Intrafamily bargaining and love

Abstract

Popular culture and common wisdom testify that the way partners in a relationship feel for one another very much depends on how they treat each other. This paper posits the hypothesis that altruism or love in a relationship is endogenous to the actions of the partners and studies how this influences allocations and efficiency in a bargaining model of household decision-making. The main results are that agents treat their partner in a kinder way than without endogenously evolving love, this leads to more equitable allocations in household decision making and greater intertemporal efficiency. There are two mechanisms at work: agents treat their partner nicely to avoid retribution by a less loving partner in the future; and they treat the partner nicely so that the kind reciprocal behavior raises their own love towards the partner, which lets them enjoy higher utility. As to love, two interpretations emerge: love is a commitment device by which couples can implement Pareto superior allocations; and love is an investment good in the sense that costly nice behavior towards the partner today may ensure higher levels of trust and efficiency in the future.

Appendix

1.1 Derivation of best responses of i to j playing O, F or W

1.1.1 Suppose j plays O

• State I, i plays S:

  Agents will enter a cycle of alternating love such that total expected utility of i is $U_{i,t}=\frac{(\beta+1)+\delta(2\beta+1)}{1-\delta^2}$.

	α i,t	α j,t	w i,t	w j,t	u i,t
t	0	0	0	1	β + 1
t + 1	1	0	1	0	2β + 1
t + 2	0	1	0	1	β + 1
t + 3	1	0	1	0	2β + 1

• State I, i plays O:

  Agents jump to the full love equilibrium within one period and total expected utility therefore is $U_{i,t}=2\beta+\delta\frac{4\beta}{1-\delta}$.

	α i,t	α j,t	w i,t	w j,t	u i,t
t	0	0	1	1	2β
t + 1	1	1	1	1	4β
t + 2	1	1	1	1	4β

• State I, i plays F:

  This leads the agents to the full love steady state, but takes longer than when i plays W. Total expected utility is $U_{i,t}=(\beta+1)+\delta(2\beta+1)+\delta^22\beta+\delta^3\frac{4\beta}{1-\delta}$.

	α i,t	α j,t	w i,t	w j,t	u i,t
t	0	0	0	1	β + 1
t+1	1	0	1	0	2β + 1
t+2	0	1	1	1	2β
t+3	1	1	1	1	4β

• State I, i plays W:

  This leads to the same situation as when i plays O.

• State III, i plays S:

  The utility outcome is the same as in state I.

• State III, i plays O:

  The outcome is the same as for strategy S.

• State III, i plays F:

  The outcome is the same as for strategy W.

• State III, i plays W:

  The utility outcome is the same as in state I.

We can now construct a strategy ranking. For state I we have:

$$ S\succ F\;\Leftrightarrow\;\frac{(\beta+1)+\delta(2\beta+1)}{1-\delta^2} >(\beta+1)+\delta(2\beta+1)+\delta^22\beta+\delta^3\frac{4\beta}{1-\delta}\; \Leftrightarrow\;\delta<\hat{\delta}_{B1} $$

(9)

$$ F\succ\{O,W\}\;\Leftrightarrow\;(\beta+1)+\delta(2\beta+1)+\delta^22 \beta+\delta^3\frac{4\beta}{1-\beta}>2\beta+\delta\frac{4\beta}{1-\delta}\; \Leftrightarrow\;\delta<\hat{\delta}_{B2}=\hat{\delta}_{A2} $$

(10)

Where $\hat{\delta}_{B1}=\frac{1-3\beta}{2\beta}+\sqrt{(\frac{1-3\beta}{2\beta})^2+ \frac{1-\beta}{\beta}}$. For state III we have:

$$ \{S,O\}\succ\{F,W\}\;\Leftrightarrow\;\frac{(\beta+1)+\delta(2\beta+1)} {1-\delta^2}>(\beta+1)+ \delta(2\beta+1)+\delta^22\beta+\delta^3 \frac{4\beta}{1-\delta}\;\Leftrightarrow\; \delta<\hat{\delta}_{B3}=\hat{\delta}_{B1}. $$

(11)

1.1.2 Suppose j plays F

• State I, i plays S:

  The outcome is the same as when j plays S and i plays S.

• State I, i plays O:

  Here, within a couple of periods the agents reach the full love steady state, total expected utility is $U_{i,t}=\beta+\delta(1+\delta)+\delta^2(2\beta+1)+\delta^3\frac{4\beta}{1-\delta}$.

	α i,t	α j,t	w i,t	w j,t	u i,t
t	0	0	1	0	β
t + 1	0	1	0	1	β + 1
t + 2	1	0	1	1	2β + 1
t + 3	1	1	1	1	4β

• State I, i plays F:

  The outcome is the same as when j plays S and i plays S.

• State I, i plays W:

  The outcome is the same as when j plays S and i plays W.

• State III, i plays S:

  Agents go to the full love equilibrium in two steps and total expected utility therefore is $U_{i,t}=(\beta+1)+\delta(2\beta+1)+\delta^2\frac{4\beta}{1-\delta}$.

	α i,t	α j,t	w i,t	w j,t	u i,t
t	0	1	0	1	β + 1
t + 1	1	0	1	1	2β + 1
t + 2	1	1	1	1	4β

• State III, i plays O:

  The outcome is the same as if i played S.

• State III, i plays F:

  The outcome is the same as when j plays S and i plays F.

• State III, i plays W:

  The outcome is the same as when j plays S and i plays W.

The strategy ranking for state I can be constructed in two steps:

$$ \{S,F\}\succ O\;\Leftrightarrow\;\frac{1}{1-\delta}>\beta+\delta(1+ \delta)+\delta^2(2\beta+1)+\delta^3\frac{4\beta}{1-\delta}\;\Leftrightarrow\; \delta<\hat{\delta}_{C1} $$

(12)

$$ O\succ W\;\Leftrightarrow\;\beta+\delta(1+\delta)+\delta^2(2\beta+1)+ \delta^3\frac{4\beta}{1-\delta}>\beta+\delta2\beta+\delta^2\frac{4 \beta}{1-\delta}\;\Leftrightarrow\; \delta<\hat{\delta}_{C2}=\frac{1-\beta}{2\beta-1} $$

(13)

Where $\hat{\delta}_{C1}$ cannot be determined analytically. It can be shown numerically, however, that $\hat{\delta}_{C1}>\hat{\delta}_{B1}$ for low levels of β and the other way around for high levels and that $\hat{\delta}_{C1}<\hat{\delta}_{A1}$ for all levels of β.

For state III, the strategy ranking is the following:

$$ \{S,O\}\succ \{F,W\}\;\Leftrightarrow\;\beta+1+\delta(2\beta+1)+ \delta^2\frac{4\beta}{1-\delta}>2\beta+\delta\frac{4\beta}{1-\delta}\; \Leftrightarrow\; \delta<\hat{\delta}_{C3}=\hat{\delta}_{C2}. $$

(14)

1.1.3 Suppose j plays W

• State I, i plays S:

  Agents go to the full love equilibrium within two periods and total expected utility therefore is $U_{i,t}=\beta+1+\delta\frac{4\beta}{1-\delta}.$

	α i,t	α j,t	w i,t	w j,t	u i,t
t	0	0	0	1	β + 1
t+1	1	0	1	1	4β
t+2	1	1	1	1	4β

• State I, i plays O:

  Agents jump to the full love equilibrium within one period and total expected utility therefore is $U_{i,t}=2\beta+\delta\frac{4\beta}{1-\delta}$.

	α i,t	α j,t	w i,t	w j,t	u i,t
t	0	0	1	1	2β
t+1	1	1	1	1	4β

• State I, i plays F:

  The outcome is the same as when i plays S.

• State I, i plays W:

  The outcome is the same as when i plays O.

• State III:

  For strategies S and W the outcomes are the same as in state I. Here strategy O is equivalent to S and F is equivalent to W.

It is immediate that $\beta+1+\delta\frac{4\beta}{1-\delta}\gtrless2\beta+\delta\frac{4\beta}{1-\delta}$ can be reduced to 1 > β which has been assumed.