9.9 Local returns to scale A local measure of the returns to scale incorporated in a production function is given by the scale elasticity eq, t ¼ @f (tk, tl )/@t Æ t/q evaluated at t ¼ 1.

a. Show that if the production function exhibits constant returns to scale, then eq,t ¼ 1.

b. We can define the output elasticities of the inputs k and l as eq, k ¼ @fðk, lÞ

@k %

k q

, eq, l ¼ @fðk, lÞ

@l %

l q

:

Show that eq,t ¼ eq, k þ eq, l

.

c. A function that exhibits variable scale elasticity is q ¼ ð1 þ k$1 l

$1

Þ

$1

:

Show that, for this function, eq,t > 1 for q < 0.5 and that eq, t < 1 for q > 0.5.

d. Explain your results from part

(c) intuitively. Hint: Does q have an upper bound for this production function?