment

54 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 54 A first success? Accounts for the main events in the data The model correctly predicts the data: corr(y, y m )=.75, corr(c, c m )=.73, corr(i, i m )=.7. BUT: corr(h, h m )=.6 Let s compute unconditional moments in the model (simulate- HP filter-compute moments)

55 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Results from Model Simulations Let s first look at data characteristics.

56 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 56 U.S. data, Variable σ( ) σ( )/σ(y) ρ(, y) ρ(, h) Auto(1) Output Consumption Services Non Durables Investment Fixed investment Durables Changes in inventories Hours worked Labor productivity

57 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 57 Now consider the results from model simulations Variable σ( ) σ( )/σ(y) ρ(, y) ρ(, h) Auto Output Consumption Investment Hours worked Labor productivity (model in yellow)

58 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models A success(?) The model correctly predicts the amplitude, serial correlation and relative variability of fluctuations It accounts for a large part of output volatility Correct ranking of the volatility of c, i, y,... Large serial correlation, although it is smaller than in the data. But

59 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 59 C and N are not volatile enough w (and r) are to too procyclical

60 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Criticisms In General The research on RBC became so successful because Propose a coherent platform to analyse growth and cycles It somewhat fails such that there is room for work Main victory: methodological (part of the toolkit of macroeconomists) Main criticisms incorrect/implausible calibration of parameters (IES, Labor

61 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 61 supply) need for sensibility analysis counterfactual prediction for some prices: real wage is strongly procyclical in the model, CRRA preferences are not compatible with the equity premium price level is too strongly countercyclical The Measure of Technological Shocks Key problem: Are technological shocks at the source of BC?

62 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 62 Prescott [1986]: Technology shocks account for 7% of output volatility, but Too volatile Little evidence of large supply shocks (except oil prices) Recessions have to be explained by technological regressions Measurement problems (contamination by demand shocks if increasing returns, imperfect competition, labor hoarding) ; in the growth accounting literature, the SR was

63 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 63 a measure of our ignorance, now it is the engine of the model The Need for Persistent Shocks Recall that log(a t ) = ρ log(a t 1 ) + ε t and ρ is large Why do we need so persistent shocks? Because the model possesses weak propagation mechanisms

64 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 64 Let s see that in details

65 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 65 IRF to A Technological Shock Technology shock

66 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 66 IRF to A Technological Shock Output Consumption Investment Capital 1 Hours worked Labor productivity (Wages)

67 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 67 IRF to A Technological Shock: Temporary vs Persistent Technology shock

68 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 68 IRF to A Technological Shock: Temporary vs Persistent Output Consumption Investment Capital 1.5 Hours worked Labor productivity (Wages)

69 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 69 IRF to A Technological Shock: Persistent vs Permanent Technology shock

70 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 7 IRF to A Technological Shock: Persistent vs Permanent Output Consumption Investment Capital 1 Hours worked Labor productivity (Wages)

71 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 71 How to improve the model: Introducing additional shocks Improving the propagation mechanisms

72 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Solving the puzzles Adding Demand Shocks Government Expenditures Change the Household s budget constraint: B t+1 + C t + I t + T t (1 + r t 1 )B t + W t h t + z t K t Government Balanced Budget: T t = G t Government expenditures are modeled as log(g t ) = ρ log(g t 1 ) + (1 ρ g ) log(g) + ε g,t

73 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 73 with ε g,t N (, σ 2 t ). G/Y =.2, ρ g =.97, σ g =.2.

74 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 74 IRF to A Government Spending Shock Government shock

75 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 75 IRF to A Government Spending Shock Output Consumption Investment x 1 3 Capital.2 Hours worked Labor productivity (Wages)

76 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 76 Technological and Government Spending Model Variable σ( ) σ( )/σ(y) ρ(, y) ρ(, h) Output Consumption Investment Hours worked Labor productivity (model in yellow)

77 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Labor indivisibility Rogerson [1984] and Hansen [1985]: work a fixed amount of hours or does not (indivisibility) Then, the commodity set is nonconvex. We convexify it by the use of lotteries The nice property of this example is that one can define a representative agent, whose preferences are different from the original agents

78 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 78 Preferences U(C it, 1 h it ) = { log(c e it ) + θ log(1 h ) if she works log(cit u) + θ log(1) if not Randomly drawn to work with probability π t = h t h EU t = π t (log(cit e) + θ log(1 h )) + (1 π t ) (log(cit u) + θ log(1) ) = π t log(cit e) + (1 π t) log(cit u) + θπ t log(1 h ) = π t log(cit e) + (1 π t) log(cit u) + θlog(1 h ) h h t Households have access to a insurance contract that pays one unit of good in unemployed, if employed.

79 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 79 They buy A it units of insurance, at unit price τ t { C e it + τ t A it + Kit+1 e (z t + 1 δ)k it + w t h if she works Cit u + τ ta it + Kit+1 u (z t + 1 δ)k it + A it if not Households cannot manipulate the probability of working Risk neutral insurance companies will make zero profit: Revenues τ t A t = Costs (1 π t )A t Therefore τ t = 1 π t Households will demand insurance so as to equalize marginal utility of consumption in the two states (employed and unemployed)

80 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 8 Therefore C e t = Cu t consumption and leisure) (because utility is additively separable in We have (assuming they make same accumulation choice): { C e t = K t+1 + (z t + 1 δ)k it + w t h (1 π t )A t if she works C u t = K t+1 + (z t + 1 δ)k it + π t A it Equalization of C u and C e gives A = wh. if not As consumption is the same and utility separable, accumulation decisions will be similar Assume that agents had initially the same wealth (in period ): they will always take similar consumption and saving

81 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 81 decisions An agent utility in t can therefore be written π t (log C t + θ log(1 h )) + (1 π t )(log C t + θ log 1) Utility collapses to with B = θ log(1 h ) h U(C t, 1 h t ) = log(c t ) Bh t The rest of the model (firms, equilibrium equations) remains unchanged

82 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 82 IRF to A Technological Shock - Indivisible Labor Model Output Consumption Investment Capital 2 Hours worked Labor productivity (Wages)

83 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 83 IRF to A Technological Shock - Indivisible Labor Model Output Consumption Investment Capital 2 Hours worked Labor productivity (Wages)

84 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 84 Indivisible Labor Model Variable σ( ) σ( )/σ(y) ρ(, y) ρ(, h) Output Consumption Investment Hours worked Labor productivity (model in yellow)

85 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 85 4 Business Cycle Accounting Chari, Kehoe and McGrattan, Business Cycle Accounting, Econometrica, Vol. 75, No. 3, Benchmark Economy Agents Problems Notation: s t are events, s t are histories, π t (s t ) is probability of s t as of period. Four wedges, that are four exogenous stochastic variables functions of the underlying random variable s t

86 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Efficiency wedge A t (s t ) 2. Labor wedge 1 τ lt (s t ) 3. Investment wedge 1/[1 + τ xt (s t )] 4. Government consumption wedge g t (s t ) Consumers maximize β t π t (s t )U(c t (s t ), l t (s t ))N t t= s t subject to c t (s t ) + [1 + τ xt (s t )]x t (s t ) = [1 τ lt (s t )]w t (s t )l t (s t ) + r t (s t )k t (s t 1 )

87 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 87 +T t (s t ) (1 + γ n )k t+1 (s t ) = (1 δ)k t (s t 1 ) + x t (s t ) Firms maximize A t (s t )F(k t (s t 1 ), (1 + γ) t l t (s t )) r t (s t )k t (s t 1 ) w t (s t )l t (s t ).

88 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Equilibrium Equilibrium is summarized by y t (s t ) = c t (s t ) + x t (s t ) + g t (s t ), (1) y t (s t ) = A t (s t )F(k t (s t 1 ), (1 + γ) t l t (s t )), (11) U lt (s t ) U ct (s t ) = [1 τ lt(s t )]A t (s t )(1 + γ) t F lt, (12)

89 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 89 U ct (s t )[1 + τ xt (s t )] = β s t+1 π t (s t+1 s t )U ct+1 (s t+1 ) { A t+1 (s t+1 )F kt+1 (s t+1 ) (13) +(1 δ)[1 + τ xt+1 (s t+1 )] } Note that one wedge (at least) enters in each of the four equations. Remark : Those wedges are reduced forms for many imperfections: Efficiency Wedge : Technological regression (?), mismeasure-

90 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 9 ment, disorganization Labor Wedge : Taxes, Unions, wages rigidities, implicit contracts,... Investment Wedge : Government Wedge : Government spending shocks, net exports shocks Credit market imperfections, 4.2 Accounting procedure Consider (y t, c t, l t, x t ).

91 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 91 One can always find a sequence of the four wedges such that the model outcome exactly matches observed (y t, c t, l t, x t ) The procedure is implemented on the US Great Depression As the government wedge plays no role for y, x and l, it is set to zero and we only consider fluctuations in y, x and l. 4.3 Results The 3 wedges are measured so that one exactly reproduces the movements of Y, I and L

92 ment along with the model s predictions for those variables when the model includes just one wedge. In terms of the data, note that labor declines 27% Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 92 FIGURE 1. U.S. output and three measured wedges (annually; normalized to equal 1 in 1929).

93 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models The efficiency wedge is in % below the 1929 level, but is back to normal level by The labor wedge remains 3% below (permanently?) 3. The investment wedge goes the wrong way Let us look at the fraction of macroeconomic movements that are explained by those 2 wedges, then by only the investment one, then by all but one wedge:

94 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models V. V. CHARI, P. J. KEHOE, AND E. R. MCGRATTAN FIGURE 2. Data and predictions of the models with just one wedge. from 1929 to 1933 and stays relatively low for the rest of the decade. Invest-

95 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models V. V. CHARI, P. J. KEHOE, AND E. R. MCGRATTAN FIGURE 3. Data and predictions of the model with just the investment wedge. In this experiment, we choose the investment wedge to be as large as it needs to be for investment in the model to be as close as possible to investment in the data, and we set the other wedges to be constants. Predictions of this model, which we call the Model With Maximum Investment Wedge, turn out to match

96 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 96 BUSINESS CYCLE ACCOUNTING 85 FIGURE 4. Data and predictions of the models with all wedges but one. B. The 1982 recession

97 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models With the efficiency wedge only: much of the downturn is accounted for, bit the recovery is too important 2. The efficiency wedge misses labor market outcome 3. With the labor market wedge only, the downturn is partially missed, but the absence of recovery is well accounted for. 4. labor market outcomes are well accounted by the labor wedge. With the 2 wedges, one gets almost all of the action (the investment wedge is not so important)

98 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models Cole and Ohanian Explanation Cole and Ohanian, New Deal Policies and the Persistence of the Great Depression: A General Equilibrium Analysis, Journal of Political Economy, 24 The recovery from the Great Depression was weak. Puzzling: the large negative shocks that some economists believe caused the downturn - including monetary shocks, productivity shocks, and banking shocks - become positive after Thesis: Roosevelt s New Deal cartelization policies, which limited competition in product markets and increased labor bar-

99 Franck Portier TSE Macro I & II Lecture 2 Real Business Cycle Models 99 gaining power, kept the economy depressed after New Deal National Industrial Recovery Act (NIRA): 1. suspended antitrust law and permitted collusion in some sectors provided that industry raised wages above market clearing levels and accepted collective bargaining with independent labor unions.