How do efficient credit markets allocate capital properly?

I've learned that efficient credit markets should theoretically allocate capital efficiently, which means that all possible investments should have the same marginal product of capital (assuming some type of concave production function).

How exactly does that work? On the surface level, it makes sense that we should allocate more money to projects that have higher returns, but what would be the mechanism by which that happens? It seems like banks would have to charge a higher interest rate in Africa - and does that happen? I think I'm having a lot of trouble developing the correct intuition here.

Under the usual assumptions of decreasing MPK and complete markets in equilibrium $$MPK_{Firm_1} = MPK_{Firm_2} = r$$. If firm i's MPK is larger than r and firm i does not face constraints with respect to scale, if will simply borrow more capital as it can make a net return of $$MPK_{Firm_i} - r$$ on every unit for which $$MPK_{Firm_i} > r$$. If $$MPK_{Firm_i} < r$$ firm i will divest until $$MPK_{Firm_i} = r$$.

There are many reasons for which this nice result can break down in reality (Deviations from the nice assumptions above). For example agency costs (Bernanke, Gertler, and Gilchrist (1996)), where an entrepreneur can only borrow up to the present value his net worth at end of the loan contract. If an entrepreneur does not have enough wealth he cannot borrow enough capital to achieve $$MPK_{Firm_i} = r$$. Hence when talking about this topic you should make your assumptions known in advance.

With regard to you question about Africa, from the point of view of the setting described above too few capital is flowing to developing countries. This fact has been known for a while and is referred to as the "Lucas paradox" or the "uphill flow of capital" a recent summary on the topic can be found here:

• Could one of the larger problems be that there are reasons why $r$ is higher for firms in the developing world compared to the developed world (which would lend to higher $MPK$ for developing nations)? – Vasting Jul 22 '19 at 12:30
• You mean lenders from developed countries require higher interest rate from borrowere based in developing countries? There could even be data on that. – Grada Gukovic Jul 22 '19 at 12:39
• Yeah - firms in developing countries facing a higher $r$ than those in developed countries - which in turn leads to a higher $MPK$ in developing countries. – Vasting Jul 22 '19 at 12:54

Why are returns assumed to be fixed percentages? Since the production function is assumed to be concave, the returns from investing an additional \$x in a firm should be a decreasing function of the amount of money already invested in the firm. If the risk is the same for firms in both countries, then at equilibrium the amount of money invested in each firm will be at such a level that, for small x, the returns on investing an additional \$x in either company are the same.

• I guess my question is more about in the real world, how is that equilibrium reached? If one firm has a higher MPK than another (assuming the same risk), then what really happens? Would banks just buy more of their debt - what is the mechanism that causes equilibrium to be reached? – Vasting Jul 17 '19 at 19:42