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Unformatted text preview: n C[ w, r, Q ] = 2 Qwr − w , then
deriving the input demand equations and finally the production function. As before, assume an
interior optimum.
a) Applying Shephard’s Lemma to the cost function, find the factor demand equations, L[w,r,Q]
and K[w,r,Q].
b) Using the two factor demand equations, show that the corresponding production function is Q
= KL + K. Hint: Try to eliminate both w and r at the same time.
Answer:
5) a) Shephard’s Lemma involves taking the partial derivatives of the long run total cost function
with respect to the factor prices: ∂C ∂ #
1
Qr
=
2 Qwr − w% = 2 Qr w −1/2 − 1 =
− 1 = L[ w, r, Q ]
&
∂w ∂w $
2
w
∂C ∂ #
1
Qw
=
2 Qwr − w% = 2 Qw r −1/2 =
= K [ w, r, Q ]
&
∂r ∂r $
2
r
b) Our goal is to eliminate the factor prices, leaving only Q[L,K]:
Qr
L +1
r
L=
−1 ⇒
=
w
w
Q K= Qw
K
w
Q
r
⇒
=
⇒
=
r
r
K
w
Q L +1
Q
=
⇒ Q = K ( L + 1) = KL + K
K
Q...
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This homework help was uploaded on 03/03/2014 for the course ECON 310 taught by Professor Whinston during the Spring '12 term at Northwestern.
 Spring '12
 Whinston

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