Existence of optimal strategies in linear multisector models with several consumption goods

We consider also the case in which \(\Gamma _\nu ={\frac{{\Gamma _\nu -\rho }}{\sigma },}\) and provide further results of existence on nonexistence.

On the contrary, constraints of this type are used by Magill (1981a), Becker et al. (1989), and Balder (1993). In Magill (1981a), Definition 4.1 and Assumption 1, p. 686 (then in Section 9, Definition 9.5 and subsequent results), allow to get the existence of what Magill calls an expansion function (Definition 5.1 and Assumption 3, p. 687, Magill 1981a) which is a key assumption for proving the existence theorem. In Becker et al. (1989), Section 4.3, the same setting of Magill (1981a), Section 9, is used. This allows to prove that the Technology Conditions (i) and (ii), p. 81, are verified and again this is a key point to prove the existence theorem. In Balder (1993), we find the Growth Condition 2.4 (p. 424) to be essential for the proof of existence (together with the compactness of \(A\left( 0\right) \)).

The condition on \(U_{1}\left( \mathbf{c}\right) \) is relevant only when \(\sigma =1\). Note that for \(\sigma \in \left( 0,1\right) \) the function \(t\rightarrow \mathrm{e}^{-\rho t}u_{\sigma }\left( \nu (\mathbf{c}_{t})\right) \) is always nonnegative so it is always semiintegrable (with the integral eventually \(+\infty \)). On the other hand, for \(\sigma >1\) the function \(t\rightarrow \mathrm{e}^{-\rho t}u_{\sigma }\left( \nu (\mathbf{c}_{t})\right) \) is always negative (and may be \(-\infty \) when \(\nu (\mathbf{c})_{t}=0\)) and again it is always semiintegrable (with the integral eventually \(-\infty \)). This means that the intertemporal utility \(U_{\sigma }\) is always well defined for \(\sigma \ne 1\). For \(\sigma =1\), the function \( t\rightarrow \mathrm{e}^{-\rho t}u_{\sigma }\left( \nu ( \mathbf{c}_{t})\right) \) may change sign so it may be not semiintegrable on \(\left[ 0,+\infty \right) \). This is the reason why we need to require that \(U_{1}\left( \mathbf{c}\right) \) is well defined to define the admissibility of \(\mathbf{c}\).

We say that commodity j is essential to the reproduction of the ith consumption good when

Note that this is always true if all components of \(\bar{\mathbf{s}}\) are very big.