Baby Busts and Baby Booms

Transcription

1 Baby Busts and Baby Booms The Fertility Response to Shocks in Dynastic Models Larry Jones 1 Alice Schoonbroodt 2 1 University of Minnesota and NBER 2 University of Southampton and CPC DGEM, REDg at CEMFI September 2010 Jones, Schoonbroodt Baby Busts and Baby Booms 1

2 Motivation Large fluctuations in fertility during the 20th century In the U.S. In other developed countries sizes differ Large: U.S., Canada, Australia Smaller: European countries Demographers: link fertility fluctuations to G-D, WWII,... (good vs. bad times, optimism, pessimism, catching up,...) Jones, Schoonbroodt Baby Busts and Baby Booms 2

3 U.S. TFR and CTFR TFR CTFR (+23) Fertility Year Jones, Schoonbroodt Baby Busts and Baby Booms 3

4 Empirical Evidence: TFP and LP Log TFP Log LP Productivity Year Jones, Schoonbroodt Baby Busts and Baby Booms 4

5 Empirical Evidence: % Deviations in TFP&LP and TFR Percent Deviations TFP LP TFR Year TFR t = TFP t 0.84 TFP t 20 + ε t Jones, Schoonbroodt Baby Busts and Baby Booms 5

6 Motivation Large fluctuations in fertility during the 20th century Demographers: link fertility fluctuations to G-D, WWII,... (good vs. bad times, optimism, pessimism, catching up,...) Economic interpretation of demographer s story: Easterlin (1961,...,2000) Neo-classical environments: Greenwood, Seshadri, Vandenbroucke (2005) Doepke, Hazan, Maoz (2007) Albanesi, Olivetti (2009)... Jones, Schoonbroodt Baby Busts and Baby Booms 6

7 This Paper Model Stochastic neo-classical growth model Population N plays the role of capital K Different ages of people similar to capital vintages Non-stochastic growth model with endogenous fertility Children cost time opportunity cost pro-cyclical Children enter utility function smooth like consumption Cyclical properties? Current fertility dependent on past fertility? Quantitative experiments: Std. Recession & U.S. BBB Cross-country: Sizes of G-D, Baby Busts and Booms Jones, Schoonbroodt Baby Busts and Baby Booms 7

8 Preview of Results Qualitatively: Fertility pro-cyclical in most cases depends on nature of costs of children all goods: pro-cyclical all time: consumption smoothing pro-cyclical opportunity cost counter-cyclical Current fertility depends negatively on last period s fertility except if only one period of working life Quantitatively, interesting magnitudes Standard Recession: not much on CTFR, maybe TFR Depression in 1930s: Baby Bust Baby Bust + high productivity: Baby Boom 1950s Continuing fluctuations dampened by productivity shocks International Evidence: size of Depression, Baby Bust and Boom Jones, Schoonbroodt Baby Busts and Baby Booms 8

9 Barro-Becker model with Age-Structure Preferences U 1 t = V 1 t + βg(n t )U 1 t+1 Laws of motion where V 1 t = 3 a=1 βa 1 u(c a t+a 1 ) Nt 1 = n t 1 Nt 1 1 is the number of births in period t 1 N a t = N a 1 t 1 N a t = 0 for a > 3 N 3 0 = 1 for a = 2, 3 Sequential substitution and g(n) = n η Preferences of Dynastic Head (initial old) U0 3 = [ 3 ] t=0 βt a=1 g(na t )u(ca t ) Jones, Schoonbroodt Baby Busts and Baby Booms 9

10 Barro-Becker model with Age-Structure Feasibility for Dynasty: t 3 a=1 Na t ca t + θ t (w 1 t )n tn 1 t 2 a=1 w a t Na t Introduce Productivity shocks (w 1 t, w 2 t ) = (γt s t w 1,γ t s t w 2 ) E(s t ) = 1 and s t are i.i.d. Costs of children Goods cost: θ t (w 1 t ) = γt θ Time cost: θ t (w 1 t ) = bw 1 t = bγ t s t w 1 [DEC] Jones, Schoonbroodt Baby Busts and Baby Booms 10

11 Dynastic Planner s Problem, P(γ, β; { N a 0}, s0 ) [ [ 3 ] ] max E 0 t=0 βt a=1 g(na t (st a ))u(ct a(st )) s 0 subject to: 3 a=1 Na t (st a )c a t (st ) + θ(s t )N 1 t+1 (st ) γ t s t 2 a=1 w a N a t (st a ); Nt+1 1 (st ) is the number of births in period t; Nt a(st a ) = N a 1 t 1 (st 1 (a 1) ) for a = 2, 3; Nt a(st a ) = 0 for a > 3; N0 a given, a = 1, 2, 3. where s t = (s 0, s 1,..., s t ) is the history of shocks. Notice non-convexity N a t ca t C a t. Jones, Schoonbroodt Baby Busts and Baby Booms 11

12 Dynastic Planner s Problem, P(1, βγ 1 σ ; { N a 0}, s0 ) [ ( max E 0 t=0 βγ 1 σ ) [ t 3 a=1 g ( Nt a(st a ) ) ( )] ] C a u t (s t ) s Nt a(st a ) 0 subject to: 3 a=1 Ca t (st ) + θ(s t )N 1 t+1 (st ) s t 2 a=1 w a N a t (st a ); Nt+1 1 (st ) is the number of births in period t; Nt a(st a ) = N a 1 t 1 (st 1 (a 1) ) for a = 2, 3; Nt a(st a ) = 0 for a > 3; N0 a given, a = 1, 2, 3. where C a t and θ t are detrended. Jones, Schoonbroodt Baby Busts and Baby Booms 12

13 Preliminary results to simplify DP Assume βγ 1 σ < 1. Assume g(n) = N η and u(c) = c1 σ 1 σ Parameter restrictions for monotonicity and concavity: 1.) B-B: 0 < 1 σ η < 1, or, 2.) G&BC: η 1 σ < 0 Let V(N 1, N 2, N 3 ; s) be maxed value in P(1,βγ 1 σ ; { N a 0}, s0 ) Then, V(N 1, N 2, N 3 ; s) is homog. of d o η in (N 1, N 2, N 3 ). Jones, Schoonbroodt Baby Busts and Baby Booms 13

14 Dynastic Planner s Problem, P(1, βγ 1 σ ; { N a 0}, s0 ) max E 0 t=0 ( βγ 1 σ ) t 3 ( a=1 N a t (s t a ) ) η ( C a t (s t ) N t a(st a ) 1 σ ) 1 σ s 0 subject to: 3 a=1 Ca t (st ) + θ(s t )N 1 t+1 (st ) s t 2 a=1 w a N a t (st a ); Nt+1 1 (st ) is the number of births in period t; Nt a(st a ) = N a 1 t 1 (st 1 (a 1) ) for a = 2, 3; Nt a(st a ) = 0 for a > 3; N0 a given, a = 1, 2, 3. where C a t and θ t are detrended. Jones, Schoonbroodt Baby Busts and Baby Booms 14

15 Preliminary results to simplify DP Assume βγ 1 σ < 1. Assume g(n) = N η and u(c) = c1 σ 1 σ 1.) B-B: 0 < 1 σ η < 1, or, 2.) G&BC: η 1 σ < 0 Then, V(N 1, N 2, N 3 ; s) is homog. of d o η in (N 1, N 2, N 3 ). Assume η = 1 σ. Then, C a c a N a = c a N a C a, a, a N 3 irrelevant use Ṽ(N1, N 2 ; s). Homogeneity implies Ṽ(N 1, N 2 ; s) = (N 2 ) 1 σ Ṽ(N 1 /N 2, 1; s) (N 2 ) 1 σ V(n; s). Jones, Schoonbroodt Baby Busts and Baby Booms 15

16 Bellman equation and FOC Bellman equation where n = N 1 /N 2, last period s fertility: V(n; s) max n ( ) 1 σ s[w 1 n+w 2 ] θ(s)n n 3 1 σ + βγ 1 σ n 1 σ E [ V(n ; s )] The FOC is given by: [ ] θ(s) (FOC) ] = βγ 3E [ˆV1 1 σ s w 1 + w 2 n θ(s)n (n, s ) 3 σ Jones, Schoonbroodt Baby Busts and Baby Booms 16

17 Bellman equation and FOC Bellman equation where n = N 1 /N 2, last period s fertility: V(n; s) max n ( ) 1 σ s[w 1 n+w 2 ] θ(s)n n 3 1 σ + βγ 1 σ n 1 σ E [ V(n ; s )] The FOC is given by: [ ] θ(s) (FOC) ] = βγ 3E [ˆV1 1 σ s w 1 + w 2 n θ(s)n (n, s ) 3 σ Jones, Schoonbroodt Baby Busts and Baby Booms 17

18 First-order Condition Goods cost (θ(sw 1 ) = θ): [ ] θ 3E ˆV 1 (n, s ) = βγ1 σ s w 1 + w 2 n θn 3 σ Time cost (θ(sw 1 ) = bsw 1 ): bw 1 s 1 σ 3E ˆV 1 (n, s ) = βγ1 σ ( w 1 (1 bn ) + w 2 n 3 ) σ Jones, Schoonbroodt Baby Busts and Baby Booms 18

19 Comparative statics results Proposition Current fertility, n (n; s) is 1. a. pro-cyclical if θ(s) = θ; b. pro-cyclical if θ(s) = bsw 1 and σ > 1; counter-cyclical if θ(s) = bsw 1 and σ < 1; 2. a. independent of the last period s fertility, n, if w 2 = 0; b. decreasing in last period s fertility, n if w 2 > 0. Thus, if σ 1 and w 2 > 0 the model generates endogenous cycles, triggered by productivity shocks. [QUANT] Jones, Schoonbroodt Baby Busts and Baby Booms 19

20 Quantitative Experiments Model period: 20 years, i.e. adult at age 20, fertile 20-40, worker 20-60, retired Calibration (to averages) Parameters for stochastic process (log ŝ t N(0,σ s )) Parameters for economic model (σ,β, w 1, w 2,γ,θ or b) Experiments Impulse response Typical recession Historical simulation: Input: sequence of shock realizations, data model Output: sequence of fertility fluctuations, model data Baseline + Sensitivityyyyy... International (statistical) evidence Jones, Schoonbroodt Baby Busts and Baby Booms 20

21 Parameterization Preset or set directly from wage and TFP data: Param. σ β w 1 w 2 γ σ s Value We experiment with 2 extreme cases: all goods vs. all time cost Calibrated to match 0.645% annual population growth: Param. θ (goods cost) b (time cost) Value Jones, Schoonbroodt Baby Busts and Baby Booms 21

22 Impulse Response and Recessions A 1% increase in productivity, s, generates: Goods Cost: a 1.7% contemporaneous increase in fertility, a 1.6% decrease 1 period later Time Cost: : a 1.0% contemporaneous increase in fertility, a 0.9% decrease 1 period later Standard Recession (e.g., 5% below trend for 2 years) These are total effects on completed fertility: not that large. Great Depression: 12% below trend for 10 years... Jones, Schoonbroodt Baby Busts and Baby Booms 22

23 How to fit model shocks to data shocks? Model period is 20 years, 4 age groups, a = 0, 1, 2, 3. Hence, fertile period should be 20 years, age 20 to 40. Hence, relevant income shock should be over 20 year period. However, this assumes that fertility is uniform age But 60% of all births occur between age 20-30, 75% age In particular, women age in the 1920s are mostly done with fertility decisions by the time the Great Depression hits. We therefore use 10 year period for shocks in baseline Jones, Schoonbroodt Baby Busts and Baby Booms 23

24 CTFR: Most affected Dynasty Decade 1910s 1930s 1950s 1970s 1990s TFP deviat Percent Deviations CTFR Model: Goods cost Model: Time cost CTFR: Data Cohort TFP Shock Birth Year of Mother + 23 years Jones, Schoonbroodt Baby Busts and Baby Booms 24

25 CTFR: All Dynasties/Cohorts Percent Deviations CTFR Model: Goods cost Model: Time cost CTFR: Data Cohort TFP Shock Birth Year of Mother + 23 years Jones, Schoonbroodt Baby Busts and Baby Booms 25

26 TFR: All Dynasties/Cohorts Percent Deviations TFR Model: Goods cost Model: Time cost TFR: Data TFP Shock Year Jones, Schoonbroodt Baby Busts and Baby Booms 26

27 Sensitivity Change shocks: Age range: 10- versus 20-year shocks Labor Productivity versus TFP Change state space: Physical capital (with w 2 = 0) 2 vs. 3 period work life (non-stochastic, assume bust) Parameter sensitivity [INT] Jones, Schoonbroodt Baby Busts and Baby Booms 27

28 10- versus 20-year TFP shocks TFP Shock, TFP Shock, Percent Deviations TFP Birth Year of Mother + 23 years Jones, Schoonbroodt Baby Busts and Baby Booms 28

29 TFR: All Dynasties/Cohorts - 10 versus 20 year shock Percent Deviations TFR Experiment Baseline TFR: Data Year Jones, Schoonbroodt Baby Busts and Baby Booms 29

30 Sensitivity Change shocks: Age range: 10- versus 20-year shocks Labor Productivity versus TFP Change state space: Physical capital (with w 2 = 0) 2 vs. 3 period work life (non-stochastic, assume bust) Parameter sensitivity [INT] Jones, Schoonbroodt Baby Busts and Baby Booms 30

31 Labor Productivity versus TFP shocks Percent Deviations LP and TFP LP Shock TFP Shock Birth Year of Mother + 23 years Jones, Schoonbroodt Baby Busts and Baby Booms 31

32 TFR: All Dynasties/Cohorts LP versus TFP Percent Deviations TFR Experiment Baseline TFR: Data Year Jones, Schoonbroodt Baby Busts and Baby Booms 32

33 Sensitivity Change shocks: Age range: 10- versus 20-year shocks Labor Productivity versus TFP Change state space: Physical capital (with w 2 = 0) 2 vs. 3 period work life (non-stochastic, assume bust) Parameter sensitivity [INT] Jones, Schoonbroodt Baby Busts and Baby Booms 33

34 Model with K (w 2 = 0, goods cost) V(N 1, K, s) = subject to: { max T C1 σ (C,N 1,K 1 σ ) + βγ1 σ E ( V(N 1, K, s ) )} TC + θn 1 + γk = sf(k, N 1 ) + (1 δ)k { } v(k, s) = max c 1 σ c,n,k 1 σ + βγ1 σ (n ) 1 σ E (v(k, s )) subject to: 0 c sf(k) γk n + (1 δ)k θn γk n (1 δ)k 0 N 0, K 0, s 0 given A 1% increase in productivity, s, generates: Baseline but w 2 = 0: a 1% increase in fertility. With capital, K : a % increase in fertility. Jones, Schoonbroodt Baby Busts and Baby Booms 34

35 Sensitivity Change shocks: Age range: 10- versus 20-year shocks Labor Productivity versus TFP Change state space: Physical capital (with w 2 = 0) 2 vs. 3 period work life (non-stochastic, assume bust) Parameter sensitivity [INT] Jones, Schoonbroodt Baby Busts and Baby Booms 35

36 TFR: All Dynasties/Cohorts w 3 = 0 vs. w 3 > 0 Percent Deviations TFR Deterministic 3 Deterministic 2 Baseline TFR: Data Year Jones, Schoonbroodt Baby Busts and Baby Booms 36

37 International Evidence Issues: We find: data availability: annual CBR and GDP for 17 countries, effects of wars: dummies or no dummies, detrending: OLS or HP filter. 1930s: Larger Baby Busts are strongly associated with larger Depressions. 1950s: Larger Baby Booms are (less strongly) associated with larger Baby Busts, if GDP deviation in the 1950s is taken into account. Jones, Schoonbroodt Baby Busts and Baby Booms 37

38 Summary of results Fertility can be either pro- or counter-cyclical, even in simple models Increase in female labor supply opportunity cost of time more important fertility less pro-cyclical? No catching up without age-structure Capital doesn t change result Interesting magnitudes of effects Current recession? Jones, Schoonbroodt Baby Busts and Baby Booms 38