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Unformatted text preview: isoquant for Q units of output.
The long run cost function is C = wL + rk = (1)(4Q) + (8)(Q) = 12Q. 4) A firm produces output (Q) using labor (L), whose factor price is w, and capital (K), whose
factor price is r. The production function is Q = KL + K. Throughout this problem assume there is
an interior optimum (with L > 0 and K > 0).
a) Find the factor demand equations, L[w,r,Q] and K[w,r,Q]. Hint: You are effectively finding
(L*,K*) for any levels of [w,r,Q] that generate an interior solution.
b) Show that the firm’s long run total cost function is C[ w, r, Q ] = 2 Qwr − w
Answer:
4) a) MRTSL, K = MPL
K
w
=
= ⇒ w( L + 1) = rK [Condition #1: Tangency]
MPK L + 1 r MPL MPK
K
w
<
⇔
<
w
r
L +1 r
(One could imagine using only cheap K inputs when the wage gets very high.)
Q
Q = KL + K = K ( L + 1) ⇒ K =
[Condition #2: Production]
L +1
#Q&
Qr
Qr
2
Combine and solve for L: w( L + 1) = r %
⇒ L +1 =
( ⇒ ( L + 1) =
$ L +1'
w
w
NOTE: Bang for the buck logic would generate L = 0 when ⇒ L[ w, r, Q ] =
⇒K= Q
=
L +1 Qr
−1
w
Q
Qw
=
= K [ w, r, Q ]
r
Qr
−1+1
w b) Plug the two factor demands into the general cost function and simplify:
" Qr % " Qw %
C = wL + rK = w $
$ w − 1' + r $ r ' = Qwr − w + Qwr = 2 Qwr − w = C[ w, r, Q ]
'$
'
#
&#
& 5) Now work problem 4 “backwards,” starting with the cost functio...
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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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