Intentional time inconsistency

Abstract

We propose a theoretical model to explain the usage of time-inconsistent behavior as a strategy to exploit others when reputation and trust have secondary effects on the economic outcome. We consider two agents with time-consistent preferences exploiting common resources. Supposing that an agent is believed to have time-inconsistent preferences with probability p,  we analyze whether she uses this misinformation when she has the opportunity to use it. Using the model originally provided by Levhari and Mirman (Bell J Econ 11(1):322–334, 1980), we determine the optimal degree of present bias that the agent would like to have while pretending to have time-inconsistent preferences and we provide the range of present-bias parameter under which deceiving is optimal. Moreover, by allowing the constant relative risk aversion class of utility form, we characterize the distinction between pretending to be naive and sophisticated.

Appendix

1.1 The proof of proposition 2

(a) Follows from Amir and Nannerup (2006).

(b) From part a, one can compute that:

$$\begin{aligned} V_{tc}\left( x\right) =\frac{\log x}{1-\delta \alpha }+\frac{\log \left( \frac{1-\delta \alpha }{2-\delta \alpha }\right) +\frac{\delta \alpha }{1-\delta \alpha }\log \frac{\delta \alpha }{2-\delta \alpha }}{1-\delta }.\end{aligned}$$

Under Assumption 2, by 7 and 8, we have

$$\begin{aligned} g_{N}\left( x\right)= & {} \frac{\left( 1-\delta \alpha \right) \left( x-g_{A_{N}}\left( x\right) \right) }{\left( 1-\delta \alpha +\beta \delta \alpha \right) } \frac{g_{A_{N}}\left( y\right) }{g_{A_{N}}\left( x\right) } =\delta f^{\prime }\left( x-g_{N}\left( x\right) -g_{A_{N}}\left( x\right) \right) \\&\quad \times \, \left( 1-g_{N}^{\prime }\left( y\right) \right) \text { where } y=\left( f\left( x-g_{N}\left( x\right) -g_{A_{N}}\left( x\right) \right) \right) . \end{aligned}$$

Using the linearity of $g_{N}\left( {}\right)$ and $g_{A_{N}}(x),$ we find $g_{N}\left( x\right) =\frac{1-\delta \alpha }{1+\beta -\alpha \delta }x$ and $g_{A_{N}}(x)=\frac{\beta \left( 1-\delta \alpha \right) }{1+\beta -\alpha \delta }x.$

(c) Under Assumption 2, by 11 and 12, we have

$$\begin{aligned} \frac{g_{S}\left( y\right) }{g_{S}\left( x\right) }&=\beta \delta \alpha \left( x-g_{S}\left( x\right) -g_{A_{S}}\left( x\right) \right) ^{\alpha -1}\left[ \left( 1-\frac{1}{\beta }\right) g_{S}^{\prime }\left( y\right) +\frac{\left( 1-g_{A_{S}}^{\prime }\left( y\right) \right) }{\beta }\right] .\nonumber \\ \frac{g_{A_{S}}\left( y\right) }{g_{A_{S}}\left( x\right) }&=\delta f^{\prime }\left( x-g_{S}\left( x\right) -g_{A_{S}}\left( x\right) \right) \left( 1-g_{S}^{\prime }\left( y\right) \right) \text { where }y=\left( f\left( x-g_{S}\left( x\right) -g_{A_{S}}\left( x\right) \right) \right) . \end{aligned}$$

(15)

Using the linearity of $g_{S}\left( {}\right)$ and $g_{A_{S}}(x),$ we find $g_{S}\left( x\right) =\frac{1-\delta \alpha }{1+\beta -\alpha \delta }x$ and $g_{A_{S}}(x)=\frac{\beta \left( 1-\delta \alpha \right) }{1+\beta -\alpha \delta }x.$

1.2 The Proof of Corollary 1

It follows from Proposition 2, as we have

$$\begin{aligned} 2g_{tc}(x)=\frac{2\left( 1-\delta \alpha \right) }{2-\delta \alpha } x<\frac{\left( 1+\beta \right) \left( 1-\delta \alpha \right) }{1+\beta -\alpha \delta }=g_{N}\left( x\right) +g_{A_{N}}(x)=g_{S}\left( x\right) +g_{A_{S}}(x). \end{aligned}$$

1.3 The Proof of Proposition 3

(a) Under case 2 and 3, agent 1’s utility is given by the following:

$$\begin{aligned} V^{1}\left( x\right)&=\log \left( g\left( x\right) \right) +\delta V^{1}\left( \left( x-g\left( x\right) -g^{A}\left( x\right) \right) ^{\alpha }\right) \text { }\\ \text {where }g\left( x\right)&=g^{S}(x)=g^{N}(x)=\frac{1-\delta \alpha }{1+\beta -\alpha \delta }x\text { and }\\ g^{A}\left( x\right)&=g_{S}^{A}(x)=g_{N}^{A}(x)=\frac{\beta \left( 1-\delta \alpha \right) }{1+\beta -\alpha \delta }x. \end{aligned}$$

By guessing that the value function has the form $A\log x+B,$ we compute

$$\begin{aligned} V^{1}\left( x\right) =\frac{\log x}{1-\delta \alpha }+\frac{\log \frac{1-\delta \alpha }{1+\beta -\alpha \delta }+\frac{\delta \alpha }{1-\delta \alpha } \log \frac{\beta \delta \alpha }{1+\beta -\alpha \delta }}{1-\delta }. \end{aligned}$$

This implies that utility maximizing $\beta$ solves the following problem : 

$$\begin{aligned} \max _{0<\beta \le 1}\log \frac{1}{1+\beta -\delta \alpha }+\delta \alpha \log \beta . \end{aligned}$$

From the first-order condition, we get $\beta =\delta \alpha .$ Note that objective function is increasing when $\beta <\delta \alpha$ and decreasing when $\beta >\delta \alpha$, i.e., $\beta =\delta \alpha$ is the unique maximization point.

(b) If the agent cannot choose $\beta ,$ she gets $V^{1}\left( x\right)$ if she acts as if she had time-inconsistent preferences and gets $V_{tc}\left( x\right)$ if she acts truly. Note that $V^{1}\left( x\right)>V_{tc}\left( x\right) \Longleftrightarrow \log \frac{1}{1+\beta -\delta \alpha }+\delta \alpha \log \beta >\log \frac{1}{2-\delta \alpha }$ . By part (a), we know that left-hand side of the equation is decreasing in $\beta$ when $\beta$ is greater than $\delta \alpha .$ Since they are equal to each other for $\beta$ equals to 1, the left-hand side is greater than the right-hand side when $\beta \in [ \delta \alpha ,1)$. Let us consider the case, such that $( \frac{1}{2-\delta \alpha }) ^{\frac{1}{\delta \alpha }} <\alpha \delta .$ Our result follows from the fact that, for any $\beta \in [( \frac{1}{2-\delta \alpha }) ^{\frac{1}{\delta \alpha } },\alpha \delta ) ,$ we have $\log \frac{1}{1+\beta -\delta \alpha } +\delta \alpha \log \beta >\delta \alpha \log \beta \ge \log \frac{1}{2-\delta \alpha }$.

1.4 The Proof of Proposition 4

(a) Under Assumption 3, by 4, we have

$$\begin{aligned} \left( \frac{g_{tc}\left( y\right) }{g_{tc}\left( x\right) }\right) ^{\sigma }=\delta \left( 1-g_{tc}^{\prime }\left( y\right) \right) \quad \text { where} \quad y=f\left( x-2g_{tc}\left( x\right) \right) . \end{aligned}$$

By imposing that $g_{tc}\left( x\right) =a_{tc}x$, we get $\left( 1-2a_{tc}\right) ^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{tc}\right) =0.$ Since $h_{tc}\left( 0\right) >0,h_{tc}\left( \frac{1}{2}\right) <0$ and $h_{tc}\left( c\right)$ is continuous in c implies that there exist $a_{tc}$$\in \left( 0,\frac{1}{2}\right)$, such that $h_{tc}\left( a_{tc}\right) =0.$ By 5, we get:

$$\begin{aligned} V_{tc}\left( x\right) =u(a_{tc}x)\frac{1}{1-\delta \left( A\left( 1-2a_{tc}\right) \right) ^{1-\sigma }}. \end{aligned}$$

(16)

(b) Under Assumption 3, by 7 and 16, we have

$$\begin{aligned} \frac{\left( \left( x-g_{N}\left( x\right) -g_{A_{N}}\left( x\right) \right) \right) ^{\sigma }}{\left( g_{N}\left( x\right) \right) ^{\sigma }}&=B\\ \left( \frac{g_{A_{N}}\left( y\right) }{g_{A_{N}}\left( x\right) }\right) ^{\sigma }&=\delta \left( 1-g_{N}^{\prime }\left( y\right) \right) \text { where }y=f\left( x-g_{N}\left( x\right) -g_{A_{N}}\left( x\right) \right) . \end{aligned}$$

By imposing that $g_{N}\left( x\right) =a_{n}x,$ and $g_{A_{N}}(x)$ is linear in x, we get $g_{A_{N}}(x)=\left( 1-a_{n}\left( 1+B^{1/\sigma }\right) \right) x$ and

$$\begin{aligned} Ba_{n}^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{n}\right) =0. \end{aligned}$$

Consider function $h_{N}\left( {}\right)$. In equilibrium, we must have $x-g_{N}\left( x\right) -g_{A_{N}}(x)>0$, i.e., $a_{n}<\frac{1}{1+B^{1/\sigma }}.$Since $h_{N}\left( a_{tc}\right) =Ba_{tc}^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{tc}\right) =\frac{\delta A^{1-\sigma }\left( \beta -1\right) a_{tc}}{1-\delta A^{1-\sigma }\left( 1-2a_{tc}\right) ^{1-\sigma }}<0,$$h_{N}^{\prime }\left( c\right) >0,$ and by assumption of the proposition $h_{N}\left( \frac{1}{1+B^{1/\sigma }}\right) >0,$ there exists unique $a_{n}$$\in \left( a_{tc},\frac{1}{1+B^{1/\sigma }}\right)$, such that $h_{N}\left( a_{n}\right) =0.$

(c) Under Assumption 3, by 11, we have

$$\begin{aligned} \left( \frac{g_{S}\left( y\right) }{g_{S}\left( x\right) }\right) ^{\sigma }&=\beta \delta A\left[ \left( 1-\frac{1}{\beta }\right) g_{S}^{\prime }\left( y\right) +\frac{\left( 1-g_{A_{S}}^{\prime }\left( y\right) \right) }{\beta }\right] .\\ \left( \frac{g_{A_{S}}\left( y\right) }{g_{A_{S}}\left( x\right) }\right) ^{\sigma }&=\delta A\left( 1-g_{S}^{\prime }\left( y\right) \right) \text { where }y=\left( f\left( x-g_{N}\left( x\right) -g_{A_{N} }\left( x\right) \right) \right) . \end{aligned}$$

By imposing that $g_{S}\left( x\right) =a_{s}x,$ and $g_{A_{S}}(x)$ is linear in x, we get $g_{A_{S}}(x)=\beta a_{s}x$ and

$$\begin{aligned} \left( 1-a_{s}-\beta a_{s}\right) ^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{s}\right) =0. \end{aligned}$$

In equilibrium, we must have $x-g_{S}\left( x\right) -g_{A_{S}}(x)>0$ i.e. $a_{s}<\frac{1}{1+\beta }.$Consider function $h_{S}\left( {}\right)$. Since $h_{S}\left( a_{tc}\right) >h_{tc}\left( a_{tc}\right) =0,$$h_{S}\left( \frac{1}{1+\beta }\right) <0$ and $h_{S}\left( c\right)$ is continuous in c,  there exists $a_{s}$$\in \left( a_{tc},\frac{1}{1+\beta }\right)$, such that $h_{S}\left( a_{s}\right) =0.$

1.5 The proof of corollary 2

Since the equilibrium strategy of a naive and a sophisticated agent does not necessarily coincide, we have two different cases to consider. Let us consider the strategic interaction with a naive agent first. We have

$$\begin{aligned} h_{N}\left( a_{n}\right) =Ba_{n}^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{n}\right) =\left( 1-c_{n}\right) ^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{n}\right) =0, \end{aligned}$$

where $c_{n}x=g_{N}\left( x\right) +g_{A_{N}}(x)=a_{n}x+\left( 1-a_{n}\left( 1+B^{1/\sigma }\right) \right) x$.Since $\left( 1-2a_{tc}\right) ^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{tc}\right) =0$, we have the relation that:

$$\begin{aligned} a_{n}>a_{tc}\Leftrightarrow c_{n}>2a_{tc}. \end{aligned}$$

By Proposition 4b, we conclude $g_{N}\left( x\right) +g_{A_{N}}(x)>2g_{tc}\left( x\right) .$Similarly, we have

$$\begin{aligned} h_{S}\left( a_{s}\right) =\left( 1-c_{s}\right) ^{\sigma }-\delta A^{1-\sigma }\left( 1-a_{s}\right) =0, \end{aligned}$$

where $c_{s}x=g_{S}\left( x\right) +g_{A_{S}}(x).$This implies that:

$$\begin{aligned} a_{s}>a_{tc}\Leftrightarrow c_{s}>2a_{tc}. \end{aligned}$$

By Proposition 4c, we conclude $g_{S}\left( x\right) +g_{A_{S}}(x)>2g_{tc}\left( x\right) .$

1.6 The pooling equilibrium and optimal value of discount factor when there is an uncertainty about agent 1’s discount factor

Suppose that agent 1’s true discount factor is $\delta _{1}$, while agent 2 believes that her discount factor is $\hat{\delta }_{1}$ with probability p. Consider the games with discount rate pairs $\left( \hat{\delta }_{1} ,\delta _{2}\right)$ and $\left( \delta _{1},\delta _{2}\right) .$Since both agents have time-consistent preferences, MPNE of the games can be found by following the steps that we define in Sect. 2.1. Let us denote MPNE by $\hat{g}_{tc}^{1}(x),\hat{g}_{tc}^{2}(x)$ and value functions by $\hat{V}_{tc}^{1}(x),\hat{V}_{tc}^{2}(x)$ for the game with discount rate $(\hat{\delta }_{1},\delta _{2}).$ Similarly, denote MPNE of the game with discount rate $( \delta _{1},\delta _{2})$ by $g_{tc}^{1}(x),g_{tc}^{2}(x)$ and the corresponding value functions by $V_{tc}^{1}(x),V_{tc}^{2}(x).$ We can define the pooling equilibrium as below:

If $\hat{V}_{tc}^{1}(x)\ge V_{tc}^{1}(x)$ and $\hat{V}_{tc}^{i}(x)\ge IC (x)$, where $i\in \left\{ 1,2\right\} ,$ there is a perfect Bayesian equilibrium $\left( \left( \tilde{s}^{1}(h,\theta ),\tilde{s} ^{2}\left( h\right) \right) ,\left( \mu \left( h\right) \right) \right)$, such that

$$\begin{aligned} \tilde{s}_{t}^{1}(h_{t},\theta =tco)&=\tilde{s}_{t}^{1}(h_{t},\theta =tin)=\hat{g}_{tc}^{1}(x)\\ \tilde{s}_{t}^{2}(h_{t})&=\left\{ \begin{array}[c]{cc} \hat{g}_{tc}^{2}(x)\text { and }\mu \left( h_{t}\right) =p &{} \quad if \quad \text { } h_{t}=\left\{ \hat{g}_{tc}^{1}(x_{k}),c_{k}^{2},x_{k}\right\} _{k=0}^{t}\\ g_{tc}^{2}(x)\quad \text { and } \quad \mu \left( h_{t}\right) =0 &{} \quad if \quad \text { }h_{t} \ne \left\{ \hat{g}_{tc}^{1}(x_{k}),c_{k}^{2},x_{k}\right\} _{k=0}^{t} \end{array} \right\} . \end{aligned}$$

Under Assumption 2, by Levhari and Mirman (1980), we have:

$$\begin{aligned} \hat{g}_{tc}^{1}(x)=\left( \frac{\delta _{2}\left( 1-\hat{\delta }_{1} \alpha \right) }{\hat{\delta }_{1}+\delta _{2}-\hat{\delta }_{1}\delta _{2}\alpha }\right) x \quad \text { and} \quad g_{tc}^{2}(x)=\left( \frac{\hat{\delta }_{1}\left( 1-\delta _{2}\alpha \right) }{\hat{\delta }_{1}+\delta _{2}-\hat{\delta } _{1}\delta _{2}\alpha }\right) x. \end{aligned}$$

Agent 1 finds the optimal value of $\hat{\delta }_{1}$ by solving the trade-off between her consumption rate and combined investment rate governed by the following maximization problem:

$$\begin{aligned} \max _{\hat{\delta }_{1}}\log c+\frac{\delta _{1}\alpha }{1-\delta _{1}\alpha } \log 1-c-d, \end{aligned}$$

where c and d represent the equilibrium consumption rates of the agents, i.e., $c=\frac{\hat{g}_{tc}^{1}(x)}{x}$ and $d=\frac{\hat{g}_{tc}^{2}(x)}{x}$. From the first-order condition, we get $\hat{\delta }_{1}=\frac{\alpha \delta _{1}\delta _{2}}{1-\alpha \delta _{1}+\alpha ^{2}\delta _{1}\delta _{2}}.$Note that objective function is increasing when $\hat{\delta }_{1}<\frac{\alpha \delta _{1}\delta _{2}}{1-\alpha \delta _{1}+\alpha ^{2}\delta _{1}\delta _{2}}$ and decreasing when $\hat{\delta }_{1}>\frac{\alpha \delta _{1}\delta _{2}}{1-\alpha \delta _{1}+\alpha ^{2}\delta _{1}\delta _{2}}$, i.e., $\hat{\delta } _{1}=\frac{\alpha \delta _{1}\delta _{2}}{1-\alpha \delta _{1}+\alpha ^{2}\delta _{1}\delta _{2}}$ is the unique maximization point.