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I am asking to myself if we must use a different discount rate on rival and non rival goods. In standard Solow-Swann model, we use generally the discount factor $\rho$.

More formally, let's give the workhorse model used in environmental economics.

$$max \int_{0}^{\infty} u(c(t),s(t)) e^{-\rho t}$$

where $c(t)$ is consumption (rival good) and $s(t)$ is the environmental quality (or amenities) which is a non rival good by definition.

Let's say that $u(c(t),s(t))$ is an additive utility function which could be defined as

$$u(c,s)=\frac{c^{1-\sigma}}{1-\sigma}+\frac{s^{1-\phi}}{1-\phi}$$

where $\phi$ and $\sigma$ are elasticities. Also, to be complete, the constraint would be as

$$\dot{s}\left(t\right)=-c(t)$$

The question that I ask to myself is why an agent (or social planner) uses the same discount rate for a rival (consumption) and non rival (environmental quality.) ? For example, I have completely different preferences for consumption and environment for future. So, how can I use the same discount $\rho$ ?

There is an interesting paper of Endres et al. (2014) where authors use individual and social discount rates which are different. In appendix B. (at page 631 of the paper), they define $A$ and $B$ where they say $A\neq B$ because they say each person treats differently rival and non rival goods.

So, is there a big caveat of growth models which contain rival and non rival goods in utility function ?

Thanks in advance for your hints.

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