Farmers Follow the Herd: A Theoretical Model on Social Norms and Payments for Environmental Services

Abstract

The economic literature on Payments for Environmental Services (PES) has studied extensively the behavioural factors that prevent farmers from signing PES contracts, even when the payments exceed the expected opportunity costs. This article provides a theoretical model of the role played by the interplay of descriptive and injunctive social norms in farmers’ decisions. When they choose to contribute voluntarily to an environmental public good, farmers may be driven by descriptive norms akin to conformity (do as the majority of their peers) as well as by injunctive norms (in line with what society expects them to do), which are the equivalent of a social injunction to act in favour of the environment. The interactions between these two social norms can yield multiple equilibria, depending on the relative weight of the descriptive norm (sensitivity to conformism) and of the injunctive norm (sensitivity to moral pressure) in the utility functions of farmers. More generally, our model can explain why social groups are sometimes trapped in low public-good-contribution equilibria, even when public subsidies to contributors are high. We make policy recommendations to help reach higher contribution equilibria, with a specific focus on the farm policy context.

Appendices

Appendix 1

Following Rege (2004), we use the replicator dynamics to represent a “virtual” learning process of trial-and error.

“The replicator dynamics say that the growth rate of the population share using a certain strategy equals the difference between the strategy’s current payoff and the current average payoff in the population (Weibull 1995, p. 73).”

In our case, the replicator dynamics is given by:

$$ \dot{x}\left( x \right) = x\left( {U_{i}^{1} \left( x \right) - \bar{U}\left( x \right)} \right) $$

where $\bar{U}\left( x \right) = xU_{i}^{1} \left( x \right) + \left( {1 - x} \right)U_{i}^{0} \left( x \right)$

$$ \dot{x}\left( x \right) = x\left( {1 - x} \right)\Delta U\left( x \right)\dot{x}\left( x \right) = x\left( {1 - x} \right)\left[ {p - c + 2\lambda \left( {2x - 1} \right)} \right] $$

Stationary states are determined by $\dot{x}\left( x \right) = 0$. Thus, there are three stationary states: $x = 0$, $x = 1$ and $x = x^{\prime} = \frac{1}{2} - \frac{p - c}{4\lambda }$.

For $0 < x < 1$, $\dot{x} > 0$ if $\Delta U = p - c + 2\lambda \left( {2x - 1} \right) > 0$ and thus if and only if $x > \frac{1}{2} - \frac{p - c}{4\lambda } = x^{\prime }$. Symmetrically, for $0 < x < 1$, $\dot{x} < 0$ if $\Delta U = p - c + 2\lambda \left( {2x - 1} \right) < 0$ and thus if and only if $x < \frac{1}{2} - \frac{p - c}{4\lambda } = x^{\prime }$. Hence, $x = x^{\prime }$ is not an asymptotically stable state because if the share of farmers who enrol in PES moves above $max\left\{ {0,x^{\prime } } \right\}$, then $x > x^{\prime }$ and $\Delta U > 0$. Therefore, more farmers will enrol in PES. This process will continue until all farmers are enrolled and the asymptotically stable state $x = 1$ is reached. Symmetrically, if the share of farmers who enrol in PES moves below $min\left\{ {1,x^{\prime } } \right\}$, then more farmers will quit the PES. This process will continue until all farmers leave the PES and the asymptotically stable state $x = 0$ is reached.

Appendix 2

Case 1: $\hat{x} \le 0 \Leftrightarrow \sigma \le 2\lambda$

The weight of the injunctive norm is not too strong relatively to the weight of the descriptive norm. In this first case $\Delta U$ is always increasing on $x \in \left[ {0,1} \right]$ and there are 3 subcases shown on Fig. 8.

Fig. 8

Farmer’s utility variation for case 1

1a) If $\Delta U > 0$ when $x = 0$ then $\Delta U > 0 \forall x \in \left[ {0,1} \right]$. Thus there is a unique Nash equilibrium in which all farmers enrol in PES ($x = 1)$.

1b) If $\Delta U < 0$ when $x = 0$ and $\Delta U > 0$ when $x = 1$ then there is a unique $x^{\prime } \in \left[ {0,1} \right]$ such that $\Delta U\left( {x^{\prime } } \right) = 0$. In that case there are three Nash equilibria: $x = 0$, $x = 1$ and $x = x^{\prime }$. However there are only two asymptotically stable states $x = 0$ and $x = 1$.

1c) If $\Delta U < 0$ when $x = 1$ then $\Delta U < 0 \forall x \in \left[ {0,1} \right]$ thus there is a unique Nash equilibrium in which no farmer enrols in PES ($x = 0)$.

Case 2: $0 < \hat{x} < 1 \Leftrightarrow 2\lambda < \sigma < 8\lambda$

The weight of the injunctive norm is not too strong and not too weak relatively to the weight of the descriptive norm. In this second case, $\Delta U$ is first decreasing until $\hat{x}$ and then increasing. There are 5 subcases, shown on Fig. 9.

Fig. 9

Farmer’s utility variation for case 2

2a) If $\Delta U_{min} > 0$ then $\Delta U > 0 \forall x \in \left[ {0,1} \right]$ thus there is a unique Nash equilibrium in which all farmers enrol in PES $(x = 1)$.

2b) If $\Delta U_{min} \le 0$ and $\Delta U > 0$ when $x = 0$ and $\Delta U > 0$ when $x = 1$ then there are two $x \in \left[ {0,1} \right] (x^{\prime }$ and $x^{\prime \prime } )$ such that $\Delta U\left( {x^{\prime } } \right) = \Delta U\left( {x^{\prime \prime } } \right) = 0$. In that case, there are three Nash equilibria: $x = x^{\prime }$, $x = x^{\prime \prime }$ and $x = 1$. However there are only two asymptotically stable states in this coordination game: $x = x^{\prime }$ and $x = 1$.

2c) If $\Delta U_{min} \le 0$ and $\Delta U < 0$ when $x = 0$ and $\Delta U > 0$ when $x = 1$ then there is a unique $x^{\prime } \in \left[ {0,1} \right]$ such that $\Delta U\left( {x^{\prime } } \right) = 0$. In that case there are three Nash equilibria: $x = 0$, $x = 1$ and $x = x^{\prime }$. However there are only two asymptotically stable states $x = 0$ and $x = 1$.

2d) If $\Delta U_{min} \le 0$ and $\Delta U > 0$ when $x = 0$ and $\Delta U < 0$ when $x = 1$ then there is a unique $x^{\prime } \in \left[ {0,1} \right]$ such that $\Delta U\left( {x^{\prime } } \right) = 0$. In that case there is a unique Nash equilibria: $x = x'$.

2e) If $\Delta U_{min} \le 0$ and $\Delta U < 0$ when $x = 0$ and $\Delta U < 0$ when $x = 1$ then $\Delta U < 0 \forall x \in \left[ {0,1} \right]$ thus there is a unique Nash equilibrium in which no farmer enrols in PES ($x = 0)$.

Case 3: $\hat{x} \ge 1 \Leftrightarrow \sigma \ge 8\lambda$

The weight of the injunctive norm is strong relatively to the weight of the descriptive norm. In this last case $\Delta U$ is always decreasing and there are 3 subcases shown on Fig. 10.

Fig. 10

Farmer’s utility variation for case 3

(3a) If $\Delta U > 0$ when $x = 1$ then $\Delta U > 0 \forall x \in \left[ {0,1} \right]$ thus there is a unique Nash equilibrium in which all farmers enrol in PES ($x = 1)$.

(3b) If $\Delta U > 0$ when $x = 0$ and $\Delta U < 0$ when $x = 1$ then there is a unique $x^{\prime } \in \left[ {0,1} \right]$ such that $\Delta U\left( {x^{\prime } } \right) = 0$. In that case there is a unique Nash equilibria: $x = x^{\prime }$.

(3c) If $\Delta U < 0$ when $x = 0$ then $\Delta U < 0 \forall x \in \left[ {0,1} \right]$ thus there is a unique Nash equilibrium in which no farmer enrols in PES ($x = 0)$.