Fish and Capital
How will a competitive industry manage fishing? The simple case is two periods, now and tomorrow, a constant price of fish P, and an interest rate r. Fish Tomorrow are Fish Left in the Sea Today, plus all the young they spawn. Economically, this is just a production function. Fish Left in the Sea Today produce Fish Tomorrow. The picture on the left shows such a function. The production function lies above the straight 45 degree line. That line would represent one fish not eaten today results in one fish to eat tomorrow. The curve shows that a fish left today results in more than one fish to eat tomorrow. The marginal product of Fish Left Today in producing Fish Tomorrow is just the slope of the production function. The two little triangles give a visual way of calculating the slope and show that the more fish one leaves, the lower the marginal product.
Now we know from economics that price times marginal product equals wage. The return selling a fish tomorrow is the price divided by the discount factor, P/(1+r). The "wage" or cost of not fishing a fish today is P. Thus
P MP /(1+r) = P
Which reduces to MP = 1+r
So the lower the interest rate, the lower MP and the higher the number of fish left in the sea.
A competitive economy leaves more reproducible resources for next period if it has a lower interest rate.