# 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.

# 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,

The first order conditions for each period

where

**(i)****Healthy members with ** From examining the first order conditions with respect to

**Healthy members with **. If

**(ii)****Healthy members with **. From

**(i)**,

**). Using the first order conditions with respect to**w t i > p t

**Healthy members with **. From

**(i)**, the first order condition with respect to

**(iii)** Adding the first order conditions with respect to

where

Now, adding the first order conditions with respect to

One has an analogous expression for the following period:

From the interpretation of the Lagrange multipliers, one has:

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

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

### A.2 Proof of Proposition 2

Every period, households take the policy scheme as given and choose a state claim

**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 *

**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 *

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:

It must therefore be that

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

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

From the private Euler equation, the inter-temporal wedge, the fact that

**Penultimate period** (

From Claim 1, households do not have an incentive to overprovide total household child care:

with equality when

Since the marginal gain from household child care is higher, household members have greater incentives to increase household child care beyond the optimal level

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

Since

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

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

From the private Euler equation, the inter-temporal wedge, the fact that

**First period** (

By following the same line of proof, it must be that

Since

*Step 2* Households accumulate zero assets. From Step 1, the household accumulates non-negative assets. The household budget constraint for all

From Claim 1, households do not have an incentive to overprovide total household child care:

with equality when

Since the marginal gain from household child care is higher, household members have greater incentives to increase household child care beyond the optimal level

The household budget constraint then becomes:

This implies that

*Step 3*. Households consume the optimal level of consumption. From Steps 1 and 2, the household accummulates zero assets:

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:

*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

### 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

Let

**Claim 3**. *The Lagrange multipliers associated with the child care constraints are zero. The first order condition with respect to *

*where *

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

where

**Claim 4**. *The qualitative features of Proposition 1 hold. (i) Using the first order conditions with respect to *

*When *

**By the same line of thought,**w t i ≥ p t .

**, 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.**w t i < p t

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

Consider an illustration with

subject to the budget constraint:

with strict inequality when ^{[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 *.

*(i) Healthy members have *.

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

*with strict inequality when *.

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

### Proof.

The first order conditions for each period

**(i)** and**(ii)** can be seen from examining the first order conditions with respect to **(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, *

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

** Proof. **

Every period, households take the policy scheme as given and choose a state claim

**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 *

**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, *

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 (**

Suppose that assets carried over to the last period are negative:

It must therefore be that

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

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

From the private Euler equation, the inter-temporal wedge, the fact that

**Penultimate period** (

From Claim 5, households do not have an incentive to overprovide total household child care:

with equality when

Since the marginal gain from household child care is higher, household members have greater incentives to increase household child care beyond the optimal level

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

Since

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

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

From the private Euler equation, the inter-temporal wedge, the fact that

**First period (**

By following the same line of proof, it must be that

Since

*Step 2*. Households accumulate zero assets.

From Step 1, the household accummulates non-negative assets. The household budget constraint for all

From Claim 5, households do not have an incentive to overprovide total household child care:

with equality when

Since the marginal gain from household child care is higher, household members have greater incentives to increase household child care beyond the optimal level

The household budget constraint then becomes:

This implies that

*Step 3*. Households consume the optimal level of consumption.

From Steps 1 and 2, the household accummulates zero assets:

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:

*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

## B Quantitative Appendix

### B.1 Effort Cost

The calibration of

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 5– Figure 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:

### Figure 6:

### Figure 7:

### B.2 US Tax and Benefit System

*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:

Federal income tax rates on taxable income | |||||||
---|---|---|---|---|---|---|---|

Tax rate | Single | Head | Married | ||||

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) | |||||||

All filing statuses | Single and Head | Married | |||||

# Kids | Phase-in | Maximum | Phase-out | Phase-out | Income | Phase-out | Income |

below 18 | rate | credit | rate | income | limit | income | limit |

0 | 7.65% | $457 | 7.65% | $7,480 | $13,460 | $12,480 | $18,470 |

1 | 34% | $3,050 | 15.98% | $16,450 | $35,535 | $21,460 | $40,545 |

2 | 40% | $5,036 | 21.6% | $16,450 | $40,363 | $21,460 | $45,373 |

45% | $5,666 | 21.6% | $16,450 | $43,352 | $21,460 | $48,360 | |

Poverty thresholds | |||||||

# Kids below 18 | |||||||

# Persons | 1 | 2 | 3 | ||||

2 | $15,030 | ||||||

3 | $17,552 | $17,568 | |||||

4 | $27,518 | $22,113 | $22,190 | ||||

5 | $27,518 | $26,675 | $26,023 |

*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:

where was equal to $4,586 (SSA 2014). I use the following formula to approximate AIME:

where ^{[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:

where

### B.3 Initial Promised Utility

Let

subject to the household budget constraint:

The above maximization problem is solved to get the initial promised utility, ^{[26]}

### B.4 Numerical Algorithm

First, define a grid over promised utility

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,

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:

### Figure 9:

### Figure 10:

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**Published Online:**2019-03-30

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