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Solution to Exercise 8.8…and beyond.
Given: Q = 10 L
.2
K
.3
; w = $1500; r = $1000.
Find the leastcost combination of L and K
to produce Q = 100.
Find the cost of producing 100.
Find shortrun total cost function
assuming that K is fixed at the level you found above. Find the longrun total cost
function.
Leastcost means a tangency between isocost and isoquant:
MRTS
LK
= W/R
For CobbDouglas production function, MRTS
LK
=
α
K/
β
L.
In this case,
α
= 0.2, and
β
=
0.3.
Therefore, .2K/>3L=1500/1000, or
K = 9/4 L .
Note that this is the equation for the
expansion path for this CobbDouglas function and the given wage and rental rates.
Plug K = 9/4 L back into the production function:
100 = 10 L
.2
(9/4)
.3
L
.3
to find the
optimal level of L for 100 units of Q to be L* = 61.4739.
Now plug back into the
expansion path relation to find optimal K* = 138.3162.
This is the leastcost
combination to produce Q = 100 units.
To find the cost associated with this combination,
use the cost equation: C = WL+RK: C = (1500)(61.4739) + (1000)(138.3162) = $230,527
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This note was uploaded on 06/09/2008 for the course BUAD 351 taught by Professor Eastin during the Spring '07 term at USC.
 Spring '07
 Eastin
 Business

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