Suppose that the production function is given by Y = K1/2 N1/2. Assume that the size of the population, the participation rate, and the unemployment rate are all constant. Is this production function characterized by constant returns to scale? Explain.
To determine if the production function has constant returns to scale, we need to examine how output changes with respect to proportional changes in inputs.
Let's consider a proportional increase in capital and labor inputs by a factor of z: K' = zK and N' = zN.
Using the original production function, we have:
Y = K1/2 N1/2
Plugging in the new inputs:
Y' = (zK)1/2 (zN)1/2
Simplifying:
Y' = z1/2 K1/2 z1/2 N1/2
Y' = zK1/2 N1/2
Notice that Y' = zY (multiply by z1/2 on both sides).
Since output increases proportionally with input, this production function exhibits constant returns to scale.
To determine if the production function is characterized by constant returns to scale, we need to examine how output changes when all inputs are proportionally increased.
Let's suppose we increase both the capital (K) and labor (N) inputs by a factor of λ. Thus, the new capital input is λK, and the new labor input is λN.
Using the given production function: Y = K^(1/2) * N^(1/2)
Now, let's substitute the new inputs into the production function:
Y' = (λK)^(1/2) * (λN)^(1/2)
= λ^(1/2) * λ^(1/2) * K^(1/2) * N^(1/2)
= λ * K^(1/2) * N^(1/2)
= λY
Based on the above calculation, we can see that Y' (the new output with increased inputs) is equal to λ times Y (the original output). This indicates that the production function exhibits constant returns to scale.
In other words, when both inputs (capital and labor) are increased by the same factor, the output increases by the same factor, implying constant returns to scale.