Algebraic manipulation of CES functions.
The standard form of a CES production function is:
1/ 


Y  A.   i X i 
 i

-(1)
Where Y is output, Xi is input of factor I and  is a substitution
parameter. A is a scale parameter and i is a share parameter for input I.
Factor inputs:
Cost minimisation implies dY/dXi = wi/P, ie real product wage equals
marginal product of input i.
Differentiating (1) wrt Xi we obtain:
Y / Xi  A i Y / X i 1   wi / P
-(2)
Doing the same for input j and using wi/wj = (wi/P)/(wj/P), we obtain:
Wi / W j  ( i /  j ).X j / X i 
1 
-(3)
Which can be rearranged as:
X i / X j  ( i /  j )1 / 1  .W j / Wi 
1 / 1 
-(4)
The elasticity of substitution, 1, between Xi and Xj is
dln(Xi/Xj)/dln(Wj/Wi)
It follows from (4) that:
  1/1  
(5a)
Or alternatively that:
  (  1) / 
(5b)
1
This is defining the elasticity of substitution so it is normally positive
It follows that when  = 1,  = 0 and the CES function becomes
indeterminate.
CES Cost Function (Price Index).
Taking equation (2) and rearranging:
X i / Y  P1 / 1  . A / 1  .i / wi 
1 / 1 
-(6)
Since CES functions have constant returns to scale, price will equal
average cost (zero profit condition). This means:
P  Wi X i / Y
i
-(7)
Substituting for Xi/Y from equation (6), we obtain:

P  Wi ( X i / Y ) A  / 1  .P1 / 1  .  i
i
1 / 1 
.wi
 /  1

i
-(8a)
Rearranging to get P onto the RHS:

1 / 1 
 /  1 
P  1/ A. i
.wi

 i

(  1) / 
-(8b)
An increase in all wi will lead to an equiproportionate increase in P.
Calibration.
Starting from rearranging equation 2, we can write:
i  A  .Si . X i / Y  
-(9)
where Si is the expenditure share of Xi:
Si=wiXi/PY
If we assume the beta ‘share’ parameters sum to 1, we can estimate A:
1/ 


A   Si .(Y / X i )  
 i

-(10)
We can also substitute for A and for Si, to allow us to write
i  wi X i1  /  w j X j1 
j
-(10a)
Own Price Elasticities.
Assuming Y is a fixed amount, the elasticity of demand for input I is
calculated from equation 2. This can be rearranged as follows:
1 / 1 
X i  Y . A / 1  .i
1 /  1
.P1 / 1  .Wi
-(11)
Differentiating wrt Wi yields the result:
X i / dWi  Y . A / 1  .i1 / 1  .P1 / 1  .wi1 /  1 (1/1   ).P / dwi .(1/ P)  (1/ wi )
-(12)
and substituting from (2) we get:
X i / dWi  X i .(1 / 1   ).P / dwi .(1 / P)  (1 / wi )
-(13)
For dP/dWi, we can use Roy’s identity:
If C=P.Y and is the cost of producing a fixed amount of Y, then
C / Wi  X i
And hence
P / dWi  X i / Y
-(14)
Substituting into (13) yields the following:
X i / dWi  X i .(1 / 1   ).X i / PY  (1 / wi )
 ( X i / Wi ).(1 / 1   )1  Si 
-(15)
Where Si denotes the expenditure share of input I in total inputs.
The own-price elasticity of demand for input I if Y is fixed is therefore:
i  (X i / Wi ).(Wi / X i )  (1/1   ).1  Si    1  Si
-(16)
It follows that when the input has a small share in total input costs, its
own-price elasticity tends towards the elasticity of substitution -, while
if its share is close to 1 its own-price demand elasticity is small.
These results are, of course, changed if output of Y is sensitive to input
costs. In that case, if the own-price elasticity of demand for Y is , then
we can write:
i   ln X i /  ln Wi   ln( X i / Y ) /  ln Wi   ln( Y ) /  ln Wi
-(17)
But the first term is the same as i when Y is fixed, ie:
i   (1  Si )   ln( Y ) /  ln Wi
-(17a)
While the second term can be rewritten:
 ln Y /  ln Wi  Y / Wi .(Wi / Y )
  .( X / Y ).(Wi / Y )
-(18)
Where  is the own-price elasticity of demand for good Y.
It follows that:
i   (1  Si i )  Si
   Si (   )
-(19)
Hence the effect of whether increasing input I’s share in good Y increases
or decreases its own-price elasticity of demand depends on whether the
elasticity of substitution in industry Y, , exceeds the own-price elasticity
of demand for good Y, . This has implications for the effects of degree
of monopoly on pricing when the inputs are imperfectly-substituting
products of rival firms (the Dixit-Stiglitz framework).
Huw Edwards 22 March 01