%\citet{bowles2008social} follows on from the above reasoning of \citet{benabou2006incentives}; it is an attempt to clarify the relationship between material and moral sentiments in an integrated framework based on non-separable preferences that isolate the marginal (crowding-in or crowding-out) effect of people's decisions to contribute to a public good based on moral values, and therefore to discriminate between cases where incentives and moral values are complements (crowding-in) or substitutes (crowding-out). They then investigate the design of optimal incentives to contribute to the public good under conditions of non-additive crowding-in and crowding-out effects. this analysis makes it possible to identify …show more content…
They consider a community of identical individuals indexed by $i = 1,...,n$ who may or may not contribute to a public project by taking action ($a^{i} \in [0,1]$) at a cost of $g(a^{i})$. The output of the project depends on each contribution, $\phi(a^{1},...,a^{n})$ and explicit incentives take the form of the subsidy $s \geq 0$ proportional to the amount contributed. Implementation of the subsidy incurs cost $c(s)$ that are increasing in the level of subsidy because higher values of $s$ increase the citizen's incentive to misrepresent the level of their contribution.
They make the assumption that payment of taxes supporting the subsidy has no effect on citizens' behaviour and can be ignored. Therefore the net social cost of the subsidy is $c(s)$.
\citeauthor{bowles2008social} create a value function to represent ethical, other-regarding and social preference influences on an individuals behaviour.
\begin{equation}\label{eq.1} v=a^{i}(\underline{v}+\lambda s) …show more content…
Under crowding in, values and incentives are complements, as increased use of incentives enhances the marginal effects of contributing on one's values and by (\ref{eq.4}) increases the effects of the subsidy on the citizen's action. Crowding out makes incentives and values substitutes, reducing the effect of incentives on citizen's behaviour. If $\lambda < -1$, which we term strong crowding out, the incentive reduces contributions. From (\ref{eq.4}) it is clear that a positive response by subjects to explicit incentives does not indicate that crowding out is absent; it indicates only that $\lambda >