Question

What kind of returns to scale do the following production functions display? (a) \(Q=\sqrt{K L}\) (b) \(Q=K_{0.3} L_{0.2}\) (c) \(Q=K+L\)

Short answer

(a) Constant returns, (b) Decreasing returns, (c) Constant returns.

Step-by-step solution

Understanding Returns to Scale

Returns to scale refer to how output changes when all inputs are changed by the same proportion. We can classify these as increasing, constant, or decreasing returns to scale.

Analyze Production Function (a): \(Q=\sqrt{KL}\)

To identify the type of returns to scale, we consider multiplying both inputs \(K\) and \(L\) by a scalar \(t\) and observe the change in output. Replace \(K\) and \(L\) with \(tK\) and \(tL\) in the function: \[Q' = \sqrt{tK \cdot tL} = \sqrt{t^2 KL} = t \sqrt{KL} = tQ\] Since the output \(Q' = tQ\), this indicates constant returns to scale.

Analyze Production Function (b): \(Q=K^{0.3} L^{0.2}\)

Change both inputs \(K\) and \(L\) by factor \(t\): \[Q' = (tK)^{0.3} (tL)^{0.2} = t^{0.3}K^{0.3} \cdot t^{0.2}L^{0.2} = t^{0.5}K^{0.3}L^{0.2} = t^{0.5}Q\] Since the revised output is \(t^{0.5}Q\) and \(0.5 < 1\), this implies decreasing returns to scale.

Analyze Production Function (c): \(Q=K+L\)

Multiply both \(K\) and \(L\) by \(t\): \[Q' = tK + tL = t(K + L) = tQ\] The resulting output is \(tQ\), indicating constant returns to scale.

Key concepts

Production Functions

Production functions are mathematical representations of how a firm transforms inputs, like capital \( K \) and labor \( L \), into outputs, such as goods or services. These functions explore the relationship between input quantities and the resulting output, providing a structure to analyze the efficiency and scalability of a production process.

Typically, production functions can take various forms, each signifying different economic interpretations about productivity and technological processes. Some common examples include linear functions, where inputs contribute to output at a constant rate, and Cobb-Douglas functions, characterized by exponents that reveal the contribution weight of each input.

By analyzing these functions, economists and business managers can deduce key metrics, such as marginal product, which is the additional output from one more unit of input, and returns to scale, which examines how output responds to a proportional increase in all inputs.

• Key to interpreting production functions are the exponents attached to inputs; these can significantly influence the type of returns to scale observed in the mathematical model.

Constant Returns to Scale

Constant returns to scale occur when a proportional increase in all inputs leads to an equally proportional increase in output. In a mathematical sense, if all inputs in a production function are multiplied by a factor \( t \), and the output is also multiplied by \( t \), the function is said to exhibit constant returns to scale.

Consider a simple linear production function \( Q = K + L \). If both capital \( K \) and labor \( L \) are increased by a factor \( t \), the new output becomes \( Q' = tK + tL = t(K + L) = tQ \). This reflects constant returns to scale since the output changes by the same proportion as the inputs.

Constant returns to scale imply an ideal situation for firms that wish to scale their operations effectively without inefficiencies or excess costs.

• This condition suggests a stable operational environment where doubling inputs results in doubling outputs, reinforcing efficient resource allocation.

Decreasing Returns to Scale

Decreasing returns to scale happen when a proportional increase in all inputs results in a less than proportional increase in output. When production is described by a function where the sum of input exponents is less than 1, such as \( Q = K^{0.3} L^{0.2} \), it typically illustrates decreasing returns.

Mathematically, this could be seen when inputs are scaled by a factor \( t \) and the resultant output \( Q' = (tK)^{0.3}(tL)^{0.2} = t^{0.5}Q \) becomes multiplied by \( t^{0.5} \), where \( t^{0.5} < t \). Here, increasing capital and labor does not lead to a 100% increase in production but rather less than this percentage.

Decreasing returns to scale can warn business managers about potential inefficiencies, signaling that simply adding more resources might not be cost-effective.

• It often suggests complexities like management challenges or resource constraints that worsen as the scale expands.