Checking for Monotonicty in translog production fucntion

[Vianny Ahimbisibwe]

Dear all,

I kindly seek for your advise.
I am trying to test for Monotonicity in a translog production function using Margin, expression ().
The Translog production function is: 𝑙𝑛𝑌 = 𝛽° + 𝛽1𝑙𝑛land + 𝛽2𝑙𝑛capitalseedling + 𝛽3𝑙𝑛labor + 𝛽4 0.5(𝑙𝑛land)^2 + 𝛽5 0.5(𝑙𝑛capitalseedling)^2 + 𝛽6 0.5(𝑙𝑛labor)^2 + 𝛽7𝑙𝑛land𝑙𝑛capitalseedling + 𝛽8𝑙𝑛land𝑙𝑛labor + 𝛽9𝑙𝑛capitalseedling𝑙𝑛labor.

Thanks to @Andrew Musau I learnt how to use margin, expression.

Below is the estimated margin of the first derivatives.

dReve_ln/dLand_ln.

margins, expression(Reve * ( _b[Land_ln]/Land + _b[ halfLand_lnsq] * (Land_ln/Land) + _b[ Land_lncapitalseedling_ln] * (capitalseedling_ln/Land) + _b[ Land_lnLrdays_Act_ln] * (Lrdays_Act/Land)))

Results obtained:

	Margin	std.Err	z	p
_cons	177469.7	37714.64	4.71	0.00

dReve_ln/dcapitalseedling_ln

margins, expression(Reve * (_b[capitalseedling_ln]/capitalseedling + _b[halfcapitalseedling_lnsq]* (capitalseedling_ln/capitalseedling) + _b[Land_lncapitalseedling_ln]* (Land_ln/capitalseedling) + _b[capitalseedling_lnLrdays_Act_ln] * (Lrdays_Act_ln/capitalseedling)))

Results obtained:

	Margin	std.Err	z	p
_cons	4.90001	1.689439	2.90	0.004

dReve_ln/d Lrdays_Act_ln

margins, expression(Reve * (_b[Lrdays_Act_ln]/Lrdays_Act + _b[halfLrdays_Act_lnsq] * (Lrdays_Act_ln/Lrdays_Act) + _b[ Land_lnLrdays_Act_ln] * (Land_ln/Lrdays_Act) + _b[ capitalseedling_lnLrdays_Act_ln] * (capitalseedling_ln/Lrdays_Act)))

Results obtained:

	Margin	std.Err	z	p
_cons	-172.1462	171.2485	-1.01	0.315

Main question is:
Based on the results above, does it mean that the production function does not qualify under the monotonicity condition?

Your advice is highly welcome

Thank you

[Andrew Musau]

Not sure how you are computing these derivatives. As you are just interested in the sign of the derivative, you should note that:

$$\text{sign}\left(\frac{\partial Y}{\partial w}\right) = \text{sign}\left(\frac{\partial lnY}{\partial lnw}\right)$$

where \(w\) is the input and \(ln\) is the natural logarithm. That being the case, just take the partial derivative of the production function w.r.t. the logged input. If each observation is a farm, then the partial derivatives are observation-specific, and you will have to check if each observation satisfies the monotonicity condition. Take a look at Chapter 4 of Kumbhakar's book on how to do this.

Reference:

S. C. Kumbhakar, H.-J. Wang, and A. Horncastle. A Practitioner’s Guide to Stochastic Frontier Analysis using Stata. Cambridge University Press, Cambridge, 2014.

[Eric Makela]

Hi Vianny. First of all, please don't be confused by Andrew's third sentence; if you are testing an aggregate production function, the results you get from your analyses are still correct for your purposes (although I haven't personally looked into testing for monotonicity). Secondly, if you post in Statalist again, you might consider using macros to make the text more readable to persons without knowledge of your specific variable-naming techniques. Lastly, you are welcome to post statistical questions in Statalist, but frankly you are likely to get more help with these questions from other sources (though you can review the work here for one review of a monotonicity test, perhaps this will guide you correctly).

[Vianny Ahimbisibwe]

Hi Andrew ans Eric,

Thank you for the tips. They helped a lot.