How to treat tax basis of capital stock

[rickecon]

In OG-USA right now, the firm's profit function is the following,

where delta**tau in the last term is the tax basis of depreciation. It represents the fraction of economic depreciation that can be deducted from the firm's tax liability at the corporate income tax rate tau**corp. One issue with this is that when the tax basis depreciation is not equal to the economic depreciation tau**delta != delta then the tax basis of aggregate capital does not equal the actual aggregate capital K**tau != K. The last term in the profit equation should really be in terms of the tax basis of aggregate capital K**tau.

Making the tax basis of aggregate capital the component that determines the degree of depreciation deductibility makes K**tau the component that firms need to keep track of. Intuitively, the tax basis of aggregate capital is a function of the entire history of actual aggregate capital stocks. This is seen by combining the law of motion for the actual aggregate capital stock,

with the law of motion for the tax basis of the aggregate capital stock.

Subtracting the actual law of motion equation from the tax basis law of motion equation and solving for K**tau_{t+1} gives the recursion that shows that the tax basis aggregate capital is a function of the history of actual capital stocks.

This renders the firm's problem dynamic. We need to update this in OG-USA.