Complements vs. Substitutes and Trends in Fertility in Dynastic Models

Transcription

1 Motivation Complements vs. Substitutes and Trends in Fertility in Dynastic Models Larry Jones 1 Alice Schoonbroodt 2 1 University of Minnesota 2 University of Southampton Humboldt Universität zu Berlin November 2008 Jones and Schoonbroodt (2008) Complements vs. Substitutes 1

2 Motivation Motivation: U.S. Crude Birth Rate CBR, (Hacker 2003), (Mitchell), (Vital Statistics) CBR hp filtered 40 CBR Year [CBR UK] Jones and Schoonbroodt (2008) Complements vs. Substitutes 2

3 Motivation U.S. Population Growth % Population Growth Rate, Annual 2.5% 2.0% 1.5% 1.0% 0.5% 0.0% Decade Jones and Schoonbroodt (2008) Complements vs. Substitutes 3

4 Question Qualitative results Motivation What accounts for the sharp decline in fertility? Two candidates often proposed in demography literature: I. Increase in productivity Industrialization and Development wages higher? urbanization? wage growth higher? II. Decrease in mortality IMR s and CMR s in particular Jones and Schoonbroodt (2008) Complements vs. Substitutes 4

5 Motivation Consistent with Economic Models? Barro-Becker Altruism Intuitive: parents care about number and utility of children U t = u(c t ) + βg(n t )U t+1 Practical: reduces to "standard" growth model with well known properties U 0 = t β t g(n t )u(c t ) = t β t N η t c 1 σ t 1 σ BUT in its simplest incarnations appears to fail at producing the fertility responses to changes in productivity and mortality as suggested by demographers. Jones and Schoonbroodt (2008) Complements vs. Substitutes 5

6 Barro-Becker Altruism Motivation Simplest incarnations of Barro-Becker model appear to fail at producing the fertility responses to changes in productivity and mortality as suggested by demographers. I. Development: An increase in Income levels no effect on fertility (balanced growth)... Income growth increases fertility in general equilibrium II. Mortality decrease: Why?... Large increase in surviving children (pop. growth)... Modest (if any) decrease in birth rates Jones and Schoonbroodt (2008) Complements vs. Substitutes 6

7 Basic Idea Household to Dynasty Comparative Statics Original Barro-Becker assumptions U t = u(c t ) + βg(n t )U t+1 For monotonicity of U t in U t+1 : g( ) 0 For monotonicity and concavity of U t in n t, g increasing and concave, g(n) = n η with η (0, 1). This implicitly assumes that U is positive. [why] Jones and Schoonbroodt (2008) Complements vs. Substitutes 7

8 Basic Idea Household to Dynasty Comparative Statics Typical growth and business cycle assumption U t = u(c t ) + βu t+1 u(c t ) = c1 σ t 1 σ U 0 = β t c1 σ t 1 σ G & BC has no problems assuming σ > 1 This leads to negative U. [why] Jones and Schoonbroodt (2008) Complements vs. Substitutes 8

9 Alternatively... Qualitative results Basic Idea Household to Dynasty Comparative Statics U t = u(c t ) + βg(n t )U t+1 For monotonicity of U t in U t+1 : g( ) 0 For monotonicity and concavity of U t in n t, either 1.) B-B: U > 0 and g increasing and concave, 2.) G&BC: U < 0 and g decreasing and convex [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 9

10 Alternatively... Qualitative results Basic Idea Household to Dynasty Comparative Statics U t = u(c t ) + βg(n t )U t+1 For monotonicity of U t in U t+1 : g( ) 0 For monotonicity and concavity of U t in n t, either 1.) B-B: U > 0 and g increasing and concave, n and U complements in utility 2.) G&BC: U < 0 and g decreasing and convex n and U substitutes in utility Preference version of Becker s Quantity/Quality tradeoff? [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 10

11 First implication Qualitative results Basic Idea Household to Dynasty Comparative Statics MU nt U t+1 = U t n t U t+1 = βg (n t ) 0 [why] Improved mortality and/or development increases U t+1 B-B: βg (n t ) > 0 complements in utility Increased U t+1 INCREASES the demand for children in t, cet. par. G&BC: βg (n t ) < 0 substitutes in utility Increased U t+1 DECREASES the demand for children in t, cet. par. Jones and Schoonbroodt (2008) Complements vs. Substitutes 11

12 Second implication Qualitative results Basic Idea Household to Dynasty Comparative Statics max {c t,n t } s.t. u(c t ) + βg(n s,t )U t+1 c t + θ b,t n b,t w t n s,t = π s,t n b,t Youth mortality ց cost of surviving children ց demand for surviving children ր size of this effect depends on desire to smooth consumption Jones and Schoonbroodt (2008) Complements vs. Substitutes 12

13 Literature Qualitative results Basic Idea Household to Dynasty Comparative Statics Simple Barro-Becker type model (with high IES) fails Augment B-B technology/constraint set Becker, Murphy, Tamura (1990): inc. ret. to human capital Doepke (2004): non-convexity in education choice Doepke (2005): sequential fertility choice (IES 2) Bar and Leukhina (2006): rural-urban (IES 1) Go to low IES similar to growth & RBC literature Mateos-Planas (2002): quantitative exercise across EU (IES 0.3) Introduce non-homothetic preferences (no BGP) Galor, Weil (2000); Greenwood, Seshadri (2002) Change direction of altruism Boldrin and Jones (2002): filial altruism/caldwell model Boldrin, De Nardi, Jones (2005): social security Jones and Schoonbroodt (2008) Complements vs. Substitutes 13

14 What we do next Qualitative results Basic Idea Household to Dynasty Comparative Statics 1. Simple model of fertility choice: Parental altruism à la B-B, BUT allow for full range of IES Stochastic aging Labor income only Intuitive analytical results Address fertility trend quantitatively 2. Add quality choice (e.g. bequests, education) 3. Add shocks to address fluctuations Jones and Schoonbroodt (2008) Complements vs. Substitutes 14

15 From Household to Dynasty Basic Idea Household to Dynasty Comparative Statics Household Problem: max {c t,n s,t } U t = u(c t ) + βg(π t + n s,t )U t+1 c t + θ b,t n b,t w t n s,t = π s,t n b,t Note certainty equivalence. Eliminate n b,t, let θ s,t = θ b,t /π s,t. Jones and Schoonbroodt (2008) Complements vs. Substitutes 15

16 From Household to Dynasty Basic Idea Household to Dynasty Comparative Statics Assume U t = u(c t ) + g(π t + n s,t )U t+1 Sequentially substitute g(x) = x η and u(c) = c1 σ 1 σ U 0 = u(c 0 ) + βg(π 0 + n s,0 ) [ u(c 1 ) + βg(π 1 + n s,1 )U 2 ] Law of M o N t+1 = t 1 k=0 (π k + n s,k ) = π t N t + N s,t Then t 1 k=0 g(π k + n s,k ) = g( t 1 k=0 (π k + n s,k )) = g(n t+1 ) Jones and Schoonbroodt (2008) Complements vs. Substitutes 16

17 Dynasty Preferences Basic Idea Household to Dynasty Comparative Statics U 0 = t β t N η t c 1 σ t 1 σ = t β t Nη+σ 1 t 1 σ C 1 σ t For monotonicity and concavity in aggregates (and p.c.), either 1.) B-B: 0 < 1 σ η < 1 (Complements) 2.) G&BC: η 1 σ < 0 (Substitutes) [HHDYN2] Jones and Schoonbroodt (2008) Complements vs. Substitutes 17

18 Dynasty constraints Qualitative results Basic Idea Household to Dynasty Comparative Statics Multiply HH constraint by N t C t + θ b,t N b,t w t N t N t+1 N s,t = π t N t + N s,t = π s,t N b,t Life Exp conditional Adulthood is 1 1 π We will assume that w t = γ t w We will assume that θ b,t = γ t θ b e.g., θ b,t = bw t (time cost) Eliminate N b,t, let θ s,t = θ b,t /π s,t. [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 18

19 Dynastic Planner s Problem Basic Idea Household to Dynasty Comparative Statics max {C t,n t+1 } t=0 t β t Nη+σ 1 t 1 σ C 1 σ t C t + θ s,t (N t+1 πn t ) w t N t with N s,t = N t+1 πn t N b,t = N s,t /π s,t θ s,t = θ b,t /π s,t [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 19

20 Fertility Measures Qualitative results Basic Idea Household to Dynasty Comparative Statics Population Growth Rate: γ N,t N t+1 N t Crude Birth Rate surviving children: CBR s,t N s,t N t +N s,t = N t+1 πn t N t +N t+1 πn t Crude Birth Rate all births: CBR b,t CBR s,t π s,t = γ N,t π 1+γ N,t π = 1 1 γ N,t π +1 [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 20

21 Basic Idea Household to Dynasty Comparative Statics Comparative Statics: Population Growth Rate First order condition for N becomes: 1 β γ N 1 η γ σ 1 + (η + σ 1) γ N = (1 σ) [ ] η wπs + π (1 σ) θ b LHS,D 0 LHS D LHS increase γ, σ<1 AI LHS increase γ, σ>1 AII D increase π s or π D 0 AII γ N AI Jones and Schoonbroodt (2008) Complements vs. Substitutes 21

22 Basic Idea Household to Dynasty Comparative Statics Sizes of effects on Population Growth LHS concave or convex: 1 β γ1 η N γσ 1 + (η + σ 1) γ N = (1 σ) [ ] η wπs + π (1 σ) θ b 5 4 LHS η 1 σ < 0 LHS η 1 σ > 0 RHS RHS change LHS,RHS γ N [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 22

23 Basic Idea Household to Dynasty Comparative Statics Summary: Population Growth Rate Proposition Comparative statics across BGP s for population growth, γ N : 1. If productivity growth, γ, increases, then B&B: if σ < 1, then γ N increases G&BC: if σ > 1, then γ N decreases 2. If child survival, π s, increases, then γ N increases; 3. If adult survival, π, increases then γ N increases; If γ N > 1, a change in π s or π has larger effects on the BGP level of γ N if σ < 1 than if σ > 1. [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 23

24 Basic Idea Household to Dynasty Comparative Statics Comparative Statics: Surviving Children Crude Birth Rate surviving children: CBR s,t N s,t N t +N s,t = N t+1 πn t N t +N t+1 πn t = γ N,t π 1+γ N,t π = 1 1 γ N,t π +1 Hence: all comparative statics of γ N carry over except wrt π WRT adult survival, π: If dγ N dπ > 1, CBR s increasing in π If dγ N dπ < 1, CBR s decreasing in π If η = 1 σ < 0, then CBR s is decreasing in π [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 24

25 Comparative Statics: All Births Basic Idea Household to Dynasty Comparative Statics Crude Birth Rate all births: CBR b,t CBR s,t π s,t Hence: all comparative statics of CBR s go through except π s WRT youth survival, π s : 1. If dcbrs dπ s 2. If dcbrs dπ s > CBRs π s, CBR increasing in π s < CBRs π s, CBR decreasing in π s The higher σ the smaller dcbrs dπ s [outline] (2.) more likely Jones and Schoonbroodt (2008) Complements vs. Substitutes 25

26 Summary of Qualitative Results Basic Idea Household to Dynasty Comparative Statics I. Productivity With SUBSTITUTES (low IES), an increase in productivity growth implies a decrease in Population Growth Rate a decrease in CBR II. Mortality With ANY IES, an increase in survival rates implies an increase in Population Growth Rate an uncertain sign for change in CBR But this effect is smaller when IES is LOW Jones and Schoonbroodt (2008) Complements vs. Substitutes 26

27 Preview of Quantitative Results Basic Idea Household to Dynasty Comparative Statics Overall with substitutability (IES 1/3) In response to changes in productivity growth and mortality as observed in the data, model accounts for about 1/2 of decrease in Population Growth Rate 2/3 of decrease in CBR Timing Increase in productivity growth most important before 1880 Decrease in mortality most important after 1880 Bust/Boom and productivity movements around trend [why] Jones and Schoonbroodt (2008) Complements vs. Substitutes 27

28 Calibration of Costs U.S. Experience since 1800 Calibration of costs of children Period length T = 20 Discount rate: β = 0.96 T Productivity growth rate: γ = T Life expectancy at age 20: π = 0.97 T Survival to adulthood: π s 1 (see next slide) Experiment with σ {0.5, 1, 3} Set η = 1 σ Given σ, calibrate θ b so that γ N = T Jones and Schoonbroodt (2008) Complements vs. Substitutes 28

29 Calibration of Costs U.S. Experience since 1800 Calibrated costs for various IES values in 1990 θ s w = 1 [ γ σ N,ann β annγ (1 σ) ann ] T π T ann θ s w σ T θs w Max CTFR [agemort2] Jones and Schoonbroodt (2008) Complements vs. Substitutes 29

30 U.S. Experience since 1800 Calibration of Costs U.S. Experience since 1800 [ ] γ N (γ,π s,π) = βγ 1 σ w θ s (π s ) + π Population Growth Rate: Model vs. Data Annual Population Growth Rate Pop Gro σ=0.5 Pop Gro σ=3.0 Pop Gro σ=1.0 Pop Gro Data [input] Time Jones and Schoonbroodt (2008) Complements vs. Substitutes 30

31 U.S. Experience since 1800 Calibration of Costs U.S. Experience since 1800 ( ) CBR b (γ,π s,π) = 1 1 π 1 s γ N (γ,π + 1 s,π) π 50 Crude Birth Rate: Model vs. Data CBR per 1000 population CBR Data CBRann σ=3.0 CBRs,ann σ=3.0 CBRann σ=1.0 CBRs,ann σ=1.0 CBRann σ= [input] Year Jones and Schoonbroodt (2008) Complements vs. Substitutes 31

32 Calibration of Costs U.S. Experience since 1800 U.S. Experience since 1800 (σ = 3) CBR: Decomposition [uk sim] CBR per 1000 population CBR (Youth Mort. only) CBR (TFP only) CBR (Life Exp only) CBR (All) Time Jones and Schoonbroodt (2008) Complements vs. Substitutes 32

33 Qualitative results Perfect foresight Child quality choice: Bequests/physical capital Education/human capital Did costs of children increase? Child-labor abolishment Compulsory education Cyclicality of fertility [conclusion] Jones and Schoonbroodt (2008) Complements vs. Substitutes 33

34 : Child quality choice, q V(q t ) = u(c t ) + βg(π t + n t )V(q t+1 ) a. n and q are complements (substitutes) if IES> 1(< 1) b. If IES low enough, productivity growth ր fertility ց in general equilibrium Complementarities in production function matter for threshold IES c. Lower IES smaller counterfactual effects of mortality [conclusion] Jones and Schoonbroodt (2008) Complements vs. Substitutes 34

35 Example: Physical capital/bequests Dynasty budget constraint becomes C t + θ s,t N s,t + K t+1 w t N t + (r t + 1 δ)k t N t+1 = π t N t + N s,t FOCs give: γ N = [β(r + 1 δ)] 1 1 η γ σ 1 η Special case: F(K t,γ t N t ) = AK α t (γ t N t ) 1 α Threshold IES 1 α < 1 [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 35

36 U.S. Experience since 1800 Crude Birth Rate: Model with K vs. Data CBR per 1000 population CBR Data CBRann =3.0 CBRs,ann =3.0 CBRann = Year Jones and Schoonbroodt (2008) Complements vs. Substitutes 36

37 : Cost of children Did costs of children increase? Child-labor abolishment Compulsory education A threefold increase in cost of children since 1800 captures the remaining 1/3 of decrease in CBR. [conclusion] Jones and Schoonbroodt (2008) Complements vs. Substitutes 37

38 Qualitative results Perfect foresight Child quality choice: Bequests/physical capital Education/human capital Did costs of children increase? Child-labor abolishment Compulsory education Cyclicality of fertility [conclusion] Jones and Schoonbroodt (2008) Complements vs. Substitutes 38

39 : Cyclical Movements in Fertility Add shocks to previous model Depressions productivity shocks Wars deaths and destruction Cyclical properties? Catching up? Easterlin s Hypothesis [details] [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 39

40 Fluctuations: Quantitative Results Productivity shocks only, IES = 1/3 0.4 Fertility Deviation Model Fertility, Time Cost Data, Detrended Model Fertility, Goods Cost Year [outline] Jones and Schoonbroodt (2008) Complements vs. Substitutes 40

41 Contribution Qualitative results Stick to Barro-Becker model and param Augment B-B technology/constraint set Threshold? Becker, Murphy, Tamura (1990), Doepke (2005), Bar and Leukhina (2006), Manuelli and Seshadri (2007),... Go to low IES similar to G & BC literature Why? Mateos-Planas (2002) Toss Barro-Becker Look at income levels with non-homothetic prefs Galor, Weil (2000), Greenwood, Seshadri (2002),... Look at mortality, soc.sec. with filial altruism Boldrin and Jones (2002),... Our alternative: Productivity GROWTH RATES and B-B w/low IES/SUBST Jones and Schoonbroodt (2008) Complements vs. Substitutes 41

42 Conclusion Qualitative results Trends and Fluctuations For low IES, an increase in productivity growth implies a substantial decrease in fertility since For low IES, counterfactual predictions in response to mortality are smaller. For low IES, fertility is procyclical bust and boom Structural Micro Direct evidence for substitutes/complements? Demography Income levels or growth rates? Jones and Schoonbroodt (2008) Complements vs. Substitutes 42