Relax assumption about capital good production

[jdebacker]

We currently assume that capital goods are produced from just one industry (namely, industry $M$). This helps greatly with the solution to the firms' problem because consumption demands from the $M-1$ industries pins down their scale (i.e., $Y_m=C_m$), and together with $r$ and $w$ yield quantities of the factors input demands.

Demand for investment is a function of the scale of firms, since it'd determined by demand for capital, $K_m$. This makes it hard to solve for the demand for investment from each industry without first knowing each industry's scale.

But Ballard et al. (1985) Chapter 4 provide an alternative approach that could work for OG-Core.

This approach involves determining investment demand from aggregate household savings (less gov't debt). The assumption is that this demand for investment is met by the supply of investment goods from firms (analogous to how we use the goods market clearing to say $Y_m=C_m$, we can now say $Y_m=C_m + I_m$). Aggregate household savings only gives us total investment. To get investment goods produced by industry $m$, we would use an input/output matrix as we do for mapping production goods to consumption goods, but in this case mapping production goods to capital goods.