## 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.

## Introduction

As a social animal, man accumulates wealth to smooth consumption and gain prestige, social status, and power in society. This captures the characteristics of wealth accumulation, that is, continual accumulation of wealth is motivated not only for the consumption reward that it brings but also for an intrinsic motive. Preference for wealth-induced status is in line with the spirit of capitalism in the sense of Weber (1958) and Keynes (1920): capitalists accumulate wealth for the sake of wealth.^{Footnote 1} Using the wealth-is-status and the spirit-of-capitalism models, many researchers have tried to explain growth, savings, and asset pricing. For example, Zou (1995) concludes that capitalist spirit contributes to the part of savings that cannot be explained by life-cycle factors and intergenerational transfers. Practically, due to incomplete or vague information, individuals may be ambiguous about labor income shocks, and thus model uncertainty arises. In the presence of model uncertainty, the ambiguity-averse consumer mistrusts the approximating model (or the reference model) and prefers a robust consumption rule. Peter (2019) finds that ambiguity-aversion induces individuals to save more.

Since ambiguity and preference for wealth status both serve as the determinants of savings, it is natural to ask whether there is some type of relationship that may produce complementarity or substitution effects on consumption. Smith (2001) shows that the spirit of capitalism reduces the rate of time preference and may increase or decrease risk aversion. Furthermore, Lou et al. (2009b) and Wang (2016) show that the capitalist spirit can influence precautionary saving related to risk preference and income shocks. In addition, Maenhout (2004) concludes that robustness can increase risk aversion. Thus, it is plausible to investigate the relationship between ambiguity and preference for wealth status that may produce potential interaction effects on consumption and saving behaviors. However, the related studies do not pay much attention to the joint acting mechanism.

By embedding ambiguity and desire for wealth into the consumption-saving model with uninsurable labor income subject to permanent shocks and borrowing constraint developed by Wang et al. (2016), this paper analytically evaluates their interactions on the consumption-saving decision. In our model, not only is the consumer ambiguous about the labor income distribution due to lack of information, but also she has the desire for wealth accumulation to enhance her social status. To highlight the effect of model uncertainty, the consumer is assumed to be ambiguity-averse in the sense that she solves the expected utility optimization problem under the worst-case scenario (using maximin criterion).

We find that the consumer’s preference for wealth and the degree of ambiguity jointly affect optimal consumption, including the consumption-income ratio, the MPC out of current income and the expected consumption growth. The model presents that the potential interaction effects on consumption are substitutional or complementary. Moreover, the intensity of substitution or complementarity is relevant to consumption variables analyzed. The results indicate that the interaction effects of ambiguity and preference for wealth play a role in the consumption-saving decision and cannot be ignored.

This paper also highlights the empirical consumption puzzles such as excess sensitivity and excess smoothness, from the perspective of ambiguity and preference for wealth. On the one hand, we conclude that ambiguity helps to resolve the excess smoothness of consumption or consumption growth observed in prior literature (see, e.g., Black, 1990; Campbell & Deaton, 1989; Ludvigson & Michaelides, 2001). Ludvigson & Michaelides (2001) show that incomplete information about the aggregate component of individual earnings may be an important factor in explaining the smoothness of aggregate consumption growth and its correlation with lagged labor-income growth. Interestingly, whether ambiguity helps to explain excess sensitivity of consumption growth is related to preference for wealth. When the preference for wealth is absent, ambiguity contributes to the excess sensitivity of consumption growth. However, the occurrence of preference for wealth makes ambiguity contribute little to the excess sensitivity of consumption growth. On the other hand, preference for wealth is unable to resolve the excess sensitivity and excess smoothness of consumption growth, but can contribute to the excess smoothness of consumption for a low degree of ambiguity, which enriches the predictions of Luo et al. (2009a) and Wang (2016).

This article is evidently related to the fast growing literature that incorporates ambiguity in consumption-saving-portfolio models. Trojani & Vanini (2002) propose a simple robust version of Merton (1971) model of intertemporal consumption and portfolio choice. Maenhout (2004) introduces model uncertainty into a dynamic portfolio and consumption problem and shows that robustness dramatically decreases the optimal fraction of wealth allocated to risky assets. Leippold et al. (2008) study asset prices under learning and ambiguous information in a Lucas exchange economy with standard power utility. Liu (2010) examines a continuous-time intertemporal consumption and portfolio choice problem for an ambiguity-averse investor with recursive preferences. Branger et al. (2013) analyze the optimal stock-bond portfolio under both learning and ambiguity aversion. Branger & Larsen (2013) analyze the portfolio planning problem in a jump-diffusion model when there is model uncertainty. Jeong et al. (2015) give an positive answer to the question whether ambiguity aversion plays an important role. Biagini & Pinar (2017) derive a closed-form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates.^{Footnote 2} Ait-sahalia & Matthys (2019) study a robust optimal consumption and portfolio choice problem where the underlying risky asset follows a Lévy process and show that ignoring uncertainty leads to significant wealth loss for the investor. Nevertheless, these studies generally model uncertainty resulting from return process of risky assets without taking labor income shocks into account. Subsequently, Luo (2017) considers the model uncertainty due to state shocks, which is an integration of income shocks and asset return shocks, and shows that robustness does not change the MPC out of perceived permanent income. In a two-period consumption-saving model, Peter (2019) considers that the agent faces ambiguity about the distribution of second-period income. Our work enriches the literature incorporating model uncertainty coming from labor income shocks.

Studies of the spirit of capitalism are another important strand of literature related to this paper. The spirit-of-capitalism models have been used to explain consumption-saving behavior and asset returns in preceding research. Zou (1995) has studied the spirit of capitalism and long-run growth and showed that a strong capitalist spirit can lead to unbounded growth of consumption and capital even under the neoclassical assumption of production technology. Bakshi & Chen (1996) examine the implications for consumption, portfolio choice and stock prices under the hypothesis that investors acquire wealth not just for its implied consumption but also for its induced status. Yang (1999) investigates testable restrictions on the time-series behavior of consumption and asset returns implied by a representative agent model with the spirit of capitalism. Smith (2001) shows that the spirit of capitalism impinges upon asset prices depending on the interaction of risk aversion, impatience, intertemporal substitution and ordinal preferences between consumption and status. Gong & Zou (2002) consider social status, the spirit of capitalism, fiscal policies and asset pricing in a stochastic model of growth. Zhang (2006) tests two models of asset pricing that feature status-seeking through accumulation of not only financial and real assets but also human capital, and finds that the spirit of capitalism hypothesis is rejected in the aggregate data. Boileau & Braeu (2007) evaluate whether the spirit of capitalism improves the ability of real business cycle (RBC) models to explain the main features of both asset returns and business cycles. Luo et al. (2009b) explore how the spirit of capitalism affects saving and consumption behavior. Luo et al. (2009a) extend the model in Luo et al. (2009b) to plausibly explain the empirical magnitude of excess smoothness. Using a consumption model with CARA preferences over consumption and wealth, Luo et al. (2009a) demonstrate that the spirit of capitalism can explain the excess sensitivity of consumption growth to anticipated income growth, but its explanation power for the excess smoothness of consumption growth to income growth depends on whether labor income is stationary. However, different from Luo et al. (2009a), when considering both the spirit of capitalism and regime switching in a consumption-saving model, Wang (2016) shows that the spirit of capitalism can help to explain the excess smoothness of consumption growth in all cases. Airaudo (2017) introduces Max Weber’s spirit of capitalism hypothesis into a benchmark Lucas tree asset pricing model by assuming that economic agents derive direct utility from wealth. Our paper particularly contributes to the literature illustrating the implication of the spirit of capitalism from the perspective of consumption behavior.

## Model setup

### Labor income, wealth dynamics and utility function

Assume that an infinitely-lived consumer receives an exogenously given perpetual stream of stochastic labor income, denoted by *Y*_{t}. The labor income is governed by

where *μ* is the expected instantaneous income growth rate, *σ* is the unconditional standard deviation of the income growth over an incremental unit of time, and \({\mathcal{B}}\) is a standard Brownian motion under the filtered probability space \(\left({{\Omega }},{\mathscr{F}},{\mathcal{P}}\right)\).

Suppose that the consumer can only have access to a risk-free asset with a positive constant interest rate *r*. Specifically, there are no other financial assets used to hedge against labor income shocks, hence labor income is uninsurable. Nevertheless, the consumer can smooth her consumption over time by saving. Let *C*_{t} denote the agent’s consumption flow at time *t*.^{Footnote 3} Then the wealth process {*W*_{t}: *t* ≥ 0} is given by

In addition, the agent faces borrowing constraint that she is not allowed to borrow against her future income, which implies that wealth is non-negative at all times, i.e., *W*_{t} ≥ 0 for all *t*. The borrowing constraint means that when *W*_{t} = 0, *C*_{t} ≤ *Y*_{t}.

Assume that utility is time-additive and the instantaneous utility function *U*(*C*_{t}, *W*_{t}) is determined by consumption flow *C*_{t} and accumulated wealth *W*_{t}. This type of utility function captures the consumer’s desire for wealth due to the resulting social status (Bakshi & Chen, 1996).^{Footnote 4} The utility function *U*(*C*, *W*) is twice differentiable in *C* and *W* and satisfies *U*_{C} > 0 (more consumption is strictly better), *U*_{W} > 0 (higher status is strictly preferred), *U*_{CC} < 0, *U*_{WW} < 0 and *U*_{CW} < 0.^{Footnote 5} For expositional convenience, *U*(*C*, *W*) takes the standard constant-relative-risk-averse (CRRA) utility form:

where *γ* > 0 is the coefficient of relative risk aversion. The parameter *λ* ≥ 0 measures the strength of preference for (absolute) wealth, the extent to which the consumer cares about wealth or the strength of spirit of capitalism. A larger *λ* implies a stronger desire for social status resulting from wealth (i.e., a stronger spirit of capitalism). If *λ* = 0, the preference for wealth disappears and the utility function degenerates into the familiar case of CRRA utility determined by consumption alone.

### Model uncertainty and the optimization problem

Different from Wang et al. (2016), there is model uncertainty/ambiguity with respect to the probability law governing the labor income process. We use robust control theory (Hansen et al., 2006; Leippold et al., 2008) featuring max-min expected utility to acquire the optimal consumption rule. Since the consumer does not have full knowledge of the distribution of labor income shocks, she would not trust the reference model (or the probability measure \({\mathcal{P}}\)) and consider an alternative probability measure \({{\mathcal{P}}}^{g}\) in order to protect herself from model errors.

Let *ξ* denote the Radon-Nikodym derivative of the alternative measure \({{\mathcal{P}}}^{g}\) with respect to the reference measure \({\mathcal{P}}\), in that

Define *ξ*_{t} as

where \({\mathbb{E}}\left[\mathop{\int}\nolimits_{0}^{t}{g}_{s}^{2}ds\right]<\infty\) and \({\mathbb{E}}\left[\mathop{\int}\nolimits_{0}^{t}{{g}_{s}}^{2}{{\xi }_{s}}^{2}ds\right]<\infty\) for all *t* > 0. Thus (5) implies

Using Girsanov’s Theorem, the process \(d{{\mathcal{B}}}_{t}^{g}=d{{\mathcal{B}}}_{t}-{g}_{t}dt\) is a standard Brownian motion under the alternative measure \({{\mathcal{P}}}^{g}\). In terms of the Brownian motion \({{\mathcal{B}}}_{t}^{g}\), we may rewrite the dynamics of labor income (1) as

According to Leippold et al. (2008), because of the ambiguity about the income process, the consumer considers an admissible set of probability measures that are close to the reference model and practically undetectable in statistics. This means that the misspecifications of the reference model, reflected by the drift distortion *g*_{t}*σ*, should be small and are constrained to

where *η* > 0 controls the size of the admissible set of probability measures different from but close to the reference model.

The ambiguity-averse consumer determines the worst-case scenario according to *η* and then maximizes her lifetime expected utility based on the worst-case scenario to obtain a robust consumption rule. *η* describes the degree of ambiguity. If *η* is larger, it means that the neighborhood of labor income drift changes is broader and the consumer is more ambiguous about labor income. It is evident that if *η* = 0, there is no belief distortion and model uncertainty.

Under the alternative measure \({{\mathcal{P}}}^{g}\), the consumer’s optimization problem can be written as

subject to the dynamics of labor income (7), wealth accumulating process (2) and probability measure constraint (8), where *ρ* > 0 is the agent’s subjective discount, and *J* (*W*, *Y*) is the indirect utility.

## Model solution

In this section, we show the solutions of the consumer’s optimization problem (9). Note that labor income *Y* and wealth *W* are state variables of our model. In order to acquire the optimal consumption, we use standard dynamic programming method to write the Hamilton-Jacobi-Bellman (HJB) equation for the consumer’s maximization problem,

subject to

Combining the objective function in (10) and the misspecification constraint (11), the worst-case belief distortion is given by

The consumer makes decision under the worse-case scenario whose Radon-Nikodym derivative with respect to the reference measure \({\mathcal{P}}\) is given by \(d\xi ={g}_{worst-case}\xi d{\mathcal{B}}\). Thus the HJB equation (10) is equivalent to

The first-order condition (FOC) with respect to consumption is:

Notice that (14) is the standard FOC for consumption, equating the marginal benefit of consumption *U*_{C} (*C*, *W*) with the marginal utility of savings *J*_{W} (*W*, *Y*). For the special case *λ* = 0, the marginal utility of consumption is independent of the consumer’s wealth *W*. More generally, in the presence of preference for wealth, *U*_{C} depends on both current consumption *C* and wealth *W*.

The consumer’s value function *J* (*W*, *Y*) takes the following form

where *b* is a coefficient to be determined and *F* (*W*, *Y*) is interpreted as the certainty equivalent wealth in *J* (*W*, *Y*) = *J* (*F*(*W*, *Y*), 0).^{Footnote 6}

With homogeneous utility and geometric labor-income process, the value function *J* (*W*, *Y*) has the homogeneity property and then the two state variables can be reduced to one that is *w* = *W*/*Y*. We use the lower case to denote the corresponding variable in the upper case scaled by contemporaneous labor income *Y*. Substituting the first-order condition with respect to consumption (14), utility function (3), value function (15) and partial derivatives of *J* (*W*, *Y*) into the objective function in (13), we get the ordinary differential equation of the scaled certainty equivalent wealth *f*(*w*).

We now discuss the boundary condition of *f*(*w*) in our solution. We first consider the right margin that *w* → *∞*. The wealth-income ratio *w* being sufficiently large means that the consumer’s wealth is so great compared to the income that she can hedge income shocks by self-insurance. Thus as *w* → *∞*, *f*(*w*) is equivalent to the sum of current wealth-income ratio and human wealth scaled by current income.^{Footnote 7} According to Luo et al. (2009b), the effective rate of interest used for discounting future labor income is *r* + *λ* in the presence of preference for wealth.^{Footnote 8} Thus, human wealth is

where the expectation operator is under the new measure \({{\mathcal{P}}}^{g}\) with a belief distortion *g*_{worst−case} from the reference measure \({\mathcal{P}}\). With *r* + *λ* > *μ* + *g*_{worst−case}*σ*, human wealth *H*_{t} is finite and given by

Let *h* = *H*/*Y* denote the human wealth scaled by current income. Hence, the right boundary condition of *f*(*w*) is \({\mathrm{lim}\,}_{w\to \infty }\left[f(w)-(w+h)\right]=0\), where \(h=\frac{1}{r+\lambda -(\mu -\sqrt{2\eta }\sigma )}\), and \({\mathrm{lim}\,}_{w\to \infty }f^{\prime} (w)=1\).

### Proposition 1

With uninsurable labor income, the consumption-income ratio *c*(*w*) is given by

where *b* is given by

and the scaled certainty equivalent wealth *f*(*w*) solves the following ordinary differential equation:

The above ODE for *f*(*w*) is solved subject to the following conditions:

where

and

Additionally, the ODE (20) for *f*(*w*) satisfies the following constraint for *c*(⋅) at *w* = 0,

The optimal consumption rule *c*(*w*) depends on both the scaled certainty equivalent wealth *f*(*w*) and its slope \(f^{\prime} (w)\) (i.e., the marginal certainty equivalent value of liquidity). Evidently, both ambiguity and preference for wealth play a vital role in determining the consumption rule. The ODE (20) describes the nonlinear certainty equivalent valuation *f*(*w*) in the region *w* ≥ 0. In the limit as *w* → *∞*, wealth completely buffers labor income shocks and

Note that the consumption constraint (24) is equivalent to the borrowing constraint *W*_{t} ≥ 0. If 0 < *c*(0) < 1, the consumption is not binding in *C*_{t} < *Y*_{t}. The left boundary condition at *w* = 0 is given by (23), which is the limit of the ODE (20). Since *c*(0) = 1 means that the consumer with zero wealth runs out of her current income, the left boundary condition is determined by (23) and *c*(0) = 1, which can be characterized by an explicit expression of *f*(0):

It is worth noting that we need not to check whether the borrowing constraint binds or not in the interior region *w* ≥ 0.^{Footnote 9}

## Quantitative analysis

In this section, in order to find out the joint influence of ambiguity and preference for absolute wealth on consumption, we explore both the pure effects and the interaction effects through quantitative analysis.

### Parameter choices

There are seven parameters in our model, including interest rate *r*, labor income parameters (*μ*, *σ*), preference parameters (*ρ*, *γ*, *λ*) and ambiguity parameter *η*. Assume that all these parameters are exogenously determined. Following Wang et al. (2016), we set the subjective discount rate *ρ* = 4% and the risk-free rate *r* = 3.5%, which implies that the agent is relatively impatient, with a difference of *ρ* − *r* = 0.5%.^{Footnote 10} The parameters related to labor income process are set as follows: the expected income growth rate *μ* = 1.5% and the volatility of income growth *σ* = 10%, which implies that the logarithmic annual income growth rate is *μ* − *σ*^{2}/2 = 1%. The risk aversion *γ* is set to 3, which is within the range of economically plausible estimates.^{Footnote 11} Following Bakshi & Chen (1996), Wang (2016) and Leippold et al. (2008), and allowing for the stability of numerical solutions, the strength of preference for absolute wealth is *λ* ∈ [0, 0.5] and the degree of ambiguity is *η* ∈ [0, 0.5]. This setting of *λ* and *η* is feasible in that our focus is to find the potential interaction effects of them on consumption. It is worth noting that as *η* and *λ* both equal 0, the results in Proposition 1 are the same as those in Wang et al. (2016) with CRRA utility. All parameters are annualized and continuously compounded.

### The optimal consumption-income ratio

#### Comparative static effect of *λ* or *η* on *c*(*w*)

Figure 1 describes the effects of shifting *λ* on the optimal consumption-income ratio *c*(*w*) for three degrees of ambiguity: *η* = 0, *η* = 0.01 and *η* = 0.03. Figure 1 shows that for different levels of *η* and *λ*, the borrowing constraint does not bind, *c*(0) < 1. Thus, relaxing the borrowing constraint and allowing the consumer to be in debt (*W* < 0) does not change the consumer’s behavior and creates no additional value for the agent. We first discuss the comparative static effects of ambiguity parameter *η* and preference parameter *λ* on consumption-income ratio.

*Ambiguity*. Comparing the values of *c*(*w*) for *λ* = 0 represented by the three dash-dot lines in Fig. 1, we see that given any *w*, *c*(*w*) decreases with *η* without the preference for wealth. For example, for *λ* = 0, and *η* = 0, *c*(0) = 0.9421, which means that the consumer with no preference for wealth and ambiguity consumes 94.21% of her current income and saves the remaining current income. As *η* increases to 0.01 or 0.03, *c*(0) decreases to 0.7340 or 0.6216. Moreover, even if the consumer cares about absolute wealth (e.g., *λ* = 0.02 or *λ* = 0.04), ambiguity still produces decreasing effect on *c*(*w*). In the presence of ambiguity, *η* determines the degree of pessimism implied by a worst-case labor income drift \(\mu -\sqrt{2\eta }\). Intuitively, with a higher degree of ambiguity, the consumer is more pessimistic about the labor income drift, and then becomes more frugal.

*Preference for wealth*. Figure 1 shows that the preference for wealth acts somewhat differently on the consumption-income ratio when compared to ambiguity. Since the consumer cares about both consumption and wealth, she chooses between current consumption and saving balancing the consumption motive and the wealth-accumulation motive for smoothing consumption and increasing absolute wealth. Smith (2001) shows that consumption and portfolio choices rely on the effective degree of risk aversion and the effective time preference rate that are endogenously determined by the spirit of capitalism (preference for wealth status). Panels A and B of Fig. 1 show that the stronger the preference for wealth is, the less the consumer consumes of her current income. The intuition behind this is straightforward. If the consumer cares more about wealth, she expects to increase her absolute wealth more quickly and hence tends to consume less of her current income. However, Panel C of Fig. 1 shows a non-monotonic effect of *λ* on *c*(*w*). For a relatively high degree of ambiguity (e.g., *η* ≥ 0.02) and at a relatively low level of liquidity *w* (e.g., *w* < 2), the preference for wealth produces insignificant effects on *c*(*w*), and even generates increasing effects on *c*(*w*). Intuitively, for a high degree of pessimism about income uncertainty and relatively low liquidity, the consumption motive predominates over the wealth-accumulation motive. That is, when wealth is poor and consumption-income ratio is low enough, consumption is more pressing than accumulating wealth. Hence, the consumption-saving decision may become irrelevant to the preference for wealth. Meanwhile, the consumer with a stronger preference for wealth wishes to get more utility through increasing consumption and raising the level of wealth. However, the motive for raising the level of wealth is restricted for low levels of consumption and liquidity. In order to maximize her utility, she may choose a higher level of *c*(*w*) to compensate for the potential utility that cannot be derived by quick wealth accumulation.

Figure 2 graphs the optimal consumption *C*_{t}. The upper and middle panels state that either the preference for wealth (spirit of capitalism) or model uncertainty has a negative effect on consumption, and the panel below states that the combination of model uncertainty and capitalist spirit can generate both overconsumption and underconsumption.

#### Interaction effect of *η* and *λ* on *c*(*w*)

We next use two approaches to examine the interaction effects of preference for wealth and ambiguity on consumption.

*Modularity of c*(*w*; *λ*, *η*) *in* (*λ*, *η*). In order to identify the potential interaction effects of preference for wealth and ambiguity, we borrow the terminology from supermodular or submodular games. If \(\frac{\partial c{(w;\lambda ,\eta )}^{2}}{\partial \lambda \partial \eta }\ge 0\ (\le \!0)\), *c*(*w*; *λ*, *η*) is supermodular (submodular) in (*λ*, *η*), which then implies a complementarity (substitution) relationship between *λ* and *η*. However, because there is no explicit solution to *c*(*w*), we cannot directly derive the expression of \(\frac{\partial c{(w;\lambda ,\eta )}^{2}}{\partial \lambda \partial \eta }\). Using the results of numerical analyses, we calculate the changes in *c*(*w*) when increasing *λ* from 0.02 to 0.04 for two given levels of liquidity, *w* = 0 and *w* = 10 (see Table 1). For *w* = 10, Δ*c*(*w*, *η*) is negative and Δ*c*(*w*, *η*) is increasing in *η*. Thus *c*(*w*, *λ*, *η*) is submodular in (*λ*, *η*), which means that preference for wealth has a stronger decreasing effect on consumption-income ratio for a lower degree of ambiguity. Therefore, they can produce a substitution effect on *c*(*w*) for a relatively high liquidity (e.g., *w* > 2). When liquidity is low, for example, *w* = 0, the effect is complex. Δ*c*(0*;* *η*) is positive and increasing in *η* for *η* > 0.02, which means *c*(0*;* *λ*, *η*) is supermodular in (*λ*, *η*) and thus implies a complementarity relationship between ambiguity and preference for wealth. For *η* < 0.02, on the contrary, Δ*c*(0, *η*) is negative and increasing in *η*, which implies a substitution relationship between them. It indicates that for a low liquidity, they may be complementary or substitutional, which depends on the degree of ambiguity.

*Iso-curves of c*(*w*) *in* (*λ*, *η*). Using iso-curves of the consumption-income ratio in the (*λ*, *η*), we continue examining the interaction between preference for wealth and ambiguity. For a given liquidity *w*, we calculate different pairs of (*λ*, *η*) that can produce the same level of *c*(*w*). Figure 3 plots the iso-curves for *w* = 0 and *w* = 10. The iso-curves in Panel A of Fig. 3 are upward or downward and show that for zero wealth, preference for absolute wealth and ambiguity may be complementary or substitutional, which depends on the level of consumption-income ratio. For a wealth-poor agent, if the consumption motive is strong and the borrowing constraint is severe, preference for absolute wealth and ambiguity produce a substitution effect on consumption-income ratio. However, with the decreasing of *c*(0), the substitution effect gradually vanishes and the complementarity effect appears. It shows that given a low level of consumption-income ratio, the decreasing effect of *η* on consumption-income ratio can offset the increasing effect of *λ*. Since the consumption-income ratio is decreasing with *η*, a small value of *c*(0) corresponds to a high degree of ambiguity. Therefore, the iso-curves give the same results as the above analysis of modularity of *c*(0; *λ*, *η*) in (*λ*, *η*). However, they interact differently when the liquidity is high. As can be seen from Panel B of Fig. 3, the iso-curves are downward, which means that there is only substitution effect. Particularly, the substitution effect on *c*(*w*) is more significant than that for the lower liquidity *w*.

### The MPC out of wealth

Using the property of homogeneity, the MPC out of wealth \(\frac{\partial C(W,Y)}{\partial W}\) can be expressed as \(c^{\prime} (w)\) (i.e. the sensitivity of consumption-income ratio to wealth-income ratio). We next investigate how ambiguity and preference for wealth influence the MPC out of wealth.

#### Comparative static effect of *η* or *λ* on \(c^{\prime} (w)\)

Given *η* = 0, *η* = 0.01 and *η* = 0.03, Fig. 4 plots the first derivative of *c*(*w*) with respect to liquidity *w* for *λ* = 0, *λ* = 0.02 and *λ* = 0.04. For *λ* = 0, as we increase *η* from 0 to 0.03, \(c^{\prime} (0)\) decreases from 0.0543 to 0.0457. In the presence of preference for wealth, \(c^{\prime} (w)\) is still decreasing with *η*. The MPC out of wealth decreasing in *η* is seemingly intuitive in that a higher degree of ambiguity may cause the consumer to have a stronger precautionary saving motive to insure against the consumption uncertainty and thus weaken her consumption motive. As can be seen from the three panels of Fig. 4, increasing the strength of preference for wealth can cause \(c^{\prime} (w)\) to decrease. For example, for *η* = 0, \(c^{\prime} (0)\) decreases from 0.0452 to 0.0372 when *λ* increases from 0.02 to 0.04. The decreasing effect of *λ* on \(c^{\prime} (w)\) means that the stronger preference for wealth the consumer shows, the lower the MPC out of wealth. This is consistent with the intuition. If the consumer has a stronger preference for wealth, she would rather increase consumption less and hold more wealth when the consumer assigns the increased wealth between consumption and wealth holding.

#### Interaction effect of *η* and *λ* on \(c^{\prime} (w)\)

Table 2 reports the changes in \(c^{\prime} (0)\), when *λ* increases from 0.02 to 0.04, for different values of *η*. It can be seen that the absolute value of \({{\Delta }}c^{\prime} (0;\eta )\) decreases with *η*, which means that the decreasing effect of *λ* on \(c^{\prime} (0)\) is stronger for a lower degree of ambiguity. For other levels of liquidity *w*, the relation between *η* and the decreasing effects of *λ* on \(c^{\prime} (w)\) is similar to that for liquidity *w* = 0. The result implies \(c^{\prime} (w;\lambda ,\eta )\) is submodular in (*λ*, *η*), i.e., \(\frac{\partial c^{\prime} {(w;\lambda ,\eta )}^{2}}{\partial \lambda \partial \eta }\, < \, 0\), and the submodularity of \(c^{\prime} (w;\lambda ,\eta )\) is independent of liquidity *w*. We thus conclude that ambiguity and preference for wealth can produce a substitution effect on the MPC out of wealth or \(c^{\prime} (w)\).

In order to further investigate the substitution effect, we calculate different pairs (*λ*, *η*) that can generate the same \(c^{\prime} (w)\), given liquidity *w*. Figure 5 plots the iso-curves of \(c^{\prime} (w)\) at *w* = 0 and *w* = 10. Given different levels of \(c^{\prime} (w)\), the iso-curves are downward which means that the substitution effect is significant. Figure 5 also shows that upon enhancing the liquidity, the substitution effect on \(c^{\prime} (w)\) gradually becomes weaker.

### The MPC out of current income

We next discuss how ambiguity and preference for wealth jointly influence the MPC out of current income. The MPC out of current income can be written as

Given different levels of *η* and *λ*, we plot the relationship between the MPC out of labor income *C*_{Y}(*W*, *Y*) and liquidity *w* in Fig. 6. Compared horizontally, Fig. 6 indicates that the MPC out of current income decreases with *η*. From the perspective of vertical comparison, the MPC out of income is decreasing in *λ* in the absence of ambiguity. But with *η* > 0, the relation between *C*_{Y} and *λ* is indefinite (Panels B and C). To further interpret this relationship, Table 3 exhibits *C*_{Y}(*W*, *Y*) with respect to *λ* and *η* at the point *w* = 5. In the case of either model uncertainty or the preference for wealth, *C*_{Y}(0, *Y*) decreases in *η* or *λ*. However, when the agent faces ambiguity and has the preference for wealth, *C*_{Y}(0, *Y*) decreases with *λ* if *η* < 0.019 while it increases with *λ* if *η* > 0.019. This result is in accordance with Fig. 6. When *η* < 0.019, Δ*C*_{Y}(*W*, *Y*; *η*) is negative and increasing in *η*, which indicates that *C*_{Y}(*W*, *Y*; *λ*, *η*) is submodular in (*λ*, *η*). On the contrary, *C*_{Y}(*W*, *Y*; *λ*, *η*) is supermodular in (*λ*, *η*) for *η* < 0.019. This shows that ambiguity and preference for wealth may cause substitution or complementarity effects on the MPC out of current income. We plot the iso-curves of *C*_{Y}(*W*, *Y*) in Fig. 7. These iso-curves are downward for high *C*_{Y}(*W*, *Y*) (or low *η*) and upward for low *C*_{Y}(*W*, *Y*) (or high *η*). In addition, the curvatures of these iso-curves are almost constant. This shows that ambiguity and preference for wealth are completely substitutional or complementary for the MPC out of current income.

Wu & Wu (2007) give a main research on the relationship between average propensity to consume and income distribution of urban households in China by using time-series data from 1985 to 2004. They find that low-income agents have a high propensity to consume while high-income agents have a relatively low propensity to consume. With respect to these empirical facts, we briefly discuss each model’s performance: (i) the ambiguity-only model does not predict the negative relation between income and MPC out of income. The presence of ambiguity lowers the expected growth rate of income from *μ* to \(\mu -\sqrt{2\eta }\sigma\) and hence reduces the income level at each time. Meanwhile, it also lowers *C*_{Y} as presented in Figure 6; (ii) for the spirit-of-capitalism-only model, *C*_{Y} decreases with *λ*, but the spirit of capitalism cannot lead to income decline. This is because the spirit of capitalism has no effect on the dynamics of the labor income. (iii) for the model with both ambiguity and the spirit of capitalism, the decreasing growth rate of income results in a lower level of income at each time *t*, but the MPC out of income *C*_{Y} may rise to a higher level (Figure 8), which is consistent with the empirical facts.

### The expected consumption growth

Applying Ito’s lemma to *C*_{t} = *c*(*w*_{t})*Y*_{t}, we obtain the consumption dynamics,

where *g*_{C}(*w*) is the expected consumption growth rate given by

and *σ*_{C}(*w*) is the volatility of the consumption given by

#### Comparative static effect of *η* or *λ* on *g*
_{C}(*w*)

We now explore the effect of *η* or *λ* on the expected consumption growth *g*_{C}(*w*). According to Eq. (29), the expected consumption growth includes three terms, which can be interpreted as the uncertainty-adjusted expected income growth, the effect due to the drift of wealth-income ratio and the effect due to the diffusion of wealth-income ratio. Model uncertainty induces the expected income growth to shift downward adjusted by \(\sqrt{2\eta }\sigma\). The degree of ambiguity *η* influences the expected consumption growth through two channels, the direct channel by the effect on the term \(-\sqrt{2\eta }\sigma\) and the indirect channel by the effect on *c*(*w*), \(c^{\prime} (w)\) and *c**″*(*w*). However, the strength of preference for wealth *λ* influences the expected consumption growth indirectly by *c*(*w*), \(c^{\prime} (w)\) and *c**″*(*w*).

Figure 9 plots *g*_{C}(*w*) for different levels of ambiguity and preference for wealth. The introduction of ambiguity or preference for wealth does not change the relationship between expected consumption growth *g*_{C}(*w*) and liquidity *w* that *g*_{C}(*w*) is positively predictable by liquidity *w*. The three dash-dot lines (*λ* = 0) in the three panels of Figure 9 tell us that ambiguity can cause the expected consumption growth to decline. The same result can be obtained when we examine the three dashed lines (*λ* = 0.02) or solid lines (*λ* = 0.04). Intuitively, the decreasing effect of ambiguity on expected consumption growth results from pessimism about the income growth and a fall in consumption.

When it comes to the effect of *λ* on *g*_{C}(*w*), we find that the relationship is not monotonic. The preference for wealth can produce a decreasing or increasing effect on consumption growth. Figure 9 shows that the effects of *λ* on *g*_{C}(*w*) are relevant to ambiguity and liquidity *w*. The decreasing effect occurs at a relatively high degree of ambiguity and a relatively low level of liquidity. When *η* increases from 0.01 to 0.03, the range of liquidity *w* in which decreasing effect occurs extends from *w* < 1.5 to *w* < 9.4. The condition of low *w* and high *η* generally corresponds to the case of low consumption-income ratio. When the consumption-income ratio is low enough, consumption is more pressing than wealth accumulation, and then the consumer with higher *λ* may choose to consume more of her current income. Nevertheless, the expected consumption growth may decrease with the strength of preference for wealth. On the contrary, for a relatively high consumption-income ratio, the stronger preference for wealth can help to increase expected income growth.

#### Interaction effect of *η* and *λ* on *g*
_{C}(*w*)

Table 4 reports the changes in *g*_{C}(*w*) when *λ* increases from 0.02 to 0.04 for *η* being 0.01, 0.02 or 0.03 and *w* being 0 or 15. Whenever *w* = 0 or *w* = 15, Δ*g*_{C}(*w*; *η*) is a decreasing function of *η*. Panel A of Table 4 shows that the preference for wealth has a decreasing effect on expected consumption growth for the three values of *η*. Because the decreasing effect on *g*_{C}(*w*) is stronger for higher *η*, *g*_{C}(*w*; *λ*, *η*) is supermodular in (*λ*, *η*). For *w* = 15, the preference for wealth has a stronger increasing effect for the lower value of *η*, which indicates that *g*_{C}(*w*; *λ*, *η*) is submodular in (*λ*, *η*). Thus *g*_{C}(*w*; *λ*, *η*) may be supermodular or submodular in (*λ*, *η*). The above analysis indicates that ambiguity and preference for wealth can generate substitution or complementarity effect on expected consumption growth.

Figure 10 plots the iso-curves of the given values of expected consumption growth for liquidity *w* = 0 and *w* = 15. As can be seen from Panel A, the three lines, including the dashed line, solid line and dash-dot line, are downward and convex to origin. The marginal substitution rate of *λ* and *η* being declining predicts that they produce a substitution effect on expected consumption growth. However, the four lines in Panel B of Fig. 10 are upward and describe a increasing effect of *λ* and a decreasing effect of *η* for high liquidity. The curvature of the iso-curves in Fig. 10 predicts that ambiguity and preference for wealth can generate substitution or complementarity effects on expected consumption growth.

#### Comparative static effect of *η* or *λ* on \(g^{\prime} (w)\)

We next explore how the sensitivity of expected consumption growth to liquidity changes with *λ* or *η*. Figure 11 plots the first derivative of *g*_{C}(*w*) with respect to *w*. The absolute value of \(g^{\prime} (w)\) determines the sensitivity of expected consumption growth to liquidity. Figure 11 shows that the sensitivity of expected consumption growth to liquidity is very small, but the influence of *λ* or *η* on \(g^{\prime} (w)\) is significant. First, as *λ* increases, the sensitivity of expected consumption growth to liquidity declines. Second, without the preference for wealth, the higher degree of ambiguity can make the expected consumption growth more sensitive to liquidity. However, when considering the preference for wealth, we find that the sensitivity of expected growth is relevant to the strength of preference for wealth. For example, as *λ* = 0.02, changing *η* hardly changes the sensitivity of expected consumption growth to liquidity. Moreover, as *λ* = 0.04, increasing *η* makes the expected consumption growth less sensitive to liquidity.

### Consumption volatility

#### Comparative static effect of *η* or *λ* on *σ*
_{C}(*w*)

We now investigate how ambiguity or preference for wealth influences consumption volatility *σ*_{C}(*w*). Figure 12 plots the standard deviation of consumption growth (consumption volatility) given different *η* and *λ*.

At *w* = 0, consumption is as volatile as income, i.e., *σ*_{C}(0; *λ*, *η*) = *σ*. Because of the borrowing constraint, the consumer with zero wealth can only consume her current income. Thus shocks to income growth completely transfer to consumption growth. As *w* → *∞*, consumption volatility *σ*_{C}(*w*) approaches zero, which means that enough wealth can hedge against shocks to income growth. As can be seen from the lines of the same line style in the three panels of Fig. 12, *η* is negatively related to the standard deviation of consumption growth. Intuitively, in the presence of model uncertainty, the consumer wishes to smooth consumption through saving channel to protect herself against model uncertainty and income risk. If the degree of ambiguity increases, the motive for smoothing consumption becomes stronger, and thus consumption is less volatile.

Each panel of Fig. 12 shows that *λ* negatively influences consumption volatility. The intuition is as follows. In the presence of preference for wealth, the consumer regards both consumption and wealth accumulation. She wishes to smooth consumption by saving channel and smooth wealth by adjusting consumption. The two motives exist simultaneously. If the consumer cares more about wealth, she has a stronger precaution motive for protecting against wealth uncertainty, and then the motive for smoothing consumption weakens. Accordingly, a stronger preference for wealth can result in larger consumption volatility.

#### Interaction effect of *η* and *λ* on *σ*
_{C}(*w*)

After examining the impact of ambiguity or preference for wealth, we next explore their interaction effect on consumption volatility. Table 5 reports the changes in consumption volatility when we change the value of *λ* from 0.02 to 0.04. Δ*σ*_{C}(*w*; *η*) is positive and increasing with *η*. That is, the increasing effect of *λ* on *σ*_{C} is stronger for a higher degree of ambiguity. Thus *σ*_{C}(*w*; *λ*, *η*) is supermodular in (*λ*, *η*), which means that the preference for wealth is complementary for ambiguity. Figure 13 describes the pairs of (*λ*, *η*) that can generate the same consumption volatility for *w* = 5 and *w* = 15. The curvature of the iso-curves in panel A is similar to that in panel B, which means that the complementary relationship does not change with liquidity. As *η* increases, the curvature of the iso-curves becomes smaller. This indicates that the complementarity effect gradually weakens when the degree of ambiguity becomes higher.

### Excess sensitivity and excess smoothness

Table 6 summarizes the comparative static effects and interaction effects of *λ* and *η* that have been analyzed above. We then direct our attention to whether ambiguity or preference for wealth can contribute to empirical consumption puzzles such as excess sensitivity and excess smoothness.

**Excess sensitivity**. Friedman’s permanent income hypothesis predicts that transitory/predicable income shocks and consumption are uncorrelated. However, subsequent empirical literature has found that consumption exhibits excess sensitivity to transitory income (see, e.g., Flavin, 1981) or that consumption growth shows excess sensitivity to income change (see, e.g., Deaton, 1986) or income growth (see, e.g., Ludvigson & Michaelides, 2001). Since the sensitivity of consumption growth to wealth-income ratio can indirectly reflect the sensitivity of consumption growth to income change, we use the sensitivity of consumption growth to wealth-income ratio as the measure of consumption growth sensitivity. Without preference for wealth, the higher degree of ambiguity causes expected consumption growth to be more sensitive to wealth-income ratio, which means that ambiguity contributes to excess sensitivity of consumption growth. However, in the presence of preference for wealth, the contribution of ambiguity to excess sensitivity of consumption growth becomes negligible, which means that the interaction effects of ambiguity and preference for wealth play a vital role in the consumption-saving decision and cannot be ignored. The sensitivity of consumption growth to wealth-income ratio decreasing in *λ* shows that preference for wealth does not help to resolve the excess sensitivity of consumption growth to income change.

**Excess smoothness**. Excess smoothness of consumption means that the standard deviation of consumption changes is less than that of income changes (see, e.g., Campbell & Deaton, 1989) or that the standard deviation of consumption growth is less than that of income growth (see, e.g., Black, 1990, Ludvigson & Michaelides, 2001). We thus use two measures of consumption smoothness: the standard deviation of consumption changes and consumption volatility. The diffusion term of *d**C* is given by \(\sigma (c(w)-wc^{\prime} (w))Yd{\mathcal{B}}\), which implies that the standard deviation of consumption changes is \(\sigma (c(w)-wc^{\prime} (w))Y=\sigma {C}_{Y}(W,Y)Y\). The comparative static effect on *C*_{Y}(*W*, *Y*) shows that ambiguity contributes to excess smoothness of consumption and the contribution of preference for wealth to excess smoothness of consumption depends on the degree of ambiguity. The consumption volatility decreasing with *η* means that ambiguity contributes to excess smoothness of consumption growth. However, preference for wealth leading to more volatile consumption means that preference for wealth provides no help for explaining the excess smoothness of consumption growth.

## Conclusion

We extend the consumption-saving model presented by Wang et al. (2016) by introducing ambiguity and preference for wealth. Using the dynamic programming method, we solve the consumer’s optimization problem with max-min expected utility to obtain a robust consumption rule. We first systematically examine the comparative static effects of ambiguity or preference for wealth on consumption. Although they can both cause decreasing effects on the MPC out of wealth, their effects on consumption-income ratio, the MPC out of current income and expected consumption growth are quite different. Ambiguity decreases consumption-income ratio, the MPC out of current income and expected consumption growth because of pessimism. However, the effects of preference for wealth on the three variables are indefinite. Preference for wealth may produce decreasing or increasing effect, which depends on the degree of ambiguity. Using two approaches, i.e., modularity and iso-curves, we further explore the potential interaction effects of ambiguity and preference for wealth on consumption. We find that ambiguity and preference for wealth may be substitutional or complementary. They can produce substitution effects on the MPC out of wealth and complementarity effects on consumption volatility. However, when it comes to consumption-income ratio, the MPC out of current income and expected consumption growth, substitution and complementarity effects both exist. It’s shown that the interaction effects of ambiguity and preference for wealth play a role in the consumption-saving decision.

In particular, we explore whether ambiguity and preference for wealth help to explain the excess sensitivity and excess smoothness of consumption growth reported in the empirical literature. We find that preference for wealth cannot provide a contribution to the excess sensitivity of consumption growth. Interestingly, whether ambiguity helps to explain excess sensitivity of consumption growth is related to preference for wealth. Without preference for wealth, ambiguity contributes to the excess sensitivity of consumption growth. However, in the presence of preference for wealth, the contribution of ambiguity to excess sensitivity of consumption growth becomes negligible. From the perspective of the excess smoothness of consumption growth, ambiguity makes a contribution to the excess smoothness of consumption growth, while preference for wealth does not. Moreover, the prediction of our model is consistent with the reported evidence that low-income agents have a high propensity to consume while high-income agents have a relatively low propensity to consume.

## Notes

The spirit of capitalism is expressed by these sentences cited from Weber (1958):

*Man is dominated by the making of money, by acquisition as the ultimate purpose of his life. Economic acquisition is no longer subordinated to man as the means for the satisfaction of his material needs. This reversal of what we should call the natural relationship, so irrational from a naïve point of view, is evidently as definitely a leading principle of capitalism as it is foreign to all peoples not under capitalistic influence*.According to the analysis of Biagini & Pinar (2017), “the investor is diffident about mean return and volatility estimates” is equivalent to “the investor is uncertain about mean return and volatility estimates”.

In this paper, “consumer” and “agent” are used interchangeably.

Bakshi & Chen (1996) assume that status (denoted by

*S*_{t}) depends on the consumer’s absolute wealth*W*_{t}and a social wealth index*V*_{t}, which represents the wealth of a typical person in the economy. Status is thus described by a function*S*_{t}=*m*(*W*_{t}*, V*_{t}), where*m*_{W}> 0 and*m*_{V}≤ 0. For ease of exposition, this paper focuses on a simple form of the status function: Wealth is status,*S*_{t}=*W*_{t}. Frank (1985) observes that human beings face constant contests for position in society and relative status often dictates who gets to receive the prizes. Cole et al. (1992) argue that wealth determines status, which in turn regulates such things as marriage patterns. In this sense, wealth can be treated as status. Subsequently, utility function incorporating wealth-is-status has been adopted by some researchers such as Zou (1994), Zou (1995), Smith (2001), Boileau & Braeu (2007), Luo et al. (2009a, b), and Wang (2016).As for the cross partial derivative,

*U*_{CW}, if Markowitz (1952) hypothesis holds, we will have*U*_{CW}< 0; otherwise,*U*_{CW}≥ 0. As an aside, Markowitz (1952) hypothesis states that an increase (or decrease) in wealth will shift an investor’s utility-of-consumption curve to the right (or the left). An interpretation of his hypothesis is that each time an investor’s wealth status changes, it essentially causes her to go back and rerank the entire consumption set, such that the wealthier the investor, the less utility from a given unit of consumption. In this paper, we assume that*U*_{CW}< 0, i.e., Markowitz hypothesis holds.The certainty equivalent wealth

*F*(*W*,*Y*) makes the consumer indifferent between two cases, one with wealth*W*and the given labor income process*Y*and the other with wealth*F*(*W*,*Y*) and no labor income henceforth (Wang et al., 2016).When investing one dollar, besides the market rate

*r*, the consumer can receive an additional psychic return by*λ*because of utility satisfaction resulted from the increase in wealth (Luo et al., 2009b).If the consumption rule with zero wealth is

*c*(0) = 1 (or*C*(0,*Y*) =*Y*), the wealth process (2) implies that the change in wealth is zero and the consumer permanently saves nothing thereafter. Otherwise, if the consumption rule with zero wealth is*c*(0) < 1 (or*C*(0,*Y*) <*Y*), the wealth process (2) indicates that the wealth will be more than zero thereafter, i.e., the borrowing constraint*W*≥ 0 never binds.Although subjective discount rate being larger than interest rate is typically required in the buffer-stock saving literature, our model does not necessarily require

*ρ*>*r*.Maenhout (2004) predicates that in the presence of robustness, risk aversion estimates based on asset prices are substantially higher than the estimates of “pure” risk aversion based on stylized experiments. In addition, Smith (2001) shows that the spirit of capitalism may increase or decrease the degree of risk aversion. This means that the effective risk aversion is endogenous and has a relation to ambiguity and preference for wealth. In our model, however, the exogenous parameter

*γ*is to measure “pure” risk aversion to a great extent rather than the effective risk aversion.

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## Acknowledgements

We thank the anonymous referees for helpful comments and suggestions. Y.W. acknowledges the support from the Postdoctoral Science Foundation of China (#2018M640370 and #2019T120325). This research is also funded by the National Social Science Foundation of China (#17CJY062).

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## 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

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

### 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*

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

*A simple computation yields*

*Define a function as*

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

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

*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

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

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

where *b*_{2} is

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

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

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

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

where Θ is given by

and *b*_{3} is given by

Figure 15 also demonstrates that the main results are robust.

### Generalizations of borrowing constraint

For a generalized description of the borrowing constraint, we assume *W*_{t} ≥ −*α**Y*_{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.

### 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

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:

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

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

For the generalized borrowing constraint, we assume that

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.

*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

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:

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

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

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:

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

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

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

For the generalized borrowing constraint, we assume that

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.

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Wang, Y., Niu, Y. & Gong, S. Robust consumption policy with the desire for wealth accumulation.
*Rev Econ Household* **20**, 993–1025 (2022). https://doi.org/10.1007/s11150-021-09551-0

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DOI: https://doi.org/10.1007/s11150-021-09551-0

### Keywords

- Capitalism
- Ambiguity
- Consumption
- Excess sensitivity
- Excess smoothness

### JEL

- D81
- E21