Question

Consider the following production function: Y=A(K+bN)α (K+dN)1−α , where Y is output, and K and N are capital and labor inputs, respectively. A,b,d and α are parameters that satisfy these conditions: A>0,b>0,d>0,0<α<1. a. Show that this production function is constant returns to scale (that is, if you multiply both K and N by a factor of c, that output is multiplied by a factor of c.)

Answer 1

The production function Y = A(K + bN)α (K + dN)1−α exhibits constant returns to scale. By multiplying inputs K and N by a constant c, the output also increases by the factor of c, demonstrating the property of constant returns to scale.

Step-by-step explanation:

The question revolves around demonstrating that a specific production function exhibits constant returns to scale. The production function given is Y = A(K + bN)α (K + dN)1−α, where Y is the output, K and N are the inputs of capital and labor, respectively, and A, b, d, and α are parameters with specified conditions.

To show constant returns to scale, we multiply both inputs by a constant c, and calculate the new output. We get the new production function: Y(c) = A(cK + bcN)α (cK + dcN)1−α. By factoring out the c from both terms and using the property that c raised to any power is still a power of c, we simplify to Y(c) = c * A(K + bN)α (K + dN)1−α, which is c times the original output. This shows that the production function has constant returns to scale because when we scale up the inputs by a factor c, the output scales up by the same factor c. To show that the production function Y=A(K+bN)α (K+dN)1−α is constant returns to scale, we need to show that multiplying both K and N by a factor of c results in the output being multiplied by the same factor of c.

Multiply K and N by c: K' = cK and N' = cN

Plug the new values of K' and N' into the production function: Y' = A(K'+bN')α (K'+dN')1−α

Simplify the expression using the rules of exponents and distribute the constant A: Y' = cα(α+(1−α))AkαNα+αbNα+dNα

Cancel out like terms and simplify: Y' = cY

This shows that multiplying both K and N by a factor of c multiplies the output Y by the same factor of c, which demonstrates constant returns to scale.