To determine whether the production function exhibits increasing, constant, or decreasing returns to scale, we need to examine the effect of proportional changes in inputs on output.
Let's consider a proportional increase in both labor (L) and capital (K) by a factor of λ. That is, the new inputs become λL and λK, respectively.
The new output can be calculated as follows:
Q' = 0.5(λL)(λK) = 0.5λ^2LK
Now let's compare this to the original output:
Q = 0.5LK
To determine the returns to scale, we can compare the proportional change in output to the proportional change in inputs.
If Q' > λQ, then the production function exhibits increasing returns to scale (IRS).
If Q' = λQ, then the production function exhibits constant returns to scale (CRS).
If Q' < λQ, then the production function exhibits decreasing returns to scale (DRS).
Let's substitute the original output into the inequality and simplify:
0.5λ^2LK > λ(0.5LK)
0.5λLK > 0.5LK
λ > 1
Therefore, we can see that when λ > 1, Q' > λQ, which means the production function exhibits increasing returns to scale (IRS). This suggests that a proportional increase in both labor and capital by a factor of λ results in a more than proportional increase in output. In other words, increasing the scale of production leads to greater efficiency and higher productivity.