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Solutions to Additional Ch 9 & 10 Problems
SCENARIO 1
Given Q(K,L) = L3 + 2KL2; in the short run, K = 4; interpret
MRTS when K = 2and L = 1; no cost info given.
MPL =Q(K,L)
L= (3L2) + 2K(2L) = 3L2 + 4KL
MPK =Q(K,L)K
= 2L2
MRTS = MPLMPK
= 3L2+4KL2L2
= 32+ 2K
L
At the point on an isoquant where K = 2 and L = 1, the MRTS =
32+ 2
(21
)= 5
2
So you can substitute 2.5 units of capital for 1 unit of labor
without changing the outputlevel.
Returns to scale:Q = Q(cK, cL) = (cL)3 + 2(cK)(cL)2 = c3(L3 +
2KL2) = c3Q(K,L)Since c3 > c, the output grew by a factor larger
than the inputs, and we have increasingreturns to scale.
Now for the short-run part...Q(L) = Q(4, L) = L3 + 2(4)L2 = L3 +
8L2
AP = Q(L)L
= (L3 + 8L2)/L = L2 + 8L
MP = Q(L)L
= 3L2 + 8(2L) = 3L2 + 16L
AP is minimized where AP =MP :L2 + 8L = 3L2 + 16L2L2 = 8LL =
4
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SCENARIO 2
Given Q(K,L) = K1/3L1/3; in the short run, K = 8; interpret MRTS
when K = 2 andL = 1; w=2 & r=3.
MPL =Q(K,L)
L= K1/3(1
3L2/3) = K
1/3
3L2/3
MPK =Q(K,L)K
= (13K2/3)L1/3 = L
1/3
3K2/3
MRTS = MPLMPK
= K1/3/(3L2/3)
L1/3/(3K2/3)= K
1/3(3K2/3)
L1/3(3L2/3)= K
L
At the point on an isoquant where K = 2 and L = 1, the MRTS =
21= 2
So you can substitute 2 units of capital for 1 unit of labor
without changing the outputlevel.
Returns to scale:Q = Q(cK, cL) = (cK)1/3(cL)1/3 = c2/3K1/3L1/3 =
c2/3Q(K,L)Since c2/3 < c, the output grew by a factor smaller
than the inputs, and we have decreasingreturns to scale.
Now for the short-run part...Q(L) = Q(8, L) = 81/3L1/3 =
2L1/3
AP = Q(L)L
= 2L1/3/L = 2/(L2/3)
MP = Q(L)L
= 2(13
)L2/3 = 2
(3L2/3)
AP is minimized where AP =MP :2
L2/3= 2
3L2/3
But this is impossible. AP can never by equal to MP , so there
is no minimum on theAP curve.
FC = rK = 3(8) = 24V C = wL = 2LBut we need to get V C in terms
of Q, so lets use the short-run production function toget L in
terms of Q:Q(L) = 2L1/3Q2= L1/3
2
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(Q2
)3= L
L = Q3
8
Now we can plug this in for L in the V C:V C = 2L = 2(Q3)/8 =
Q3/4TC = FC +V C = 24+Q3/4 (sometimes we label this cost STC to be
specifically short-term)
AFC = FC/Q = 24/Q
AV C = V C/Q = Q3/4Q
= Q2/4
ATC = AFC + AV C or STC/Q = 24/Q+Q2/4
SMC = STCQ
= 34Q2 (also = SV C
Q)
ATC is minimized where ATC = SMC:24Q+ 1
4Q2 = 3
4Q2
24Q= 1
2Q2
48 = Q3
Q = 48(1/3) 3.6So the ATC hits its minimum at approximately Q =
3.6.(Note that AV C also hits its minimum when its equal to SMC.
But its impossible tohave 1
4Q2 = 3
4Q2, except in the trivial case of Q = 0, so there is no minimum
cost.)
Heres an extra challenge, back in the long-run environment: Try
to find the formula forthe long-run expansion path. This is the
relationship between K and L as the firm getslarger.
You can find this using the formulaMPLw
= MPKr
(K1/3)/(3L2/3)2
= (L1/3)/(3K2/3)
3K1/3
6L2/3= L
1/3
9K2/3
K = 23L
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SCENARIO 3
Given Q(K,L) = K2LK3; in the short run, K = 2; interpret MRTS
when K = 2 andL = 8; w=1 & r=2.
MPL =Q(K,L)
L= K2
MPK =Q(K,L)K
= 2KL 3K2
MRTS = MPLMPK
= K2
2KL3K2
At the point on an isoquant where K = 2 and L = 8,MRTS = 2
2
2(2)(8)3(2)2 =4
3212 =15. So you can substitute 0.2 units of capital for 1 unit
of
labor without changing the output level.
Returns to scale:Q = Q(cK, cL) = (cK)2(cL) (cK)3 = c3(K2LK3) =
c3Q(K,L)Since c3 > c, the output grew by a factor larger than
the inputs, and we have increasingreturns to scale.
Now for the short-run part...Q(L) = Q(2, L) = 22L 23 = 4L 8
AP = Q(L)L
= (4L 8)/L = 4 (8/L)
MP = Q(L)L
= 4
AP is minimized where AP =MP :4 (8/L) = 4But this is impossible.
AP can never by equal to MP , so there is no minimum on theAP
curve.
FC = rK = 2(2) = 4V C = wL = 1L = LBut we need to get V C in
terms of Q, so lets use the short-run production function toget L
in terms of Q:Q = 4L 8L = 1
4(Q+ 8) = Q
4+ 2
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Now we can plug this in for L in the V C:V C = L = Q
4+ 2
TC = FC+V C = Q4+6 (sometimes we label this cost STC to be
specifically short-term)
AFC = FC/Q = 4/QAV C = V C/Q = (Q
4+ 2)/Q = 1
4+ 2
Q
ATC = AFC + AV C or STC/Q = 14+ 6
Q
SMC = STCQ
= 14(also = SV C
Q)
ATC is minimized where ATC = SMC:14+ 6
Q= 1
4
But this is impossible. ATC can never by equal to SMC, so there
is no minimum on theATC curve.(Note that AV C also hits its minimum
when its equal to SMC. But its impossible tohave 1
4= 1
4+ 2
Q, so there is no minimum cost.)
Heres an extra challenge, back in the long-run environment: Try
to find the long-runcost curve, like we do in Question 13 of the
practice problems in Aplia, or Question 12of the graded problems.
Youll get some awkward numbers/exponents because I wasntthinking of
this additional piece when I set it up, but bear with me...
First we have to find the long-run expansion path. This is the
relationship between Kand L as the firm gets larger. You can find
this using the formulaMPLw
= MPKr
K2
1= 2KL3K
2
2
2K2 = 2KL 3K25K2 = 2KLK = (2/5)L
Now we use this, in combination with the long-run production
function, to get K and Lin terms of Q.Q = K2LK3 = (2
5L)2L (2
5L)3 = 2
125L3
So L = (125Q/2)1/3 = 5(Q/2)1/3.Plug this back in the long-run
expansion path to getK = 2
5L = 2
5(5(Q/2)1/3) = 2(Q/2)1/3
Long-term total cost in terms of K and L is simplyLTC = rK + wL
= 2K + LWe can use the L and K formulas we found above to get
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LTC = 2[2(Q/2)1/3
]+
[5(Q/2)1/3
]= 9(Q/2)1/3
Now you could get LAC by dividing by Q, or LMC by taking the
derivative wrt Q.
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