Please help with this Production Technology question!

[Nupur]

Please show the working.

[Dreyfus]

a+b<1 is the sufficient condition for diminishing returns to scale

[Nupur]

Yeah I know that!  But I'm asking Diminishing Technical Rate of Substitution (TRS) and not diminishing returns to scale. Returns to scale is when you scale all the factors by some positive constant 't' but Technical Rate of Substitution means when you increase x1 input by one unit then by how much x2 will decrease to be on the same isoquant. Am I right? If it is so then after we've calculated TRS (=(-)MP of x1/MP of x2) and we take d/dx1(TRS) then according to me it comes positive but then it can't be diminishing TRS. I hope I'm making some sense??

[kangkan]

Hi Nupur...a prod function will have dimnishing TRS as long as it its well behaved and convex...the given function is a cobb duouglus functio which is always well behaved and convex for positive values of a,b and A.

:)

[kangkan]

hi vaibhav..but the isoquant of a cobb douglus finction is always convex right?

[Dreyfus]

@kangkan...OK my bad....I took that function mistakenly and applied the wrong technique!:'(

[Nupur]

Yes that is indeed the answer Kangkan- decreasing TRS for all values of a, b and C. So if we are given a production function other than a Cobb Douglas one then how do we check if TRS is diminishing or not?

[Nupur]

But you at least tried! Thanks Vaibhav. And Kangkan :)

[kangkan]

and if you take C=1 and a=1.5 and b=-3(since a+b<0),then the production x1^1.5* x^-3 doesnt come out convex :(

[Dreyfus]

To check whether TRS diminishing or not dTrs/dx1<0
dTRS/dx1=d(f1/f2)/dx1  using quotient rule and chain rule
I'll get dTRS/dx1={f2²*f11-2f1*f2*f12+f1²*f22}/f2³
fi is partial derivative
f2>0 therefore denominator is positive
For numerator to be negative.... f11, f22 has to be negative and f12 has to be positive and it make sense also!
Now impose the conditions .......for a,b C

[AS]

Dimnishing RTS is equivalent to dRTS/dx1<0  that is strictly  quasi-concave functions *(convex isoquants) for cobb douglas a b>1

[AS]

a+b>1 is required

[Akshay Jain]

The ans is correct...for all positive values of a,b nd c....the diminishing MRTS
will be true....cobb douglas technology gives diminishing MRTS provided dat the elasticity of output w.r.t. Each input is positive....

[Nupur]

That reminds me of an elasticity question I had to ask! It is:
D(p)= 1000-P+ln M. Find price elasticity of demand when P=2 and M= 500. Answer is -2. But I don't get the correct answer!

[Dreyfus]

I m getting -0.002!

[Nupur]

Yes I'm getting the same! This is a question from Varian Workbook. And answer given in it is -2. Similar
Y answer is not matching for income elasticity also as well as for other values of p and M!!

[Siddhant]

MRTS = (a/b)(x2/x1). Clearly, as x1 will increase this will fall, for all positive values of a, b and C.

[Dreyfus]

Nupur...read that question again....its ln(D(p)) not D(p)!

[Nupur]

Oh god! Can't believe thrice I read it wrong!! I'm so sorry for wasting time!