Update foreign private capital K^f specification
With the addition of government capital K_g to the model, I am worried about our derivation and specification in the OG-Core master branch of foreign capital inflow to the country K^f. In the documentation, the current specification is derived in equations (73), (74), and (75) in the Private capital market clearing section of the Market Clearing chapter.
Currently, the open-economy domestic private capital demand K^{d,r*} is determined by (i) using the open-economy interest rate r* to solve for (ii) the open-economy capital-labor ratio, then (iii) multiply the open-economy capital-labor ratio by L to get the implied open-economy capital demand.
This approach of deriving a capital-labor ratio from r* only works in the model without government capital K_g=0. Furthermore, in this approach in step (iii), the L multiplied by the capital-labor ratio is based off of the market determined interest rate and not the open-economy rate.
I propose two alternative approaches, and I favor the latter approach (2).
- In the presence of government capital, we could calculate
K^{d,r*}by calculating (a) the open-economyY/Kgivenr*, then back out the open economyK^{d,r*}asY / (Y/K). - We could assume that
K^fis a fixed percentzeta_Kof total capitalK. This means the private capital market clearing conditionK = K^d + K^fis equivalent to sayingK^f = zeta_K * KandK^d = (1 - zeta_K) * K.
I think that both approaches (1) and (2) will work. Approach (1) is most analogous to what we are currently doing, and it incorporates K_g > 0 into the solution. However, approach (2) is more simple. And its specification is more closely connected to the way we would calibrate zeta_K.
cc: @jdebacker