Gambling to leapfrog in status?

Abstract

This paper tests our theoretical prediction that households with positional concerns use gambling to attempt leapfrogging in the social hierarchy. We rely on household data that is representative for Germany and proxy the households’ positional concerns by their expenditures for conspicuous consumption. Our empirical results strongly indicate that households who care about status are not only more likely to participate in gambling but also to invest more in gambling.

Appendix: Details on the theoretical model

The preferences of the representative household are represented by the function:

$$T=u(x)+v(y)+gw(S), $$

(9)

where x and y are the consumption levels of the positional and the non-positional good, and S is the relative standing. We assume $u'>0>u''$, $v'>0>v''$, and both $\lim _{x\rightarrow 0}u'=\infty$ and $\lim _{y\rightarrow 0}v'=\infty$. The scalar $g\ge 0$ represents the relative importance given to status, where relative standing is determined by

$$S\equiv x-\bar{x}, $$

(10)

with $\bar{x}$ as the average level of absolute consumption of the positional good.

The household with income I may participate in a lottery, paying Bl with probability 1 − p for an investment of l, where B > 1 (i.e., the investment influences only the payout of the lottery). Lotteries usually have a negative expected payoff, $EV=(1-p)(Bl-l)-pl$, such that

$$EV_l=(1-p)(B-1)-p<0. $$

(11)

We distinguish the level of available income left for consumption depending on whether the winning state M or the no win state N materializes:

$$ I_M=I+Bl-l$$

(12)

$$ I_N=I-l. $$

(13)

The household seeks to

$$\begin{aligned} \max _{y_j,l} ET&=p[u(I_N-y_N)+v(y_N)+gw(I_N-y_N-\bar{x})]\nonumber \\&\quad+(1-p)[u(I_M-y_M)+v(y_M)+gw(I-y_M-\bar{x})], \end{aligned}$$

(14)

where $x_j=I_j-y_j$, $j=M,N$. We obtain

$$ ET_{y_N}=p\left[v'_N-u'_N-gw'_N\right]=0$$

(15)

$$ET_{y_M}=(1-p)\left[v'_M-u'_M-gw'_M\right]=0$$

(16)

$$ET_{l}=(1-p)(B-1)\left[u'_M+gw'_M\right]-p\left[u'_N+gw'_N\right]\le 0 $$

(17)

$$l\times ET_l=0 $$

(18)

where $v'_j$ is a shorthand for $v'(y_j)$ and so on.

We are interested in heterogeneity regarding the level of g. In this regard, we arrive at our first observation.

Lemma 1

Households with negligible positional concerns (i.e., households for which g → 0 holds) will not participate in a lottery with negative expected value.

This follows from the fact that the household spends more on x in state M than in state N, diminishing utility with respect to the good x, and (11). As a next step, we turn to households with a non-negligible weight g. When the household chooses to invest in the lottery, the condition $ET_l=0$ together with (11) implies

$$\frac{(1-p)(B-1)}{p}=\frac{u'_N+gw'_N}{u'_M+gw'_M}<1, $$

(19)

such that

$$ 0<u'_N-u'_M<g\left[ w'_M-w'_N\right] . $$

(20)

This allows us to conclude:

Lemma 2

Households who invest in a lottery with negative expected value must have status utility w that is sufficiently strictly convex.

We summarize as follows.

Proposition 1

Households who attach more importance to relative standing are more likely to gamble.

Next, we present results from a comparative-statics analysis for subjects that do participate in the lottery. Our research question concerns how household investment in the lottery varies with their ambition for favorable status positions. In the following, we will disregard equilibrium effects on the level of comparison consumption $\bar{x}$. The comparative-static properties of the model follow from

$$\left( \begin{array}{*{20}c} ET_{y_Ny_N} & 0 & ET_{y_Nl} \\ 0 & ET_{y_My_M} & ET_{y_Ml} \\ ET_{ly_N} & ET_{ly_M} & ET_{ll} \end{array}\right) \left( \begin{array}{c} dy_N \\ dy_M \\ dl \end{array}\right) = \left( \begin{array}{c} -ET_{y_Ng} \\ -ET_{y_Mg} \\ 0 \end{array}\right) dg. $$

(21)

The determinant of the 3 × 3 matrix on the left-hand side will be denoted H in our subsequent argumentation, and is supposed to be negative by the sufficient second-order conditions.

From the first-order conditions and the assumption that the sufficient second-order conditions are fulfilled, we obtain

$$ ET_{y_My_M}=(1-p)\left[ v''_M-u''_M-gw''_M\right] <0 $$

(22)

$$ET_{y_Ny_N}=p\left[ v''_N-u''_N-gw''_N\right] <0 $$

(23)

$$ET_{ll}=p\left[ u''_N+gw''_N\right] +(1-p)(B-1)^2\left[ u''_M+gw''_M\right] <0$$

(24)

$$ ET_{y_Nl}=p\left[ u''_N+gw''_N\right] $$

(25)

$$ET_{y_Ml}=-(1-p)(B-1)\left[ u''_M+gw''_M\right] $$

(26)

$$ ET_{y_Ng}=-pw'_N<0 $$

(27)

$$ ET_{y_Mg}=-(1-p)w'_M<0 . $$

(28)

We are interested in the expenditures for lotteries of status-oriented households, and therefore seek to interpret:

$$\begin{aligned} \frac{dl}{dg}= A\left\{ (1-p)(B-1)\left[ u''_M+gw''_M\right] \frac{v''_N-u''_N-gw''_N}{v''_M-u''_M-gw''_M}-p\left[ u''_N+gw''_N\right] \frac{w'_N}{w'_M}\right\} \end{aligned}$$

(29)

where $A=\left\{ p(1-p)w'_M\left[ v''_M-u''_M-gw''_M\right] \right\} H^{-1}>0$.

An increase in the importance attached to relative standing implies that both the beneficial comparison in the winning state of the world and the disadvantageous comparison in the losing state of the world have a greater impact on well-being. The former comparison gets even more favorable as a consequence of a greater investment in the lottery, whereas the latter one becomes more unfavorable. We have concluded in Lemma 2 that households who invest in the lottery must have strictly convex status utility. This can be used for the interpretation of (29), because the first term in the parentheses will be positive (due to $u''_M+gw''_M>0$) and the second one will go to zero for w sufficiently convex. Our second prediction follows therefrom.

Proposition 2

Households who participate in the lottery will invest the more in the lottery, the more importance they attach to relative standing for w sufficiently convex.