The two sides of envy

Abstract

The two sides of envy, destructive and constructive, give rise to qualitatively different equilibria, depending on the economic, institutional, and cultural environment. If investment opportunities are scarce, inequality is high, property rights are not secure, and social comparisons are strong, society is likely to be in the “fear equilibrium,” in which better endowed agents underinvest in order to avoid destructive envy of the relatively poor. Otherwise, the standard “keeping up with the Joneses” competition arises, and envy is satisfied through suboptimally high efforts. Economic growth expands the production possibilities frontier and triggers an endogenous transition from one equilibrium to the other causing a qualitative shift in the relationship between envy and economic performance: envy-avoidance behavior with its adverse effect on investment paves the way to creative emulation. From a welfare perspective, better institutions and wealth redistribution that move the society away from the low-output fear equilibrium need not be Pareto improving in the short run, as they unleash the negative consumption externality. In the long run, such policies contribute to an increase in social welfare due to enhanced productivity growth.

Appendix

Detailed Form of Lemma 2 The detailed version of Eq. (12) is:

• 1. If $AK_2\geqslant \widehat{A}_{21}$, then

  $$\begin{aligned} C_2^{*}(C_1)={\left\{ \begin{array}{ll} AK_2, &{} \hbox {if} \quad C_1\geqslant \widehat{C}_{11}; \\ \theta C_1+(AK_2)^{1/\sigma }, &{} \hbox {if} \quad \widehat{C}_{12}\leqslant C_1<\widehat{C}_{11}; \\ C_1\cdot \frac{1}{\tau \theta }, &{} \hbox {if} \quad \widehat{C}_{13}\leqslant C_1<\widehat{C}_{12}; \\ C_2^d(C_1), &{} \hbox {if} \quad C_1<\widehat{C}_{13}, \end{array}\right. } \end{aligned}$$

  where

  $$\begin{aligned} \widehat{C}_{11}\equiv \frac{AK_2-(AK_2)^{1/\sigma }}{\theta },\! \quad \widehat{C}_{12}\equiv \frac{\tau \theta (AK_2)^{1/\sigma }}{1-\tau \theta ^2}, \quad \!\widehat{C}_{13}\equiv \frac{\tau \theta }{1-\tau \theta ^2}\left( \frac{1+\theta ^2}{2}AK_2\right) ^{1/\sigma }, \end{aligned}$$

  and $C_2^d(C_1)$ is implicitly given by

  $$\begin{aligned} C_2-\theta C_1=\phi \cdot \left( \frac{C_1+\theta C_2}{C_2}\right) ^{1/\sigma }, \quad \phi \equiv \left( \frac{1+\theta ^2}{2\theta (1+\tau )}AK_2\right) ^{1/\sigma }. \end{aligned}$$

  (26)

• 2. If $AK_2\in [\widehat{A}_{22},\widehat{A}_{21})$, then

  $$\begin{aligned} C_2^{*}(C_1)={\left\{ \begin{array}{ll} AK_2, &{} \hbox {if} \quad C_1\geqslant \widehat{C}_{14}; \\ C_1\cdot \frac{1}{\tau \theta }, &{} \hbox {if} \quad \widehat{C}_{13}\leqslant C_1<\widehat{C}_{14}; \\ C_2^d(C_1), &{} \hbox {if} \quad C_1<\widehat{C}_{13}, \end{array}\right. } \end{aligned}$$

  where $\widehat{C}_{14}\equiv \tau \theta AK_2$.

• 3. If $AK_2\in (\widehat{A}_{23}, \widehat{A}_{22})$, then

  $$\begin{aligned} C_2^{*}(C_1)={\left\{ \begin{array}{ll} AK_2, &{} \hbox {if} \quad C_1\geqslant \widehat{C}_{14}; \\ \widetilde{C}_2^{d}(C_1), &{} \hbox {if} \quad \widehat{C}_{15}\leqslant C_1<\widehat{C}_{14}; \\ C_2^d(C_1), &{} \hbox {if} \quad C_1<\widehat{C}_{15}, \end{array}\right. } \end{aligned}$$

  where $\widetilde{C}_2^{d}(C_1)$ is implicitly given by

  $$\begin{aligned} C_1=\theta C_2\cdot \left( \frac{1+\tau }{AK_2}\cdot C_2-1\right) \end{aligned}$$

  and $\widehat{C}_{15}$ solves $C_2^d(\widehat{C}_{15})=\widetilde{C}_2^d(\widehat{C}_{15})$.

The threshold $\widehat{C}_1$ from Lemma 2 is defined as $\widehat{C}_1\equiv \min \{\widehat{C}_{13},\hat{C}_{14}\}$.

Detailed Form of Lemma 3 The detailed version of Eq. (15) is:

• 1. If $AK_1\geqslant \widehat{A}_{11}$, then

  $$\begin{aligned} C_1^{*}(C_2)={\left\{ \begin{array}{ll} \widetilde{C}_1^d(C_2), &{} \hbox {if} \quad C_2\geqslant \widehat{C}_{22}; \\ C_1^d(C_2), &{} \hbox {if} \quad \widehat{C}_{21}\leqslant C_2<\widehat{C}_{22}; \\ \theta C_2+(AK_1)^{1/\sigma }, &{} \hbox {if} \quad C_2<\widehat{C}_{21}, \end{array}\right. } \end{aligned}$$

  where

  $$\begin{aligned} \widetilde{C}_1^d(C_2)\equiv \frac{1+\tau }{\tau }AK_1-\theta C_2, \quad \widehat{C}_{21}\equiv \frac{(AK_1)^{1/\sigma }}{\theta (\tau -1)}, \end{aligned}$$

  the function $C_1^d(C_2)$ is implicitly given by

  $$\begin{aligned} C_1-\theta C_2=\psi \cdot \left( \frac{C_1}{C_1+\theta C_2}\right) ^{1/\sigma }, \quad \psi \equiv \left( \frac{1+\tau }{\tau }AK_1\right) ^{1/\sigma }, \end{aligned}$$

  (27)

  and $\widehat{C}_{22}$ solves $C_1^d(\widehat{C}_{22})=\widetilde{C}_1^d(\widehat{C}_{22})$.

• 2. If $AK_1\in [\widehat{A}_{12},\widehat{A}_{11})$, then

  $$\begin{aligned} C_1^{*}(C_2)={\left\{ \begin{array}{ll} \widetilde{C}_1^d(C_2), &{} \hbox {if} \quad C_2\geqslant \widehat{C}_{24}; \\ AK_1, &{} \hbox {if} \quad \widehat{C}_{23}\leqslant C_2<\widehat{C}_{24}; \\ \theta C_2+(AK_1)^{1/\sigma }, &{} \hbox {if} \quad C_2<\widehat{C}_{23}, \end{array}\right. } \end{aligned}$$

  where

  $$\begin{aligned} \widehat{C}_{23}\equiv \frac{AK_1-(AK_1)^{1/\sigma }}{\theta }, \quad \widehat{C}_{24}\equiv \frac{AK_1}{\tau \theta }. \end{aligned}$$

• 3. If $AK_1<\widehat{A}_{12}$, then

  $$\begin{aligned} C_1^{*}(C_2)={\left\{ \begin{array}{ll} \widetilde{C}_1^d(C_2), &{} \hbox {if} \quad C_2\geqslant \widehat{C}_{24}; \\ AK_1, &{} \hbox {if} \quad C_2< \widehat{C}_{24}. \end{array}\right. } \end{aligned}$$

The threshold $\widehat{C}_2$ from Lemma 3 is defined as $\widehat{C}_2\equiv \min \{\widehat{C}_{21},\hat{C}_{24}\}$.

Detailed Form of Proposition 1 The unique subgame perfect equilibrium $(C_1^{*},C_2^{*})$ of the envy game is determined as follows.

• 1. If $AK_1\geqslant \widehat{A}_{11}$ and $AK_2\geqslant \widehat{A}_{21}$, then:

  • (a) If $k\geqslant \tilde{k}$, it is the KUJ equilibrium (16);

  • (b) If $\hat{k}\leqslant k<\tilde{k}$, it is the fear equilibrium (19);

  • (c) If $k<\hat{k}$, it is the destructive equilibrium implicitly defined by

    $$\begin{aligned} {\left\{ \begin{array}{ll} C_1^{*}=\min \{C_1^d(C_2^{*}),\widetilde{C}_1^d(C_2^{*})\};\\ C_2^{*}=C_2^d(C_1^{*}). \end{array}\right. } \end{aligned}$$

  The threshold values of $k$ are given by

  $$\begin{aligned} \tilde{k}\equiv \left[ \frac{\theta (\tau -1)}{1-\tau \theta ^2}\right] ^{\sigma }, \quad \hat{k}\equiv \tilde{k}\cdot \frac{(1+\theta ^2)}{2}. \end{aligned}$$

  (28)

• 2. If $\widehat{A}_{12}\leqslant AK_1<\widehat{A}_{11}$ and $AK_2\geqslant \widehat{A}_{21}$, then:

  • (a) If $AK_1\geqslant \widehat{C}_{11}$, it is the full-time KUJ equilibrium (18);

  • (b) If $AK_1<\widehat{C}_{11}$ and $m_1(AK_1)\leqslant AK_2<m_2(AK_1)$, it is the KUJ equilibrium (17), where

    $$\begin{aligned} m_1(AK_1)\equiv \left[ \frac{(1-\theta ^2)AK_1-(AK_1)^{1/\sigma }}{\theta }\right] ^{\sigma }, \quad m_2(AK_1)\equiv \left[ \frac{(1-\tau \theta ^2)AK_1}{\tau \theta }\right] ^{\sigma }; \end{aligned}$$

  • (c) If $AK_2<m_1(AK_1)$, it is the KUJ equilibrium (16);

  • (d) If $m_2(AK_1)\leqslant AK_2<2m_2(AK_1)/(1+\theta ^2)$, it is the fear equilibrium (20);

  • (e) If $AK_2\geqslant 2m_2(AK_1)/(1+\theta ^2)$, it is the destructive equilibrium in case 2 of (21).

• 3. If $AK_1<\widehat{A}_{11}$ and $\widehat{A}_{22}\leqslant AK_2<\widehat{A}_{21}$, then:

  • (a) If $k\geqslant \tau \theta$, it is the full-time KUJ equilibrium (18);

  • (b) If $k<\tau \theta$ and $AK_2<2m_2(AK_1)/(1+\theta ^2)$, it is the fear equilibrium (20);

  • (c) If $AK_2\geqslant 2m_2(AK_1)/(1+\theta ^2)$, it is the destructive equilibrium in case 2 of (21).

• 4. If $AK_1<\widehat{A}_{11}$ and $\widehat{A}_{23}<AK_2<\widehat{A}_{22}$, then:

  • (a) If $k\geqslant \tau \theta$, it is the full-time KUJ equilibrium (18);

  • (b) If $k<\tau \theta$, it is the destructive equilibrium implicitly defined by

    $$\begin{aligned} {\left\{ \begin{array}{ll} C_1^{*}=\widetilde{C}_1^d(C_2^{*});\\ C_2^{*}=\min \{C_2^d(C_1^{*}),\widetilde{C}_2^d(C_1^{*})\}. \end{array}\right. } \end{aligned}$$