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

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.

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,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 Y−Y _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 (Y−Y _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 Y−v<Y−γ a Y−v<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.