Bequest Session anchez-Romero 1 Miguel S´ 1 Wittgenstein Centre (IIASA, VID/¨ OAW, WU), Vienna Institute of Demography/Austrian Academy of Sciences 11th November 2014 (Tenth Meeting of Working Group on Macroeconomic Aspects of Intergenerational Transfer: International Symposium on Demographic Change and Policy Response)
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Goals 1 Previous model for estimating bequest • Drawback #1: The bequest model was deterministic • Drawback #2: Unrealistic profiles early in life 2 Proposing a new model for estimating bequest • The new model should be consistent with economic theory and rig o rous with the demographic setup - Dynamic General Stochastic Economic (DGSE) model populated by overlapping generations - Stochasticity comes from the risk of mortality rather than through productivity or income shocks 3 Pending research questions • Assessment of the role of bequests vs inter-vivos intergenerational transfers as sources of wealth • Relation to annuitization of wealth 2 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Model background • Time is discrete • Individuals are assumed to receive a stream of income over their lifecycle { y x } ω x =0 and to make decisions about consumption/saving • Individuals face mortality risk • Let π x be the conditional probability of surviving to age x and ℓ x = � x − 1 u =0 π u be the probability of surviving from birth to age x • Let θ x be a random variable that denotes whether the parent of an individual of age x is alive ( s ) or dead ( d ), i.e. θ x ∈ { s , d } • Let θ x = ( θ 0 , θ 1 , . . . , θ x ) represent the history of the variable θ up to age x 3 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Transition probabilities: P ( θ x +1 | θ x ) In a stable population the probability that the parent of an individual of age x dies is characterized by the following Markovian process: ℓ θ π θ ℓ θ � � � � � � 0 x +1 x x = · 1 − ℓ θ 1 − π θ 1 − ℓ θ 1 x +1 x x � ω − x u =0 e − nu f u ℓ u + x +1 with ℓ θ 0 = 1 and π θ x = , � ω − x u =0 e − nu f u ℓ u + x where θ ℓ is the survival probability of the parent of an individual at age a ℓ a is a the survival probability to age a ω is the maximum longevity n is the population growth rate f a is the fertility rate at age a 4 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Average bequest received Per capita bequest received b x = ℓ θ x (1 − π θ x ) E[ B x ] In a stable population the average bequest received at age u is given by ω − u � E [ B u ] = P( A u = x ) E ( B| A u = x ) x =0 with e − nx f x d x + u P( A u = x ) = , � Ω − u x =0 e − nx f x d u + x ∞ a ( θ x + u ) � � E � E ( B| A u = x ) = P ( H x + u = h ) , 1 + h h =0 where • A u is the random variable ‘Age of death of a parent of an individual of age u ’ • d x = ℓ x − ℓ x +1 is the fraction of deaths between ages x and x + 1 • H x is the number of additional heirs at age x • E [ a ( θ x )] = a ( θ x ) d P( θ x ) is the mean financial wealth at age x � 5 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Household problem A household head of age x > x 0 maximizes the following conditional expected utility with respect to consumption ( c ): � � θ x +1 �� V [ a ( θ x )] = U [ c ( θ x )] + βπ x � P ( θ x +1 | θ x ) , V a θ x +1 ∈{ s , d } s.t. the budget constraint � a ( θ x +1 ) = R [ a ( θ x ) + y x + τ x − c ( θ x )] + R E( B x ) If ( θ x +1 , θ x ) = ( d , s ) , a ( θ x +1 ) = R [ a ( θ x ) + y x + τ x − c ( θ x )] Otherwise . a ( · ) ≥ 0 is the financial wealth (borrowing constraint) R > 1 is the capitalized interest rate E[ B x ] is the average bequest received at age x y x is the endowment at age x τ x is the transfer at age x 6 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Optimal consumption/saving decision The optimal consumption path is characterized by the following Euler equation U c [ c ( θ x )] = R βπ x � π θ x U c [ c ( s , θ x )] + (1 − π θ x ) U c [ c ( d , θ x )] � If θ x = s , U c [ c ( θ x )] = R βπ x U c [ c ( d , θ x )] If θ x = d , Note that there exists saving for precautionary motive when θ x = s (Jensen’s inequality) 7 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Assumptions (target: maximum bequest-output ratios) 1 Stable populations that differ by their fertility and mortality schedules (data collected from UNPD, World Population Prospects: The 2012 Revision ) 2 The consumption of children is supported by parents 3 The labor income profile is that of a developed country ( to maximize savings for retirement motive ) 4 The financial wealth of an individual without offspring is taxed at 100% and distributed ( τ x ) according to the expected bequest profile 5 Two strong demographic assumptions: At the aggregate level the total number of offspring at age x of an individual at age u is assumed to be given by the sum of N u − x independent Bernoulli random variables, where N u − x is assumed to be distributed according to a Poisson of parameter f u − x 6 Fixed interest rate r = 5% and no productivity growth g = 0% 8 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Underlying demographic data 0,35 1,0 0,9 0,30 0,8 0,25 Survival probability 0,7 Fertility rate 0,6 0,20 0,5 0,15 0,4 TFR 7.49, e0 46.9 0,3 0,10 TFR 6.57, e0 47.5 TFR 5.57, e0 53.8 0,2 TFR 4.49, e0 62.3 TFR 3.47, e0 66.6 0,05 TFR 2.46, e0 70.4 0,1 TFR 1.63, e0 74.7 TFR 1.34, e0 83.7 0 0 0 10 20 30 40 50 60 0 20 40 60 80 100 Age Age Figure: Fertility ( f x ) and survival ( ℓ x ) profiles. Source: UNPD, World Population Prospects: The 2012 Revision . 9 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work High mortality prevents splitting the wealth among too many heirs 6 TFR=7.49, e0=46.9 TFR=6.57, e0=47.5 TFR=5.57, e0=53.8 TFR=4.49, e0=62.3 TFR=3.47, e0=66.6 (conditional on having at least one offspring) TFR=2.46, e0=70.4 TFR=1.63, e0=74.7 5 Japan 2012 Expected number of offspring 4 3 2 1 20 30 40 50 60 70 80 90 100 Age Figure: Expected number of offspring of a parent at age x (conditional on being one of the offspring) 10 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work The per capita bequest inflow shifts to the right the higher the proportion of bequest given to spouses 1,0 All to offspring (relative to avg. lab. inc. btw. 30−49) 2/3 to offspring, 1/3 to spouse All to the spouse 0,8 Per capita bequest inflow 0,6 0,4 0,2 0 0 10 20 30 40 50 60 70 80 90 100 110 Age Figure: Per capita bequest inflow over the lifecycle 11 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work Per capita bequest inflows and outflows could be as large as the labor income earned in advanced countries 1,4 1,4 TFR 6.57, e0 47.5 TFR 6.57, e0 47.5 (relative to the average labor income TFR 4.49, e0 62.3 (relative to the average labor income TFR 4.49, e0 62.3 1,2 TFR 2.46, e0 70.4 1,2 TFR 2.46, e0 70.4 Japan 2012 Japan 2012 Per capita bequest outflow Per capita bequest inflow between 30 and 49) between 30 and 49) 1,0 1,0 0,8 0,8 0,6 0,6 0,4 0,4 0,2 0,2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Figure: Per capita bequest inflows across the lifecycle by fraction of bequest given to offspring 12 / 17
Outline Household decision problem Solution of the model Assumptions Results Discussion and future work The bequest received increases more than proportionally with declines in TFR due to uncertainty TFR 6.57, e0 47.5 TFR 4.49, e0 62.3 bequest received relative to bequest received relative to 15 15 avg. lab. inc. btw 30−49 avg. lab. inc. btw 30−49 Married parent Widow/er parent 10 10 5 5 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Age Age TFR 2.46, e0 70.4 Japan 2012 bequest received relative to bequest received relative to 15 15 avg. lab. inc. btw 30−49 avg. lab. inc. btw 30−49 10 10 5 5 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Age Age Figure: Bequest received at death of the parent, by marital status, E[ B x ] 13 / 17
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