5
Cost function approach:
Let r = (r1, …, rn) be the vector of prices for inputs x = (x1, …, xn). The cost function is defined as
C(r, y, t) = r’ xc(r, y) = minx {r’ x: y = f(x, t)},
where xc(r, y, t) are the cost-minimizing input demand functions that solve the above cost minimization problem.
* Cobb-Douglas cost function
For a given technology t, the cost function C(r, y, t) associated with a Cobb-Douglas production function takes the form
C(r, y, t) = A yk ,
where ai > 0,ai = 1, where any of the parameters (A, k, a1, …, an) can change with technology t. For a given t, the cost function can be equivalently written as
ln(C) = ln(A) + k ln(y) + ai ln(ri).
It is associated with a technology that is homogenous of degree (1/k). And it exhibits unitary Allen elasticities of substitution (sij = 1 for i ¹ j). Thus, it is not a flexible functional form. From Shephard’s lemma, xic = ∂C/∂ri, and the associated cost minimizing input demand functions are
xic = A yk ai /ri, i = 1, …, n,
or
ln(xic) = ln(A) + k ln(y) + (ai-1) ln(ri) + aj ln(rj).
In the case of general technological change, any of the parameters (A, k, a1, …, an) can change with technology t. It means that technological change can affect productivity A, the scale effect parameter k, and the price responsiveness parameters (a1, …, an).
As a special case, the parameters (k, a1, …, an) can be constant, the only parameter affected by technological change being A, with A= A(t). This corresponds to Hicks-neutral technical change, where technical change is just an intercept shifter for ln(C) or ln(xc).
* Translog cost function
For a given technology t, the translog cost function C(r, y, t) takes the form
ln(C) = a0 + ay ln(y) + ai ln(ri) + ½ aij ln(ri) ln(rj)
+ ayi ln(y) ln(ri) + ½ ayy [ln(y)]2.
It is associated with a general technology that imposes no a priori restriction on the Allen elasticity of substitution. Thus, it is a flexible functional form. It includes the Cobb-Douglas cost function as a special case when ayi = aij = ayy = 0 for all i and j = 1, …, n.
In the general case, any of the parameters a’s can change with technology t.
1/ Theoretical restrictions
symmetry: ¶2ln(C)/¶ln(r)2 = symmetric, implying that
aij = aji for all i ¹ j,
generating (n2 – n)/2 restrictions.
homogeneity: C(r, ×) is linear homogenous in prices r, implying that
¶ln(C)/¶ln(ri) = 1 (Euler equation),
In the translog case, we have
¶ln(C)/¶ln(ri) = ai + ayi ln(y) + aij ln(rj), i = 1, …, n,
implying that
[ai + ayi ln(y) + aij ln(rj)] = 1, for all y and r.
This gives the following homogeneity restrictions
ai = 1
ayi = 0
aij (= aji) = 0, for all j = 1, …, n.
2/ Special cases
For a given technology t, we have:
- homothetic technology: obtained when ayi = 0. (This follows from the fact that, under a homothetic technology, C(r, y) = K(y) × H(r), or ln(C(r, y)) = ln(K(y)) + ln(H(r)), implying that ∂ln(C)/∂ln(r) is independent of y).
- homogenous technology: obtained when ayy = 0, and ayi = 0, for all i, where 1/ay measures the degree of homogeneity (e.g., ay = 1 corresponding to CRS). (This follows from the fact that, under a homogenous technology of degree k, C(r, y) = y1/k × H(r), or ln(C(r, y)) = (1/k) ln(y) + ln(H(r)), implying that ∂ln(C)/∂ln(y) =1/k, i.e. that ∂ln(C)/∂ln(y) is equal to 1/k and is independent of (r, y)).
- Cobb-Douglas technology: obtained when ayy = 0, ayi = 0 (since the Cobb-Douglas technology is homogenous), and aij = 0 for all i and j (since the Cobb-Douglas technology implies unitary Allen elasticities of substitution; see below).
3/ Estimation
From Shephard’s lemma ¶C/¶ri = xic, we have
¶ln(C)/¶ln(ri) = (¶C/¶ri)(ri/C) = ri xic/C.
Let wi = ri xic/C denote the i-th cost share. It follows that, under cost minimizing behavior,
wi = ¶ln(C)/¶ln(ri)
always.
In the translog case, holding technology t constant, this gives the following cost share specification
wi = ai + ayi ln(y) + aij ln(rj), i = 1, …, n.
In conducting econometric analysis, an error term is typically added to the above model, yielding the following econometric model
wi = ai + ayi ln(y) + aij ln(rj) + ei, i = 1, …, n,
where ei is a random variable with mean zero and finite variance. This is a system of n equations that can be estimated using standard estimation methods.
Note that, by definition, we have wi = 1. This implies that the dependent variables are linearly dependent, meaning that associated variance of the e’s is singular. This is often handled by simply dropping an equation before estimation, and thus estimating a system of (n-1) equations. The parameters of the equation dropped can then be recovered from the homogeneity restrictions mentioned above. In addition, the parameter estimates are invariant to the equation dropped when the model is estimated by the maximum likelihood estimation method.
4/ Elasticities
For i ¹ j, we have
¶ln(xic)/¶ln(rj) = (¶xic/¶rj)(rj/xi)
= (¶2C/¶ri¶rj)(rj/xi) (from Shephard’s lemma)
=
= [¶2ln(C)/¶ln(ri)¶ln(rj)][C/(rixi)] + [¶ln(C)/¶ln(ri)][¶ln(C)/¶ln(rj)][C/(rixi)]
= aij/wi + wi wj/wi, where wi = ¶ln(C)/¶ln(ri).
This gives the following cross-price elasticities of the cost minimizing input demand functions xic
¶ln(xic)/¶ln(rj) = aij/wi + wj, for all i ¹ j.
Being homogeneous of degree zero in prices, xic must satisfy
¶ln(xic)/¶ln(ri) = -¶ln(xic)/¶ln(rj) (from Euler equation)
= -[aij/wi + wj].
But-aij = aii from the linear homogeneity of the cost function in r, andwj = 1 - wi. This gives the following own-price elasticities of the cost minimizing input demand functions xic
¶ln(xic)/¶ln(ri) = aii/wi + wi – 1, for all i = 1, …, n.
Note: We have seen that the Allen elasticities of substitution (AES) between inputs i and j are
sij = [¶ln(xic)/¶ln(rj)]/wj.
For i ¹ j, this gives
sij = [aij/wi + wj]/wj
or
sij = aij/(wi wj) + 1, for all i ¹ j.
Note: In the Cobb-Douglas case where sij = 1 for all i ¹ j, this implies aij = 0 for all i ¹ j. In addition, we have seen that the linear homogeneity of the cost function in r gives the restrictionaij = 0, or aii = -aij. In the Cobb-Douglas case, this implies that aii = 0. In other words, unitary Allen elasticities of substitution (sij = 1 for all i ¹ j) implies that aij = 0 for all i, j = 1, …, n (e.g., under a Cobb-Douglas technology).
5/ The Case of Technological Change
When technology t changes, then the translog cost function can be written as
ln(C) = a0 + at t + ay ln(y) + ai ln(ri)
+ ½ aij ln(ri) ln(rj) + ayi ln(y) ln(ri) + ½ ayy [ln(y)]2
+ ait ln(ri) t + ayt ln(y) t.
With wi = ri xic/C denoting the i-th cost share under cost minimizing behavior, we have
wi = ¶ln(C)/¶ln(ri)
or, under a translog cost specification,
wi = ai + ayi ln(y) + aij ln(rj) + ait t,
i = 1, …, n.
It follows that technological change affects the i-th cost share if and only if ait ≠ 0. Thus, technological change is:
· Hicks neutral if ait = 0 for all i = 1, …, n.
· Biased toward using the i-th input if ait > 0
· Biased against using (or toward saving) the i-th input if ait < 0.
Note: Under constant return to scale (CRS). Under a linear homogenous/CRS technology, we have C(r, y, t) = y × H(r, t), or ln(C(r, y, t)) = ln(y) + ln(H(r, t)), implying that ∂ln(C)/∂ln(y) = 1, i.e. that ∂ln(C)/∂ln(y) is equal to 1 and is independent of (r, y, t)). Under a translog specification and CRS, it follows that
∂ln(C)/∂ln(y) = ay + ayi ln(ri) + ayy ln(y) + ayt t = 1,
for all (r, y, t). This implies that ay = 1, ayi = 0 for all i, ayy = 0, and ayt = 0. Thus, under technological change, the CRS translog cost function is
ln(C) = a0 + at t + ln(y) + ai ln(ri)
+ ½ aij ln(ri) ln(rj) + ait ln(ri) t,
or
ln(C/y) = a0 + at t + ai ln(ri)
+ ½ aij ln(ri) ln(rj) + ait ln(ri) t,
The theoretical restrictions are:
symmetry: ¶2ln(C)/¶ln(r)2 = symmetric, implying that
aij = aji for all i ¹ j,
generating (n2 – n)/2 restrictions.
homogeneity: C(r, ×) is linear homogenous in prices r, implying that
¶ln(C)/¶ln(ri) = 1 (Euler equation),
In the CRS translog case, we have
¶ln(C)/¶ln(ri) = ai + aij ln(rj) + ait t, i = 1, …, n,
implying that
[ai + aij ln(rj) + ait t] = 1, for all (r, t).
This gives the following homogeneity restrictions
ai = 1
aij (= aji) = 0, for all j = 1, …, n,
ait = 0.
From Shephard’s lemma, the i-th cost share (wi = ri xic/C) can be written as
wi = ¶ln(C)/¶ln(ri),
or, in the CRS translog case,
wi = ai + aij ln(rj) + ait t,
i = 1, …, n.
In the CRS translog case, the rate of technical change can be measured as
∂ln(f)/∂t = -∂ln(C)/∂t
= -[at + ait ln(ri)].
Given the i-th cost share wi = ai + aij ln(rj) + ait t, it follows that ∂wi/∂t = ait, implying that technological change affects the i-th cost share if and only if ait ≠ 0. Thus, technological change is:
· Hicks neutral if ait = 0 for all i = 1, …, n.
· Hicks-biased if ait ≠ 0 for at least one i = 1, …, n.
o Biased toward using the i-th input (or i-th input-using) if ait > 0
o Biased against using the i-th input (or i-th input-saving) if ait < 0.
* Generalized Leontief cost function
For a given technology t, the generalized Leontief cost function C(r, y) takes the form
C(r, y, t) = h(y) gij (ri)1/2 (rj)1/2 + g(y) ai ri.
It is associated with a general technology that imposes no a priori restriction on the Allen elasticity of substitution. Thus, it is a flexible functional form. It includes the Leontief technology as a special case (see below).
1/ Theoretical restrictions
symmetry: ¶2C/¶r2 = symmetric, implying that
gij = gji for all i ¹ j,
generating (n2 – n)/2 restrictions.
homogeneity: C(r, y) is already linear homogenous in prices r. Thus there is no additional homogeneity restriction to impose.
2/ Special cases
For a given technology t:
- homothetic technology: obtained when ai = 0 for all i.
- Leontief technology: obtained when gij 0 for all i ¹ j. This gives
C(r, y) = g(y) ai ri.
It implies that ¶2C/¶r2 = 0, yielding sij = 0. This is the Leontief technology, with fixed proportions, and zero possibilities of substitution among inputs.
Note: Again, the effects of technological change can be introduced by allowing any of the parameters to vary with t.
3/ Estimation
From Shephard’s lemma, we have
¶C/¶ri = xic.
In the generalized Leontief case, this gives the following cost minimizing input demand functions
xic = h(y) gij (rj/ri)1/2 + ai g(y), i = 1, …, n.
In conducting econometric analysis, an error term is typically added to the above model, yielding the following econometric model
xic = h(y) gij (rj/ri)1/2 + ai g(y) + ei, i = 1, …, n,
where ei is a random variable with mean zero and finite variance. This is a system of n equations that can be estimated using standard estimation methods. (Note that, in this case, there is no problem with the singularity of the variance of the e’s, and thus no need to drop an equation).
4/ Elasticities
For i ¹ j, we have
¶ln(xic)/¶ln(rj) = (¶xic/¶rj)(rj/xi).
This gives the following cross-price elasticities of the cost minimizing input demand functions xic
¶ln(xic)/¶ln(rj) = ½ h(y) gij (ri rj)-1/2 rj/xi, for all i ¹ j.
Being homogeneous of degree zero in prices, xic must satisfy
¶ln(xic)/¶ln(ri) = -¶ln(xic)/¶ln(rj) (from Euler equation)
= -½ h(y) [gij (ri rj)-1/2 rj/xi].
This gives the following own-price elasticities of the cost minimizing input demand functions xic
¶ln(xic)/¶ln(ri) = -½ h(y) [gij (rj/ri)1/2 /xi], for all i = 1, …, n.