Question

We have introduced the definitions of constant return to scale (CRS), increasing return to scale (IRS) and decreasing return to scale (DRS) to the Cobb-Douglas production functions. Explain the following production functions are CRS, IRS or DRS. Suppose a production function specified the national output (Y) as a function of capital (K) and labor (L) as below. The function is a CRS, IRS or DRS?

Y = A * K¹/² * L²/³

Answer 1

The production function Y = A * K¹⁄₂ * L²⁄₃ exhibits constant returns to scale (CRS) since the sum of the exponents of K and L is 1. This means that an increase in all inputs leads to a proportional increase in output, which aligns with the flat portion of the long-run average cost curve where economies of scale are exhausted.

Step-by-step explanation:

To determine whether the production function Y = A * K¹⁄₂ * L²⁄₃ exhibits constant returns to scale (CRS), increasing returns to scale (IRS), or decreasing returns to scale (DRS), we need to consider the sum of the exponents of capital (K) and labor (L).

If the sum of the exponents equals 1, the function is CRS. If it is greater than 1, it is IRS, and if it is less than 1, it is DRS.

In the given Cobb-Douglas production function, the exponents of K and L add up to (¹⁄₂ + ²⁄₃) = ⅓ + ⅔ which simplifies to 1.

Therefore, the production function Y = A * K¹⁄₂ * L²⁄₃ exhibits constant returns to scale (CRS).

In the context of the long-run average cost curve (LRAC), the middle portion or the flat portion of the curve around Q3 indicates a range where economies of scale have been exhausted, which means the average cost of production will not change as the scale of production is increased or decreased.

This is reflective of a CRS situation.

The analogy for this in perfectly competitive markets can be seen with the long-run supply (LRS) curve, which is flat in a constant-cost industry where the average costs remain unchanged despite expansion in output, as shown in Figure 8.8 (a).