## Abstract

The sudden acquisition of a large sum of money, known as “wealth shock,” can have unanticipated negative consequences, and actually cause greater unhappiness in its so-called beneficiaries. There is extensive economic literature describing these negative consequences on a macro-economic level, but there is no coherent theoretical model that describes the various consequences of wealth shock on a micro-economic level. To explain both the short- and long-term effects of an exogenous monetary shock (for example, winning a lottery) on individual happiness, this paper offers a novel dynamic equilibrium model of human happiness. A dynamic equilibrium model is best suited for this purpose, because happiness is a dynamic process. The proposed model captures both short- and long-term effects, and describes an equilibrium in which a person’s experienced utility and happiness is improved after the sudden wealth shock, and why at the saddle point, life can become sadder and more miserable. The conditions detrimental to winners’ happiness include reducing the amount of time and effort they allocate to preserving their stock of hedonic capital.

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## Notes

- 1.
William “Bud” Post who won USD 16.2 million in the Pennsylvania lottery in 1988. http://www.businessinsider.com/lottery-winners-who-lost-everything-2013-12?op=1.

- 2.
Evelyn Adams who won the New Jersey lottery twice, in 1985 and 1986, raking in USD 5.4 million (Doll 2012).

- 3.
“Wild” Willie Seeley of Manahawkin, New Jersey who won USD 3.8 million. http://www.nbcnews.com/news/other/drama-nonstop-powerball-winner-wild-willie-wants-his-old-life-f8C11251444.

- 4.
Jack Whittaker of West Virginia won USD 314.9 million in, 2002 http://www.thewire.com/national/2012/03/terribly-sad-true-stories-lotto-winners/50555/.

- 5.
A winner of the Florida lottery, quoted in Ugel (2007).

- 6.
By 2017 more than NKr 8bn ($1tn) had accumulated in Norway’s oil fund (Norges Bank Investment Management 2017). The purpose of this and similar “boom funds” is to invest the revenues outside of the local market, in order to minimize the appreciation of the currency and guarantee the economic prosperity of future generations.

- 7.
Based on Frey and Stutzer (2014, p. 940) intrinsic needs reflect: (1) “a need for relatedness,” (2) “a need for competence” and (3) the “desire for autonomy.” The attributes related to extrinsic desires represent the desire for “material possessions, fame, status or prestige,” which are obtained using pecuniary resources. Frey and Stutzer (2104) explain that each activity and good has attributes related to both intrinsic and extrinsic needs, but some are more intrinsic by nature.

- 8.
- 9.
- 10.
A more reasonable assumption would be that the relative productivity between working hours and productive leisure eventually decreases when \(T_{t}^{w}\) increases and when \(L_{t}^{\Pr o}\) decreases. However, this assumption is not required here in order to avoid a corner solution because \(T_{t}^{w}\) also expends the budget set, allowing increased consumption in which utility function is concave.

- 11.
We note that without loss of generality, it could be assumed that \(\alpha ,\beta ,\) and \(\gamma\) equal unity. Nevertheless, it will be useful not to do so to allow for comparative static.

- 12.
The only implication of a multiplicative utility function is that the marginal rate of substitution (MRS) between two utilities is independent of the third one (“weak separability”). This is a reasonable assumption; for example, the MRS between the pecuniary utility and the non-pecuniary utility is unlikely to depend on the level of passive utility. Note that the form in (5) need not imply that any one of the goods is vital for the agent; for example: let \(U_{t}^{P} (c_{t} + \gamma C_{t - 1} ) = e^{{c_{t} + \gamma C_{t - 1} }}\), \(U_{t}^{NP} (\delta H_{t - 1} + L_{t}^{\Pr o} + \zeta T_{t}^{W} ) = e^{{H_{t - 1} + L_{t}^{\Pr o} + \zeta T_{t}^{W} }}\), and \(U_{t}^{Pas} (L_{t}^{{\text{P} as}} ) = e^{{L_{t}^{{\text{P} as}} }}\), and \(\alpha = \beta = \varepsilon = 1\) then \(U_{t}\) = \(e^{{c_{t} + \gamma C_{t - 1} + H_{t - 1} + L_{t}^{\Pr o} + \zeta T_{t}^{W} + L_{t}^{{\text{P} as}} }}\). In this form, it is easy to see that any of the goods might be zero without zeroing the overall utility.

- 13.
Concavity follows from the standard assumption of decreasing marginal utility.

- 14.
The first part of

*Oblomov*by Ivan Goncharov (1859) describes a man who desires only to spend all his day in bed or lying on a sofa, meaning his \(\beta \to 1.\) - 15.
The linear formalization of the utility function is achieved by taking the natural logarithm and defining \(u_{t}^{P} : = \ln U_{t}^{P} ,\;\;u_{t}^{NP} : = \ln U_{t}^{NP} ,\;\) and \(u_{t}^{Pas} : = \ln U_{t}^{Pas} .\) Note that taking an order-preserving transformation of the target function does not affect the location of which it attains its maximum (Varian 2010, p. 55).

- 16.
In a steady state, capital goods are naturally proportionate to consumption, and hedonic capital is proportionate to active leisure (given by \(C_{t} = \frac{{\gamma c_{t} }}{1 - \gamma }\) and \(H_{t} = \frac{{\delta L_{t}^{\alpha } }}{1 - \delta }\), respectively). Thus, this conclusion follows directly from the concavity of the utility functions.

- 17.
It emerges that equation (ii) depends only on the relative productivity of work and productive leisure for producing hedonic capital (see “Appendix 1” for details).

- 18.
Research has revealed that more time devoted to internet use is positively associated with loneliness and negatively correlated to life satisfaction (Stepanikova et al. 2010). “Heavy” internet use in specific domains, such as gaming, is a predictor of lower levels of happiness and life satisfaction (Mitchell et al. 2011). The negative effect of consuming passive leisure may be due to lack of hedonic capital. These individuals do not invest time and creative effort in gaining productive capital, and therefore lack stimulating alternatives for leisure time.

- 19.
We use this anecdotal evidence, and additional cases presented below, to support the predictions of our model. We emphasize that while our model is consistent with all of our anecdotal evidence, there are other plausible explanations that are not included in the model. Future research might propose an alternative model that could also explain this evidence coherently.

- 20.
Naturally, there are other possible explanations for the “end of the party and back to reality.” One example is rising aspirations (Easterlin 2001). The winner might think that having more money would change dramatically his or her life for the better but increased aspirations can lead to disappointment. Other explanations might be the envy of friends or their inability to cope with the

*winner’s*increased demands, or and winner’s economic inability to “make ends meet.” The wealth shock is not enough to satisfy all winner's material desires. - 21.
They demonstrate that winners of USD 50,000 to USD 150,000 are more likely to file for bankruptcy three to five years after winning than winners of smaller prizes.

- 22.
Note that replacing, according equation (1) (which holds for every

*t*), \(J_{t+1}(C_{t+1},H_{t+1},W_{t+1})\) with \(u_{t+1}^{p}(C_{t+1}+c_{t+1})+u_{t+1}^{PAS}(L_{t+1}^{Pas})+u_{t+1}^{NP}(H_{t+1}+\zeta _{1}L_{t+1}^{Pro}+\zeta _{2}(\bar{T}-L^{Pas}-L^{Pro})+\rho \cdot J_{t+2}(C_{t+2},H_{t+2},W_{t+2})\) and continue in this manner (i.e. replacing \(J_{t+2}\) in the last equation according to equation (1) with a formal containing \(J_{t+2}\) and so forth) infinitely many times, one obtains the original maximization problem \(\max \sum _{0}^{\infty }\rho ^{t}U_{t}(.)\). - 23.
For the necessary and sufficient conditions of an Hamiltonian problem see, for example, Acemoglu (2008, p. 324).

- 24.
Note that the amount of wealth in the steady state path must also be constant for otherwise it will diverage. If, for example \(W_{t}>W_{t-1}\), then, \(W_{t+1}>W_{t}\) must also be the case due to higher interest rate gains (namely, because \(W_{t}\cdot r>W_{t-1}\cdot r\)).

- 25.
Note that the amount of wealth in the steady state path must also be constant for otherwise it will diverage. If, for example \(W_{t}>W_{t-1}\), then, \(W_{t+1}>W_{t}\) must also be the case due to higher interest rate gains (namely, because \(W_{t}\cdot r>W_{t-1}\cdot r\)).

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## Acknowledgements

We would like to thank the research authority at the School of Business Administration in the Academic College of Management Academic Studies (Israel) for financial support.

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## Appendices

### Appendix 1: Proof of Theorem 1

Let \(\zeta :=\zeta _{2}\) and define the productivity of productive leisure by \(\zeta _{1}\), then the Hamiltonian of the problem at any time *t* is given by:

- (1)
\(J_{t}=\alpha u_{t}^{p}(C_{t}+c_{t})+\beta u_{t}^{PAS}(L_{t}^{Pas})+(1-\alpha -\beta )u_{t}^{NP}(H_{t}+\zeta _{1}L_{t}^{Pro}\zeta _{2}(\bar{T}-L^{Pas}-L^{Pro}))+\rho \cdot J_{t+1}(C_{t+1},H_{t+1},W_{t+1})\)).

Subject to the laws of motion:

- (2)
\(W_{t}=W_{t-1}(1+r)+w(\bar{T}-L^{Pas}-L^{Pro})\)

- (3)
\(C_{t}= C_{t-1}+c_{t}\)

- (4)
\(H_{t}=H_{t-1}+L_{t}^{Pro}+\zeta T_{t}^{w}\).

where \(u_{t}^{p}+u_{t}^{PAS}+u_{t}^{NP}\) is, in terms of the Hamiltonian problem, the value function. *W*, *C*, and *H* are the state variables and \(L^{Pas}, L^{Pro}\), and *c* are the control variables, and \(J_{t+1}(C_{t+1},H_{t+1},W_{t+1}\)) is the Hamiltonian at time \(t+1\).^{Footnote 22} Replacing equations (2)–(4) into the Hamiltonian function, we obtain:

Differentiating with respect to the state variable, \(c_{t}\), \(L_{t}^{Pas}\), and \(L_{t}^{Pro}\), we obtain:^{Footnote 23}

- (I)
\(\alpha \cdot \frac{\partial u_{t}^{p}}{\partial c_{t}}+\rho (\gamma \cdot \frac{dJ_{t+1}}{dC_{t+1}}-\frac{dJ_{t+1}}{dW_{t+1}})=0.\)

- (II)
\(\beta \cdot \frac{\partial u_{t}^{PAS}}{\partial L_{t}^{Pas}}+\rho (-\delta \zeta _{2}\cdot \frac{dJ_{t+1}}{dH_{t+1}}-\frac{dJ_{t+1}}{dW_{t+1}}\cdot w)=0.\)

- (III)
\((1-\alpha -\beta )\frac{\partial u_{t}^{NP}}{\partial L_{t}^{Pro}}+\rho (\delta (\zeta _{1}-\zeta _{2})\frac{dJ_{t+1}}{dH_{t+1}}-\frac{dJ_{t+1}}{dW_{t+1}}\cdot w)=0.\)

Differentiating with respect to the state variables, \(C_{t}\) and \(H_{t}\), we obtain:

- (IV)
\(\frac{\frac{dJ_{t}}{dC_{t}}=}{{\partial u_{t}^{p}}{\partial c_{t}}+\rho \gamma \cdot \frac{dJ_{t+1}}{dC_{t+1}}}\)

- (V)
\(\frac{dJ_{t}}{dH_{t}}=(1-\alpha -\beta )\frac{\frac{\partial u_{t}^{NP}}{\partial L_{t}^{Pro}}}{\zeta _{1}-\zeta _{2}}+\rho (\delta \cdot \frac{dJ_{t+1}}{dH_{t+1}}-\frac{dJ_{t+1}}{dW_{t+1}}\cdot w).\)

We now look for the steady state, that is, for a saddle point in which the state variables are constant overtime. First note that since in the steady state \(C_{t}=C_{t+1}\), \(c_{t}=c_{t+1}\) must also be the case, and hence \(\frac{\partial u_{t}^{p}}{\partial c_{t}}=\frac{\partial u_{t+1}^{p}}{\partial c_{t+1}}\) holds.

Now, iterating equation (IV) infinitely many times, we obtain: \(\frac{dJ_{t}}{dC_{t}}=\varSigma _{i=t}^{\infty }\rho ^{i-t}\cdot \frac{ \partial u_{t}^{p}}{\partial c_{t}}\), which is independent of *t*. This shows that in steady state \(\frac{dJ_{t}}{dC_{t}}=\frac{dJ_{t+1}}{dC_{t+1}}\) for every *t*. In the exact same way, it is possible to show that in steady state \(\frac{ dJ_{t}}{dH_{t}}=\frac{dJ_{t+1}}{dH_{t+1}}\) and \(\frac{dJ_{t}}{dW_{t}}=\frac{ dJ_{t+1}}{dW_{t+1}}\) for every *t*.

Replacing \(\frac{dJ_{t}}{dC_{t}}=\frac{dJ_{t+1}}{dC_{t+1}}\) and \(\frac{dJ_{t} }{dH_{t}}=\frac{dJ_{t+1}}{dH_{t+1}}\) in equations (IV) and (V), respectively, we obtain:

- (VI)
\(\frac{dJ_{t}}{dC_{t}}=\alpha \frac{\frac{\partial u_{t}^{p}}{\partial c_{t}}}{1-\rho \gamma }.\)

- (VII)
\(\frac{dJ_{t}}{dH_{t}}=(1-\alpha -\beta )\frac{\frac{\partial u_{t}^{NP}}{\partial L_{t}^{Pro}}}{(\zeta _{1}-\zeta _{2})(1-\rho \delta )}.\)

Replacing \(\frac{dJ_{t}}{dC_{t}}\) and \(\frac{dJ_{t}}{dH_{t}}\) from equation (VI) and (VII) into equations (I)-(III), we obtain:

- (VIII)
\(\frac{\alpha \cdot \frac{\partial u_{t}^{p}}{\partial c_{t}}}{1-\rho \gamma }=\rho \cdot \frac{dJ_{t+1}}{dW_{t+1}}\).

- (IX)
\(\frac{\beta \cdot \partial u_{t}^{PAS}}{\partial L_{t}^{Pas}}\cdot \frac{\zeta _{1}(1-\rho \delta )-\zeta _{2}}{(\zeta _{1}-\zeta _{2})((1-\rho \delta )}=\rho w\cdot \frac{dJ_{t+1}}{dW_{t+1}}\).

- (X)
\(\frac{(1-\alpha -\beta )\frac{\partial u_{t}^{NP}}{\partial L_{t}^{Pro}}}{1-\rho \delta }=\rho w\cdot \frac{dJ_{t+1}}{dW_{t+1}}\). Dividing (VIII) by (X) we obtain equation (i) in Theorem 1, and dividing (IX) by (X) equation (ii) is obtained, where \(\zeta :=\frac{\zeta _{2}}{\zeta _{1}}\). This completes the proof.

### Appendix 2

See Table 1.

### Appendix 3: An numerical example

In this appendix, we provide a closed form solution to a special case of our model. First, we follow Graham and Oswald (2010) by assuming that the utility function is logarithmic. Merely for the sake of expositional simplicity, we also assume that \(\gamma =\delta\) (i.e. that the capital goods and the hedonic capital are depreciating in the same rate), \(\zeta =0\) (i.e. that there is no value from work *per se*), and, prior to wining, we assume that \(W_{0}=0\); further, we normalize \(w=1\) .

Replacing these parameters in equations (i) and (ii) in Theorem 1, we obtain the following amount of \(c_{t}\), \(L_{t}^{Pas}\), and \(L_{t}^{pro}\) in steady state:^{Footnote 24}

The amount of capital goods and hedonic capital is then given by:

Finally, the overall utility is given by:

We first stimulate case 1 in which winning does not cause any change in the decision utility’s parameters. The exogenous shock in wealth (denoted by \(A>0\)) increases wealth from \(W_{t}=0\) to \(W_{t+1}=A\). The new steady state amounts are then given by:^{Footnote 25}

and the utility level in the steady state is increased to:

We now proceed with what we find to be the most interesting scenario in which \(\varepsilon\) is reduced in the decision utility (to a level denoted by \(\varepsilon ^{\prime }\)) but experienced utility does not change (as well as the other parameters),

The overall utility is then given by:

\(V_{t}=(\ln \frac{\bar{T}(1-\rho \delta (\alpha -\varepsilon ^{\prime }-\alpha \varepsilon ^{\prime })}{(1-(\alpha +\varepsilon ^{\prime })\rho \delta )(1-\delta )}+\ln (Ar))^{\alpha }\cdot (\ln \frac{\bar{T}\beta }{1-(\alpha +\varepsilon ^{\prime })\rho \delta }+\ln (Ar))^{\beta }+(\ln\)\(\frac{\bar{T}\alpha \varepsilon ^{\prime }\rho \delta }{1-(\alpha +\varepsilon ^{\prime })\rho \delta }+\ln (Ar))^{\varepsilon }\). Note that the power \(\varepsilon\) is used rather than \(\varepsilon ^{\prime }\) because there is no change in experienced utility’s parameters. The question of interest is for which set of parameters, the agent is experiencing a reduction in level of utility, after winning. To answer this question, we fix \(\bar{T}=1\), \(Ar=10\), \(\rho ,\delta =0.9\), \(\alpha =\frac{1}{3},\)\(\beta =1-\alpha -\varepsilon\), and we plot (see Fig. 1) \(U_{t}\) and \(V_{t}\) for different values of \(\varepsilon ^{\prime }<\varepsilon\):

The levels of experienced utility prior to winning (dashed line),and after winning (solid line) as a function of \(\varepsilon\) for different values of \(\varepsilon ^{\prime }\).

As one can observe from the figure, for large enough \(\varepsilon\), there is a small enough \(\varepsilon ^{\prime }\) such that \(V_{t}<U_{t}\), and as \(\varepsilon\) grows larger, a smaller reduction in \(\varepsilon ^{\prime }\) is sufficient for this. That is, as the hedonic capital becomes relatively more important prior to winning, a smaller change in the decision utility’s parameters is sufficient to create a reduction of experienced utility after winning.

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Sherman, A., Shavit, T. & Barokas, G. A Dynamic Model on Happiness and Exogenous Wealth Shock: The Case of Lottery Winners.
*J Happiness Stud* **21, **117–137 (2020). https://doi.org/10.1007/s10902-019-00079-w

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### Keywords

- Wealth shock
- Dynamic model
- Hedonic capital
- Decision utility
- Experience utility