Show that the Cobb-Douglas production [tex]Q =A K^2 L^{1-a}[/tex], where [tex]$Q$[/tex] is total output, [tex]$K$[/tex] is capital stock, [tex]$L$[/tex] is labour stock and [tex]$A$[/tex] and [tex]$\alpha$[/tex] are positive constants, exhibit constant returns to scale.
Answer (1)
Multiply both inputs K and L by a constant factor t, resulting in new inputs tK and tL.
Substitute tK and tL into the production function: Q ′ = A ( t K ) a ( t L ) 1 − a .
Simplify the expression: Q ′ = t ( A K a L 1 − a ) = tQ .
Since Q ′ = tQ , the Cobb-Douglas production function exhibits constant returns to scale. Q ′ = tQ
Explanation
Understanding the Problem We are given the Cobb-Douglas production function Q = A K a L 1 − a , where Q is the total output, K is the capital stock, L is the labor stock, and A and a are positive constants. We want to show that this production function exhibits constant returns to scale. This means that if we multiply both inputs (capital and labor) by a constant factor t , the output will also be multiplied by the same factor t .
Multiplying Inputs by a Constant Let's multiply both inputs K and L by a constant factor 0"> t > 0 . The new capital stock is t K and the new labor stock is t L . We can now calculate the new output Q ′ by substituting these new inputs into the production function: Q ′ = A ( t K ) a ( t L ) 1 − a
Simplifying the New Output Now, let's simplify the expression for Q ′ :
Q ′ = A ( t a K a ) ( t 1 − a L 1 − a ) Q ′ = A t a t 1 − a K a L 1 − a Q ′ = A t a + 1 − a K a L 1 − a Q ′ = A t 1 K a L 1 − a Q ′ = t ( A K a L 1 − a )
Conclusion Since Q = A K a L 1 − a , we can substitute Q back into the equation for Q ′ :
Q ′ = tQ This shows that when we multiply both inputs by a constant factor t , the output is also multiplied by the same factor t . Therefore, the Cobb-Douglas production function Q = A K a L 1 − a exhibits constant returns to scale.
Final Answer Thus, we have shown that the Cobb-Douglas production function exhibits constant returns to scale.
Examples
In economics, the Cobb-Douglas production function is often used to model the relationship between inputs (capital and labor) and output in an economy. Understanding constant returns to scale is crucial for analyzing how changes in the scale of production affect output. For example, if a company doubles its capital and labor inputs, a production function with constant returns to scale would predict that the company's output will also double. This concept is important for making decisions about investments and production levels.
