Jang-Ting Guo

Lecture 2-1

Real Business Cycle Model (I) Kydland and Prescott (1982), (1990) Long and Plosser (1983) Hansen (1985) Objective: Use the neoclassical growth model to explain the observed patterns of aggregate business cycle fluctuations Model: A one-sector optimal growth model with variable labor supply and subject to a technology shock The Baseline Model (Divisible Labor Economy)

1)

n1t+χ  ∞ t Max E 0 ∑ β (log c t − A ), A > 0, and χ ≥ 0 t =0 1 + χ  

2)

c t + k t +1 ≤ (1 − δ ) k t + y t , k 0 = k 0

3)

yt = stk t n t

4)

α

1− α

st = sρt −1 ν t , 0 < ρ < 1, s0 = s0 , ν t independent N (1, σ ν ) ⇒ E ( st ) = 1

1

Next, form the Lagrangian

[

]

n1t+χ α 1−α L = E 0 ∑ β {log c t − A +λ t s t k t n t + (1 − δ)k t − c t − k t +1 } t =0 1+ χ ∞

t

The factor markets are assumed to be perfectly competitive. Thus the first-order conditions of firms’ profit maximization problem are w t = MPn = (1 − α)

yt nt

and rt = MPK = α

yt kt

First-order conditions yt nt

5)

Act n χt = w t = (1 − α)

6)

 1 1 y  = E t β (1 − δ + α t +1 )  ct k t +1   c t +1

7)

k t +1 = (1 − δ ) k t + st k αt n1t −α − c t , k 0 = k 0

8)

st +1 = sρt ν t +1 , s0 = s0

2

From equation 5), nt can be expressed as a function of ct, kt and st: 9)

A c t − α 1+χ nt = [ ] α 1 − α st k t

Substituting 9) into 6), 7), and 8), the equilibrium is characterized by a first-order non-linear stochastic dynamical system in {c t , k t , st } with two initial conditions {k 0 , s 0 } given Solution Method 1. Find the steady state of the dynamical system: {c , k ,1} 2. Define the transformed variables:  k t − k  ~  st − 1 ~  νt − 1 ~c =  c t − c , ~ = k  , st =  , ν t =   t t c k 1 1         3. Log-Linearize 6) -- 8) around the steady state and obtain the transformed dynamical system: ν t +1  ~ ~ ~  ct   ct +1  w c  ~ ~  k  = A k  + B t +1 , ~ 10) k 0 , ~s0 are given, t t +1 k      w t +1  ~ ~ s s  t   t +1   s   w t +1  where w xt+1 = Et [xt+1 ] − xt+1, for x = c,k,s. 3

4. Since there is no externality or distortion of any kind in this economy, the rational expectations equilibrium is a unique saddle path. In this 3x3 case with two predetermined variables, the uniqueness requires λ1 < 1 < λ 2 < λ 3

Solve the stable root λ1 forward ⇒ a linear restriction that relates ct to kt and st, which defines the stable branch of the saddle path 5. Therefore, the model can be represented by:

11)

~ ~  k t +1  a 11 a 12   k t   0  ~ ~ ~  =  , {k o , so } given   ~s  + ~ ν 0 ρ s   t   t +1   t +1  

Notice that it becomes a uncoupled 2nd order dynamical system with two real eigenvalues ( = a 11 and ρ) 6. Find additional linear equations which specify how the transformed output, investment, labor hours, and average productivity (= y/n) relate to the state variables ~ {k t , ~st }

4

Calibration and Simulation Calibration: a methodology for choosing parameter values based on evidence from growth observations and micro studies In our divisible labor economy: (1) β = quarterly discount factor = 0.99 (2) δ = quarterly deprecation rate of capital = 0.025 (3) α = capital share of national income = 0.36 (4) χ = inverse of labor supply elasticity = 0.5 (5) A = 3.45 ⇒ hours worked at the steady state is 1/3 (6) ρ = 0.95 and σ ν = 0.00712 ∈ [0.007,0.01], which are calibrated by looking at the Solow residuals of US data 12)

log st = log y t − 0. 36 log k t − 0. 64 log n t

Simulation (1) Given the parameter values, find the steady state k (2) Start the dynamical system (11) from {k 0 , s0 } = {k ,1} ~ (Alternatively, set {k 0 , ~s0 } = {0,0}) (3) Generate a sequence of {~ ν t }Tt =1 from N (0,σ ν ) ~ (4) Iterate equation (11) to get {k t , ~st }Tt =1 ~ (5) Obtain {~ct , ~y t , it , ~ n t , ~zt }Tt =0 (z = y/n) from step 4 & 6 above (6) Compare selected moments of the simulated time series with the US data at the over the business cycle

5

Detrending the data

Lucas: Business cycles are deviations from trend Hodrick-Prescott Filter A time series y t = τ t ( trend) + d t (business cycle) , where the trend component {τ t } is the solution to the problem: 13)

Min {τ t }

λ T−1 1 T 2 2 ( ) ( ) ( ) [ ] y − τ + τ − τ − τ − τ , λ >0 ∑ t ∑ t +1 t t t −1 t = t = 2 t 1 T T

The first term is the sum of squared deviations, dt; the second term penalizes variations in the growth rates of the trend component where a larger λ results in a smoother trend. For quarterly data, λ = 1600 6

Hansen conducted 100 simulations on the model; each simulation consists of 115 periods, which is the same number of periods as the US sample (55.3-84.1) All the data series, taken from Citibase, are seasonally adjusted and logged Both the artificial and actual time series have been passed through the H-P filter Quantitative Results U.S. Time Series (55.3-84.1) Series S.D.(%) CORR. Output 1.76(1.00) 1.00 Consumption 1.29(0.73) 0.85 Investment 8.60(4.89) 0.92 Hours 1.66(0.94) 0.70 Productivity 1.18(0.67) 0.68

The Model (Divisible Labor) S.D.(%) CORR. 1.35(1.00) 1.00 0.42(0.31) 0.89 4.24(3.14) 0.99 0.70(0.52) 0.98 0.68(0.50) 0.98

The standard deviations and correlations with output reported in column 4 and 5 are sample means of statistics computed for each of 100 simulations For variable x, the number in the parentheses is its relative standard deviation to output

7

Things Consistent with the Data (1) Consumption fluctuates less and investment more than output (2) There is a positive co-movement of output, consumption, investment, hours, and productivity over the business cycle, i.e., they are procyclical Minor Discrepancies (1) Consumption and investment are less volatile (2) The correlation between investment, hours, productivity and output are too high Major Problems: (1) It is necessary to increase σ ν by 30 percent (from 0.00712 to 0.00929) to match the output fluctuations in the model economy with the actual data ( = 1.76) (2) Hours series is too smooth (3) Data: Model: Why?

σn = 1. 4, CORR ( n, w ) = 0. 07 σw σn = 1. 0, CORR ( n, w ) = 0. 93 σw ⇒

Indivisible Labor Economy 8

Jang-Ting Guo

Lecture 2-2

Real Business Cycle Model (II) Hansen's Divisible Labor Economy ∞

1)

Max E 0 ∑ β t (log c t + A log l t ), A > 0

2)

c t + k t +1 ≤ (1 − δ ) k t + st k t n t , k 0 = k 0

3)

nt + l t = 1

t =0

α

1− α

st = sρt −1 ν t , 0 < ρ < 1, s0 = s0 ,

4)

ν t independent N (1, σ ν ) ⇒ E ( st ) = 1

Two Main Problems (1) The required standard deviation of σ ν is too high ( = 0.00929) to match the output fluctuations in the model with the actual data ( = 1.76)

σn = 1. 4, CORR ( n, w ) = CORR ( n, AP) = 0. 07 σw σn = 1. 0, CORR ( n, w ) = CORR ( n, AP) = 0. 93 Model: σw (Hours series is too smooth)

(2) Data:

1

Reasons 1)

Cross-section studies suggest that the labor supply elasticity is rather low

2)

A single technology shock shifts the labor demand curve

However in U.S. data, variations of labor input come about more by fluctuations in employment (number of individuals worked) than from fluctuations in hours per worker Implication: Most People will work full time or not at all, that is, the labor supply is indivisible Explanation: there is a fixed cost of traveling to work The Indivisible Labor Economy of Hansen (1985) 5)

U (c t ,l t ) = log c t + A log l t , A > 0

Assume that in each period households supply n 0 ( a constant ∈[ 0,1]) units of labor or none at all ⇒ the model generates fluctuations in the number of employed workers over the business cycle Implications (1) Unemployment is introduced (2) The labor supply elasticity is increased 2

It follows that l = 1 − n 0 or 1, so the consumption set for this economy is non-convex ( a big problem for applying the second welfare theorem!!) Solution: Following Rogerson (1988), we convexify the consumption set by requiring individuals choose employment lottery (a probability of working = π t ) rather than hours worked A random draw of the lottery determines whether the agent will be employed or unemployed Since all households are identical, they all choose the same π t Thus there is a new commodity, which is the contract between firms and households that commits each household to work n 0 hours with probability π t The contract is being traded (not actual hours of work), so households get paid whether it works or not ⇒ Firms are providing complete unemployment insurance Cf. In Appendix of Hansen (1985): Households only get paid for the work they actually do; unemployed individuals get paid nothing by the firms. But households have access to an insurance market where individuals will choose to be full insured in equilibrium

3

Denote n t as (expected) per capita hours worked in period t, which is given by n t = π t n 0 The period utility function of the representative agent can be derived form (5) using the assumption that individual households choose lotteries to maximize expected utility 6) U (c t , n t ) = π t [log c t + A log(1 − n 0 )] + (1 − π t )[log c t + A log(1)] = log c t + n t A

log(1 − n 0 ) n0

= log c t − Bn t , where B = − A

log(1 − n 0 ) n0

Therefore we can solve the indivisible labor model as if it were a divisible labor model with different instantaneous utility function by replacing (1) with (6) s.t. (2)-(4) Note: Independent of the willingness of individuals to substitute leisure over time, the elasticity of substitution between leisure in any two periods is infinite for the representative agent ⇒ horizontal labor supply curve. The model is consistent with large amounts intertemporal substitution at the aggregate level and relatively little intertemporal substitution at the individual level

4

Three Other Alternatives: Hansen and Wright (1992) 1. Nonseparable Leisure: Kydland and Prescott (1982) Instantaneous utility depends on a weighted average of current and past leisure in a nonseparable pattern

8)

∞

∞

i =0

i =0

U (c t , L t ) = log c t + A log L t , where L t = ∑ a i l t −i , ∑ a i = 1 a i+1 = (1 − η)a i for i = 0,1, 2, .... ∞, and a 0 is given

The contribution of past leisure to Lt decays geometrically at rate η. Since Lt now provides utility, individuals are more willing to intertemporally substitute by working more in some periods and less in other, i.e., the labor supply elasticity is increased Parameterization: a 0 = 0. 35, η = 0.10, others same as those in the Divisible Labor model Results:

σ y = 1.51,

σn = 1. 63, σw

7

CORR ( n, w ) = 0.80

2. Stochastic Government Spending: Christiano and Eichenbaum (1992) log( g t +1 ) = (1 − λ ) log( g ) + λ log( g t ) + µ t , λ ∈ ( 0,1) µ t independent N ( 0, σ µ )

9)

New resource constraint: c t + i t + g t = y t

Assume that: a) µ t is independent of the technology shock b) g t is financed by lump-sum taxation c) g t enters neither the utility function nor the production function An increase in g t is a pure drain on output ⇒ Negative wealth effect induces households to work more since leisure is a normal good ⇒ shifting the labor supply curve while technology shocks shift labor demand curve Parameterization: λ = 0. 96, σ µ = 0. 021, g = 0. 22 , others same as those in the Divisible Labor model Results:

σ y = 1. 24,

σn = 0. 90, σw

8

CORR ( n, w ) = 0. 49

3. Household Home Production: Benhabib, Rogerson, and Wright (1991)

U (c t , l t ) = log c t + A log l t , A >0 10)

c t = [aceMt + (1 − a )c eHt ]1/ e l t = 1 − n Mt − n Ht

Notice that n Mt and n Ht are perfect substitutes while elasticity of substitution between c Mt and c Ht is 1/(1-e)

11)

y Mt = sMt k θMt n1Mt−θ , θ ∈ (0,1) y Ht = sHt k ηHt n1Ht− η , η ∈ (0,1)

sMt = sρMt −1ε Mt , 0 < ρM < 1, ε M0 = ε M0 M

sHt = sρHt −1ε Ht , 0 < ρH < 1, ε H0 = ε H0 ε Mt is N (1, σ M ), and ε Ht is N (1, σ H ) γ = cor ( ε Mt , ε Ht ) H

12)

9

13)

k t = k Mt +k Ht , k 0 is given k t +1 = (1 − δ ) k t + i t y Mt = c Mt + i t y Ht = c Ht

Note: all new capital is produced in the market sector Parameterization:

( A , a ) ⇒ n M = 0. 33, and n H = 0. 28; c θ = 0. 36, and η = 0. 08 ⇒ H c = 1 / 4; M ρM = ρH = 0. 95, and σ M = σ H = 0. 007; e = 0.8, and γ = 2 / 3

Results:

σ y = 1. 71,

σn = 1. 92, σw

10

CORR ( n, w ) = 0. 49