ECONOMIC ENVIRONMENT

In this section, the baseline model that is calibrated to the U.S. economy in numerical examinations is presented. The underlying economic environment is a discrete-time overlapping generations world. In this economy, an individual's life is divided into three parts, childhood, adulthood before retirement, and adulthood after retirement.

During childhood an individual makes no economic decisions, hence age is counted only for adulthood in the model. It is assumed that 1 period corresponds to 5 years and an individual lives up to 12 periods. This period setting is made for easing computational burdens.

An adult before retirement, who is between age 0 (age 25–29 in real life) and age 6 (age 55–59 in real life), is called a young adult. A young adult receives labor and capital incomes, a lump-sum transfer from the government, and a bequest on his parent's death, if any, and makes decisions on consumption and assets holdings for the next period. He has a child at age 1 (age 30–34). His parent, who is 6 periods older than him, leaves him assets at death as a bequest, because of the joy of giving and uncertain lifetime. He supplies a fixed amount of time (normalized to 1) for work each period. Earnings depend on his ability settled before becoming an adult, his age (work experience), and a stochastic shock, which reflects all the stochastic changes in earnings during the working period. During this period, the agent faces two types of risks, labor earnings uncertainty and uncertainty about his parent's living status.

When turning age 7 (age 60–64 in real life), he retires and starts to face a positive probability of death for each year (including age 7) of his remaining life. An adult after retirement is called an old adult. He receives capital income and social security benefits, and makes decisions on consumption and asset holding for the next period. His social security receipt depends on the average earnings during his working period, which is a good approximation of the current U.S. social security system. After age 11 (age 80–84 in real life) he dies for certain. Figure 1 presents the generational structure of the model. The following subsections describe the model in detail.

Generational structure.

Consumer's Maximization Problem

An individual derives utility from the flow of his consumption when alive and from a bequest transferred to his child on his death. Note that he cares about the bequest left to his child but not about his child's consumption. That is, he has “impure altruism.” As mentioned in the introduction, if the individual is assumed to care about the consumption of his child, and the both agents maximize their utilities as different economic units, strategic interactions across generations would arise in the model. This would increase the complexity of the analysis significantly, hence this simpler assumption is adopted.⁷

One of the models considered in Hendricks (2002) assumes perfect altruism. However, he imposes unrealistic assumptions on informational structures in order to avoid computational difficulties associated with strategic interactions between a parent and his child. That is, a parent is assumed not to know anything about his child's states and the child at the beginning of life learns exactly how much he will inherit from the parent and can borrow the present value of the future inheritance. Hence, if the child receives a large amount of bequest in future, he faces a very weak borrowing constraint and consequently accumulates little wealth. Laitner (2001) also constructs a dynamic general equilibrium model with perfect altruism. In his model, an individual does not face temporary earnings shocks, has access to actuarially fair annuities and life insurance (hence only intended bequests exist), and makes intervivos transfers to his child until the child's borrowing constraint is lifted and saves the remaining transfers for his bequest. This setting allows the model to be solved numerically. Nishiyama (2002) also considers a model with perfect altruism. The setting of the model is more realistic than the above two papers and hence strategic interactions arise between a parent and his children. The computational difficulty is resolved by assuming that an individual lives for four periods at most and his earnings ability is not correlated with that of his parent.

Let c_j and a_j be his consumption and assets at age j. Denote his momentary utility function from consumption by U(·) and the one from bequest by F(·). Let the discount factor on future utilities be β. Then his expected lifetime utility is given by the following expression:

The expectation operator E₀ is attached to the above expression to indicate that as of period zero there exist risks other than the one associated with uncertain lifetime (mentioned below). An individual survives until age j−1 with probability

. With probability p_j−1, he survives the next period and obtains the utility from consumption c_j. With probability 1−p_j−1, he dies before becoming age j and obtains utility from leaving a bequest a_j. After age 11, he dies with certainty and obtains utility from bequest a₁₂.

Assume that functions U(·) and F(·) are strictly increasing and strictly concave. Further assume that lim_c→0U′(c)=∞ and lim_b→0F′(b)<κ<∞, where κ is a positive real number. The latter assumption is imposed in order to allow zero bequests for a portion of the population.

The individual chooses consumption and assets plans in order to maximize his expected utility subject to the following constraints:

where a_j+1 represents the asset holding at age j+1 and b_j is the bequest received from his parent at age j, if any. Furthermore, r, w, and l_j are the interest rate, the wage rate per efficiency unit of labor, and the effective labor supply at age j, respectively. Finally, τ_c, τ_l, and τ_k are tax rates on consumption, labor income, and capital income, respectively; tr is a lump-sum transfer received during his working periods; and soc_j denotes his social security receipt at age j. He faces a borrowing constraint each period so that he has to keep nonnegative assets each period, which also implies that he cannot leave a negative bequest to his child.⁸

There is a large literature on endogenous borrowing constraints, but due to computational complication, the simpler exogenous constraint is assumed.

Because an individual may receive a bequest on his parent's death, he needs to form an expectation on his future bequest receipt to solve the maximization problem. Assume that he has full access to the necessary information to predict his future bequest accurately, that is, the information needed to solve the parent's maximization problem after retirement (death can occur only after retirement). In order to simplify numerical computations, it is further assumed that the young individual solves his decision problem after observing his parent's current decisions.

Earnings Process

The agent's effective labor supply l_j depends on his inherited and acquired ability before becoming an adult (earnings ability), his age (work experience), and a stochastic shock (earnings shock). Assume the following functional form for the effective labor supply:

where θ is the earnings ability, ϕ(j) is an age-dependent deterministic component, and η_j is the earnings shock.

The earnings ability, θ, captures all abilities inherited or acquired before becoming an adult, which would depend on his innate ability, nurture, education, and family and environments he grew up in, although none of these dependencies are explicitly modeled. θ is time-invariant throughout an individual's life, but it is correlated with his parent's earnings ability and follows the following stochastic process:

where θ_p is the earnings ability of his parent and ε is a stochastic shock to the process, which follows an i.i.d. normal distribution.⁹

Because decisions on human capital investment are not explicitly modeled, this process captures all the sources of the intergenerational correlation of earnings including genetic transmission of ability and effects of market incompleteness on human capital investment.

The time-varying earnings shock, η_j, captures all the shocks to earnings after an individual starts working, which would include changes in his employment status, his job performance, health condition, the performance of the company he works for, and so on. The shock is assumed to follow the following AR(1) process in logs,

where η_j−1 is the earnings ability in the previous period (at age j−1) and υ_j is a stochastic shock to the process, which follows an i.i.d. normal distribution.

Recursive Formulation

The above maximization problem is reformulated in a recursive way so that decision problems and state variables at every stage of life are stated clearly. This recursive formulation is used for solving the model numerically.

Young adult's Problem I (while his parent is alive).

A young adult, who is between age 0 and age 5 and has a living parent, earns labor and capital incomes, and may receive a bequest next period if his parent dies. His dynamic programming problem at age j (0≤j≤5) reads as follows,

In equation (9),

is his expected discounted welfare at age j;

and

are the next period's welfare when his parent is alive and dies, respectively; β is the discount factor on future utilities; p_j+6 is the conditional probability that his age j + 6 parent survives the next period; and E is the expectation operator conditional on the information available in the current period. In the budget constraint (10), c is consumption, l is effective labor supply, a is assets (variables with superscript ‘′’ denote variables for the next period). The wage rate per effective labor supply is denoted by w and the interest rate is denoted by r. Tax rates on consumption, labor income, and capital income are expressed by τ_c, τ_l, and τ_k, respectively; and tr is a lump-sum transfer from the government. The effective labor supply l is dependent on the time-invariant earnings ability θ, the age-dependent deterministic component ϕ(j), and the stochastic component η, which follows an AR(1) process in logs [equations (11) and (13)]. The average effective labor supply up to this period,

, is a component of his state vector because the future social security benefit is dependent on it. A vector of state variables for his parent, s^p, is needed to predict his bequest receipt (the exact component of the vector is explained later) and the transition function for this vector is denoted ϒ(·) [equation (15)]. Vector S represents the aggregate state of the world and

characterizes the evolution of the aggregate state [equation (16)].

Young adult's Problem II (after his parent dies).

In the period his parent dies, the young adult at age j (1≤j≤5) receives a bequest and solves the following problem:

In this formulation,

is his welfare in the next period and b is the bequest received from his parent, which depends on his parent's state s^p. Because the young adult is assumed to have enough information to know his parent's states after retirement, b coincides with the parent's optimal asset holding at age j+6 (determined at age j+5) when the parental state is s^p.

From the next period on, the young adult solves the following problem:

Maximization problem at age 6.

In the next period (at age 7), he becomes a retiree, starts receiving a social security payment and facing a positive probability of death. By this age his parent is dead with certainty (see Figure 1), and now he has to care about the event of his child inheriting his assets.

If his parent is dead before the previous period, he solves the following problem:

subject to equations (10), (11), (14), and (16), where p₆ is the agent's conditional probability of surviving the next period,

is the welfare at age 7 if he is still alive, and F(a′) is the utility associated with leaving bequest a′ if he dies. Note that function J⁷ does not depend on θ and η anymore.

If his parent was alive in the previous period, he solves the problem below:

Old adult's problem.

After retirement, the individual receives social security benefits, and faces a positive probability of death. An old adult at age j (7≤ j≤11) solves the following problem:

In the above formulation,

is the welfare in the next period when he is still alive and

denotes his social security receipt, which depends on his average earnings during his working life,

. Note that the individual dies before becoming age 12 with probability 1, so that p₁₁=0.

From the formulation of the maximization problem, it is now clear that the vector of parental state variables needed for a child to predict the bequest from his parent is

where j^p,

, and a^p′ are the age, the lifetime average effective labor supply, and the next period's assets of his parent, respectively. The variable a^p′ is the current state variable because the child makes decisions after observing his parent's decisions by assumption.

Final Goods Production

The firm hires capital and labor and solves the profit maximization problem,

where O(·) denotes the CRS final good production function, k and l denote aggregate capital and efficiency labor, respectively, and δ is the depreciation rate of physical capital. From the first-order conditions the real interest rate and the wage rate are expressed as functions of the capital-efficiency labor ratio.

Governmental Policies

The government taxes labor income, capital income, and consumption to finance the lump-sum transfer, social security, and the nontransfer policy, which is the consumption of the final good by the government and is assumed not to affect individuals' utilities. The government's budget constraint reads

where tr is the total lump-sum transfer, soc is the total social security payment, and c_g is the consumption of the final good by the government.

RECURSIVE COMPETITIVE EQUILIBRIUM

The analyses in later sections focus on a stationary recursive competitive equilibrium in which decision problems are recursive, and the aggregate state of the world and governmental policies are time-invariant. The followings are definitions of a recursive competitive equilibrium and of a stationary recursive competitive equilibrium.

DEFINITION

A Recursive competitive equilibrium for the economy consists of

1. value functions
   , and
2. decision rules

1. for a young adult with a living parent,
2. for a young adult who has just lost his parent,
3. for a young adult whose parent died by the previous period,
4. for an old adult,

1. price functions w and r,
2. governmental policies τ_l, τ_k, τ_c, tr,
   , and c_g,
3. the law of motion for the parent's state of the world
   and
4. the law of motion for the aggregate state of the world
   , Ψ, where S_V is the joint distribution of
   and s^p for young adults with alive parents;
   is that of the same variables for young adults who have just lost their parents; S_W is the joint distribution of
   and a for young adults whose parents died by the previous period; and S_J is the joint distribution of
   and a for old adults, such that
5. An age j (0≤j≤5) young adult with a living parent solves problem (9), with the maximized value function given by V^j and the decision rules by
   and
   ;
6. An age j (1≤j≤6) young adult who have just lost his parent solves problem (17), with the maximized value function given by
   and the decision rules by
   and
   (when he is age 6, he solves problem 20);
7. An age j (2≤j≤6) young adult whose parent died by the previous period solves problem (22), with the maximized value function given by W^jand the decision rules by
   and
   (when he is age 6, he solves problem 20);
8. An age j (7≤j≤11) old adult solves problem (17), with the maximized value function given by J^j and the decision rules by
   and
   ;
9. The law of motion for the parent's state of the world
   , is composed of the following set of equations:
   
   A young adult's prediction of the future bequest in state s^p is also determined by (27);
10. The firm solves problem (24), with the first-order conditions given by r=O₁(k, l)−δ and w=O₂(k, l), where k and l are equal to
    
11. The government's budget constraint is satisfied, i.e.
    
12. The law of motion of the aggregate state of the world is consistent with the individual decisions, the firm's choices, and the governmental policies, and evolves according to
    

DEFINITION

A stationary recursive competitive equilibrium is a recursive competitive equilibrium in which

and the governmental policies are time-invariant.

COMPLETING MARKETS

In the baseline economy described earlier, an individual self-insures against earnings and lifetime uncertainties by accumulating wealth for a precautionary reason. When there are markets to insure against these risks, precautionary savings and accidental bequests disappear. In order to examine changes in the distribution of wealth and aggregate capital accumulation in the presence of such insurance markets, two hypothetical economies, the one with complete annuity markets, and the one without earnings uncertainty, are constructed in this section.

Economy with Complete Annuity Markets

Suppose that there are complete annuity markets so that an individual can insure against his uncertain time of death. Let

units of consumption of the next period be the gross return of one unit of the annuity security if the individual survives and nothing otherwise.¹⁰

In this way, the relative price of this security in terms of consumption is 1. Clearly, there are other equivalent ways to set up this market where the relative price is not one.

Let a_I denote the amount of assets that is annuitized. Then, at any point in time, if he were to die, the nonannuitized portion of his assets, a, will be bequeathed to his child, while the annuitized portion, a_I, will go to finance insurance claims of survivors.

Then the budget constraints of an individual over age 6 become

The return of the annuity is higher than normal assets due to p_j−1<1. Hence, in the absence of altruism, the individual will hold all of his assets in the annuitized form.

CALIBRATION

In this section, the model's functional forms are specified and parameter values are set for numerical simulations. The parameters are set based on existing empirical works, if available. Otherwise, they are set so that the simulated economy produces statistics that resemble those of the U.S. economy when policy parameters are set based on actual policies.

Final Good Production and Preferences

Final good production function: The function is assumed to be of the standard Cobb-Douglas type:

The parameter α is set to 0.36, following most works in quantitative macroeconomics.

Depreciation rate for physical capital δ: The annual rate of depreciation is usually set between 0.08 and 0.10 in the quantitative business cycle literature. Choosing the midpoint of these estimates and noting that each period in the model corresponds to 5 years in real life, the depreciation rate for numerical simulations is set to be 1−(0.91)⁵.

Utility functions: The utility function for final good consumption is assumed to be of the CRRA variety:

The bequest utility function is given by

The presence of 1 in this expression permits some individuals to leave no bequests.

Preference parameters σ, b₁ and b₂, and the discount factor on future utilities β are set so that the simulated model economy delivers a good overall match of the U.S. wealth distribution. The parameter values chosen for simulations are presented in the next section, when simulation results of the baseline model are discussed.

Earnings Process

Remember that the effective labor supply of an individual at age j is given by the following function,

where θ is the earnings ability (time-invariant), ϕ(j) is the deterministic component at age j, and η is the time-variant stochastic component (earnings shock). The earnings shock η follows

The earnings ability θ is correlated across generations in the following manner,

where θ_p is the earnings ability of his parent.

The age-dependent deterministic component ϕ(j) is specified based on the empirical estimates by French (2000). He uses the PSID and the PSID validation study for the years 1978–1987 to estimate the stochastic process of labor earnings. His estimate of the deterministic component is

The AR processes for the innate ability and for the earnings ability are discretized based on the Tauchen (1986)'s procedure. The parameters ρ,

, ζ and

are difficult to set based on the existing empirical work; hence, they are pinned down so that the simulated model produces the distribution of earnings and intergenerational correlations of earnings close enough to the corresponding statistics of the U.S. economy. The chosen parameter values are presented in the next section.

Governmental Policies

Recall that the government collects its revenue by levying taxes on labor income, capital income, and consumption. The tax revenues are then used to finance lump-sum transfer, the social security system, and the nontransfer policy, which is the governmental consumption of the final good and is assumed not to affect individuals' utilities. The government's budget constraint reads

where tr is the total lump-sum transfer, soc is the total social security payment, c_g is the spending on the nontransfer policy, and τ_c, τ_l, and τ_k are tax rates on consumption, labor income, and capital income.

Social Security System: The social security payment is based on average life-time labor earnings

. Therefore, agents with different earnings histories receive different amounts of social security benefits. Assume that the social security benefit of an old adult is determined by the following function:

The parameter ς is set to be 0.4. Then a retired worker gets 40% of the average of his lifetime labor earnings, which is close to the value (42%) people with average earnings would receive in the U.S. economy, according to the information of the Social Security Administration. However, the actual social security system is progressive, so that people with higher earnings receive less than proportionally. In order to take into account this feature of the actual system, the computed average lifetime effective labor supply

is adjusted accordingly.¹²

In particular,

is adjusted so that workers in particular positions of the earnings distribution receive the same proportions of their average lifetime earnings as they would when they retire before age 62 in the U.S. economy. The targeted workers are those who receive earnings at the national average level, 45% of the average earnings, and 160% of the average earnings steadily throughout their lives. These particular workers are chosen as their benefits examples are found in a document of the social security administration. The referred U.S. replacement ratios are the averages of the ratios for the years 1990–1999.

Tax rates: The tax rates on labor income τ_l, capital income τ_k, and consumption τ_c are set to be equal to 0.2887, 0.398, and 0.0523, respectively, following Mendoza, Razin, and Tesar (1994), who computed effective tax rates on factor incomes and consumption using national accounts and revenue statistics. The values are the averages of their estimates for the years 1980–1999.

Lump-sum transfer: The value of the lump-sum transfer to the working population tr is chosen so that the ratio of the transfer to GDP in the baseline economy becomes 0.031, which is the average of the ratios in the U.S. economy for the years 1980–1999. The U.S. transfers used to compute the ratios are sums of all the non-educational governmental transfers excluding social security, Medicare, and retirement benefits.

Comparisons of baseline economies under different parameterization.

The procedure of the recalibration is as follows. For parameter ζ, three different values 0.3, 0.4, and 0.5 are tried. Given ζ, the variance of the disturbance term to the earnings shock process,

, is adjusted so that the cross-sectional distribution of earnings remains unchanged. As for the coefficient of relative risk aversion σ, three values 3.0, 4.0, and 5.0 are tried. For a given value of σ, the parameters of the bequest function b₁ and b₂ are adjusted so that the baseline model under the new parameterization yields the same Gini coefficient for the wealth distribution of the whole population. Note that this does not mean that other dimensions of the wealth distribution remain unchanged with the new parameters. Table 14 shows the chosen values of the bequest parameters for 9 combinations of ζ and σ, including the original parameterization, ζ=0.4 and σ=4.0.

Table 15 presents selected statistics of the earnings distributions when the correlation coefficient of the earnings shock process, ζ, is set 0.3, 0.4, and 0.5. Because the unconditional distribution of the earnings shock is kept constant, the distributions of earnings for the whole population and for each skill group do not change. In contrast, the distribution of lifetime earnings becomes more unequal with larger ζ, since the earnings shock at age 0, which is drawn from the same unconditional distribution, becomes more crucial in determining the shocks at later ages. Obviously, the period-by-period correlation of log earnings rises and the intergenerational correlation of log-lifetime earnings falls with higher ζ.

Selected statistics of the distributions of wealth, bequests, and consumption for different values of ζ when σ=4.0 are summarized in Table 16. Similar qualitative results are found for σ=3.0 and 5.0 as well. Other things being equal, higher persistence of the earnings shocks (larger ζ) increases inequalities of lifetime earnings and, thus, life-cycle savings and decreases precautionary savings by the young, contributing to higher overall wealth inequality.³⁰

Table 17 shows selected statistics of the distributions of wealth, bequests, and consumption for different values of σ when ζ is set 0.4. Qualitative results remain unchanged when ζ=0.3 and 0.5. Other things being equal, the higher value for the coefficient of relative risk aversion raises precautionary savings and thus reduces wealth inequality among the working population. The higher σ also implies a lower rate of intertemporal substitution and hence a flatter age-consumption profile, increasing life-cycle savings for post-retirement periods and savings for bequests.³¹

Robustness of results on distributions.

Changes in the distributions of wealth, bequests, and consumption in the three hypothetical economies in comparison to the baseline economy are qualitatively the same in almost all the dimensions under all the parameterization. Quantitatively, as detailed here, differences in the rates of change are observed.

Economy with complete annuity markets: In all the nine cases, almost all the measures of inequalities worsen compared to the baseline economy. When the coefficient of relative risk aversion σ is higher, rates of increase in wealth and bequests inequalities become smaller. By contrast, raising the correlation coefficient of the earnings shock process ζ has mostly ambiguous effects on rates of change of the distributional measures.

Economy without earnings uncertainty: In the economy without earnings uncertainty, the proportion of households without wealth goes up except for the retired population, and all the other inequality measures fall for all the parameter combinations. When σ is higher, rates of decrease of the inequality measures become smaller except proportions of individuals without wealth. In contrast, increasing ζ raises rates of decrease of the inequality measures aside from proportions of people without wealth (except for the retired population), whose rates of increase fall.

Economy without altruism: In all the cases, wealth and bequests inequalities decline compared to the baseline economy. In contrast, the effect on consumption inequality is ambiguous, declining when σ=3.0 and 4.0 but rising when σ=5.0, and the rates of change are very small. Rates of decrease of wealth and bequests inequalities are larger when the coefficient of relative risk aversion σ is higher, except for proportions of households without wealth or bequests. The rates of decline are greater when the correlation coefficient of the earnings shock process ζ is smaller.

Robustness of results on aggregate capital and bequests accumulation.

Finally, robustness is checked with respect to rates of change in aggregate capital and bequests of the three hypothetical economies in comparison to the baseline economy. Table 18 presents aggregate capital and bequests in the economy with complete annuity markets in all the nine cases. As before, the values are normalized so that capital and bequests in the baseline economy are equal to 1 for each parameterization. In all the cases, both capital and bequests decrease with the latter decreasing to a greater extent. When the coefficient of relative risk aversion σ is higher, the decrease in capital is greater and the decrease in bequests is smaller, except for one case. The decreases are larger when the correlation coefficient of the earnings shock process ζ is higher. The rate of capital decline ranges from 1.0%, when σ=3.0 and ζ=0.3, to 12.2%, when σ=5.0 and ζ=0.5, while the rate of bequests decline ranges from 25.7%, when σ=5.0 and ζ=0.3, to 42.3%, when σ=3.0 and ζ=0.5.

Table 19 shows capital and bequests in the economy without earnings uncertainty. Aggregate capital declines in all the cases, whereas bequests decrease in six cases and increase in three cases. Capital decreases more and bequests are more likely to decline when σ is greater except when σ=4.0 and ζ=0.3. As for ζ, there are no such clear-cut tendency. The rate of capital decline ranges from 3.8%, when σ=3.0 and ζ=0.5, to 10.4%, when σ=5.0 and ζ=0.5, whereas the rate of bequests decline ranges from −1.7%, when σ=4.0 and ζ=0.3, to 2.5%, when σ=5.0 and ζ=0.5.

Finally, Table 20 presents capital and bequests in the economy without altruism. Under every parameterization, both capital and bequests decline greatly, and the declines are larger when σ is larger and ζ is smaller. The rate of capital decline ranges from 9.9%, when σ=3.0 and ζ=0.5, to 30.0%, when σ=5.0 and ζ=0.3, while the rate of bequests decline ranges from 18.5%, when σ=3.0 and ζ=0.5, to 45.1%, when σ=5.0 and ζ=0.3.

Comparisons of the three hypothetical economies show that capital decline is always largest in the economy without altruism and bequests decline is always smallest in the economy without earnings uncertainty. Other rankings depend on parameter values, but in six out of the nine cases, capital decline is larger in the economy with complete annuity markets than in the economy without earnings uncertainty, and again in six out of the nine cases, bequests decline is larger in the economy without altruism than in the economy with complete annuity markets. Differences in the rates of change of aggregate capital among the three economies are smallest when σ=3.0 and ζ=0.5 and largest when σ=5.0 and ζ=0.3, whereas differences in the rates of change of bequests are smallest when σ=3.0 and ζ=0.3 and largest when σ=5.0 and ζ=0.3 or when σ=3.0 and ζ=0.5.

ALGORITHM TO COMPUTE THE STEADY STATE DISTRIBUTION

Step 1: Enter the jth iteration with a guess for the real interest rate r^j, and the wage rate w^j. Also set the interval for assets to be [0, a_max].

Step 2 (Solving the Dynamic Programming Problem): Solve the problem backward starting from the problem at age 11.

2–1 (Old Adult's Problem): Given the prices, solve the old adult's maximization problem (17) at age 11 and obtain the value function

and the corresponding decision rules. Given the value function at age 11,

, solve the old adult's maximization problem (17) at age 10 and obtain the value function

and the corresponding decision rules. Continuing in this way, solve the maximization problems up to age 7. The solution of the problem at age 7 gives the value function

and the associated decision rules.

2–2 (Problem at Age 6):

(a) When the parent died before the previous period: Given the value function at age 7,

solve the maximization problem (20) and obtain the value function

and the associated decision rules.

(b) When the parent has just died: Given the value function, solve the maximization problem (17) and obtain the value function

and the decision rules.

2–3 (Young Adult's Problem):

(a) When the parent died before the previous period: Given

(1≤j≤5), solve the problem (22) and obtain

and the corresponding decision rules.

(b) When the parent has just died: Given

(0≤j≤5), solve the problem (17) and obtain

and the decision rules.

(c) When the parent is alive: Given

(0≤j≤5) and

(0≤j≤4), solve the problem (9) and obtain

and the decision rules.

Step 3 (Monte Carlo Simulation): Pick an initial child-parent pair, who are at age 0 and age 6, respectively. Starting from this pair, perform a Monte Carlo simulation for one lineage based on the decision rules computed above. Continue the simulation for large enough numbers of generations. Based on the simulation, obtain the distribution of assets and efficiency labor. Because the model has an ergodic property, the computed distribution remains the same if the simulation is performed for many different lineages. See the next subsection for detailed procedures.

Step 4 (Adjustment of the interval for assets): Check if the distribution of assets does not have a large mass at the maximum level of assets, a_max. If it has, increase the value of a_max and resolve the maximization problems and redo the Monte Carlo simulation (go back to Step 2).

Step 5 (Update of the prices and the convergence check): Based on the distributions of the state variables, compute aggregate capital k^j and efficiency labor l^j. Substituting these values into the firm's first-order conditions, obtain the implied real interest rate r^j and wage rate w^j. Stop if the price differences from the last iteration are small enough. If not, go back to Step 1 with new guesses for the prices. For the updated prices, the weighted averages of the currently used prices and the newly computed prices might be used.

MONTE CARLO SIMULATION

Step 1 (Initial states of the initial child-parent pair): Start the simulation with the initial child-parent pair, whose ages are 0 and 6, respectively. Their initial states must be set.

Assume that the parent of the age 6 adult's parent is dead before the previous period. Then the age 6 adult's state variables are the only information needed to solve his maximization problem (20). His initial assets level

and average labor productivity

are set arbitrary,³²

are drawn randomly from the underlying (unconditional) distributions. Based on the initial state variables, the decision rules for his current consumption and next period's assets holdings,

, and the transition rules for his state variables are determined.

The state variables of the young adult are the ones needed to solve the problem (9). His initial asset level

is set to be zero, the variables θ₀ and

are drawn randomly from the underlying (unconditional) distributions, and the average labor productivity \smash{

} is set to be

Because he is assumed to make decisions after observing his parent's decisions, his remaining state variables (the parental state) are the age of his parent,

and

. The parental state is needed to predict possible bequests receipt. Based on the initial state variables, the decision rules for his consumption and assets

, and the transition rules for the states are determined.

Step 2 (Determination of the initial old adult's living status): Because the old adult faces a positive death probability, his living status for the next period must be determined. His living status is set based on the survival probability. If he dies when turning age 7, the amount of bequest left to the young adult is equal to

.

Step 3 (Initial young adult before age 6 and his parent): One period has passed. If the old adult is still alive, he solves the same maximization problem as in the previous period. Given his current states

and

the decision rules and the transition rules for his states are set.

If the old parent is still alive, the young adult solves the same problem as in the previous period. If not, he receives bequest

and solves the problem (17). The value of

is drawn randomly from the underlying unconditional distribution, which together with the stochastic process (13) determines

. His new average labor productivity

is determined based on (12). The other state variables are set as in the previous period. Given his states, the decision rules and the transition rules for the states are determined. A similar process is continued until the young adult reaches age 6.

Step 4 (Initial young adult after age 6 and his child): Now the young adult is at age 6. At this age, he has a child (age 0) and his parent is already dead. He solves the problem (20) if his parent died before this period and solves the problem (17) if his parent has just died. His child's initial state variables must be set. The initial value for assets

is zero, the variable

is drawn randomly from the underlying unconditional distribution, and the initial ability

must be determined based on the parent's ability

and the stochastic process (\ref{Innate ability}). The initial average labor productivity is given by

The new age 0 individual solves the problem (9).

Step 5 (Remaining generations):Step 3 and Step 4 are repeated for a large enough number of generations, say 150,000 generations. The state variables of the first 15,000 generations are discarded in order to remove effects of initial conditions, and by using the variables for the remaining generations, the distributions of the states over the population are computed.