In economics, the Cobb-Douglas production function P(x, y) = K xα y1−α,where 0 < α <...

In economics, the Cobb-Douglas production function

 P(x, y) = K xα y1−α,

where 0 < α < 1, is supposed to measure the productive capacity of an economic entity (such as the United States) as a function of capital x and labor y. Show that P is a solution to the differential equation

Equation (3.4.9) is often called Euler’s6 equation. The function P has the desirable property (from an economist’s point of view) that it exhibits constant returns to scale: If x and y are replaced by tx and ty for some constant t, then the value of P is scaled, up by the factor t.

6 Leonhard Euler (1707−1783) was born in Switzerland and spent most of his life at the Academies of Sciences in St. Petersburg and Berlin. He produced mathematics at a prodigious rate throughout his life, as evidenced by the fact that his papers constituted at least half of the journals of both academies for many years after his death. Many mathematical notions, equations, and theorems bear his name.