finding the returns to scale of a production function

Question

Find the returns to scale for he following production function- $(x_1 + 1)^.5 (x_2)^.5$

My attempt, Let $f(x_1,x_2)=(x_1 + 1)^.5 (x_2)^.5$ And $g(x_1,x_2)=(x_1)^.5 (x_2)^.5$ Now, $(x_1 + 1)^.5 (x_2)^.5$ $>$ $(x1)^.5 (x_2)^.5$ Therefore, $f(x_1,x_2)>g(x_1,x_2)$ Or, $f(tx_1,tx_2)>g(tx_1,tx_2)$ for some t>0 $>(tx_1)^.5 (tx_2)^.5$ $=t(x_1)^.5 (x_2)^.5$ $=tg(x_1,x_2)$ The function g exhibits constant returns to scale. The function f being bigger than it should exhibit increasing returns to scale. The answer given, however, is decreasing returns to scale. Where am I wrong? Thank you in advance.

Answer 1

What you have proven is that $$f(tx_1, tx_2) > tg(x_1,x_2)$$

which translates "scaled $f$ is higher than some other, homogeneous function $g$ scaled by the same factor". This does not prove anything about the returns to scale related to $f$, although I can see why it may appear otherwise.

You can go in reverse, starting from

$$tf(x_1,x_2) = t^{0.5}t^{0.5}f(x_1,x_2)=...$$

and compare to $f(tx_1, tx_2)$.

Also beware about the following: what are the permissible values for $t$ when returns are not constant? Are you sure we should obtain the result for $t>0$, or maybe we examine the property for a smaller interval?