The Welfare Effects of Social Insurance Reform in the Presence of Intergenerational Transfers

A.1 Computation Algorithm for a Stationary Equilibrium

The computation of the steady-state equilibrium requires the following steps:

1. Guess respective initial values for r, τ _ss , and κ₁. Since the labor supply L is determined exogenously, making an initial guess for r is equivalent to guessing an initial value of K. Use the first-order conditions from the firm’s profit maximization problem to obtain the implied values for the relative factor prices w (where w t = ( 1 − α ) A t ( r + δ α ) α α − 1 ). Substitute r and w into the budget constraint of a household with age-j children:

   (16) m c j s + d c j + 7 p + a j + 1 + ζ h = e j + [ 1 + r ( 1 − τ k ) ] a j + κ ,

   where

   (17) e j = ( w ϵ j μ j z s m + w ϵ j + 7 z p ) ( 1 − τ l − τ s s )   if  j = 1 w ϵ j μ j z s ( m − ξ h ) ( 1 − τ l − τ s s ) + d S S , if  j = 2,3 , … , 7 ,

   (18) κ = κ 1 e j + κ 2

   (19) κ 2 = max { 0 , ( m + d ) c ̲ + ζ h − [ e j + a j [ 1 + r ( 1 − τ k ) ] + κ 1 e j } .

   Parameters: m, ζ, ξ, τ _k , τ _l , SS replacement rate, ϵ _j , z _p , z _s , μ _j , c ̲ are exogenously calibrated parameters as shown in Table 7. d = {0, 1}, h = {0, 1}, z _s , z _p = {H, L}, μ = {0.36, 1.0, 2.7} follows the transition matrix as described in the data calibration section.

2. In the inner loop, make an initial guess for β and solve the following finite horizon problem of a household:

   (20) V j ( x ) = max c s , c f , a ′ m U ( c s ) + d U ( c p ) + β E t V j + 1 ( x ′ ) ,

   subject to Equations (23)–(26), a′ ≥ 0, c _s ≥ 0, c _p ≥ 0, where

   (21) V j + 1 ( x ′ ) = V j + 1 d ′ , h ′ , a ′ , z p ′ , z s ′ , μ ′   if  j = 1,2 , … , 6 m V 1 1 , h ′ , a ′ m , z s ′ , z s s ′ , μ ′   if  j = 7 ,

   Make an initial guess for the value function V₀. Use the value function iteration method to solve the maximization problem of the household and obtain the policy function and value function.

3. Compute the stationary distribution of wealth and abilities across households. Guess the initial distribution for household X₀(x) (assume that all agents enter the economy with zero assets and use a uniform distribution). Iterate on Equations (22) and (23):

   (22) X j + 1 d ′ , h ′ , a ′ , z p , z s , μ ′ = 1 ( 1 + n ) j ∑ d , h , a , μ : a ′ Ω ( μ , μ ′ ) Γ ( h , h ′ ) Λ ( d , d ′ ) X j ( x ) , for  j = 1,2 , … , 6 ,

   (23) X 1 1,1 , a ′ , z s , z s ′ , μ ′ = m ∑ d , h , a , μ , z p , : a ′ Ω ̄ ( μ ) Π z s , z s ′ X 7 ( x ) , for  j = 7 ,

   until the absolute value between ‖X_j+1 − X _j ‖ < 0.0001. Obtain the steady-state invariant distribution X*.

4. Compute the new aggregate capital stock K₁ = ∑_j,xa _j (x)X _j (x) (where x = (d, h, a, z _p , z _s ,μ)), using X* and the policy and value function. Calculate the aggregate supply of labor L = ∑_t,x[ϵ _j μ _j z _s (m − ξh) + ϵ_j+7z _p (1 − h)]X _j (x).

   Update the value of κ₁ and τ _ss based on Equations (24) and (25):

   (24) ∑ j , x κ 1 e j X j ( x ) = τ l w L + τ k r K − G ,

   (25) τ s s = ∑ j = 2 7 ∑ x d ( S S j + κ 2 ) X j ( x ) ∑ x ( m w ϵ 1 μ 1 z s + w ϵ 8 z p ) X 1 ( x ) + ∑ j = 2 7 ∑ x [ ( w ϵ j μ j z s ) ( m − ξ h ) ] X j ( x ) .

5. Compute the value of the new interest rate (say r_new), using r = R − δ = αA(K/L)^α−1 − δ that corresponds to the new aggregate capital stock and aggregate supply of labor.

   Stop if ‖r − r_new‖ < 0.0001, otherwise, return to step 2 and update until convergence.

6. Manually calibrate β. Compute the aggregate household savings rate using the steady-state capital stock and output. If the savings rate is lower than 30.86%, increase β, and vice versa. Return to step 2 until the target savings rate is reached.

A.2 Computation of Welfare Gain

A household’s preferences over consumption over the household’s life cycle is modelled as the expected discounted lifetime utility:

where c s j and c p j are the consumption of the children and the parent in the baseline economy at age j (of the children), respectively, and X _j (x) is the age-dependent measure of the household.

The welfare gain from policy changes is measured by the percentage change in a household’s lifetime consumption λ (the Consumption Equivalent Variation), such that the household is indifferent between the baseline economy and the economy with policy changes:

where c s j ̃ and c p j ̃ denote the consumption of the children and the parent at age j (of the children), respectively, under the counterfactual policy changes. To isolate the effect of a change in the policy, the computation is done for the system in the steady state.

Let V represent the value function in the baseline economy in its steady-state equilibrium:

and let V ̃ denote the value function from the counterfactual experiments:

Recall that the utility function is assumed to take the form of:

The left-hand side of Equation (27) then becomes:

Rearranging Equation (31), we have:

Substitute Equations (29) and (32) into Equation (27), we obtain:

Thus, λ can be calculated as follows:

A.3 Computation of Inter-Vivos Transfers

In the model, inter-vivos transfers between the parent and the children are embedded implicitly in the budget constraint as they pool resources to solve the household’s common maximization problem. Alternatively, I can obtain the path of inter-vivos transfers using the optimal path of consumption and asset holdings by following certain assumptions outlined in Fuster, İmrohoroğlu, and İmrohoroğlu (2003).

I first assume that the initial assets of a household are owned by the parent and individuals do not own assets when they enter the model economy. Let a t p represent the assets of the parent, where t is the age of the household, and let a t s denote the assets of each child. The assets of the parent and the child are determined by the following budget constraints:

and

where c t p and c t s are the optimal consumption of the parent and the children, respectively, and nt _t is the net intervivos transfer from the parent to each of his m children. A positive nt _t means the parent gives more to the children while a negative nt _t means a parent receives more from the children. A government transfer of κ₂ to the household is separated between the parent and the children to mimic how much the government would pay to each member of the household if they lived separately. The separation is as follows: κ 2 p = max 0 , d c ̲ − e t p + a t p [ 1 + r ( 1 − τ k ) ] + κ 1 e t p , and κ 2 s = max 0 , c ̲ + ζ h m − e t s + a t s [ 1 + r ( 1 − τ k ) ] + κ 1 e t s . The sum of κ 2 p and m κ 2 s is the same as κ₂.

Net transfers are computed as follows. I assume that when an individual’s consumption is larger than the sum of his after-tax income and lump-sum transfers, he will receive a transfer from another member of the household to cover his consumption. As a consequence, his assets for the next period will be zero. For example, if at time t, the child faces:

then the parent will give a transfer to the child for him to finance his consumption. The amount of transfer is:

and the child’s asset for the next period will be zero:

All assets of the household in the next period will be held by the parent (i.e. a t p = a t ).

Similarly, I assume that the parent will receive transfers from the children when c t p is beyond what the parent can finance himself using his own resources, and nt _t will be negative in this case.

I set nt _t = 0 if neither the parent nor the child needs a transfer to finance their consumption, and the distribution of assets among the household members is determined based on the separate budget constraints of the parent and the children (Equations (35) and (36)).