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Unformatted text preview: Intermediate Microeconomics Monika Thomas
Summer 2007 Solutions: Problem Set 3 8.3 Starting with the tangency condition, we have MPL_w MPK—7
K_2
'f‘i
K==2L Substituting into the production function yields Q=LK
Q=L(2L)
L: Q i 2 Plugging this into the expression for K above gives K=2 2
2 Finally, substituting these into the total cost equation results in were
a TC=J® 8.4 and average cost is given by _§g_1000Q—30Q2+Q3
Q Q
AC=1000—30Q+Q2 AC Graphically, average cost is a
V)
O U
Q)
bl)
a
L.
3 <1 Minimum efﬁcient scale occurs where the average cost curve reaches a minimum,
Q = 15 for this cost function. 8.12 a) From the production function we see that Q = 5x/Z , so the amount of
Q2 labor required to produce Q is given by L = The short run total cost 2
function is C = 25L + 20K 2 25+ 20(5) = 100 + Q. 9.7 7 First, ﬁnd the minimum of A VC by setting A VC = SMC . AVC=K=91
Q Q
AVC=Q
Q=2Q
Q=0 The minimum level of AVC is thus 0. When the price is O the ﬁrm will
produce 0, and for prices above 0 ﬁnd supply by setting P = SMC . P=2Q
Q=%P Thus,
s(P) = %P b) Market supply is found by horizontally summing the supply curves of the
individual ﬁrms. Since there are 20 identical producers in this market, market supply is given by
S (P) = 20s(P)
S(P) =10P c) Equilibrium price and quantity occur at the point where S (P) = D(P). 10P=110~P
P=10 Substituting P =10 back into D(P) implies equilibrium quantity is
Q = 100. So at the equilibrium, P =10 and Q = 100. 9.17 The longrun equilibrium price in a perfectly competitive equilibrium equals the
minimum level of longrun average cost. This is given as $5 per ton. Each producer supplies a quantity of output equal to the point at which longrun
average is minimized. This is given as 2 million tons per year. Market demand at the longrun equilibrium price of $5 per ton is equal to 205 — 5 = 200 million tons
per year. This implies that there must be 100 active ﬁrms in the longrun
equilibrium because (200 million tons per year)/(2 million tons per year per ﬁrm)
= 100. 9.18 In a longrun equilibrium all ﬁrms earn zero economic proﬁt implying P = AC
and each ﬁrm produces where P = MC . Thus, 40—12Q+Q2 =40—6Q+%Q2
Q = 9
So each individual ﬁrm produces Q = 9 , and the long—run equilibrium price must
beP = 40 —12(9) + 92 =13. Since D(P) = 2200 ~100P , D(P) = 2200 «100(13)
D(P) = 900 If each ﬁrm produces 9 units, the market will have 100 ﬁrms in equilibrium. 9.24 Since an individual ﬁrm will supply where P = SMC , P = 4Q Q=%P
Assuming a ﬁrm will supply for any positive price this implies 5(P) = %P .
Graphically we have 0 20 40 Producer surplus for an individual ﬁrm is given by area A in the ﬁgure above
which is %(200)50 = 5000 . Since all ﬁrms are identical, overall producer surplus will be 100(5000) = 500, 000. ...
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 Summer '08
 JustinMarion
 Supply And Demand, producer, Average cost, longrun equilibrium price

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