Question

modelling Production Functions-

Q=2L-.01L^2+3K-.02K^2

L=55 K=55

compare this to L=50 K=50

Calculate output elasticity (% changeQ)/(%change inputs)what does it tell you about returns to scale for this firm and why?

Answer 1

Q=2L - .01L²+ 3K- .02K²

Scenario 1 : When L=55 and K=55

Q = (2*55) - 0.01*(55)² + 3(55) – 0.02*(55)²

= 110 – 30.25 + 165 – 60.5

= 184.25

Scenario 2 : L=50 and K=50

Q = (2*50) - 0.01*(50)² + 3(50) – 0.02*(50)²

= 100 – 25 + 150 – 50

= 175

Initial Inputs = L+ K = 55+55 = 110

New Inputs = L+ K = 50+50 = 100

Output Elasticity = (% change in Q)/(%change in inputs)

= [(175 - 184.25 )/ 184.25] / [( 100 – 110)/ 110]

= 0.050203 / 0.0909

= 0.55224

This means as the inputs decrease by 9.09% (10/110), the output decrease by 5.02%. therefore, it is a situation of decreasing returns to scale.

This can be understood easily why understanding that a 9.09% increase in inputs would be to only a 5.02% increase on output which is a smaller proportionate change as compared to change in inputs. So it is decreasing returns to scale. The output elasticity is less than 1, i.e. 0.55224 which shows decreasing returns.