A firm has a production function f(x, y)= 1.80(x^0.80 + y^0.80)³ whenever x 0 and y 0.When the amounts of both inputs are positive, this firm has A)increasing returns to scale if x + y 1 and decreasing returns to scale otherwise. B)decreasing returns to scale. C)constant returns to scale. D)increasing returns to scale. E)increasing returns to scale if output is less than 1 and decreasing returns to scale if output is greater than 1.

Question

A firm has a production function f(x, y)= 1.80(x^0.80 + y^0.80)³ whenever x > 0 and y > 0.When the amounts of both inputs are positive, this firm has A)increasing returns to scale if x + y > 1 and decreasing returns to scale otherwise. B)decreasing returns to scale. C)constant returns to scale. D)increasing returns to scale. E)increasing returns to scale if output is less than 1 and decreasing returns to scale if output is greater than 1.

Quizplus · Accepted Answer

The production function exhibits constant returns to scale if f(tx, ty)=t^1.8f(x,y) for all t>0. To check if the production function has increasing, decreasing or constant returns to scale, we can calculate f(2x,2y) and compare it with 2f(x,y) as follows: f(2x,2y) = 1.80(2x)^0.8 + 1.80(2y)^0.8 = 3.24x^0.8 + 3.24y^0.8 2f(x,y) = 2[1.80x^0.8 + 1.80y^0.8] = 3.60x^0.8 + 3.60y^0.8 Comparing these two expressions, we see that f(2x,2y) < 2f(x,y) if x^0.8 + y^0.8 > 1, f(2x,2y) > 2f(x,y) if x^0.8 + y^0.8 < 1, and f(2x,2y) = 2f(x,y) if x^0.8 + y^0.8 = 1. Note that x^0.8 + y^0.8 is the exponent of the inputs in the production function, and it represents the elasticity of output with respect to inputs, measured at a point. If this elasticity is greater than 1, we get increasing returns to scale (IRTS), if it is equal to 1, we get constant returns to scale (CRTS), and if it is less than 1, we get decreasing returns to scale (DRTS). Therefore, the answer is (D) increasing returns to scale. This means that if the firm doubles its inputs, it will more than double its output, and hence it can benefit from scale economies.

A Firm Has a Production Function F(x, Y)= 1