I learned that when there is decreasing returns to scale, the average cost is always increasing.
But the professor told us today that the other way around might not always be true. So if average cost is increasing, it might not necessarily mean that there is decreasing returns to scale.
The production function q = min {K, L^(1/2)}$q = \min \{K, L^{(1/2)}\}$ was one he gave us as something to think about as a possible counterexample.
we assumed that the input prices are 2 for capital and 1 for labor.
If we solve for this using cost minimization, we get C(q) = 2q + q^2$C(q) = 2q + q^2$ and the AC(q) = 2 + q$AC(q) = 2 + q$ which is an increasing function.
And if I input kK, kL$kK, kL$ in the production function where k > 1$k > 1$ , then the function becomes
q = min {kK, (kL)^(1/2)} = min {kK, k^(1/2) * L^(1/2)}.$q = \min \{kK, (kL)^{1/2}\} = \min \{kK, k^{1/2} * L^{1/2}\}.$
How do I know what returns to scale this is showing?
if the min function was simply q = min { K, L }$q = \min \{ K, L \}$, then I could definitely tell that this is showing constant returns to scale since min {kK, kL} = k*min{K,L}$\min \{kK, kL\} = k*\min\{K,L\}$, but how do you mathematically solve for returns to scale for a function like the one above, q = min {K, L^(1/2)}$q = \min \{K, L^{1/2}\}$ ?
I don't know what step to take next after arriving at min {kK, k^(1/2) * L^(1/2)}$\min \{kK, k^{1/2} * L^{1/2}\}$. I don't know what to extract out of the min function from this point on.
I would appreciate advice on how to continue to determine the returns to scale for this kind of example.
