Question

A firm's production function is f(x,y) = xayb (x to the power of a times y to the power of b). Consider the following 10 statements: The production function never exhibits constant returns to scale The production function never exhibit decreasing returns to scale The production function always exhibits constant returns to scale The production function always exhibits increasing returns to scale The production function always exhibits constant returns to scale if a = b The production function always exhibits increasing returns to scale if a > b The production function always exhibits increasing returns to scale if a + b ≥ 1 The production function always exhibits decreasing returns to scale if a + b < 1 The production function always exhibits constant returns to scale if a + b = 0 The production function always exhibits constant returns to scale if a + b = 1 Which statement(s) is/are correct? Select one: a. 2, 7, 9 b. 1, 6 c. 8, 10 d. 4 e. 2, 5, 6 f. 2, 6, 10 g. 1, 6, 8 h. 3 i. 1, 7, 8 j. 1, 6, 7

Answer 1

The correct answer is (c) 8, 10

A production function exhibits increasing returns to scale if f(tx , ty) > t*f(x,y) for all t > 1

A production function exhibits decreasing returns to scale if f(tx , ty) < t*f(x,y) for all t > 1

A production function exhibits constant returns to scale if f(tx , ty) = t*f(x,y) for all t > 1

here f(x,y) = x^ay^b ,

This function will exhibit increasing returns to scale if

f(tx , ty) > t*f(x,y) => (tx)^a(ty)^b > tx^ay^b => t^{(a + b)}x^ay^b > tx^ay^b for all t > 1

and t^{(a + b)}x^ay^b > tx^ay^b if a + b > 1

hence this function exhibit increasing returns to scale if (a + b) > 1

This function will exhibit decreasing returns to scale if

f(tx , ty) < t*f(x,y) => (tx)^a(ty)^b < tx^ay^b => t^{(a + b)}x^ay^b < tx^ay^b for all t > 1

and t^{(a + b)}x^ay^b < tx^ay^b if a + b < 1

hence this function exhibit decreasing returns to scale if (a + b) < 1

This function will exhibit Constant returns to scale if

f(tx , ty) = t*f(x,y) => (tx)^a(ty)^b = tx^ay^b => t^{(a + b)}x^ay^b = tx^ay^b for all t > 1

and t^{(a + b)}x^ay^b = tx^ay^b if a + b > 1

hence this function exhibit increasing returns to scale if (a + b) = 1

hence the correct statements are ;

(8) The production function always exhibits decreasing returns to scale if a + b < 1

(10) The production function always exhibits constant returns to scale if a + b = 1