Pray where?
Using Equation 3, we have: Percentage growth in GDP = Growth in total factor productivity + (Output of elasticity of capital) × (Growth in capital stock) + (1 − Output of elasticity of capital) × (Growth in labor input) = 0.6% + (0.3 × 3.5%) + (0.7 × 0.4%) = 1.93%
no the formula used is a aproximation of the real formula. This means that it will hold only for very small changes.
The real formula is
Y = A * K^alpha * L^Beta
The constant return to scal implied that Beta = 1- alpha
Which in fact means that, if both K and L raise by same X% amount, Y will raise by that X% amount.
ex:
Y = A * ( x *K)^alpha * ( x *L)^Beta where x = 1+X%
Knowing that BEta = 1-alpha you can derive it like this :
Y = A * x^alpha * x^(1-alpha) * K^(alpha) * L^(1-alpha)
which is in fact equal to
Y = A * x^1 * K^Alpha * L^(1*alpha ) i.e. x^(1-alpha+alpha) = x^1
which you can see like :
Y = x * ( A * (K)^alpha * (L)^Beta )
ass you can see Y will have increase by exactly the factor x
if A was increase by another factor, both factor would have a coumpund effect. like x*y or ex : 1.03 * 1.05 doesnt equal 0.03 + 0.05
finally, if K and L increase by the same amount, TFP do not increase or decrease and constant return to scale old ( Alpha = 1-beta ) then Y will have exactly increase by the X amount.
Yeah, exactly. If %change in TFP = 0 when constant returns to scale holds, why is that 0.6% and not just 0? And I get that it’s a small difference in this particular problem, but it’s a big difference in others (%TFP=2.78%, etc.)
I understand the logic, and I understand the math. But if what you explained is the case, and %change in TFP = 0 when constant returns to scale holds, then why do we not set %change in TFP equal to zero when we solve?
[quote=“padniaki”]
as I said, this is the approximation formula. the real value would have been different since it would have compounded like this :
1.006 * 1.035^0.3 * 1.004^0.7
this is the real formula, what you are using is the aproximation formula, which is the one CFA wants u to use.
because change in TFP has a direct impact on the change in Y
so if you use the aproximation formula, you need to take it into consideration.
aprrox : 0.6% + (0.3 × 3.5%) + (0.7 × 0.4%) = 1.93%
real : (1.006 * 1.035^0.3 * 1.004^0.7 ) -1 = 1.928%
From CFAi
Taking first differences of Equation 2 and utilizing the fact that, for small changes in any variable x
(Institute 125)
Institute, CFA. Level III 2013 Volume 3 Capital Market Expectations, Market Valuation, and Asset Allocation. John Wiley & Sons P&T, 6/18/2012. .
what this means is that very important thing :
if you want to use the aproximation formula and have the exact same increase as the real formula, %TFP must be equal to 0 and increase in K and L must be the same
also at at the bottom of the page
As a result (of constant return to scale ), if both capital and labor change by a percentage x, then the total change in output is αx + (1 − α)x = x. The use of constant returns to scale is predicated on empirical results from several large econo- mies over various time periods during the 19th and 20th centuries.
(Institute 125)
Institute, CFA. Level III 2013 Volume 3 Capital Market Expectations, Market Valuation, and Asset Allocation. John Wiley & Sons P&T, 6/18/2012. .
1.93 vs. 1928 what is the difference that you are harping about?
you still used 0.6% for change in TFP didn’t you - where you used 0.6% or 1.006…
you did no use 0, did you, if constant returns to scale meant 0 should be used for %TFP.
… let use bigger number then.
%TFP = 10% % K = 7% % Y = 1% alpha:0.9 beta:0.1
1.10 * 1.07^0.9 * 1.01^0.1 -1 = 17.20%
10 + 0.7*0.9 + 0.1*1 = 16.4%
_ As I said earlier _ this is the compounding effect… the bigger the number the bigger the difference
for this to hold :
As a result (of constant return to scale ), if both capital and labor change by a percentage x, then the total change in output is αx + (1 − α)x = x. The use of constant returns to scale is predicated on empirical results from several large econo- mies over various time periods during the 19th and 20th centuries.
CFAI page 125 book 3
the ONLY way. is that %TFP is 0. it doesnt mean you cant use it for aproximation.
% TFT = 0 is the final answer guys. stop confusing others.
Anyone who assumes TFP is zero because of constant returns to scale is going to get it wrong. Constant returns to scale only assumes the relationship that labor and capital have on GDP. The fact that constant returns to scale is assumed is the reason we are able to calculate TFP. The point is that x amount of labor and y amount of capital should produce z amount of output. TFP is the difference between the real output and the theoretical output. That is the only way that TFP is calculated, after the fact. So if we assume increasing or decreasing returns to scale, we will get a different TFP. There is no other way to calculate TFP other than using the known capital and labor stock, and the assumed constant returns to scale exponents and the known GDP from the time in question. Basically, constant returns assumption is required to get TFP number. If someone knows of another way to get TFP please post it but I dont think there is. Google; Roubini, Total factor productivity measure
Anything constant means rate of change is zero. That’s how a mathematical derivative is define.
rate of change=0 means constant.% tfp = 0 is the assumption
you all are still debating this???
Yes because these posters are wrong. Constant applies only to the change based on capital and labor. That is why the TFP is also known as the Solow Residual. Its a residual after plugging in all the assumptions and values. Check Roubinis lecture slides on this topic. If you think that constant returns to scale implies zero TFP growth, please explain how you find the TFP value. He assumes constant returns to scale for labor and capital and finds a positive value for TFP.
Apologies for bringing this topic back from the dead, but I just ran into the Schweser constant returns question on the practice exam.
So to reiterate. Schweser is wrong correct? If schweser were right we would assume 0.0% instead of 0.6%. But constant returns to scale only refers to the Alpha vs Beta relationship, so you can use 0.6% as a constant value.
TFP is the residual when calculating GDP growth. It is calculated by GDP growth - labor growth* elast. - capital growth *(1-elast. labor) or vice versa for elasticity of capital. Constant returns to scale only imply the relationship between labor elasticity and capital elasticity. You should google this and see how it is calculated because there are some wrong explanations in this thread. And yes scwheser gets things wrong. I distinctly remember one time when they described core satellite as requiring 50% or more in index funds which is completely incorrect per CFAI.
In the article I wrote on the Cobb-Douglas production function (http://financialexamhelp123.com/cobb-douglas-production-function/) I touch on constant returns to scale, as well as increasing and decreasing returns to scale. It may clear up some of the misconceptions here.
^ u really need to get some ads on there 
I think this is a point that is confusing some people. You can have a positive value for TFP growth and constant returns to scale. Assuming constant returns to scale does not mean that TFP will be zero in the given equation. I understand TFP to be measured by Solow only as an after the fact measurement from the other measureable variables involved. Which is why in macro you can measure and interchange L and K as the means of production but A cannot be substituted for either. Its exogenous.