The Cobb-Douglas Production Function is given by q=x1αx2β, which matches with (III). This is a specific functional form that relates output to inputs through exponents alpha and beta.
Constant returns to scale occurs when scaling both inputs by a factor t results in output also scaling by exactly t. Mathematically, this is expressed as f(tx1,tx2)=t⋅f(x1,x2), which matches with (IV).
Increasing returns to scale happens when scaling inputs by factor t produces an output increase greater than t times the original output. This is shown as f(tx1,tx2)>t⋅f(x1,x2), matching with (I).
Decreasing returns to scale occurs when scaling inputs by factor t results in output increasing by less than t times. This is represented by f(tx1,tx2)<t⋅f(x1,x2), matching with (II).
Therefore, the correct matching is: (A) - (III), (B) - (IV), (C) - (I), (D) - (II).