Equivalence of inada conditions and non-negativity constraints?

Question

In a standard constrained utility maximization problem with an agent's preferences defined over good(s), does the imposition of Inada conditions on the utility function preclude us from adding non-negativity constraints while setting up the Lagrangean? The latter seem redundant because the Inada conditions will guarantee an interior solution. Thanks!

Answer 1

In a standard growth model, people don't put non-negativity constraints as you have mentionned. Theoretically, concavity of the Hamiltonian (or Lagrangian in discrete time) is the sufficient condition for the optimality of the program. Just for some additional details, the non-negativity constraints are so usefulin some subfields of economics, like in environmental economics. For example, if you want to put a ceiling for stock of pollution in atmosphere as $\overline{P}$. You can use it if you want that economy do not pass beyond this critical threshold. So, basically you can make as

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

s.t

$$\dot{K}=F(K)-c\\ \dot{P}=\epsilon K - \delta P$$

where $\epsilon$ is the emission rate from the use of capital and $\delta$ is the decay rate of pollution.

$$\mathcal{H} = u(c,P) + \lambda (F(K)-c) - \mu (\epsilon K - \delta P) + \alpha (\overline{P} - P) $$

The constraint $\alpha$ is a non-negativity constraint which will ensure that pollution level will not pass beyond the threshold $\overline{P}$. As long as the pollution level is below the threshold, $\alpha$ will be non-binding (and will be equal to zero).

Hint : You can look at the Kuhn-Tucker conditions in order to understand better the role of non-negativity constraint.