In the Cobb–Douglas production function, Q = AK^aL^b, increasing returns to scale occurs when:

Correct option is A

The Cobb–Douglas production function is expressed as:

Q = AK^aL^b​

where:

Q represents the output

A is a constant of proportionality (technology factor)

K is the capital input

L is the labour input

​$a$ and $b$ are the output elasticities of capital and labor, respectively (how much output changes when capital and labour is increased respectively, keeping the other thing constant)

​Returns to Scale:
Returns to scale measure how the output changes when both inputs (K and L) ​are scaled proportionally.
1. a+b>1:Increasing Returns to Scale (IRS)

• A proportional increase in both inputs results in a greater-than-proportional increase in output.
• Example: Doubling both $K$ and $L$ increases output by more than double.

2. $a+b=1$: Constant Returns to Scale (CRS)

A proportional increase in inputs results in an equal-proportional increase in output.

3. $a+b<1$: Decreasing Returns to Scale (DRS)

​Output increases by a smaller proportion than the increase in inputs.

​Why $a+b>1$ implies IRS:
When $a+b>1$, a 1% increase in both $K$ and $L$ results in a more than 1% increase in $Q$, indicating that the production process benefits from higher economies of scale.

Information Booster:

• Cobb–Douglas Function Applications: Widely used in economics to model production in firms and industries.
• Interpretation of Parameters $a$ and $b$:
  • $a$: Measures the contribution of capital to output.
  • $b$: Measures the contribution of labor to output.
• Significance of Returns to Scale:
  • Helps businesses decide how to allocate resources efficiently.
  • Indicates the level of technological efficiency in a production process.
• Examples of Increasing Returns to Scale:
  • Mass production industries, where economies of scale reduce costs.

Additional Knowledge:

(b) $K+L>1$:

• $K$ and $L$ are the quantities of capital and labor, not their elasticities. Returns to scale depend on $a$ and $b$, which represent the relative contributions of $K$ and $L$.
• Misinterpreting this option leads to the false idea that returns to scale depend on the magnitude of inputs, not their elasticities.

(c) $a+b<1$:

• This represents Decreasing Returns to Scale (DRS).
• In DRS, output increases by a smaller proportion than inputs, typically occurring in scenarios with resource constraints or inefficiencies.

(d) $K+L<1$:

• Similar to (b), this refers to input quantities, not elasticities.
• It fails to consider the role of proportional elasticities ($a$ and $b$) in determining returns to scale.

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