Kwan Choi, International Trade, Spring 2009
Unit Cost Function
Consider unit cost of good 1. Choose
aL1 and aK 1
to
min c( w, r ) = waL1 + raK 1
Subject to:
1 − F (aL1 , aK 1 ) = 0.
The Lagrangian function associated with this problem is:
ℒ( aL1 , aK 1 , w, r, λ ) = waL1 + raK 1 + λ [1 − F (aL1 , aK 1 )]. (1)
FOCs are:
w − λ FL = 0,
r − λ FK = 0,
1 − F (aL1 , aK 1 ) = 0.
It follows that
−
daK 1 w FL
= =
,
daL1 r FK which is a well-known
condition. The solutions are aK 1 ( w, r ) and aL1 ( w, r ).
Substituting these into the objective function, we get
c( w, r ) = aL1 ( w, r ) w + aK 1 ( w, r ) r.
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Note that multiplying all factor prices by λ does not affect
the wage-rent ratio, and hence does not affect a ’s. Thus,
ij
c(λ w, λ r ) = aL1 ( w, r )λ w + aK 1 ( w, r )λ r = λ c( w, r ).
That is, the unit cost function is HD (1).
Shephard’s Lemma
∂c
⎛ wdaL1 + rdaK 1 ⎞
= aL1 + ⎜
⎟ = aL1 ,
∂w
dw
⎝
⎠
∂c
⎛ wdaL1 + rdaK 1 ⎞
= aK 1 + ⎜
⎟ = aK 1 ,
∂r
dr
⎝
⎠
which is also consistent with HD(1) of the unit cost
function c(w,r),
c( w, r ) =
∂c
∂c
w + r = aL1 ( w, r ) w + aK 1 ( w, r ) r.
∂w
∂r
(2)
Then marginal functions, aLj(w,r) and aKj(w,r) are HD(0),
i.e.,
2
∂aLj
∂w
∂aKj
∂w
w+
w+
∂aLj
∂r
∂aKj
∂r
r = 0,
r = 0,
(3)
which holds if, and only if
a Lj ( w, r ) = a Lj (λ w, λ r ),
a Kj ( w, r ) = a Kj (λ w, λ r ),
Total Cost Approach
Usually, Shephard’s lemma is expressed for total cost,
rather than for unit cost.
Let C ( w, r ) = wL + rK .
Then Shephard’s Lemma states:
∂C
∂C
= L,
= K.
∂w
∂r
Also, total cost is HD(1) in factor prices,
C ( w, r ) =
∂C
∂C
w+
r = wL + rK .
∂w
∂r
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(4)
Then the first derivatives, i.e., factor demand functions
L(w,r) and K(w,r), are HD(0) in factor prices.
That is,
∂L
∂L
w+
r = 0, ( 0i L )
∂w
∂r
∂K
∂K
(
,
)
K wr =
w+
r = 0, ( 0i K )
∂w
∂r
L( w, r ) =
(5)
which holds if, and only if L( w, r ) = L(λ w, λ r ) .
Cost minimization
● If we index labor demand by industries, for instance,
L1(w,r) and K1(w,r).
Cost minimization implies wL1 + rK1 is minimized, and
hence wdL1 + rdK1 = 0. Thus, for instance,
w
∂L1 ( w, r )
∂K ( w, r )
+r 1
= 0.
∂w
∂w
Likewise,
w
∂L( w, r )
∂K ( w, r )
+r
= 0.
∂r
∂r
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CONCAVITY OF COST FUNCTION C ( w, r )
(This property was used in Choi “Factor growth and
Equalized Factor Prices,” IREF 2008.
● The indirect cost function, c( w, r ) = aL1w + aK 1r ,
obtained by minimizing the cost function which is
linear in factor prices is now concave in factor prices.
● From any given point (w,r), a change in a factor price
has a linear impact on the cost, if aij ’s are constant,
i.e., if no adjustment is made. (This is the case of the
Leontief technology)
● If
aij ’s respond to a change in factor prices to
minimize cost, then c( w + Δw, r + Δr ) will be
below the cost with the Leontief technology, as
illustrated in Figure 1.
Leontief Technology
Y = min i (Vi / ai ) = min( K / α , L / β ).
● Isoquants have kinks, not necessarily L-shaped. At
every kink, there is no input substitution for a small
change in factor prices.
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Reason for concavity. With Leontief technology, C(w,r) is
linear in factor prices. With all other production functions,
aij’s respond to changes in factor prices and reduce the cost
below what it would have cost under the Leontief
technology.
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Properties of Short Run Cost Functions
One input, T (land) is fixed.
C ( w, r, T ) = wL + rK + sT
Then C ( w, r, T ) is concave in (w,r) for the same reason.
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Extra note
(1) Production function,
Y = F ( K , L, T ) , is HD(1). ⇒
FK K + FL L + FT T = mY = Y .
(6)
(2) If F(K,L) is HD(1), then Marginal Products are HD(0).
Differentiate (6) with respect to K,
FKK K + FLK L + FTK T + FK = FK .
Or
FKK K + FLK L + FTK T = 0i FK = 0.
The same is true with MPL.
Divide (6) by T. Average product of land is:
FK ( K / T ) + FL ( L / T ) + FT = Y / T .
Is the APL homogenous of degree 1 in nonland inputs? No.
In the two variable case, holding T constant, raising K
increase APK first but lowers it later.
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