Giving and volunteering over a lifecycle

Abstract

Charitable giving in the US is substantial, contributing 2% of the annual GDP. I develop a lifecycle model of warm-glow giving for consumers who derive utility from both acts of giving and volunteering, and explore the general equilibrium characteristics of an economy of these pro-social consumers. By separating the charitable deduction rate from the income tax rate as well as identifying non-separable utility not only between consumption and charitable giving, but also between giving and volunteering, the model unambiguously determines the direction of welfare from any particular change in a tax system, illuminating the role of policy in the private provision of public goods. Subject to mortality risks, the consumers are enabled to endogenously choose their retirement age, revealing salient features regarding lifecycle giving/volunteering on top of consumption/leisure behaviors in a calibrated OLG equilibrium. A simulation result shows that given an income tax rate, an increase in deduction rate increases not only giving but also output and consumption, due to greater labor supply. Reasonable parameterization of the model confers the highest level of welfare for the tax rate of about 23% given an equal rate for the income tax and charitable deduction.

Appendix

This appendix introduces a tractable version of the warm glow model so that a closed form solution can be obtained. Consider an agent who gets utility from the act of giving in the form of both goods and time as described in Section 2. The agent who is subject to mortality risk Q_t may live up to T-period and each period he has one unit of time endowment, as well as a stream of productivity measured in efficiency units. The optimization problem of the agent at t = 0 is,

$$Ma{x}_{\{{x}_{t},{g}_{t},{l}_{t},{v}_{t}\}}\mathop{\sum }\limits_{t=0}^{T}{\beta }^{t}{Q}_{t}u({x}_{t},{g}_{t},{l}_{t},{v}_{t})$$

subject to

$${x}_{t}+(1-{t}_{g}){g}_{t}+{b}_{t+1}=(1-{t}_{w})[w{e}_{t}(1-{l}_{t}-{v}_{t})]+[1+(1-{t}_{r})r]{b}_{t}+{B}_{t}^{y}$$

(A1)

$$\begin{array}{l}{x}_{t},\,{g}_{t},\,{l}_{t},\,{v}_{t}\ge 0,1\ge {l}_{t}+{v}_{t}\ge 0\\ {b}_{0}=0,{b}_{T+1}=0\end{array}$$

where x_t is consumption, g_t is charitable giving, l_t is leisure, and v_t is volunteering. Rewrite the resource constraint to get

$${x}_{t}+(1-{t}_{g}){g}_{t}+(1-{t}_{w})w{e}_{t}[{l}_{t}+{v}_{t}]+{b}_{t+1}=(1-{t}_{w})w{e}_{t}+[1+(1-{t}_{r})r]{b}_{t}+{B}_{t}^{y}$$

(A2)

Let M_t ≡ x_t + (1 − t_g)g_t + (1 − t_w)we_t[l_t + v_t]. Then the maximization problem is decomposed into two sub-problems:

(i) Intra-temporal optimization

$$V\left(M,we\right)=Ma{x}_{x,l,g,v}u\left(x,g,l,v\right)$$

(A3)

subject to

$$\begin{array}{l}x+(1-{t}_{g})g+(1-{t}_{w})we[l+v]=M\\ x,g,l,v\ge 0,\,1\ge l+v\ge 0\end{array}$$

(ii) Inter-temporal optimization

$$Ma{x}_{{M}_{t,}{b}_{t+1}{{\mbox{}}}}\mathop{\sum }\limits_{t=0}^{T}{\beta }^{t}{Q}_{t}V\left({M}_{t}{{\mbox{}}},w{e}_{t}\right)$$

(A4)

s u b j e c t t o

$$\begin{array}{l}{M}_{t}+{b}_{t+1}=(1-{t}_{w})w{e}_{t}+[1+(1-{t}_{r})r]{b}_{t}+{B}_{t}^{y}\\ {b}_{0}=0,{b}_{T+1}=0\end{array}$$

By solving the first problem, one gets an optimal consumption-giving and leisure-volunteering amount as a function of total resource level M committed to spend each time. Let us assume CRRA utility function that satisfies the Inada condition:

$$u(x,g,l,v)=\left\{\begin{array}{l}\frac{1}{1-\gamma }{[{c}^{\eta }{h}^{1-\eta }]}^{1-\gamma }\,{{\mbox{if}}}\,\gamma \,\ne\, 1\\ \\ \ln \,{{\mbox{}}}{c}^{\eta }{h}^{1-\eta }\;{{\mbox{if}}}\,\gamma =1\end{array}\right\}$$

(A5)

where c = x^θg^1−θ represents the total consumption of goods, and the total hours not spent on labor is h = l^σv^1−σ, with η ϵ(0, 1), θ ϵ(0, 1), σϵ(0, 1) and γ > 0. From the first order condition, it is satisfied that

$$\frac{x}{(1-{t}_{g})\theta }=\frac{g}{(1-\theta )}$$

(A6)

$$\frac{l}{\sigma }=\frac{v}{(1-\sigma )}$$

(A7)

These conditions dictate the relative importance of the two arguments in each utility component of goods and hours: c = x^θg^1−θ and h = l^σv^1−σ. Giving (g) and volunteering (v) can be written in terms of consumption (x) and leisure (l), or vice versa. Rewriting the agent’s intra-temporal problem in terms of (x, l) for

$$Ma{x}_{x,l}\frac{D}{1-\gamma }{\left({x}^{\eta }{l}^{1-\eta }\right)}^{1-\gamma }$$

(A8)

subject to

$$\begin{array}{l}\left[1+\frac{1-\theta }{\theta }\right]x+(1-{t}_{w})we\left[1+\frac{1-\sigma }{\sigma }\right]l=M\\ \sigma \ge l\ge 0\end{array}$$

for a constant $D={({A}^{{}^{\eta }}{B}^{1-\eta })}^{1-\gamma }$, where $A={\left(\frac{1-\theta }{(1-{t}_{g})\theta }\right)}^{1-\theta }$ and $B={\left(\frac{1-\sigma }{\sigma }\right)}^{1-\sigma }.$ Notice that the leisure is constrained by a fixed variable σ. Once again, the resource constraint can be reduced to a simpler form of $\,\frac{1}{\theta }x+(1-{t}_{w})we(\frac{1}{\sigma })l=M.$ With this preliminary work, the model is analyzed via intra-temporal and inter-temporal optimization.

A. Solving the Intra-temporal optimization

$$V\left(M,we\right)=Ma{x}_{x,l}\frac{D}{1-\gamma }{\left({x}^{\eta }{l}^{1-\eta }\right)}^{1-\gamma }$$

(A9)

subject to

$$\begin{array}{l}\frac{1}{\theta }x+(1-{t}_{w})we(\frac{1}{\sigma })l=M\\ x\ge 0,\sigma \ge l\ge 0\end{array}$$

Using the resource constraint, $\frac{1}{\theta }x+(1-{t}_{w})we(\frac{1}{\sigma })l=M,$ the optimal consumption and leisure levels are derived for any specific resource level and wage rate. Because it is satisfied that (1 − t_w)we < (1 − η)M when leisure binds, the solution is determined by the threshold level ${M}^{* }\equiv \frac{(1-{t}_{w})we}{1-\eta }.$ Given we, depending on whether the leisure is binding or not, the optimal choice is:

$$x\left(M,we\right)=\left\{\begin{array}{l}\theta \eta M\,\,if\,\,M\le {M}^{* }\\ \theta [M-(1-{t}_{w})we]\,\,if\,\,M \,>\, {M}^{* }\end{array}\right\}$$

(A10)

$$l\,\left(M,we\right)=\left\{\begin{array}{l}\sigma \left(1-\eta \right)\frac{M}{(1-{t}_{w})we}\,\,if\,M\le {M}^{* }\\ \sigma \,\,if\,\,M \,>\, {M}^{* }\end{array}\right\}$$

(A11)

Because $g=\frac{(1-\theta )}{(1-{t}_{g})\theta }x$ and $v=\frac{(1-\sigma )}{\sigma }l,$ it is also true that

$$g\left(M,we\right)=\left\{\begin{array}{l}(1-\theta )\eta \frac{M}{(1-{t}_{g})}\,\,if\,\,M\le {M}^{* }\\ \frac{1-\theta }{1-{t}_{g}}[M-(1-{t}_{w})we]\,\,if\,\,M \,>\, {M}^{* }\end{array}\right\}$$

(A12)

$$v\left(M,we\right)=\left\{\begin{array}{l}(1-\sigma )\left(1-\eta \right)\frac{M}{(1-{t}_{w})we}\,\,if\,\,M\le {M}^{* }\\ {{\mbox{}}}1-\sigma \,\,if\,\,M \,>\, {M}^{* }\end{array}\right\}$$

(A13)

The indirect utility function of the optimization is given by

$$V\left(M,we\right)=$$

$$\left\{\begin{array}{c}{{\mbox{}}}\frac{D}{1-\gamma }{\left[{\left(\theta \eta \right)}^{\eta }{\left(\frac{\sigma \left(1-\eta \right)}{(1-{t}_{w})we}\right)}^{1-\eta }M\right]}^{1-\gamma }\,if\,M\le {M}^{* }\\ \frac{D}{1-\gamma }{\left[{\theta }^{\eta }{[M-(1-{t}_{w})we]}^{\eta }{\sigma }^{1-\eta }\right]}^{1-\gamma }\,if\,M \,>\, {M}^{* }\end{array}\right\}$$

(A14)

B. Solving the Inter-temporal optimization

Let the time-indexed resource level be ${M}_{t}=\frac{1}{\theta }{x}_{t}+\left(\frac{(1-{t}_{w})w{e}_{t}}{\sigma }\right){l}_{t}$ and let $V\left({M}_{t},w{e}_{t}\right)$ be the indirect utility function obtained from the analysis in section A. Then the maximization problem for the agent over time is

$$Ma{x}_{{M}_{t,}{b}_{t+1}{{\mbox{}}}}\mathop{\sum }\limits_{t=0}^{T}{\beta }^{t}{Q}_{t}V\left({M}_{t},w{e}_{t}\right)$$

(A15)

s u b j e c t t o

${M}_{t}+{b}_{t+1}=(1-{t}_{w})w{e}_{t}+[1+(1-{t}_{r})r]{b}_{t}+{B}_{t}^{y}$

b₀ = 0, b_T+1 = 0

The first order condition is

$${V}_{M}\left({M}_{t},w{e}_{t}\right)=\frac{{\lambda }_{0}}{{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}}$$

(A16)

where λ₀ is the Lagrange multiplier of the optimization at time 0 and ${V}_{M}=\frac{\partial V}{\partial M}.$ Given a value of we_t, because the derivative of the value function ${V}_{M}\left({M}_{t},w{e}_{t}\right)$ is strictly decreasing in M_t, there exists an inverse function over the domain. Let ${{\Psi }}={V}_{M}^{-1}\left({M}_{t},w{e}_{t}\right)$ be the inverse function. From the first order condition above, it is true that

$${M}_{t}={V}_{M}^{-1}\left(\frac{{\lambda }_{0}}{{Q}_{t}{[\beta \left(1\right.+(1-{t}_{r})r]}^{t}},w{e}_{t}\right)={{\Psi }}\left(\frac{{\lambda }_{0}}{{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}},w{e}_{t}\right)$$

(A17)

Plugging M_t back into the constraint to get

$$\mathop{\sum }\limits_{t=0}^{T}\frac{{{\Psi }}\left(\frac{{\lambda }_{0}}{{{Q}_{t}[\beta \left(1\right.+(1-{t}_{r})r]}^{t}},w{e}_{t}\right)}{{[1+(1-{t}_{r})r]}^{t}}=\mathop{\sum }\limits_{t=0}^{T}\frac{(1-{t}_{w})w{e}_{t}+{B}_{t}^{y}}{{[1+(1-{t}_{r})r]}^{t}}$$

(A18)

By solving for λ₀, the optimal M_t is obtained: ${M}_{t}={{\Psi }}\left(\frac{{\lambda }_{0}^{* }}{{Q}_{t}{[\beta \left(1\right.+(1-{t}_{r})r]}^{t}},w{e}_{t}\right),$ where ${\lambda }_{0}^{* }$ is the value solved from the constraint equation.

Next, regarding the choice of working hours related to retirement, notice that the agent freely chooses his optimal working hours, and thus optimal leisure/volunteer hours each period. The paper explores the conditions under which the agent chooses either to retire early or postpone entering into the labor market. Revisiting the equation (A16), ${V}_{M}\left({M}_{t},w{e}_{t}\right)=\frac{{\lambda }_{0}}{{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}},$ the condition can be rewritten into

$$D{\left[{\left(\theta \eta \right)}^{\eta }{\left(\frac{\sigma \left(1-\eta \right)}{(1-{t}_{w})w{e}_{t}}{{\mbox{}}}\right)}^{1-\eta }\right]}^{1-\gamma }{M}_{t}^{-\gamma }=\frac{{\lambda }_{0}}{{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}}$$

(A19)

or

$${M}_{t}={\left(\frac{{\lambda }_{0}}{{\kappa }_{t}{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}}\right)}^{-\frac{1}{\gamma }}$$

(A20)

where ${\kappa }_{t}=D{\left[{\left(\theta \eta \right)}^{\eta }{\left(\frac{\sigma \left(1-\eta \right)}{(1-{t}_{w}){we}_{t}}\right)}^{1-\eta }\right]}^{1-\gamma }$. Combining this with the constraint gives the solution for λ₀, which is

$${\lambda }_{0}^{* }={\left(\frac{\mathop{\sum }\nolimits_{t = 0}^{T}\frac{(1-{t}_{w})w{e}_{t}+{B}_{t}^{y}}{{[1+(1-{t}_{r})r]}^{t}}}{\mathop{\sum }\nolimits_{t = 0}^{T}{\kappa }_{t}^{\frac{1}{\gamma }}{Q}_{t}^{\frac{1}{\gamma }}{\left[\frac{{\left[\beta \left(1+(1-{t}_{r})r\right)\right]}^{1/\gamma }}{1+(1-{t}_{r})r}\right]}^{t}}\right)}^{-\gamma }={\left(\frac{\mathop{\sum }\nolimits_{t = 0}^{T}\frac{(1-{t}_{w})w{e}_{t}+{B}_{t}^{y}}{{[1+(1-{t}_{r})r]}^{t}}}{\mathop{\sum }\nolimits_{t = 0}^{T}{\kappa }_{t}^{\frac{1}{\gamma }}{Q}_{t}^{\frac{1}{\gamma }}{\left[\frac{1}{{\phi }^{* }}\right]}^{t}}\right)}^{-\gamma }$$

(A21)

where $\frac{1}{{\phi }^{* }}=\frac{{\left[\beta \left(1+(1-{t}_{r})r\right)\right]}^{1/\gamma }}{1+(1-{t}_{r})r}.$ Therefore, the optimal resource commitment schedule for work and partial leisure/volunteer, {σ > l > 0, 1 − σ > v > 0}, or no work and full leisure/volunteer, {σ = l > 0, 1 − σ = v > 0} is:

$${M}_{t}^{{{\mbox{}}}}=\left\{\begin{array}{l}{{\mbox{}}}{\left(\frac{{\lambda }_{0}^{* }}{{\kappa }_{t}{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}}\right)}^{-\frac{1}{\gamma }}{{\mbox{}}}if\,\,{M}_{t}\le {M}_{t}^{* }{{\mbox{}}}\ \iff \ \frac{{\lambda }_{0}^{* }}{{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}}{{\mbox{}}}\ge {q}_{t}^{* }\\ {\left(\frac{{\lambda }_{0}^{* }}{{\overline{\kappa }}_{t}{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}}\right)}^{-\frac{1}{\overline{\gamma }}}+(1-{t}_{w})w{e}_{t}{{\mbox{}}}\;if\,\,{M}_{t} \,>\, {M}_{t}^{* }\ \iff \ \frac{{\lambda }_{0}^{* }}{{Q}_{t}{[\beta \left(1+(1-{t}_{r})r\right)]}^{t}} \,<\, {q}_{t}^{* }\end{array}\right\}$$

(A22)

where ${q}_{t}^{* }={\kappa }_{t}{\left(\frac{(1-{t}_{w}){we}_{t}}{1-\eta }\right)}^{-\gamma }=\overline{\kappa }{\left(\frac{(1-{t}_{w}){we}_{t}}{1-\eta }\right)}^{-\overline{\gamma }}$ with $\overline{\kappa }=D\eta {\theta }^{\eta (1-\gamma )}{\sigma }^{{}^{(1-\eta )(1-\gamma )}}$ and $\overline{\gamma }=1-\eta \left(1-\gamma \right).$