Robust consumption policy with the desire for wealth accumulation

Abstract

We study the optimal consumption-saving problem for consumers who are ambiguous about labor income shocks and have a preference for (absolute) wealth. It is shown that the preference for wealth interacts with the degree of ambiguity in non-trivial ways in determining the optimal consumption. The model predicts that ambiguity and preference for wealth can produce substitution or complementarity effects on consumption. In addition, we find that the interactions of ambiguity and desire for wealth generate different implications for MPCs. We then show that these interactions significantly influence the contribution of ambiguity to excess sensitivity of consumption growth and the contribution of preference for wealth to excess smoothness of consumption.

Appendices

Appendix: Proof of proposition

This appendix provides technical details for the main results of the paper. The homogeneity property of the value function holds for the cases in our paper. Therefore, we conjecture that the value function is given by (15). Further, we write the certainty equivalent wealth as F(W, Y) = f(w)Y. Additionally, we have

$${J}_{W}={b}^{1-\gamma }{(f(w)Y)}^{-\gamma }f^{\prime} (w)\ ,$$

(A.1)

$${J}_{Y}={b}^{1-\gamma }{(f(w)Y)}^{-\gamma }\left(f(w)-wf^{\prime} (w)\right)\ ,$$

(A.2)

$${J}_{WW}={b}^{1-\gamma }{(f(w)Y)}^{-1-\gamma }\left(f(w)f^{\prime\prime} (w)-\gamma {(f^{\prime} (w))}^{2}\right)\ ,$$

(A.3)

$${J}_{WY}={b}^{1-\gamma }{(f(w)Y)}^{-1-\gamma }\left(-wf(w)f^{\prime\prime} (w)-\gamma f^{\prime} (w)\left(f(w)-wf^{\prime} (w)\right)\right)\ ,$$

(A.4)

$${J}_{YY}={b}^{1-\gamma }{(f(w)Y)}^{-1-\gamma }\left({w}^{2}f(w)f^{\prime\prime} (w)-\gamma {\left(f(w)-wf^{\prime} (w)\right)}^{2}\right)\ .$$

(A.5)

Using the FOC (14) and the expression (A.1), we obtain (18). Substituting (18), the value function (15), (A.1), (A.2) and (A.5) into the HJB equation (13), and simplifying, we obtain the ODE (20). Now we turn to analyze the boundary conditions.

First, substituting w = 0 into ODE (20), we have (23). Second, the borrowing constraint implies 0 < c(0) ≤ 1. Finally, when the wealth-income ratio w approaches infinity, non-diversifiable risk no longer matters for consumption. Therefore, the certainty equivalent wealth is given by

$$\mathop{\mathrm{lim}}\limits_{w\to \infty }f(w)=w+h\ .$$

(A.6)

Corollary 1

In the limit w → ∞, the optimal consumption-income ratio c(w) with ambiguity and/or capitalist spirit is always lower than that with neither of them.

Proof With ambiguity and/or capitalist spirit, consumption-income ratio c(w) is

$$c(w)={b}^{1-\frac{1}{\gamma }}\left(w+h\right)-\lambda w=\left(r+\lambda +\frac{\rho -(r+\lambda )}{\gamma }\right)\left(w+\frac{1}{r+\lambda -\mu +\sqrt{2\eta }\sigma }\right)-\lambda w\ .$$

(A.7)

If neither ambiguity nor the spirit of capitalism exists, consumption-income ratio c^nf(w) is

$${c}^{nf}(w)=\left(r+\frac{\rho -r}{\gamma }\right)\left(w+\frac{1}{r-\mu }\right)\ .$$

(A.8)

A simple computation yields

$$c(w)-{c}^{nf}(w)=-\frac{\lambda w}{\gamma }+\left(\frac{r+\lambda +\frac{\rho -(r+\lambda )}{\gamma }}{r+\lambda -\mu +\sqrt{2\eta }\sigma }-\frac{r+\frac{\rho -r}{\gamma }}{r-\mu }\right)\ .$$

(A.9)

Define a function as

$$N(x)=\frac{x+\frac{\rho -x}{\gamma }}{x-\mu }\ .$$

(A.10)

The first-order derivative of N(x) with respect to x

$$N^{\prime} (x)=\frac{-\gamma [\rho +(\gamma -1)\mu ]}{{[\gamma (x-\mu )]}^{2}}\, < \, 0$$

(A.11)

implies that N(x) decreases with x. Thus we have

$$N(r)=\frac{r+\frac{\rho -r}{\gamma }}{r-\mu }>N(r+\lambda )=\frac{r+\lambda +\frac{\rho -(r+\lambda )}{\gamma }}{r+\lambda -\mu }>\frac{r+\lambda +\frac{\rho -(r+\lambda )}{\gamma }}{r+\lambda -\mu +\sqrt{2\eta }\sigma }\ .$$

(A.12)

Thus c(w) − c^nf(w) < 0 for all w.

Alternative specification of U(C, W)

In this Appendix, we attempt to use other forms of U(C, W) to demonstrate the robustness of the main results. Similarly, these forms capture the concept of capitalist spirit and satisfy Markowitz hypothesis.

Case 1: In this case, the form of utility function U(C, W) is given by

$$U(C,W)=\frac{{C}^{1-\gamma }}{1-\gamma }{W}^{-\lambda }\ ,$$

(B.1)

where λ ≥ 0 when γ ≥ 1, and λ < 0 otherwise. ∣λ∣ measures the agent’s concern with her social status or measures her spirit of capitalism. The larger the parameter ∣λ∣, the stronger the agent’s spirit of capitalism or concern for social status.

Similarly, let

$${J}_{2}(W,Y)=\frac{{({b}_{2}F(W,Y))}^{1-\gamma }}{1-\gamma }=\frac{{({b}_{2}f(w)Y)}^{1-\gamma }}{1-\gamma }$$

(B.2)

denote the agent’s value function. In a similar way, the ODE for f(w) is obtained as follows:

$$\begin{array}{l}0=\left[\frac{\gamma {b}_{2}^{1+\frac{\lambda }{\gamma }-\frac{1}{\gamma }}{w}^{-\frac{\lambda }{\gamma }}f{(w)}^{\frac{\lambda }{\gamma }}{\left(f^{\prime} (w)\right)}^{1-\frac{1}{\gamma }}}{1-\gamma }-\frac{\rho }{1-\gamma -\lambda }+\left(\mu -\sqrt{2\eta }\sigma \right)-\frac{(\gamma +\lambda ){\sigma }^{2}}{2}\right]f(w)\\ +\, f^{\prime} (w)+\left[r-\left(\mu -\sqrt{2\eta }\sigma \right)+(\gamma +\lambda ){\sigma }^{2}\right]wf^{\prime} (w)+\frac{{\sigma }^{2}{w}^{2}}{2}\left(f^{\prime\prime} (w)-(\gamma +\lambda )\frac{f^{\prime} {(w)}^{2}}{f(w)}\right)\ ,\end{array}$$

where b₂ is

$${b}_{2}={\left[\frac{1-\gamma }{\gamma }\left(\frac{\rho }{1-\gamma -\lambda }-r\right)\right]}^{\frac{\gamma }{\gamma +\lambda -1}}\ .$$

(B.3)

Figure 14 states that the results in the main text are robust.

Fig. 14

Optimal consumption-income ratio c(w). The utility function is given as Eq. (B.1). The baseline parameters are r = 3.5%, μ = 1.5%, σ = 10%, ρ = 4%, and γ = 3. A The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0) and (1, 0). B The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0) and (0, 0.05). C The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0.05) and (1, 0.05)

Case 2: In this case, the form of utility function U(C, W) is given by

$$U(C,W)=\frac{{\left({C}^{\theta }{W}^{1-\theta }\right)}^{1-\gamma }}{1-\gamma }.$$

(B.4)

The utility function in Eq. (B.4) embodies the feature of the spirit of capitalism 0 < θ < 1. If the spirit of capitalism is absent (θ = 1), then the utility function in Eq. (B.4) reduces to the standard form.

Let

$${J}_{3}(W,Y)=\frac{{({b}_{3}F(W,Y))}^{1-\gamma }}{1-\gamma }=\frac{{({b}_{3}f(w)Y)}^{1-\gamma }}{1-\gamma }$$

(B.5)

denote the agent’s value function. By a similar way, the ODE for f(w) is obtained as follows:

$$\begin{array}{l}0=\left(-\frac{\rho }{1-\gamma }+\mu -\sqrt{2\eta }\sigma -\frac{\gamma {\sigma }^{2}}{2}\right)f(w)+{{\Theta }}{w}^{\frac{-(1-\theta )(1-\gamma )}{\theta (1-\gamma )-1}}f{(w)}^{-\frac{\gamma }{\theta (1-\gamma )-1}}f^{\prime} {(w)}^{\frac{\theta (1-\gamma )}{\theta (1-\gamma )-1}}+f^{\prime} (w)\\ +\,\left(r+\lambda -\left(\mu -\sqrt{2\eta }\sigma \right)+\gamma {\sigma }^{2}\right)wf^{\prime} (w)+\frac{{\sigma }^{2}{w}^{2}}{2}\left(f^{\prime\prime} (w)-\gamma \frac{f^{\prime} {(w)}^{2}}{f(w)}\right)\ ,\end{array}$$

(B.6)

where Θ is given by

$${{\Theta }}={\left(\frac{{b}_{3}^{1-\gamma }}{\theta }\right)}^{\frac{1}{\theta (1-\gamma )-1}}\left(\frac{1}{\theta (1-\gamma )}-1\right),$$

(B.7)

and b₃ is given by

$${b}_{3}={\theta }^{\frac{1}{1-\gamma }}{\left[\frac{\rho -r(1-\gamma )}{\frac{1}{\theta }-(1-\gamma )}\right]}^{\frac{\theta (1-\gamma )-1}{1-\gamma }}.$$

(B.8)

Figure 15 also demonstrates that the main results are robust.

Fig. 15

Optimal consumption-income ratio c(w). The utility function is given as Eq. (B.4). The baseline parameters are r = 3.5%, µ = 1.5%, σ = 10%, ρ = 4%, and γ = 3. A The two lines represent the optimal consumption-income ratio for two consumers with (θ, η) being (1, 0) and (0.75, 0). B The two lines represent the optimal consumption-income ratio for two consumers with (θ, η) being (1, 0) and (1, 0.05). C The two lines represent the optimal consumption-income ratio for two consumers with (θ, η) being (1, 0.05) and (0.75, 0.05)

Generalizations of borrowing constraint

For a generalized description of the borrowing constraint, we assume W_t ≥ −αY_t for all t, where α is a non-negative constant. Figure 16 plots c(w) as a function of w for different constraints: α = 2, 3, and 5. In the three cases, the agent tends to underconsume if she cares about either model uncertainty or social status. If the agent pursues social status and faces ambiguity, c(w) increases at first and then decreases. Hence, without loss of generality, we take the value of α as zero. This setting does not alter the main results.

Fig. 16

Optimal consumption-income ratio c(w). The consumer is allowed to borrow α times her current income. The baseline parameters are r = 3.5%, µ = 1.5%, σ = 10%, ρ = 4%, and γ = 3. A, D, G Given α is 2, 3, or 5, the two lines in each panel represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0) and (0.02, 0). B, E, H Given α is 2, 3, or 5, the two lines in each panel represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0) and (0, 0.03). C, F, I Given α is 2, 3, or 5, the two lines in each panel represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0.03) and (0.02, 0.03)

Wealth shocks

Case 1: Interest rate risk. In the baseline model, we have assumed the agent only faces labor income risk. However, in reality, the agent also faces substantial risk for holding financial wealth that would significantly affect her optimal consumption and saving decision. Thus we explore the implications of wealth shocks for consumption dynamics in the model with both labor income risk and interest rate risk. In this case, the dynamics of wealth process are given by

$$d{W}_{t}=\left(r{W}_{t}+{Y}_{t}-{C}_{t}\right)dt+{\sigma }_{w}{W}_{t}d{{\mathcal{B}}}_{t}^{W},$$

(D.1)

where ${{\mathcal{B}}}^{W}$ is a standard Brownian motion that summarizes interest rate risk. For ease of exposition, interest rate risk is instantaneously independent of labor income risk.

Using the standard principle of optimality, we may write the HJB equation as follows:

$$\begin{array}{l}\rho J(W,Y)=\mathop{\max }\limits_{C>0}\mathop{\min }\limits_{g}U(C,W)+(rW+Y-C){J}_{W}(W,Y)+(\mu +g\sigma )Y{J}_{Y}(W,Y)\\ +\,\frac{{\sigma }^{2}{Y}^{2}}{2}{J}_{YY}(W,Y)+\frac{{\sigma }_{w}^{2}{W}^{2}}{2}{J}_{WW}(W,Y).\end{array}$$

(D.2)

By using the homogeneity property of the value function, which holds for all the cases in our paper, we write the value function as

$$J(W,Y)=\frac{{(bF(W,Y))}^{1-\gamma }}{1-\gamma }=\frac{{(bf(w)Y)}^{1-\gamma }}{1-\gamma }\ .$$

(D.3)

Substituting (D.3) into the HJB equation and simplifying, we obtain

$$\begin{array}{l}0=\left[\frac{\gamma {b}^{1-\frac{1}{\gamma }}{\left(f^{\prime} (w)\right)}^{1-\frac{1}{\gamma }}-\rho }{1-\gamma }+\left(\mu -\sqrt{2\eta }\sigma \right)-\frac{\gamma {\sigma }^{2}}{2}\right]f(w)+f^{\prime} (w)\\ +\,\left[r+\lambda -\left(\mu -\sqrt{2\eta }\sigma \right)+\gamma {\sigma }^{2}\right]wf^{\prime} (w)+\frac{\left({\sigma }^{2}+{\sigma }_{w}^{2}\right){w}^{2}}{2}\left(f^{\prime\prime} (w)-\gamma \frac{f^{\prime} {(w)}^{2}}{f(w)}\right)\ .\end{array}$$

(D.4)

For the generalized borrowing constraint, we assume that

$${W}_{t}\ge -\alpha {Y}_{t}\ .$$

(D.5)

We set the parameter α = 2, and the other model parameters remain the same as their baseline levels. Figure 17 exhibits the main results when considering a shock to wealth process. The results remains unchanged.

Fig. 17

Optimal consumption-income ratio c(w). Wealth shocks are considered. The baseline parameters are r = 3.5%, µ = 1.5%, σ = 10%, ρ = 4%, and γ = 3. A The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0) and (0.02, 0). B The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0) and (0, 0.03). C The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0.03) and (0.02, 0.03)

Case 2: Partial insurance. In order to enable inclusion of partial insurance, we introduce an additional asset which is correlated with the labor income process. That is, we assume that the dynamics of the price of this newly introduced asset are given by

$$d{S}_{t}={S}_{t}\left(rdt+{\sigma }_{S}d{{\mathcal{B}}}_{t}^{S}\right)\ ,$$

(D.6)

where σ_S is the volatility parameter and ${{\mathcal{B}}}^{S}$ is a standard Brownian motion. ${{\mathcal{B}}}^{S}$ is allowed to be correlated with ${\mathcal{B}}$ with correlation coefficient ρ_ys. Let π_t denote the fraction of financial wealth allocated to this risky asset. Then, wealth W accumulates as follows:

$$d{W}_{t}=(r{W}_{t}+{Y}_{t}-{C}_{t})dt+{\sigma }_{S}{\pi }_{t}{W}_{t}d{{\mathcal{B}}}_{t}^{S}\ .$$

(D.7)

In addition, the dynamics of labor income can be rewritten as

$$\frac{d{Y}_{t}}{{Y}_{t}}=\mu dt+\sigma \left({\rho }_{ys}d{{\mathcal{B}}}_{t}^{S}+\sqrt{1-{\rho }_{ys}^{2}}d{{\mathcal{B}}}_{t}^{N}\right)\ ,$$

(D.8)

where ${{\mathcal{B}}}^{N}$ is also a standard Brownian motion that captures the idiosyncratic risk of labor income. It is reasonable to hypothesize that the agent is ambiguous about the motion ${{\mathcal{B}}}^{N}$. Using the same method in the main text, the income process in the presence of model uncertainty becomes

$$\frac{d{Y}_{t}}{{Y}_{t}}=\left(\mu +\sqrt{1-{\rho }_{ys}^{2}}{g}_{t}\sigma \right)dt+\sigma \left({\rho }_{ys}d{{\mathcal{B}}}_{t}^{S}+\sqrt{1-{\rho }_{ys}^{2}}d{{\mathcal{B}}}_{t}^{Ng}\right)\ ,$$

(D.9)

where $d{{\mathcal{B}}}_{t}^{Ng}=d{{\mathcal{B}}}_{t}^{N}-{g}_{t}dt$ is a standard Brownian motion under the alternative measure.

Using the standard principle of optimality, we may write the HJB equation as follows:

$$\begin{array}{l}\rho J(W,Y)=\mathop{\max }\limits_{C,\pi }\mathop{\min }\limits_{g}U(C,W)+(rW+Y-C){J}_{W}(W,Y)+\left(\mu +\sqrt{1-{\rho }_{ys}^{2}}g\sigma \right)Y{J}_{Y}(W,Y)\\ +\,\frac{{\pi }^{2}{\sigma }_{S}^{2}{W}^{2}}{2}{J}_{WW}(W,Y)+{\rho }_{ys}{\sigma }_{S}\sigma \pi WY{J}_{WY}(W,Y)+\frac{{\sigma }^{2}{Y}^{2}}{2}{J}_{YY}(W,Y)\ ,\end{array}$$

(D.10)

subject to $\frac{1}{2}{g}^{2}\le \eta$. Let

$$J(W,Y)=\frac{{(bF(W,Y))}^{1-\gamma }}{1-\gamma }=\frac{{(bf(w)Y)}^{1-\gamma }}{1-\gamma }$$

(D.11)

denote the agent’s value function. Using the homogeneity property and the FOCs for consumption and portfolio allocation, we obtain the following decision rules:

$${g}^{* }=-\sqrt{2\eta }\ ,$$

(D.12)

$$c(w)={b}^{1-\frac{1}{\gamma }}f(w){\left(f^{\prime} (w)\right)}^{-\frac{1}{\gamma }}-\lambda w\ ,$$

(D.13)

$$\pi (w)=\frac{\rho \sigma }{{\sigma }_{S}}\left(1-\frac{\gamma f(w)f^{\prime} (w)}{w\left(\gamma {(f^{\prime} (w))}^{2}-f(w)f^{\prime\prime} (w)\right)}\right).$$

(D.14)

After simplifying, we have the following ODE for f(w):

$$\begin{array}{l}0=\left[\frac{\gamma {b}^{1-\frac{1}{\gamma }}{\left(f^{\prime} (w)\right)}^{1-\frac{1}{\gamma }}-\rho }{1-\gamma }+\left(\mu -\sqrt{2\eta \left(1-{\rho }_{ys}^{2}\right)}\sigma \right)-\frac{\gamma {\sigma }^{2}}{2}\right]f(w)\\ +\,\left[\left(r+\lambda -\left(\mu -\sqrt{2\eta \left(1-{\rho }_{ys}^{2}\right)}\sigma \right)\right)w+1\right]f^{\prime} (w)+\left(1-{\rho }_{ys}^{2}\right)\gamma {\sigma }^{2}wf^{\prime} (w)\\ +\,\frac{\left(1-{\rho }_{ys}^{2}\right){\sigma }^{2}{w}^{2}}{2}\left(f^{\prime\prime} (w)-\gamma \frac{f^{\prime} {(w)}^{2}}{f(w)}\right)+\frac{{\rho }_{ys}^{2}{\gamma }^{2}{\sigma }^{2}f(w)}{2\left(\gamma -\frac{f(w)f^{\prime\prime} (w)}{{\left(f^{\prime} (w)\right)}^{2}}\right)}\ .\end{array}$$

(D.15)

For the generalized borrowing constraint, we assume that

$${W}_{t}\ge -\alpha {Y}_{t}\ .$$

(D.16)

Figure 18 plots c(w) when ρ_ys = 0.5. Only ambiguity or only the spirit of capitalism has a negative effect on consumption. However, when considering ambiguity and the spirit of capitalism simultaneously, c(w) first increases and then declines. Undoubtedly, the results in the main text are robust.

Fig. 18

Optimal consumption-income ratio c(w). The labor income risk can be partially insured by holding the risky asset with correlation coefficient ρ_ys = 0.5. The baseline parameters are r = 3.5%, µ = 1.5%, σ = 10%, ρ = 4%, and γ = 3. A The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0) and (0.02, 0). B The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0) and (0, 0.05). C The two lines represent the optimal consumption-income ratio for two consumers with (λ, η) being (0, 0.05) and (0.02, 0.05)