Christine Ho

# Abstract

The possibility of engaging in household child care may exacerbate the incentives of parents and grandparents to falsely claim disability benefits as households also get to save on formal child care costs. This paper considers a multi-generational family model with persistence in privately observed shocks and presents an efficient implementation case for subsidizing formal child care costs of the disabled. An implementation of the optimal scheme that consists of capped formal day care subsidies, non-linear income taxation and asset-testing is proposed. Simulations based on a parametrization that targets key features of the US labor and child care markets suggest that day care subsidies may lead to sizeable cost savings.

JEL Classification: H21; H24; H31; J14; J22

# Acknowledgements

I would like to thank James Banks, David Blau, Samuel Belinski, Richard Blundell, Tomoki Fujii, Nicolas Jacquet, Claus Kreiner, Guy Laroque, Kathleen McGarry, Costas Meghir, Nicola Pavoni, Ian Preston, Thomas Sargent, Conny Wunsch, seminar attendees at University College London, Sciences Po., Boston College, University of Essex, University of St. Gallen, Academia Sinica, Singapore Management University, University of Macau, IIPF, ESWC, PET, EEA-ESEM, and RES conferences, IAREP Workshop, and LSE Public Economics presentation for insightful comments and suggestions. All mistakes remain my own.

# Appendix

## A Theoretical Appendix

### A.1 Proof of Proposition 1

To keep the notation simple, I drop the constrained optimal subscript, , and loosely use the notation s˜>s to indicate that agents in state s˜ may mimic agents in state s. Let λtst1 denote the Lagrange multiplier associated with the promise keeping constraint (1) for agents who truthfully declared to be in state st1 in the previous period, λ˜tst1,s˜t1 denote the Lagrange multipliers associated with the threat keeping constraints (2) for agents who declared to be in state st1 in the previous period when they were actually in state s˜t1St1, and ηtst,s˜t denote the Lagrange multipliers associated with the incentive compatibility constraints (3) of agents who declare to be in state st when they are actually in state s˜tSt.

The first order conditions for each period 0<t<T from problem (4) are:

ctst:πtst,st1ζtπt,λt,λ˜t,ηt+s˜t>stζ˜tπt,λ˜t,ηtuctst=0,ltist:πtst,st1wtiζtπt,λt,λ˜t,ηt+s˜t>stζ˜tπt,λ˜t,ηtvihtist+ltist0,htist:πtst,st1ptζtπt,λt,λ˜t,ηt+s˜t>stζ˜tπt,λ˜t,ηtvihtist+ltist0,Vt+1st:πtst,st1GVt+1(st)Vt+1st,V˜t+1st,s˜t+ζtπt,λt,λ˜t,ηt=0,V˜t+1st,s˜t:πtst,st1GV˜t+1(st,s˜t)Vt+1st,V˜t+1st,s˜t+ζ˜tπt,λ˜t,ηt=0,

where

ζtπt,λt,λ˜t,ηt=λtst1πtst,st1+s˜t1>st1λ˜tst1,s˜t1πts˜t,s˜t1Is˜t=st+s˜t<stηts˜t,st,ζ˜tπt,λ˜t,ηt=s˜t1>st1λ˜tst1,s˜t1πts˜t,s˜t1Is˜t>stηtst,s˜t.

Is˜t>st is an indicator function taking a value of 1 if it is privately optimal for those in state s˜t>st to declare to be in state st, and a value of 0 otherwise. Is˜t=st is an indicator function taking a value of 1 if an agent who was previously untruthful happens to be in state st in the current period, and a value of 0 otherwise. By incentive compatibility, such agents will find it optimal to be truthful and declare state st. They therefore get continuation utility V˜t+1st,st=Vt+1st.

(i)Healthy members with wtipt. From examining the first order conditions with respect to ltist and htist, one can rule out cases with both ltist=0 and htist=0, when agents are healthy and πtst,st1>0, since vi0=0 and wtipt>0. I now show that ltist>0 and htist=0. Suppose to the contrary that ltist=0 and htist>0. Then, it would be possible to decrease htist by ϵ>0 and increase ltist by ϵ such that the total effort of member i is the same. The promise keeping, threat keeping and incentive compatibility constraints are still satisfied, while the government’s expected costs decrease by πtst,st1wtiptϵ0. Thus, ltist=0 and htist>0 cannot be optimal. The same argument applies for cases where ltist>0 and htist>0. It must therefore be that ltist>0 and htist=0 for healthy members with wtipt.

Healthy members with wti<pt. If htist is an interior solution, then the first order condition with respect to htist is satisfied with equality whereas the first order condition with respect to ltist is satisfied with strict inequality since wti<pt. Thus, htist>0 and ltist=0. Suppose to the contrary that htist=0 and ltist>0. Then, it would be possible to increase htist by ϵ>0 and decrease ltist by ϵ such that the total effort of member i is the same. The promise keeping, threat keeping and incentive compatibility constraints are still satisfied, while the government’s expected costs decrease by πtst,st1ptwtiϵ>0. Thus, htist=0 and ltist>0 cannot be optimal. It must therefore be that htist>0 and ltist=0 for healthy members with wti<pt, as long as all child care needs have not yet been met. Healthy household members with wti<pt may work on the labor market only when all child care needs have been met through household child care since there are no cost savings from reallocating effort from labor to child care.

(ii)Healthy members with wtipt. From (i), ltist>0 and htist=0. The first order condition with respect to ltist is satisfied with equality while the first order condition with respect to htist is satisfied with inequality (strict inequality when wti>pt). Using the first order conditions with respect to ctst, ltist, htist and rearranging, one gets:

uctstwti=vietistanductstptvietist.

Healthy members with wti<pt. From (i), the first order condition with respect to htist is satisfied with equality as long as all child care needs have not yet been met (i.e. interior solution), while the first order condition with respect to ltist is satisfied with strict inequality. Using the first order conditions with respect to ctst, ltist, htist, and rearranging, one gets:

uctstwti<vietistanductstpt=vietist.

(iii) Adding the first order conditions with respect to Vt+1st and Vt+1st,s˜t,s˜t>st, and taking into account the fact that Is˜t=st+Is˜t>st=Is˜tst, one gets:

πtst,st1GVt+1stVt+1st,V˜t+1st,s˜t+s˜t>stGV˜t+1st,s˜tVt+1st,V˜t+1st,s˜t=ψtπt,λt,λ˜t,ηt,

where ψtπt,λt,λ˜t,ηt=ζtπt,λt,λ˜t,ηt+s˜t>stζ˜tπt,λ˜t,ηt. Using the first order condition with respect to ctst, one gets:

(11)1uctst=GVt+1stVt+1st,V˜t+1st,s˜t+s˜t>stGV˜t+1st,s˜tVt+1st,V˜t+1st,s˜t.

Now, adding the first order conditions with respect to ctst for all stSt, and taking into account the fact that  stSts˜tstπts˜t,s˜t1Is˜tst=1, one gets:

stStπtst,st1uctst=λtst1+s˜t1>st1λ˜tst1,s˜t1.

One has an analogous expression for the following period:

st+1St+1πt+1st+1,stuct+1st+1=λt+1st+s˜t>stλ˜t+1st,s˜t.

From the interpretation of the Lagrange multipliers, one has:

(12)st+1St+1πt+1st+1,stuct+1st+1=GVt+1stVt+1st,V˜t+1st,s˜t+s˜t>stGV˜t+1st,s˜tVt+1st,V˜t+1st,s˜t.

The inverse Euler equation follows from equations (11) and (12):

1uctst=st+1St+1πt+1st+1,stuct+1st+1.

Applying Jensen’s inequality to the inverse Euler equation, one then gets the inter-temporal wedge between current and future marginal utilities of consumption:

uctst<st+1St+1πt+1st+1,stuct+1st+1.

### A.2 Proof of Proposition 2

Every period, households take the policy scheme as given and choose a state claim s˜tSt, consumption, household child care, and assets that maximize utility. From the decentralized household problem (6), the function It=ptihtis˜tih˜tis˜t,st when ihtis˜tih˜tis˜t,st>0, and It=0 otherwise. Under the suggested implementation in Proposition 2, the cost of formal child care faced by the household is positive only if total household child care is lower than the optimal level (i.e., the cost of formal child care is beyond the free day care cap). Income taxes Tts˜t=iwtiltis˜tcts˜t if  A˜ts˜t1,st10, and Tts˜t=iwtiltis˜tcts˜t+1+rA˜ts˜t1,st1 otherwise.

Claim 1. Households have no incentives to engage in total household child care that is higher than the optimal level. Suppose to the contrary that households choose ih˜tis˜t,st>ihtis˜t. Then, households do not get any formal child care cost savings (in fact, they give up part of the free day care for which they are eligible) but need to exert costly effort on household child care. By reducing the effort of any member by ϵ>0, the household can save on effort cost and benefit from free day care worth ptϵ. Thus, ih˜tis˜t,st>ihtis˜t cannot be optimal.

Claim 2. Households claiming a state in which optimal free day care is full time will engage in the optimal level of individual household child care. This applies to households consisting of a combination of healthy members with high earnings capacity wtipt and disabled members only. Such households benefit from full time free day care fts˜t=ptnt. Household members have no incentives to engage in higher than optimal household child care, h˜tis˜t,st>htis˜t=0, as effort is costly and there is no scope to save on formal child care costs. In addition, h˜tis˜t,st cannot be below htis˜t=0 for all members. Household members will therefore engage in the constrained optimal level of household child care: htis˜t=0 for all i.

I now get to the gist of the proof and show that households will accumulate zero assets and engage in the optimal level of household child care. The decentralized choices coincide with the constrained optimal allocations and are therefore incentive compatible.

Step 1. Households accumulate non-negative assets. The proof is done by contradiction using backward induction.

Last period (t=T).

Suppose that assets carried over to the last period are negative: A˜Ts˜T1,sT1<0. The last period’s budget constraint is given by:

c˜Ts˜T,sT+IT=cTs˜T+1+rA˜Ts˜T1,sT1.

It must therefore be that c˜Ts˜T,sT<cTs˜T since A˜Ts˜T1,sT1<0 and IT0.

For such household choice to be optimal, it must also be that the private Euler equation holds:

uc˜T1s˜T1,sT1=sTSTπTsT,sT1uc˜Ts˜T,sT.

Now, from Proposition 1(iii), there is an inter-temporal wedge at the optimal consumption levels:

ucT1s˜T1<sTSTπTsT,sT1ucTs˜T.

From the private Euler equation, the inter-temporal wedge, the fact that c˜Ts˜T,sT<cTs˜T and from concavity of u, it must therefore be that c˜T1s˜T1,sT1<cT1s˜T1.

Penultimate period (t=T1)

From Claim 1, households do not have an incentive to overprovide total household child care: ih˜T1is˜T1,sT1ihT1is˜T1. I now show that households do not have an incentive to underprovide total household child care: ih˜T1is˜T1,sT1ihT1is˜T1. Suppose to the contrary that ih˜T1is˜T1,sT1<ihT1is˜T1. From Proposition 1(ii), the consumption-child care margins at the optimal allocations are given by:

ucTs˜T1pT1vilTis˜T1+hT1is˜T10,

with equality when wT1i<pT1 and ihT1is˜T1<nT1. It follows that if c˜T1s˜T1,sT1<cT1s˜T1, then

uc˜T1s˜T1,sT1pT1vilT1is˜T1+hT1is˜T1
>ucT1s˜T1pT1vilT1is˜T1+hT1is˜T1.

Since the marginal gain from household child care is higher, household members have greater incentives to increase household child care beyond the optimal level hT1is˜T1. Thus, it cannot be that ih˜T1is˜T1,sT1<ihT1is˜T1. The household must therefore engage in the optimal level of total household child care ih˜T1is˜T1,sT1=ihT1is˜T1.

The penultimate period’s budget constraint is then given by:

c˜T1s˜T1,sT1+A˜Ts˜T1,sT1=cT1s˜T1+1+rA˜T1s˜T2,sT2.

Since c˜T1s˜T1,sT1<cT1s˜T1 and A˜Ts˜T1,sT1<0, it must be that A˜T1s˜T2,sT2<0.

For such household choice to be optimal, it must also be that the private Euler equation holds:

uc˜T2s˜T2,sT2=sT1ST1πT1sT1,sT2uc˜T1s˜T1,sT1.

Now, from Proposition 1(iii), there is an inter-temporal wedge at the optimal consumption levels:

ucT2s˜T2<sT1ST1πT1sT1,sT2ucT1s˜T1.

From the private Euler equation, the inter-temporal wedge, the fact that c˜T1s˜T1,sT1<cT1s˜T1 and from concavity of u, it must therefore be that c˜T2s˜T2,sT2<cT2s˜T2.

First period (t=0)

By following the same line of proof, it must be that A˜1s˜0,s0<0 from the second period’s budget constraint, and that c˜0s˜0,s0<c0s˜0 from the Euler equations between the first and second periods. In addition, ih˜0is˜0,s0=ih0is˜0 from the household’s consumption-child care margins in the first period. The first period’s household budget constraint is given by:

c˜0s˜0,s0+A˜1s˜0,s0=c0s˜0.

Since c˜0s˜0,s0<c0s˜0, it must be that A˜1s˜0,s0>0, which is a contradiction. Thus, it must be that households accumulate non-negative assets.

Step 2 Households accumulate zero assets. From Step 1, the household accumulates non-negative assets. The household budget constraint for all t<T may therefore be rewritten as:

c˜ts˜t,st+It+A˜t+1s˜t,st=cts˜t.

From Claim 1, households do not have an incentive to overprovide total household child care: ih˜tis˜t,stihtis˜t. I now show that households do not have an incentive to underprovide total household child care: ih˜tis˜t,stihtis˜t. Suppose to the contrary that ih˜tis˜t,st<ihtis˜t. From Proposition 1(ii), the consumption-child care margins at the optimal allocations are given by:

ucts˜tptviltis˜t+htis˜t0,

with equality when wti<pt and ihtis˜t<nt. It follows that if ih˜tis˜t,st<ihtis˜t, then c˜ts˜t,st<cts˜t from the budget constraint since A˜t+1s˜t,st0. We therefore have

uc˜ts˜t,stptviltis˜t+htis˜t>ucts˜tptviltis˜t+htis˜t.

Since the marginal gain from household child care is higher, household members have greater incentives to increase household child care beyond the optimal level htis˜t. Thus, it cannot be that ih˜tis˜t,st<ihtis˜t. The household must therefore engage in the optimal level of total household child care: ih˜tis˜t,st=ihtis˜t.

The household budget constraint then becomes:

c˜ts˜t,st+A˜t+1s˜t,st=cts˜t.

This implies that c˜ts˜t,st<cts˜t whenever A˜t+1s˜t,st>0 and c˜ts˜t,st=cts˜t whenever A˜t+1s˜t,st=0. Thus, A˜t+1s˜t,st>0 cannot be optimal as the household gets a lower stream of consumption compared to the case when A˜t+1s˜t,st=0. The household will therefore choose to accumulate zero assets.

Step 3. Households consume the optimal level of consumption. From Steps 1 and 2, the household accummulates zero assets: A˜ts˜t1,st1=0. The household budget constraint may therefore be rewritten as:

c˜ts˜t,st=cts˜t.

Households thus consume the optimal level of consumption.

Step 4. Individual household members engage in the optimal level of household child care.

From Step 2, households engage in the optimal level of total household child care: ih˜tis˜t,st=ihtis˜t. In addition, from Claim 2, individual members in households claiming a state in which optimal free day care is full time will engage in the optimal level of household child care: h˜tis˜t,st=htis˜t=0,i. Now consider a household that claims a state in which optimal free day care is less than full time: ftst˜=ptntihtis˜t. This applies to healthy single parent or single grandparent households with low earnings capacity wti<pt. From Proposition 1(i), it is optimal for such members to engage in strictly positive household child care htis˜t>0. As there is only one adult member, individual household child care is equal to total household child care, which is optimally implemented through the optimal free day care.

Step 5. The decentralized allocations are incentive compatible.

Steps 1 to 4 show that for any state claim, household choices coincide with the optimal allocations. It follows that the promise keeping (1), threat-keeping (2) and incentive compatibility (3) constraints of the government problem (4) hold. In other words, the household will choose to claim its true state. Thus, a scheme with subsidized day care, non-linear income taxation and asset-testing implements the constrained optimal allocations ctst,ltst,htst,t,st.

### A.3 Government Problem with Hidden Household Child Care

This section solves the government problem (7) and show that the results from Proposition 1 still hold in this context. Let λtst1 denote the Lagrange multiplier associated with the promise keeping constraint for agents who truthfully declared to be in state st1 in the previous period, λ˜tst1,s˜t1 denote the Lagrange mutipliers associated with the threat keeping constraints for agents who declared to be in state st1 in the previous period when they were actually in state s˜t1St1, ηtst,s˜t denote the Lagrange multipliers associated with the incentive compatibility constraints of agents who declare to be in state st when they are actually in state s˜tSt, ϕtist denote the Lagrange multipliers associated with the child care constraints of agents who are honestly in state st, and ϕ˜tist,s˜t denote the Lagrange multipliers associated with the child care constraints of agents who declare to be in state st when they are actually in state s˜tSt.

Let Is˜t>st is an indicator function taking a value of 1 if it is privately optimal for those in state s˜t>st to declare to be in state st, and a value of 0 otherwise. Is˜t=st is an indicator function taking a value of 1 if an agent who was previously untruthful happens to be in state st in the current period, and a value of 0 otherwise. By incentive compatibility, such agents will find it optimal to be truthful and declare state st. They therefore get household child care h˜tist,st=htist and continuation utility V˜t+1st,st=Vt+1st.

Claim 3. The Lagrange multipliers associated with the child care constraints are zero. The first order condition with respect to h˜tist,s˜t is given by:

s˜t1>st1λ˜tst1,s˜t1s˜t>stπts˜t,s˜t1Is˜t>sts˜t>stηtst,s˜tf.o.c.˜st,s˜tjis˜t>stϕ˜tjst,s˜tu′′btstptntih˜tist,s˜tpt2s˜t>stϕ˜tist,s˜ts.o.c.˜st,s˜t0,

where f.o.c.˜st,s˜t and s.o.c.˜st,s˜t are, respectively, the private first and second order conditions with respect to child care: f.o.c.˜st,s˜t=ubtstptntih˜tist,s˜tptviltist+h˜tist,s˜t and s.o.c.˜st,s˜t=u′′btstptntih˜tist,s˜tpt2vi′′ltist+h˜tist,s˜t. From Khun-Tucker conditions, ϕ˜tist,s˜t0. In particular, if h˜tist,s˜t=0, then the child care constraint is non-binding and ϕ˜tist,s˜t=0. If h˜tist,s˜t>0, then the child care constraint is binding and ϕ˜tist,s˜t0. Private optimality then implies that f.o.c.˜st,s˜t=0 and s.o.c.˜st,s˜t<0. Since utility is concave, it must therefore be that ϕ˜tist,s˜t=0i,st,s˜tSt. A similar line of proof shows that ϕtist=0i,stSt.

The government’s first order conditions are then given by:

btst:πtst,st1ζtπt,λt,λ˜t,ηt+s˜t>stζ˜tπt,λt,λ˜t,ηtuctst=0,ltist:πtst,st1wtiζtπt,λt,λ˜t,ηtvihtist+ltist+s˜t>stζ˜tπt,λ˜t,ηtvih˜tist,s˜t+ltist0,htist:ζtπt,λt,λ˜t,ηtuctstptvihtist+ltist0,h˜tist,s˜t:ζ˜tπt,λ˜t,ηtuc˜tst,s˜tptvih˜tist,s˜t+ltist0,Vt+1st:πtst,st1GVt+1(st)Vt+1st,V˜t+1st,s˜t+ζtπt,λt,λ˜t,ηt=0,V˜t+1st,s˜t:πtst,st1GV˜t+1(st,s˜t)Vt+1st,V˜t+1st,s˜t+ζ˜tπt,λ˜t,ηt=0,

where

ζtπt,λt,λ˜t,ηt=λtst1πtst,st1+s˜t1>st1λ˜tst1,s˜t1πts˜t,s˜t1Is˜t=st+s˜t<stηts˜t,st,ζ˜tπt,λ˜t,ηt=s˜t1>st1λ˜tst1,s˜t1πts˜t,s˜t1Is˜t>stηtst,s˜t.

Claim 4. The qualitative features of Proposition 1 hold. (i) Using the first order conditions with respect to btst and ltist, and summing across the first order conditions with respect to htist and h˜tist,s˜ts˜t>st, we have, respectively:

(13)ζtπt,λt,λ˜t,ηtuctstwtvihtist+ltist+s˜t>stζ˜tπt,λt,λ˜t,ηtuc˜tst,s˜twtviltist+h˜tist,s˜t0,
(14)ζtπt,λt,λ˜t,ηtuctstptvihtist+ltist+s˜t>stζ˜tπt,λt,λ˜t,ηtuc˜tst,s˜tptviltist+h˜tist,s˜t0.

When wtipt and (13) is satisfied with equality, (14) will be satisfied with strict inequality. It must therefore be that ltist>0 and htist=0 for healthy members with wtipt. By the same line of thought, htist>0 and ltist=0 for healthy members with wti<pt, as long as all child care needs have not yet been met. Similar lines of proof as in Appendix A.1 may then be used to show that (ii) and (iii) also hold.

### A.4 Multi-Member Households with Low Earnings Capacity

Consider an illustration with T=1 and I=2. Suppose that a household consisting of two healthy members (state s) claims that member i=1 is disabled. Suppose that w2<p. I now show that the exacerbation of incentives to mimic the disabled still exists despite the household benefiting from free day care of fs=pnh2s. This is because the falsely disabled member i=1 has incentives to engage in household child care activities, which would enable the healthy member i=2 to spend less effort on household child care. Let ϵ0 be the increase (decrease) in household child care of member i=1 (i=2). The mimicker household solves:

Maxϵucs˜,sv1ϵv2l2s˜+h2s˜ϵ,

subject to the budget constraint: c˜s˜,s=w2l2s˜Ts˜, where income tax Ts˜=w2l2s˜cs˜. Thus, c˜s˜,s=cs˜. The private first order condition of the household is given by:

v1ϵ+v2l2s˜+h2s˜ϵ0,

with strict inequality when ϵ=0. But then, since v10=0 and l2s˜+h2s˜>0 from Proposition 1(i), we cannot have strict inequality. It must therefore be that ϵ>0.[23] It follows that the utility that the mimicker household gets in deviation is higher than the constrained optimal one, which still exacerbates the incentive constraint.

### A.5 Child Care Production

This section presents Proposition 3 and Proposition 4, which are analogous to Proposition 1 and Proposition 2. The same notation as in Appendix A.1 and A.2 is used for the proofs.

### Proposition 3.

Let ctst,ltst,htst solve the government problem (10).

(i) Healthy members have ltist0andhtist0.

(ii) The consumption-labor and consumption-child care margins of healthy members are, respectively, given by:

uctstwtivietistanductstptFtKtivietist,

with strict inequality when ltist=0 and htist=0, respectively.

(iii) The inter-temporal savings wedge is as in the main model.

### Proof.

The first order conditions for each period  0<t<T from problem (10) are as in Appendix A.1 with the first order condition with respect to htist modified as follows:

πtst,st1ptFtKtiζtπt,λt,λ˜t,ηt+s˜t>stζ˜tπt,λ˜t,ηtvihtist+ltist0.

(i) and(ii) can be seen from examining the first order conditions with respect to c, l, and h. The proof of (iii) is as in the main model.

### Proposition 4.

The following scheme implements the constrained optimum from the government problem (10) for single adult households and for multi-member households with employed and unemployed members only or disabled members only.

(i) Subsidized formal day care. Households benefit from free day care capped at the optimal level of formal child care, t,st:

ftst=ptFtst.

(ii) Non-linear income taxes and asset-testing. The government imposes net taxes and asset-testing, t,st:

Ttst=iwtiltistctstifAtst10iwtiltistctst+1+rAtst1ifAtst1>0.

Proof.

Every period, households take the policy scheme as given and choose a state claim s˜tSt, consumption, household child care, and assets that maximize utility. From the decentralized household problem analogous to (6), the function It>0 when Kts˜tK˜ts˜t,st>0, and It=0 otherwise. Kts˜t denote the optimal level of household child care associated with state s˜t, and K˜ts˜t,st denote household child care provided by a household in state st that declares to be in state s˜t. Under the suggested implementation in Proposition 4, the cost of formal child care faced by the household is positive only if total household child care is lower than the optimal level (i.e., the cost of formal child care is beyond the free day care cap). Income taxes Tts˜t=iwtiltis˜tcts˜t if A˜ts˜t1,st10, and Tts˜t=iwtiltis˜tcts˜t+1+rA˜ts˜t1,st1 otherwise.

Claim 5.Households have no incentives to engage in total household child care that is higher than the optimal level. Suppose to the contrary that households choose K˜ts˜t,st>Kts˜t. Then, households do not get any formal child care cost savings (in fact, they give up part of the free day care for which they are eligible) but need to exert costly effort on household child care. By reducing the effort of any member by ϵ>0, the household can save on effort cost and benefit from free day care worth ptFtKtiϵ. Thus, K˜ts˜t,st>Kts˜t cannot be optimal.

Claim 6.Households claiming a state in which optimal free day care is full time will engage in the optimal level of individual household child care. Since such households benefit from full time free day care, household members have no incentives to engage in higher than optimal household child care, h˜tis˜t,st>htis˜t=0, as effort is costly and there is no scope to save on formal child care costs. In addition, h˜tis˜t,st cannot be below htis˜t=0 for all members. Household members will therefore engage in the constrained optimal level of household child care: htis˜t=0 for all i.

I now get to the gist of the proof and show that households will accumulate zero assets and engage in the optimal level of household child care. The decentralized choices coincide with the constrained optimal allocations and are therefore incentive compatible.

Step 1. Households accumulate non-negative assets. The proof is done by contradiction using backward induction.

Last period (t=T)

Suppose that assets carried over to the last period are negative: A˜Ts˜T1,sT1<0. The last period’s budget constraint is given by:

c˜Ts˜T,sT+IT=cTs˜T+1+rA˜Ts˜T1,sT1.

It must therefore be that c˜Ts˜T,sT<cTs˜T since A˜Ts˜T1,sT1<0 and IT0.

For such household choice to be optimal, it must also be that the private Euler equation holds:

uc˜T1s˜T1,sT1=sTSTπTsT,sT1uc˜Ts˜T,sT.

Now, from Proposition 3(iii), there is an inter-temporal wedge at the optimal consumption levels:

ucT1s˜T1<sTSTπTsT,sT1ucTs˜T.

From the private Euler equation, the inter-temporal wedge, the fact that c˜T(s˜T,sT)<cT(s˜T) and from concavity of u, it mus therefore be that c˜T1(s˜T1,sT1)<cT1(s˜T1).

Penultimate period (t=T1)

From Claim 5, households do not have an incentive to overprovide total household child care: K˜T1s˜T1,sT1KT1s˜T1. I now show that households do not have an incentive to underprovide total household child care: K˜T1s˜T1,sT1KT1s˜T1. Suppose to the contrary that K˜T1s˜T1,sT1<KT1s˜T1 . From Proposition 3(ii), the consumption-child care margins at the optimal allocations are given by:

ucT1s˜T1pTFT1KT1ivilT1is˜T1+hT1is˜T10,

with equality when hT1is˜T1>0 and KT1s˜T1<nT1. It follows that if c˜T1s˜T1,sT1<cT1s˜T1, then

uc˜T1s˜T1,sT1pT1FT1KT1ivilT1is˜T1+hT1is˜T1
>ucT1s˜T1pT1FT1KT1ivilT1is˜T1+hT1is˜T1.

Since the marginal gain from household child care is higher, household members have greater incentives to increase household child care beyond the optimal level hT1is˜T1. Thus, it cannot be that K˜T1s˜T1,sT1<KT1s˜T1. The household must therefore engage in the optimal level of total household child care K˜T1s˜T1,sT1=KT1s˜T1.

The penultimate period’s budget constraint is then given by:

c˜T1s˜T1,sT1+A˜Ts˜T1,sT1=cT1s˜T1+1+rA˜T1s˜T2,sT2.

Since c˜T1s˜T1,sT1<cT1s˜T1 and A˜Ts˜T1,sT1<0, it must be that A˜T1s˜T2,sT2<0.

For such household choice to be optimal, it must also be that the private Euler equation holds:

uc˜T2s˜T2,sT2=sT1ST1πT1sT1,sT2uc˜T1s˜T1,sT1.

Now, from Proposition 3(iii), there is an inter-temporal wedge at the optimal consumption levels:

ucT2s˜T2<sT1ST1πT1sT1,sT2ucT1s˜T1.

From the private Euler equation, the inter-temporal wedge, the fact that c˜T1s˜T1,sT1<cT1s˜T1 and from concavity of u, it must therefore be that c˜T2s˜T2,sT2<cT2s˜T2.

First period (t=0)

By following the same line of proof, it must be that A˜1s˜0,s0<0 from the second period’s budget constraint, and that c˜0s˜0,s0<c0s˜0 from the Euler equations between the first and second periods. In addition, K˜0s˜0,s0=K0s˜0 from the household’s consumption-child care margins in the first period. The first period’s household budget constraint is given by:

c˜0s˜0,s0+A˜1s˜0,s0=c0s˜0.

Since c˜0s˜0,s0<c0s˜0, it must be that A˜1s˜0,s0>0, which is a contradiction. Thus, it must be that households accumulate non-negative assets.

Step 2. Households accumulate zero assets.

From Step 1, the household accummulates non-negative assets. The household budget constraint for all t<T may therefore be rewritten as:

c˜ts˜t,st+It+A˜t+1s˜t,st=cts˜t.

From Claim 5, households do not have an incentive to overprovide total household child care: K˜ts˜t,stKts˜t. I now show that households do not have an incentive to underprovide total household child care: K˜ts˜t,stKts˜t. Suppose to the contrary that K˜ts˜t,st<Kts˜t. From Proposition 3(ii), the consumption-child care margins at the optimal allocations are given by:

ucts˜tptFtKtiviltis˜t+htis˜t0,

with equality when htis˜t>0 and Kts˜t<nt. It follows that if K˜ts˜t,st<Kts˜t, then c˜ts˜t,st<cts˜t from the budget constraint since A˜t+1s˜t,st0. We therefore have

uc˜ts˜t,stptFtKtiviltis˜t+htis˜t>ucts˜tptFtKtiviltis˜t+htis˜t.

Since the marginal gain from household child care is higher, household members have greater incentives to increase household child care beyond the optimal level htis˜t. Thus, it cannot be that K˜ts˜t,st<Kts˜t. The household must therefore engage in the optimal level of total household child care: K˜ts˜t,st=Kts˜t.

The household budget constraint then becomes:

c˜ts˜t,st+A˜t+1s˜t,st=cts˜t.

This implies that c˜ts˜t,st<cts˜t whenever A˜t+1s˜t,st>0 and c˜ts˜t,st=cts˜t whenever A˜t+1s˜t,st=0. Thus, A˜t+1s˜t,st>0 cannot be optimal as the household gets a lower stream of consumption compared to the case when A˜t+1s˜t,st=0. The household will therefore choose to accumulate zero assets.

Step 3. Households consume the optimal level of consumption.

From Steps 1 and 2, the household accummulates zero assets: A˜ts˜t1,st1=0. The household budget constraint may therefore be rewritten as:

c˜ts˜t,st=cts˜t.

Households thus consume the optimal level of consumption.

Step 4. Individual household members engage in the optimal level of household child care.

From Step 2, households engage in the optimal level of total household child care: K˜ts˜t,st=Kts˜t. In addition, from Claim 6, individual members in households claiming a state in which optimal free day care is full time will engage in the optimal level of household child care: h˜tis˜t,st=htis˜t=0,i. This applies to households in which all members are disabled. Now, consider a household with healthy employed and unemployed members only, that claims a state in which optimal free day care is less than full time. From Proposition 3(ii), the consumption-child care margins are not distorted such that individual household child care is equal to total household child care, which is optimally implemented through the optimal free day care.

Step 5. The decentralized allocations are incentive compatible.

Steps 1–4 show that for any state claim, household choices coincide with the optimal allocations. It follows that the promise keeping (1), threat-keeping (2) and incentive compatibility (3) constraints of the government problem (10) hold. In other words, the household will choose to claim its true state. Thus, a scheme with subsidized day care, non-linear income taxation and asset-testing implements the constrained optimal allocations ctst,ltst,htst,t,st.

## B Quantitative Appendix

### B.1 Effort Cost

The calibration of αki is done as follows. First, define a grid of possible values over α. Then, for each household structure, find the labor supply predicted by the the household’s maximization problem under the US welfare system for healthy workers, lkiα, where α is a vector of grid points associated with the effort cost parameters of all household members. I solve each household’s utility maximization problem by taking into account the US Social Security and Federal Taxes and EITC. I then minimize the sum of squares of the distance between labor supply predicted by the household’s maximization problem and average weekly labor hours from the CPS, lkiˆ:

αk=argminiElkiαlkiˆ2.

As an external validity check, I report the profiles of labor supply for healthy workers under the US tax and benefit system, and averaged over all child compositions in Figure 5Figure 7. As can be seen from Figure 5, the simulated hours replicate the life cycle profiles of labor supply for working parents very closely. The life cycle profiles of adult members in households with a grandparent are also closely replicated although slightly overestimated, especially for grandfathers in Figure 6 and Figure 7.

### Figure 5:

Parent households labor supply under the US tax and benefit system.

Note:  Solid lines represent average hours from CPS and dashed lines represent average hours from the simulated household problem.

### Figure 6:

Grandparent households labor supply under the US tax and benefit system.

Note: Solid lines represent average hours from CPS and dashed lines represent average hours from the simulated household problem. Grandparents are retired in periods 3 and 4.

### Figure 7:

Intergenerational households labor supply under the US tax and benefit system.

Note:  Solid lines represent average hours from CPS and dashed lines represent average hours from the simulated household problem. Grandparents are retired in periods 3 and 4.

Taxes and Tax Credits. Social Security taxes are calculated as 6.2% of the first $106,800 earnings (SSA 2010). Taxable income is computed as gross earnings minus exemptions and deductions. Deductions are$5,700 for singles, $8,400 for household heads, and$11,400 for married couples. Each individual and dependent also gets personal exemptions of $3,650. Federal income tax brackets that are based on taxable income are reported in Table 7. ### Table 7: 2010 US tax and benefit system. Federal income tax rates on taxable incomea Tax rateSingleHeadMarried 10%<$8,375<$11,950<$16,750
15%$8,375–$34,000$11,950–$45,500$16,750–$68,000
25%$34,000–$82,400$45,500–$117,650$68,000–$137,300
28%$82,400–$171,850$117,650–$190,550$137,300–$209,250
33%$171,850–$373,650$190,550–$373,650$209,250–$373,650
35%$373,650 and above$373,650 and above$373,650 and above Earned income tax credit (EITC)b All filing statusesSingle and HeadMarried # KidsPhase-inMaximumPhase-outPhase-outIncomePhase-outIncome below 18ratecreditrateincomelimitincomelimit 07.65%$4577.65%$7,480$13,460$12,480$18,470
134%$3,05015.98%$16,450$35,535$21,460$40,545 240%$5,03621.6%$16,450$40,363$21,460$45,373
345%$5,66621.6%$16,450$43,352$21,460$48,360 Poverty thresholdsc # Kids below 18 # Persons123 2$15,030
3$17,552$17,568
4$27,518$22,113$22,190 5$27,518$26,675$26,023
1.  Sources:  a. http://www.moneychimp.com. b. Historical Earned Income Tax Credit Parameters, Tax Policy Center. Phase-out income for married filing jointly status computed by author based on phase-out rate and income limit. c. U.S. Census Bureau.

Federal income tax brackets depend on a tax payers filing status. I assume that households with a single parent or grandparent file taxes under the single status when there are no children present in the household and file under the head of household status when there are children below 18 present. Married households, on the other hand, file jointly for taxes irrespective of presence of children. For intergenerational households with children aged below 18, I assume that the grandparent files as head of household while the parent files under the single status. If the grandparent is disabled or retired, then the parent files as the head of household. To qualify as head of household, one must be unmarried, provide for more than half of housing expenses, and have a qualifying dependent who may be a descendant aged below 18 or a disabled relative of any age (Inland Revenue Service).

The EITC is a refundable tax credit designed for lower income working families. The phase-in rate, maximum credit, phase-out rate and income limits depend on the number of children aged below 18 in the household. The income limits also depend on a tax payers filing status. The EITC schedule is given Table 7. Furthermore, working families may also benefit from a non-refundable CTC of $1,000 per child, which is phased-out at the rate of$50 for each additional $1,000 earned above$110,000 [$75,000] for married couples [others]. Child Care Subsidies. The CDCTC is a non-refundable tax credit program available to working families with children under 13. The CDCTC has a tax credit rate of 20% to 35% of child care expenses up to a cap of$3k for families with one child and $6k for families with two or more children (Tax Policy Center, 2010). The 35% credit rate applies to families with annual gross income of less than$15k, and declines by 1% for each $2k of additional income until it reaches a constant tax credit rate of 20% for families with annual gross income above$43k.

The CCDF is a block grant fund managed by states within certain federal guidelines. CCDF subsidies are available as vouchers or as part of direct purchase programs to working families with children under 13 and with income below 85% of the state median income. I set the CCDF rate to 90% which is the recommended subsidy rate under Federal guidelines although there are variations across states. I take into account the fact that only a certain proportion of eligible households received the CCDF subsidy: 39%, 24%, and 5% of potentially eligible children living in households, respectively, below, between 101 to 150%, and above 150% of the poverty threshold and below the CCDF eligibility threshold of 85% of state median income (DHHS 2014). US median household income was $51,144 in 2010. The poverty thresholds are summarized in Table 7. Social Security Benefits. To be eligible for disability benefits, one must have worked for at least 5 out of the 10 most recent years with the benefits being permanent thereafter. SSDI benefits are based on the age at which one becomes disabled and Average Indexed Monthly Earnings (AIME). I assume that if a person is disabled, that person is disabled at the start of the period and the relevant AIME is a summary of earnings from the previous periods. SSDI benefits are automatically converted to social security retirement benefits when the recipient is past the retirement age of 65. Retired non-disabled grandparents also receive social security retirement benefits based on their AIME. Social security retirement and disability benefits are computed as follows: 0.9AIMEkt1iifAIMEkt1i0,d1SSDIkti=0.9d1+0.32AIMEkt1id1ifAIMEkt1id1,d20.9d1+0.32d2d1+0.15AIMEkt1id2ifAIMEkt1i>d2, where d1 and d2 are bend points. In 2010, d1 was equal to$761 andd2 was equal to $4,586 (SSA 2014). I use the following formula to approximate AIME: AIMEkti=12AIMEkt1i+minssbaset,ykti, where ssbaset is the social security base wage of$106,800. I assume that parents are not eligible for SSDI in period t=0 while grandparents are eligible to claim disability benefits in the first period.[24] The relevant AIME if a grandparent is disabled in the first period is approximated from average earnings of individuals aged 45–49 with the same gender and marital status.

SSI is a means-tested program that provides benefits to low income individuals aged above 65 and to the disabled. The definition of disability is the same as under SSDI although there are no contribution requirements under SSI. It is possible to receive both SSI and SSDI if income is sufficiently low.[25] To be eligible for SSI, countable resources need to be less than $2k for an individual and$3k for a couple (Morton 2014). I use household assets as the measure of resources. SSI benefits are reduced one-for-one for income. SSI benefits are approximated as follows:

SSIkti=max0,SSIˉktiSSDIkti,

where SSIˉkti is the maximum SSI benefits. In 2010, the maximum monthly benefits available to a single individual and to a couple were, respectively, $674 and$1,011.

### B.3 Initial Promised Utility

Let Ht denote a given history of health and disability states, s0,...,st and ΠtHt=π0s0π1s1,s0...πtst,st1 denote the probability that the family experiences this history path. The initial promised utility, V, is set equal to the expected lifetime utility under the US tax and benefit system according to household members’ actual health and disability status:

V=Maxc,h,lt=04HtβtΠtHtuctHtivihtiHt+ltiHt,

subject to the household budget constraint:

t=04HtΠtHtctHt+Itpt,nt,ihtiHt,fta=t=04HtΠtHtiwtiltiHtTtaHt.

Itpt,nt,ihtiHt,fta represents subsidized formal child care costs, fta represent the cap on formal child care costs subsidized, and TtaHt are net income taxes under the actual US tax and benefit system described in Appendix B.2.

The above maximization problem is solved to get the initial promised utility, Vk, associated with each of the 73 family types considered in the quantitative exercise. In line with the theoretical interpretation, Vk, captures the targeted welfare of each family type given the current system’s generosity level.[26]

### B.4 Numerical Algorithm

First, define a grid over promised utility V. Starting from period t=4, for each possible disability state reported in the previous period and for each grid point, Vj, find the allocations that minimize expected costs subject to the promise keeping and incentive compatibility constraints, and find the threatened utility Vˆj that can be delivered through the threat keeping constraints. Given the solved allocations for each grid point, find the expected costs of the government, Gj.

In period t = 3, for each possible disability state reported in the previous period and for each grid point, use cubic spline interpolation to approximate the government’s continuation values [threatened continuation values] based on the grid of expected costs [threatened utilities] derived in the previous step. Solve for the allocations that minimize expected costs subject to the promise keeping and incentive compatibility constraints, and find the threatened utility that can be delivered through the threat keeping constraints. Given the solved allocations for each grid point, find the expected costs of the government.

Those steps are repeated until period t = 0. Given the calibrated initial promised utility, Vk , the optimal allocations for each possible disability history can then be computed.

In the government problem with hidden household child care, the problem is solved in a similar fashion with the imposition of additional child care constraints based on agents’ private first order conditions. This follows the first-order approach described in Section 3.3.

### Figure 8:

Single fathers optimal allocations.

Note:  Top panels report optimal allocations averaged over all households. Bottom panels report allocations with (τ=1) and without (τ=0) formal day care subsidies, averaged among households with children.

### Figure 9:

Married grandparents optimal allocations.

Note: Top panels report optimal allocations averaged over all households. Bottom panels report allocations with (τ=1) and without (τ=0) formal day care subsidies, averaged among households with children. Grandparents are retired in the last two periods.

### Figure 10:

Grandmothers and fathers optimal allocations.

Note: Top panels report optimal allocations averaged over all households. Bottom panels report allocations with (τ=1) and without (τ=0) formal day care subsidies, averaged among households with children. Grandparents are retired in periods 3 and 4. Grandmothers have zero probability of becoming disabled in periods 3 and 4 if they were previously healthy.

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Published Online: 2019-03-30

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