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Old money, the nouveaux riches and Brunhilde’s marriage strategy

An Erratum to this article was published on 20 September 2016

Abstract

In a courtship game, wasteful conspicuous spending may provide information about some components of a suitor’s income. But conspicuous spending may be costly not only for the potential husband but also for the woman: it reduces the wealth of the man she may marry. In the optimal contractual arrangement, the bride’s cost moderates the threshold value of the conspicuous spending that she requires for marriage. We also find that a sound observable financial background (‘old money’) benefits both the suitor and the woman, and reduces wasteful spending on status goods. Furthermore, we analyze how a change in the intensity with which the suitor seeks the woman may affect the equilibrium pattern of conspicuous spending.

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Fig. 1
Fig. 2

Notes

  1. The character of Brunhilde (and the episode we allude to here) in ‘The Song of the Nibelungs’ differs from a very similar character in Richard Wagner’s opera cycle ‘The Ring of the Nibelung’ or the Volsunga Saga. Therefore, depending on the mythology referred to, she may also be spelled Brunhild, Brünnhilde, Brynhild or Prunhilt.

  2. Marriage may be about more than money, income, or wealth. The resource motive, however, finds much support among evolutionary biologists (e.g., Trivers 1972). They emphasize the resource capacity that the husband may bring into a marriage and which benefits the couple’s offspring. We follow this tradition, disregarding love and affection as marriage motives for our analysis here.

  3. The economic theory on status consumption highlights the instrumental role of conspicuous consumption for attracting a better marriage partner. This instrumental aspect of status lies behind many models of status-seeking. De Fraja (2009) explicitly links utility maximization to the biological problem of fitness maximization. A man faces a trade-off between investing in his survival, and conspicuous consumption that signals his quality and thus increases his matching probability. Much of the theory emphasizes the role of status goods as signals of income (Bagwell and Bernheim 1996; Corneo and Jeanne 1997b; Frank 1985a, 1985b; Ireland 1994, 1998, 2001; Glazer and Konrad 1996; Moav and Neeman 2012) often with consideration of the role of the income of potential grooms in the context of marriage matching.

  4. Matching, marriage, and partnership is a complex issue with many aspects. For instance, men may incur debt to provide a dishonest signal of their desirability as a mate (Gallup and Frederick 2010). Kruger (2008) finds that men who spend more than they save are likely to have more sex partners compared to more frugal men. Conspicuous goods may signal not the desirable qualities of a partner but rather the opposite: interest in status goods is triggered by feelings of powerlessness (Rucker and Galinsky 2008, 2009) or a need to restore one’s self-worth (Sivanathan and Pettit 2010). These and many other aspects are beyond the scope of the analysis here, which focuses on one important information problem.

  5. Screening by Brunhilde for a strong husband, as in the epic poem ‘The Song of the Nibelungs’, is also costly for Brunhilde if their fight dilutes his (and also her) strength, or if he or she gets hurt while fighting.

  6. Whereas, in our model, conspicuous consumption declines with observable income, in Moav and Neeman (2012) conspicuous consumption declines with observable human capital. They argue that the poor and the nouveaux riches do not hold diplomas or professional titles and therefore rely on conspicuous consumption to signal their success.

  7. See, e.g., Burdett and Coles (1997, 1999). Browning et al. (2014) provide a broad treatment of family economics, including matching theory.

  8. Pioneering contributions include Veblen (1899), Rae (1834), Hirsch (1976), and Frank (1985b). For surveys see McAdams (1992) and Truyts (2010).

  9. In Feltovich et al. (2002), apart from the endogenously chosen signal, the receiver observes some noisy information about the sender. This extra information is unknown to the sender when he chooses his signal. Equilibria are found in which medium types signal to distinguish themselves from low types. In contrast, high types choose to countersignal, i.e., they do not signal as they are confident that they will not be seen as low types.

  10. This display has proven an evolutionarily beneficial courtship strategy. For a comprehensive survey of consumer behavior from an evolutionary perspective, see Griskevicius and Kenrick (2013), who discuss so-called fundamental motives such as attaining status, and acquiring and keeping a mate. Pan and Houser (2011) also summarize evidence from experimental economics and evolutionary psychology explaining gender differences in pro-social behavior.

  11. This finding is substantiated by a field experiment on a Chinese online dating website where women of all income levels visited profiles of high-income males more often, and where women’s visits to these profiles were an increasing function of their own income (Ong and Wang 2015).

  12. In a face-to face interaction with a suitor, she may communicate this reaction in a slightly more subtle way. Depending, however, on the culture, and also thinking of online dating platforms and TV shows, she may indeed be explicit about the contract.

  13. The non-material benefit from marriage is given, common knowledge, and identical for all suitors. It is also unaffected by their income. Experimental evidence, however, finds that men primed with a large sum of money adjust their mating strategy; that is, they increase their dating requirements—particularly for physical attractiveness (Yong and Li 2012). Similarly, evidence from lonely heart advertisements suggests that men with more resources make higher demands about physical attractiveness (Bereczkei et al. 1997; Waynforth and Dunbar 1995). Candidates who differ in income should therefore also differ in their preference for B. If B can freely observe a, the heterogeneity does not invalidate the analysis here. Relaxing these assumptions leads to a two-sided search and screening problem that we leave for future research.

  14. Several motives can drive this preference. B may simply enjoy consumption. Another important motive that is prominent in much of the literature on marriage (see, e.g., Edlund 2006for a review) is the desire to provide resources for raising children.

  15. For γ>0, B dislikes the successful suitor’s spending and would like to keep it low, because it reduces what is left for the couple if they marry. A successful suitor’s income becomes the joint consumption of the married couple; one interpretation is that these resources are used to raise children and children are a pure public good for them. In a more general consideration, a suitor’s present value of income may yield a higher or lower utility to him if he marries than if he does not marry. The assumption that c has the same effect on his utility is mainly for notational convenience. This income net of conspicuous spending also affects B’s utility, and it may do so either more or less strongly. Though we assume that the monetary amount affects B’s payoff directly, the results do not change qualitatively if B’s payoff is scaled by a positive factor. The analysis also includes the two extreme cases where B also bears the full screening costs (γ=1) and where B bears no screening costs at all (γ=0). In this latter case, the problem reduces to a standard problem.

  16. Welfare considerations for alternative marriage decisions as functions of Y are less straightforward. We assume that income Y becomes a public good in a marriage. So the welfare effects of marriage for the two players in question depend on their reservation utilities, in particular, on whether these entail marrying someone else or staying single.

  17. We rule out divorce. Were income revealed after marriage and B could costlessly divorce from a husband who turns out to have low income; divorce would, in this extreme case, resolve the information problem and lead to different outcomes.

  18. Other factors not modeled here explicitly, may also enter into B’s default utility v. She may earn some income on her own, which may increase over time, and in turn, increase v as B ages.

  19. Waynforth and Dunbar (1995) find that whereas men become more demanding with age, women become less demanding. Bereczkei et al. (1997) find, however, that the proportion of women demanding traits associated with high wealth and high status is constant across age groups. Pawlowski and Dunbar (1999) conclude that female advertisers who try to present themselves as younger than they really are tend to be more demanding in what they look for in a prospective partner.

  20. Formally, she can require an impossible \(\hat {c}>Y_{O}+1\).

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Acknowledgment

We thank Daniel S. Hamermesh, Marco Pagano, Fangfang Tan, participants at the CSEF seminar in Naples, and participants at the Family Economics Workshop at Royal Holloway University in London for helpful comments. We also thank Yue Yu for research assistance. The editor and reviewers gave much useful advice. Of course, the usual caveat applies.

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Correspondence to Kai A. Konrad.

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Conflict of interest:

The authors declare that they have no conflict of interest.

Funding:

Amihai Glazer and Kai A. Konrad declare that they received no funding for this study. Anne- Kathrin Bronsert received a scholarship from the Max Planck Society for doing her Ph.D. at the Max Planck Institute for Tax Law and Public Finance.

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Responsible editor: Alessandro Cigno

An erratum to this article is available at http://dx.doi.org/10.1007/s00148-016-0617-9.

Appendix: Proof of proposition 1

Appendix: Proof of proposition 1

We use backward induction to solve for the equilibrium and start with the decision problem of the suitor in the continuation game once B has made an offer. Suppose B accepts the suitor if and only if \(c\in \mathcal {M}\) for a given set \(\mathcal {M}\). Let the income of the suitor be Y = Y O + Y U . The consumption choice \(c\in \mathcal {M}\) yields him a payoff

$$Y-\frac{c}{Y}+a. $$

The consumption choice \(c\notin \mathcal {M}\) yields him a payoff

$$Y-\frac{c}{Y}. $$

Among all \(c\notin \mathcal {M}\) the payoff-maximizing choice is c=0. Among all \(c\in \mathcal {M}\), the payoff-maximizing choice is the smallest feasible element \(c\in \mathcal {M}\). Denote this smallest consumption level by \(\hat {c }\). The choice between c=0 and \(c=\hat {c}\) depends on Y. Define

$$ Y(\hat{c})=\frac{\hat{c}}{a}. $$
(12)

The suitor chooses c=0 if \(Y<Y(\hat {c})\) and \(c=\hat {c}\) if \(Y\geq Y(\hat {c})\). Note also that \(Y^{\prime }(\hat {c})=1/a>0\), and that \(Y^{\prime \prime }(\hat {c})=0\).

Turn now to B’s choice. She can reject all suitors,Footnote 20 yielding her a payoff \( w_{\varnothing }=v\). She can accept all suitors, in which case the sequentially rational behavior of the suitors leads to \(\hat {c}=0\). Her expected benefit is then \(w_{1}=Y_{O}+E[Y_{U}]=Y_{O}+\frac {1}{2}\). Lastly, she can choose \(\hat {c}\) to apply a mechanism that makes positive shares of suitors self-select into c=0 and into \(c=\hat {c}\). As follows by the sequentially rational behavior of suitors, such a selection mechanism is characterized by a critical \(\hat {c}\) with \(Y(\hat {c})\in [Y_{O},Y_{O}+1]\) and maximizes

$$\begin{array}{@{}rcl@{}} w_{\hat{c}} &=&(Y(\hat{c})-Y_{O})v+{\int}_{Y(\hat{c})}^{Y_{O}+1}(z-\gamma \hat{ c})dz \\ &=&(Y-Y_{O})v+(\frac{(Y_{O}+1)^{2}-Y^{2}}{2})-\gamma aY(Y_{O}+1-Y) \end{array} $$

by a choice of Y, making use of \(\hat {c}=aY(\hat {c})\). The first term in the first line says that B’s payoff equals v with the probability that \( Y<Y(\hat {c})\). The second line calculates the integral using the distribution assumption about Y U . The first-order condition for a local maximum of \(w_{\hat {c}}\) is

$$ \frac{\partial w_{\hat{c}}}{\partial Y}=v-Y+2\gamma aY-\gamma aY_{O}-\gamma a=0. $$
(13)

Using Eq. (12), this is equivalent to Eq. (11). Note further that

$$ \frac{\partial^{2}w_{\hat{c}}}{(\partial Y)^{2}}=-(1-2\gamma a). $$
(14)

Hence, the payoff \(w_{\hat {c}}\) is concave in Y for a≤1/(2γ). This is where (3) is used. The solution for Eq. (13) makes sense only for feasible \(Y(\hat {c})=\frac {\hat {c}}{a}\in [Y_{O},Y_{O}+1]\) requiring \(\frac {v-\gamma a(Y_{O}+1)}{1-2\gamma a} >Y_{O}\) and \(\frac {v-\gamma a(Y_{O}+1)}{1-2\gamma a}<Y_{O}+1\), which can be transformed into v∈[(1−γ a)Y O + γ a,(Y O +1)(1−γ a)]. For v smaller than the lower limit of this interval, B prefers to admit all suitors with this Y O unconditionally; for v larger than the upper limit, she prefers to reject all suitors with this Y O . Note that the lower limit corresponds to H 3(Y O ) and the upper limit corresponds to H 1(Y O ).

So far, we characterized the optimal separating contract under the condition that it is optimal for B to set a positive, but not prohibitive, threshold \(\hat {c}\). Recall that B has three potentially optimal actions: outright reject (\(\mathcal {M}=\varnothing \)), outright accept with \(\hat {c}=0\), and the best non-trivial contract offer with \(\hat {c}\). The maximal payoffs for these three actions are given by

$$\begin{array}{@{}rcl@{}} w_{\varnothing }(v,Y_{O}) &=&v. \\ w_{1}(v,Y_{O}) &=&Y_{O}+\frac{1}{2}\text{,} \\ w_{\hat{c}}(v,Y_{O}) &=&\max_{Y\in [Y_{O},Y_{O}+1]}\left[ (Y-Y_{O})v+{\int}_{Y}^{Y_{O}+1}(z-\gamma Ya)dz\right] . \end{array} $$
(15)

We can now study B’s optimal choice as a function of Y O and v. Figure 1 helps to sort out matters.

Rejecting the suitor with observed income component Y O independent of his conspicuous consumption is superior to active screening if v>H 1(Y O ), as has already been shown, and a separating contract, where it exists, is superior to outright rejection for values of v close to, but below, H 1(Y O ). Note that H 1(Y O ) is exactly the point at which \(\hat {c}\) reaches its upper corner solution and \(Y(\hat {c})=\allowbreak Y_{O}+1\).

The hyperplane H 2(Y O ) in Fig. 1 represents combinations (v,Y O ) for which \( w_{1}=w_{\varnothing }\), which can also be expressed as v = Y O +(1/2). It separates all combinations (v,Y O ) for which \(w_{\varnothing }>w_{1}\) (upper-left) from those with \(w_{\varnothing }<w_{1}\) (lower-right). The two hyperplanes H 1 and H 2 intersect for a value of observed income

$$\hat{Y}_{O}=\frac{1-2\gamma a}{2\gamma a}>0. $$

At the intersection, B is indifferent among all three alternatives.

To limit further the area of possible non-trivial separating contracts, note that such contracts are strictly dominated by outright acceptance for all (v,Y O ) for which \( Y(\hat {c}(Y_{O}))\leq Y_{O}\). This condition yields a further hyperplane H 3, which determines the combinations v and Y O for which \(Y(\hat {c }(Y_{O}))=Y_{O}\). For all (v,Y O ) combinations below this line, the separating contract is inferior to outright acceptance. Unlike H 1 and H 2, however, this line only provides a sufficient condition.

Hyperplanes H 1, H 2, H 3 and the vertical line through \((v(\hat { Y}_{O}),\hat {Y}_{O})\) span seven regions A,B,C,D,F,K, and L, for which the following partial order is established. In region A, she outrightly rejects, as rejection dominates active screening and outright acceptance. In region F, she chooses outright acceptance, as \(w_{\hat {c}}<w_{\varnothing }\) and \(w_{\varnothing }<w_{1}\) in this region. For regions B,C,D,K, and L, she will not outrightly reject. Whether the optimal separating contract or outright acceptance yields a higher payoff needs to be considered more closely. A necessary condition for the separating contract not to be dominated by outright acceptance with c=0 is that (v,Y O ) lies to the upper-left of H 3. Accordingly, outright acceptance with c=0 occurs in regions K and L.

So we turn to regions B, C, and D. Consider some \(\tilde {Y}_{O}>\hat {Y}_{O}\) and go to the point \((H_{1}(\tilde {Y}_{O}),\tilde {Y}_{O})\) vertically above \(\tilde {Y}_{O}\) on H 1. A reduction in v leaves w 1 unchanged. But it reduces \(w_{\hat {c}}\), as

$$ \frac{dw_{\hat{c}}}{dv}=Y-Y_{O}>0. $$
(16)

where, by the envelope theorem, \(\frac {\partial w_{\hat {c}}}{\partial Y} \frac {\partial Y}{\partial v}=0\). The inequality YY O >0 always holds in an active screening equilibrium above H 3. The condition (16) shows that if v is decreasing between H 1 and H 3, then \(w_{\hat {c} }\) is strictly monotonically decreasing.

For \(Y_{O}>\hat {Y}_{O}\), consider the point \((H_{1}(\tilde {Y}_{O}),\tilde {Y}_{O})\) vertically above \(\tilde {Y}_{O}\) on H 1. Consider a decrease in v starting from this point. At this point, \(w_{\hat {c}}=H_{1}(\tilde {Y} _{O})=w_{\varnothing }<w_{1}\). A decrease in v further reduces \(w_{\hat {c}} \), but keeps w 1 constant. Accordingly, \(w_{\hat {c}}<w_{1}\) for all combinations (v,Y O )∈C, establishing that B outrightly accepts with \(\hat {c}=0\) for combinations (v,Y O ) in region C.

For \(Y_{O}\in [0,\hat {Y}_{O})\), consider again a point \((H_{1}(\tilde { Y}_{O}),\tilde {Y}_{O})\) vertically above \(\tilde {Y}_{O}\) on H 1. Consider a decrease in v starting from this point. At this point, \(w_{\hat { c}}=w_{\varnothing }=H_{1}(\tilde {Y}_{O})>w_{1}\). A decrease in v decreases \(w_{\hat {c}}\), but keeps w 1 constant. A decrease in v reduces \(w_{\hat {c}}-w_{1}\). Once we reach \(H_{2}(\tilde {Y}_{O})\), we know that \(w_{\hat {c}}>w_{\varnothing }\) at this point (we are below H 1). Moreover, we know that \(w_{\varnothing }=w_{1}\) at this point (which lies on H 2). Accordingly, \(w_{\hat {c}}>w_{1}\), implying that she will actively screen for all combinations (v,Y O )∈B. If, for given \(\tilde {Y}_{O}\), v is further reduced below \(H_{2}(\tilde {Y}_{O})\), then \(w_{\hat {c}}\) decreases further and eventually falls below w 1. For instance, for \( v=H_{3}(\tilde {Y}_{O})\) the strategy of outright accepting (implying that c=0) is superior to choosing the \(\hat {c}\) that makes a suitor with Y = Y O just indifferent about spending this \(\hat {c}>0\). By monotonicity and the intermediate-value theorem, there is exactly one v between \(H_{2}(\tilde {Y}_{O})\) and \(H_{3}(\tilde {Y}_{O})\) such that \(w_{\hat {c}}=w_{1}\) for this v. By this principle, we can construct a critical level of v for every \(Y_{O}\in [0,\hat {Y}_{O})\). These critical levels yield a fourth hyperplane H 4(v) which is the dashed line in Fig. 1. All [v,Y O ] between H 1 and H 4 and for \(Y_{O}\in [0,\hat {Y}_{O})\) describe combinations of (v,Y O ) for which she uses a separating contract; for all combinations below H 4, she chooses outright acceptance that yields a choice c=0.

Lastly, we characterize H 4. The condition (8) determines (v,Y O ) for which \(w_{\hat {c}}\) (left-hand side) is equal to w 1 (right-hand side). It separates the range \(w_{\hat {c}}>w_{1}\) from \(w_{\hat {c }}<w_{1}\). We already showed that it has the property H 2(Y O )>H 4(Y O )>H 3(Y O ) for \(Y_{O}\in [0,\hat {Y}_{O}),\) and it passes through the intersection of H 1 and H 2. Furthermore, it has a positive slope. Note that w 1 is invariant to changes in v, but increases with Y O . As H 4 is an indifference surface with \(w_{1}=w_{ \hat {c}}\), for a proof that its slope is indeed positive we consider the slope of this locus. Using the envelope theorem again and solving (YY O )d v+(−v + Y O +1−γ a Y−1)d Y O =0 for this slope yields

$$ \frac{dv}{dY_{O}}=-\frac{Y_{O}-\gamma aY-v}{Y-Y_{O}}. $$
(17)

As \(Y(\hat {c}(Y_{O}))\) must exceed Y O for separating contracts not to be strictly dominated by outright acceptance, the denominator is positive. Furthermore, Y O γ a Yv<Yγ a Yv<0 as \(v>Y-\gamma \hat {c}\left (Y_{O}\right ) \), which is implied by the characterization (11) together with the condition stated above that v∈[(1−γ a)Y O + γ a,(Y O +1)(1−γ a)]. Hence, the slope (17) is positive for all Y O in the relevant range.

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Bronsert, AK., Glazer, A. & Konrad, K.A. Old money, the nouveaux riches and Brunhilde’s marriage strategy. J Popul Econ 30, 163–186 (2017). https://doi.org/10.1007/s00148-016-0610-3

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  • DOI: https://doi.org/10.1007/s00148-016-0610-3

Keywords

  • Marriage
  • Courtship game
  • Conspicuous consumption
  • Status

JEL Classification

  • J12
  • D82