# A closed-form solution to a model of two-sided, partial altruism

Abstract: This paper presents a closed-form characterization of the allocation of resources in an overlapping generations model of two-sided, partial altruism. Three assumptions are made: (i) parents and children play Markov strategies, (ii) utility takes the CRRA form, and (iii) the income of children is stochastic but proportional to the saving of parents. In families where children are rich relative to their parents, saving rates—measured as a function of the family's total resources—are higher than when children are poor relative to their parents. Income redistribution from the old to the young, therefore, leads to an increase in aggregate saving.

## Summary (1 min read)

### 1. INTRODUCTION

- Financial support from Banco de Portugal, Fundação Para a Ciência e a Tecnologia and Fundación Ramón Areces is gratefully acknowledged.
- Let us fix the child’s income and examine what happens as the parent’s income increases.

### 2.1. Preferences

- The environment is that of a standard OLG model with partial altruism [see Laitner (1988)].
- Each family member lives for two periods.
- In the following period, that child will raise her grandchild.
- Parent and child value each other’s utility but by less than they value their own.

### 2.2. A Dynasty

- Equations (5) and (6) characterize the preferences of old and young agents, respectively.
- Let us now turn to other aspects of the extended family.
- Parental income is assumed nonstochastic, for simplicity.
- Old agents in period t have income atR, where at represents assets accumulated from period t − 1, which yields the exogenous return R.6 Each agent moves in two adjacent periods.
- Both parent and child know the entire past history of endowment realizations, transfers, and asset accumulation decisions made by other dynasty members.

### 3. A PARAMETRIC EXAMPLE

- I assume that the income of the child is stochastic but proportional to her parent’s saving.
- When transfers are positive (either low or high realizations of the shock z), optimal consumption and savings are expressed as a constant fraction of the family’s total resources, a(R + z).
- In the no-transfer regime, saving rates are decreasing functions of λ, as this parameter also measures the intensity of altruism regarding the children of the current savers.

### 4. CONCLUSION

- The most important assumption in overcoming the typical breakdown of value-function concavity—a consequence of partial altruism—was the proportionality of parent’s and child’s income.
- Partial altruism suggests that accurate measurement of wealth inequality must take into account the degree of persistence of income across generations.
- This is trivially true of the parent’s own marginal utility from consumption.
- This modification would have no implications for Markov perfect strategies or the results of the parametric example of Section 3, and the simultaneous choice of transfers and savings is therefore maintained here.

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### "A closed-form solution to a model o..." refers background in this paper

...Models of the family are especially important in analyzing topics that involve resource redistribution across generations, such as the implementation of social security programs; altruism has been a recurrent paradigm in this context [see, e.g., Barro (1974), Becker (1974), Laitner (1979, 1988)]....

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### Additional excerpts

...Samuelson (1958)....

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