a)
Prove that the given production function exhibits constant return to scale (6 marks).
1
2
1
2
2
0
L
K
MP
L
and
1
2
3
2
0
L
MP
K
L
L
b)
Show that,
when
w
is the labour wage and
r
is the capital rent, the optimal ratio between
capital and labour is
1
4
K
L
. (6 marks)
In equilibrium, the optimal bundle requires
L
K
MP
w
MRTS
MP
r
With
1
1
2
2
1
1
2
2
2
1
2
;
2
8
4
L
K
K
L
K
w
MP
MP
MRTS
L
r
L
K
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The optimal bundle requires
1
4
K
L
c)
Compute the optimal number of labour and capital used to produce 16 units of output. (6
marks)
Optimal bundle is one equation. The second equation is
1/2
1/2
(
,
)
4
Q K L
K
L
=32.
Substitute (1) into (2) we get 2L = 16 or L = 16 and K = 4.
d)
Suppose capital is fixed at
4
K
in the short run. Derive (if exists) the short run total cost, short
run average total cost, short run marginal cost, short run variable cost and short run fixed cost
as a function of Q
. (7 marks)
When K is fixed at 4 in the short run, the short run production function becomes:
Q = 8L
1/2
or L = Q
2
/64
SRTC(Q) = rK + wL = 4*8 + 2*Q
2
/64 = 32 + Q
2
/32
SRATC(Q)=32/Q + Q/32
SRMC(Q)=Q/16
SRVC(Q) = Q/32
SRFC = 32
e)
Use a diagram to
prove
that the long run total cost is never greater than the short run total cost.
(5marks)
Another way to show this is as per figure 1022 on page 299 where long run average cost curve
is an envelope of all the short run average cost curves.
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 Fall '12
 Danvo
 Microeconomics, Supply And Demand, Tom

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