The marginal product of capital is given as:

MP_K = a*a₀K^(a-1)L^b

This can also be written as:

MP_K = a*(a₀K^aL^b)/K = a*y/K

Similarly, the marginal product of labor can be written as:

MP_L = a*(a₀K^aL^b)/L = b*y/L

The competitive profit-maximizing use of inputs calls for inputs to be paid the value of their marginal product. Let the price of y = $1 and a + b = 1.

show that if capital and labor were paid the value of their marginal product, that the total revenue is exhausted. That is, you want to show:

K*MP_K + L*MP_L = y(since the price of y is $1, it has been dropped)