# Capital and investment dynamics

Capital invested in a given region is a bundle obtained using the same CES nesting structure as for intermediate consumption. However, the product composition of both bundles differs according to the data, while the composition by geographical origin for each product is unique.

Installed capital is assumed to be immobile. This confers investment an important role, as the only adjustment device for capital stock. This “putty-clay” hypothesis is important because it implies that capital stock adjusts gradually. The sectoral allocation of capital can thus be sub-optimal, and the corresponding loss is interpreted as an adjustment cost for the economy. In addition, these assumptions imply that the rate of return to capital varies across sectors after the initial year.

## Investment allocation

FDIs are not considered explicitly in MIRAGE-e, though they are embodied implicitly in the current account (taken from EconMap). In practice, a single generic formalization encompassing both domestic and foreign investment is used, though only the domestic \(r,r\) elements are used. This leaves the possibility to update the outdated version, which includes explicit FDIs. Total investment is given by the current account: \[SAV_{s,t} REV_{s,t} = \sum_{i,r} P^{INVTOT}_{s,t} INV_{i,r,s,t} + WGDPVAL_t . CABal_{s,t},\] and investment level is function of the initial savings pattern, of the present capital stock, and of the sectoral rate of return to capital, with an elasticity \(\alpha\), rewritten as (see Capital and Investment Dynamics (Outdated)): \[INV_{i,r,s}= B_r a_{i,r,s} Capital_{i,s} \mathrm{e}^{\alpha W^{Capital}_{i,s} / P^{INVTOT}_{s,t} - \delta_r}.\]

The value for \(\alpha\) has been calibrated to \(\alpha=40\) (Bchir et al. 2002):

“Since \(\alpha\) cannot be calibrated, two static models were built, corresponding to short and long-run versions of MIRAGE. We applied the same shocks to both of them and chose \(\alpha\) so that half the adjustment of capital stocks towards the long run would be made in around 4 years, for a variety of small commercial shocks. It gave the value \(\alpha=40\).

## Capital goods

Total investment \(INV_{r,t}^{TOT}\) is made of capital goods \(KG_{i,r,t}\), following a CES specification: \[KG_{i,r,t} = a^{KG}_{i,r} INV^{TOT}_{r,t} \left(\frac{P^{INVTOT}_{r,t}}{P^{KG}_{i,r,t}}\right)^{\sigma_{KG}},\] \[P^{INVTOT}_{r,t} INV^{TOT}_{r,t} = \sum_i P^{KG}_{i,r,t} KG_{i,r,t}.\]

and capital goods are subject to a specific taxation: \[P^{KG}_{i,r,t} = P^{DEMTOT}_{i,r,t}\left(1+tax^{KG}_{i,r,t}\right).\]

## References

*CEPII Working Paper*2002-17. http://www.cepii.fr/CEPII/fr/publications/wp/abstract.asp?NoDoc=160.