A Dynamic Model on Happiness and Exogenous Wealth Shock: The Case of Lottery Winners

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.

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 $t+1$.²² Replacing equations (2)–(4) into the Hamiltonian function, we obtain:

$$\begin{aligned} J_{t}&= {} \alpha u_{t}^{p}(C_{t}+c_{t})+\beta u_{t}^{Pas}(L_{t}^{Pas})+(1-\alpha -\beta )u_{t}^{NP}(H_{t}+\zeta _{1}L_{t}^{Pro}+\zeta _{2}(\bar{T}-L^{Pas}-L^{Pro})) \\&\quad+\rho \cdot J_{t+1}(\gamma (C_{t}+c_{t}),\delta (H_{t}+\zeta _{1}L_{t}^{Pro}+\zeta _{2}(\bar{T}-L^{Pas}-L^{Pro})),W_{t}-c_{t}+w(\bar{T}-L_{t}^{Pas}-L_{t}^{Pro})). \end{aligned}$$

Differentiating with respect to the state variable, $c_{t}$, $L_{t}^{Pas}$, and $L_{t}^{Pro}$, we obtain: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.

Table 1 Symbols and their meanings

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:²⁴

$$\begin{aligned}&c_{t}=(\bar{T}+Ar)\left(1-\frac{\beta +\alpha \varepsilon \rho \delta }{1-(\alpha +\varepsilon )\rho \delta }\right).\\&L_{t}^{Pas}=\frac{\beta \bar{T}}{1-(\alpha +\varepsilon )\rho \delta }.\\&L_{t}^{pro}=\frac{\alpha \varepsilon \rho \delta \bar{T}}{1-(\alpha +\varepsilon )\rho \delta }. \end{aligned}$$

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

$$\begin{aligned}&C_{t}=\frac{\bar{T}(1-\rho \delta (\alpha -\varepsilon -\alpha \varepsilon ))}{(1-(\alpha +\varepsilon )\rho \delta )(1-\delta )},\\&H_{t}=\frac{\bar{T}\beta }{(1-(\alpha +\varepsilon )\rho \delta )(1-\delta )}. \end{aligned}$$

Finally, the overall utility is given by:

$$\begin{aligned} U_{t}=(\ln \frac{\bar{T}(1-\rho \delta (\alpha -\varepsilon -\alpha \varepsilon )}{(1-(\alpha +\varepsilon )\rho \delta )(1-\delta )})^{\alpha }\cdot (\ln \frac{\bar{T}\beta }{1-(\alpha +\varepsilon )\rho \delta })^{\beta }+(\ln \frac{\bar{T}\alpha \varepsilon \rho \delta }{1-(\alpha +\varepsilon )\rho \delta })^{\varepsilon }. \end{aligned}$$

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:²⁵

$$\begin{aligned}&c_{t}=(\bar{T}+Ar)(1-\frac{\beta +\alpha \varepsilon \rho \delta }{1-(\alpha +\varepsilon )\rho \delta }).\\&L_{t}^{Pas}=\frac{\beta (\bar{T}+Ar)}{1-(\alpha +\varepsilon )\rho \delta }.\\&L_{t}^{pro}=\frac{\alpha \varepsilon \rho \delta (\bar{T}+Ar)}{1-(\alpha +\varepsilon )\rho \delta }, \end{aligned}$$

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

$$\begin{aligned} V_{t}=(\ln \frac{\bar{T}(1-\rho \delta (\alpha -\varepsilon -\alpha \varepsilon )}{(1-(\alpha +\varepsilon )\rho \delta )(1-\delta )}+\ln (Ar))^{\alpha }\cdot (\ln \frac{\bar{T}\beta }{1-(\alpha +\varepsilon )\rho \delta }+\ln (Ar))^{\beta }+(\ln \frac{\bar{T}\alpha \varepsilon \rho \delta }{1-(\alpha +\varepsilon )\rho \delta }+\ln (Ar))^{\varepsilon } \end{aligned}$$

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 $\varepsilon ^{\prime }<\varepsilon$:

Fig. 1

Experienced utility prior and after winning

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.