## Abstract

This paper analyzes the effects of changes in relative bargaining power within two-member households participating in labor and product markets. The most striking effects occur when household members differ in individual preferences and enjoy positive leisure-dependent externalities. For instance, a global change in relative bargaining power where the hardworking member becomes more influential in each working class household can render the working class worse off. Moreover, we show that restrictions on labor supply can prevent hard workers from exerting too much pressure on their hedonistic partners to work more. A restriction on individual labor supply improves welfare of the working class population, which adds a new twist to the literature on why working hours are limited in many European countries.

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## Notes

Note that if the group externality depended on the leisure time of the other household member, individual decisions about working and leisure time would not be efficient for the household.

In the current formulation, leisure of either household member constitutes a local public good. A further possibility could be that the externalities depend on the time household members can spend together, that is, on the minimum of the individual leisure times. The qualitative behavior of this type of externality is quite similar to the variable externalities studied next but more cumbersome to analyze.

The employment effects of such work week restrictions, if any, have been modest and ambiguous. The empirical evidence tends to reject the idea that reducing work hours will help to decrease unemployment (see Börsch-Supan 2002; Entorf et al. 1992; Hunt 1998). Possibly, the hourly productivity and hourly wage of those employed went up. Since many of the indirect labor costs, like mandatory employer health insurance contributions, are independent of hours worked and wages paid, the full cost of employment per hour increased significantly. As a consequence, substitution of capital for labor and relocation of production to low cost countries became even more attractive than before. Unemployment remained constant at best. At worst people got laid off.

Too strong externalities would imply a low labor supply and fairly large leisure consumption to begin with. This would translate into a small marginal utility of leisure so that the strong positive labor supply response causes only a weak negative welfare effect.

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## Acknowledgements

We would like to thank the referees for beneficial critique and suggestions. We are grateful to Clive Bell, Stephan Imhof, Christoph Schmidt, participants in the World Congress of the Econometric Society in Seattle, and seminar audiences in Berlin and Heidelberg for their helpful comments.

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*Responsible editor:* Alessandro Cigno

## Appendices

### Appendix 1: The bargaining environment

As a rule, there is a surplus to be divided by household members, for instance a surplus relative to the outside opportunities they would have as single individuals. The creation of such a surplus constitutes the rationale for household formation in the first place. The disappearance of any surplus can cause dissolution of the household. If in our context individuals have the same preferences for private consumption and leisure as household members, represented by

then the term \(G_1^h+G_2^h\) constitutes a potential surplus resulting from formation of household *h*. But how will any potential surplus be divided among household members? Browning et al. (2006, p. 6) list a number of different approaches to model intra-household bargaining. They further state—and we concur—that there is no broad consensus which particular model to use. We follow them and many others and assume collective rationality of households. As mentioned in Section 2, serious objections have been raised against this assumption. But both the theoretical and the empirical literature appear to be split in this matter: Browning and Chiappori (1998, p. 1245) claim “support for our view that the collective model is a viable alternative to the unitary model.”

Given that one assumes collective rationality, a particular efficient household decision in our context can always be obtained as the outcome of maximizing a utilitarian welfare function (Eq. 1) subject to the budget constraint (Eq. 2) and non-negativity constraints. While the weights *α* and 1 − *α* are taken as exogenous by the household, they are not necessarily exogenous or constant over time. The literature (e.g., Browning et al. 2004, 2006; Chiappori and Ekeland 2006) tends to distinguish between so-called distribution factors and prices as variables that influence intra-household balance of bargaining power. We follow Basu (2006) and distinguish between endogenous, denoted *x*, and exogenous, denoted *z*, determinants of intra-household bargaining power. The labor incomes \(wl_1^h\) and \(wl_2^h\) would constitute endogenous factors and might be part of *x*. Exogenous factors could be non-labor income(s), legal provisions, the sex ratio in the marriage market, individual wealth at the time of household formation, etc. In the context of Basu (2006), *x* = *x*(*α*) and \(\alpha=\widehat{\alpha}(x,z)\). Basu considers two conceivable scenarios: First, given *z*, the values of *α* and *x* are endogenously and simultaneously determined, where *α* is a fixed point of the composed mapping \(\alpha \mapsto \widehat{\alpha}(x(\alpha),z)\) and *x* = *x*(*α*). Second, still *x* = *x*(*α*), but *α* adjusts to endogenous factors with a time lag: *x*
_{
t
}, *z*
_{
t
}, and *α*
_{
t
} follow a dynamic process in discrete time *t* where \(x_t=x(\alpha_t), \alpha_t=\widehat{\alpha}(x_{t-1}, z_t)\). The latter scenario is the more plausible one.

Several qualifications are warranted in our context: (a) Some variables which are exogenous for the household, like the wage rate, are endogenously determined in the economy. (b) When we take a snapshot of the economy from time to time, some or all of the *α*’s in various households may have changed, since the exogenous factors *z*
_{
t
} (and possibly the endogenous factors *x*
_{
t
} as well once homogeneity is lost) may affect bargaining power in different households differently over time. (c) The economy may experience other changes over time, for instance technological progress. In our context, the latter could be implemented by considering variations in the coefficient *β* of the production function. Technological progress simply means an increase in *β*. In general, such an increase, ceteris paribus, need not benefit all consumers, but it does in our model. If an individual benefits from a change of bargaining power, then the effect is enhanced by technological progress. In contrast, if an individual would suffer from a change of bargaining power without technological progress, then this effect and the impact of technological progress on the individual’s welfare mitigate or offset each other.

### On income pooling

Prima facie, maximization of Eq. 1 subject to the household budget constraint (Eq. 2) suggests a unitary model of the household and income pooling. However, the distinction between presence and absence of income pooling cannot simply be reduced to a distinction between “one-pot” and “two-pot” households, the presence or absence of a common budget constraint. Income pooling stricto sensu means a common budget constraint plus constancy of the function \(\widehat{\alpha}\). While we have not modeled any function \(\widehat{\alpha}\) explicitly, its implicit assumption explains a possible shift of intra-household bargaining power over time and the conceivable absence of income pooling despite Eqs. 1 and 2.

### Widespread shifts of intra-household bargaining power

Here we mention just a few examples of how distribution factors and consequently relative bargaining power in households might change. Changes in divorce law can change the outside options of many household members. Duration and maturity of the partnership may erode (or accentuate) differences in bargaining power (and income pooling). Namely, the buildup of sizeable durable public goods, in particular housing, could erode differences in bargaining power and enhance income pooling—as may the buildup of mutual trust. On the other hand, differences in lifetime income may widen over time and cause more asymmetry in bargaining power. Hence, with an aging population, there might be widespread shifts of bargaining power. Such shifts might be mitigated or reinforced by legal changes, like changes in the spousal or survivor benefits of retirement systems. Not only past and current earnings but also future earning prospects can influence a person’s intra-household bargaining power. For instance, less job security in traditional manufacturing may affect relative bargaining power in the respective households. Finally, in many cases, it may be difficult to separate cause and effect if there is a feedback loop of the form *x* = *x*(*α*), \(\alpha=\widehat{\alpha}(x,z)\), which constitutes a challenge for theoretical and even more so for empirical investigations.

### Appendix 2: Proof of Proposition 4

We shall explore the relationship in Proposition 3 for special parameter values which imply \(\frac{\partial \, U^h_1}{\partial \, \alpha} < 0\). In particular, suppose that *k*
_{2} is very small, *T* is sufficiently large, and \(g_1^h=g_2^h=:g\) for all *h*. Then we obtain approximately:

where we have used that

If, in addition, *g* + *k*
_{1} is sufficiently close to one, the second term can be neglected. If, moreover, *g* < 1/2, the first term is negative and therefore \(\partial U^h_1/\partial \alpha <0\). Therefore, an increase in relative importance or power harms the first household member. This validates our claim.

Let us first discuss the assumptions made during the proof of the claim and then try to assess the result. We now return to the ∧-notation with re-normalized utility coefficients. Then, \(\hat{g}_1^h=\hat{g}_2^h=\hat{g}\) corresponds to \(g_1^h=g_2^h=:g\), with \(\hat{g}=g/(1+g)\). The condition \(\hat{g}<1/2\) amounts to *g* < 1, which means that the externality is less important than the own commodity consumption and leisure consumption combined. By choosing *g* and *k*
_{1} close to one, one obtains \(\hat{g}+\hat{k}_1\) close to one. A very small *k*
_{2} yields a very small \(\hat{k}_2\). Hence, the conditions are met if both the hardworking and the hedonistic trait are very pronounced and the externality is almost as important as consumption of the composite good and leisure combined.

Compared with an increase of *α* in Eq. 1, an increase of \(\hat{\alpha}\) in Eq. 14 has two additional effects: The leisure term \(\hat{g}_1^h \ln (T-l_2^h)\) weighs more heavily in the household’s objective function, whereas the leisure term \(\hat{g}_2^h \ln (T-l_1^h)\) has less weight. This is immediately reflected in the first member’s optimal labor supply. Without the variable externality, the dependence on *α* assumes the form − *α*(1 − *k*
_{1}) in Eq. 5. With the variable externality, the dependence on \(\hat{\alpha}\) takes the form \(-\hat{\alpha}(1-\hat{k}_1-\hat{g}_1^h-\hat{g}_2^h)\) in Eq. 17. Under the assumptions made to demonstrate the claim, the latter equals approximately \(\hat{\alpha}/2\). Hence, if workaholism is very pronounced and the externalities are strong, but not too strong, then an increase of \(\hat{\alpha}\) has a strong positive effect on the first household member’s labor supply.^{Footnote 5}

Notice that for \(\hat{g}_1^h < 1/2\), \(\hat{g}_2^h < 1/2\) and sufficiently small \(\hat{k}_2\), \(1-\hat{g}_1^h-\hat{g}_2^h-\hat{k}_2 > 0\) holds. If so, the second members’ labor supply goes up, while their composite good consumption goes down in response to an increase of \(\hat{\alpha}\). Thus, there exist model parameter values such that a global shift of power or priorities within households makes everyone in population *I* worse off.

With respect to a local change in relative bargaining power, say in household *h*, an increase of \(\hat{\alpha}\) from \(\hat{\alpha}_*\) to \(\hat{\alpha}^*\) harms all members of population *I* not belonging to household *h*. If again \(1-\hat{g}_1^h-\hat{g}_2^h-\hat{k}_2 > 0\), then the second member of household *h* will be harmed as well. The change can be detrimental to the first household member’s welfare, too. The easiest way to arrive at this conclusion is to consider ceteris paribus an increase in *n*. The labor supply effect for the individual is independent of *n*. But the wage effect and, hence, the effect on this individual’s composite good consumption become arbitrarily small as *n* goes to infinity. Hence, for sufficiently large *n*, the net effect of a given change from \(\hat{\alpha}_*\) to \(\hat{\alpha}^*\) on the first household member’s utility is negative. Therefore, there are model parameter values such that a local shift of power or priorities within a household makes everyone in population *I* worse off.

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Gersbach, H., Haller, H. “Hard workers” and labor restrictions.
*J Popul Econ* **25**, 469–494 (2012). https://doi.org/10.1007/s00148-010-0352-6

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DOI: https://doi.org/10.1007/s00148-010-0352-6

### Keywords

- Household behavior
- General equilibrium
- Labor supply

### JEL Classification

- D10
- D50
- J22