Why are Hayekian triangles the shape of a triangle? Why not two axes with an irregular line between them?
Let's pretend the yield curve doesn't exist and there's actually a linear relationship between time and the interest rate. That still doesn't guarantee that it should be a triangle. Although in that case the marginal cost of capital is linear over time, the marginal cost of capital at the point that it is equal to the marginal product of capital need not have this linear relationship with time.
Another way of posing the question is, why can't higher order processes be very heavy in the addition of factors of production, lower order processes be very light in the addition of factors of production (but perhaps time intensive). Or vice versa to such an extreme extent that the capital structure is hyperbolic rather than triangular.
This seems to make a big difference when we get around to:
1. Empirically looking at changes in the capital structure, as Andrew Young does in his forthcoming RAE article, and
2. Thinking about how the capital structure rebalances during a downturn.
And yet I've never heard anyone really ask this question - why is the Hayekian triangle a triangle? I'm not sure if there is a good answer, and if the answer is "there's no reason for it to be", I'm still not quite sure what all the implications of that are.