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Sample HW 6 answers
1 a)
Q=200
The SR expansion path
indicates that we will
be at an inefficient
point on the isoquants
for Q<200. Output=L
when L<200. Output
cannot get higher than
200 in the SR
L=200
Notice the SR expansion path shows no increase in output, even as L is increased. You can do this
without input prices because efficient production is always at the “point” of the isoquant.
b) Q = 2K = L K=Q/2 and L=Q
LTC = 2K + 4L ---> LTC = 2(Q/2) + 4Q = 5Q
LAC = 5 and LMC = 5
STC=2K+4L as well, but Q=2K=L. As noted in the SR expansion path, changing L cannot increase Q, but it
can decrease Q. STC=2K+4L=200+4Q for Q200. (Note on grading: Give 2 points for LR costs, 1 point
for SR costs).
c) Economies of Scale relates the change in cost as output changes. In the SR, there are ultimate
diseconomies of scale for Q>200 since Q cannot exceed 200. Given the STC, as Q falls SAC goes up so
there are economies of scale for Q<200. In the LR, economies of scale are measured at efficient points.
Since the expansion path is linear, and we know this production function has CRS, that means there are
Constant economies of scale as well. (Note on grading – getting the LR economies of scale is sufficient).
2a) Q = RI ---> R = Q/I
Short Run Variable Cost = VC = wR = 100*Q/I = 100*Q/1000 = 0.1Q
Short Run Average Variable Cost = 0.1
Short Run Total Cost = FC + VC = 1*1000 + 0.1Q = 1000 + 0.1Q
Short Run Average Fixed Cost = FC/Q = 1000/Q
Short Run Marginal Cost = dTC/dQ = 0.1
SRTC for Q = 10000 is 1000 + 0.1(10000) = 2000
SRTC for Q = 12,000 is 1000 + 0.1(12000) = 2200
b) MRTS = I/R =100/1
I = 100R
We know Q = RI which means Q = 100R2 or Q = I2/100 ---> R = Q1/2/10 and I = 10Q1/2
using the cost function 100R + I = LTC ---> 100(Q1/2/10) + 10Q1/2 =LTC ---> LTC = 20Q1/2
LTC for Q = 10,000 is 2000 and LTC for Q = 12000 is 2190.89.
The difference comes from the fact that in short run the firm cannot adjust the amount of I it could use.
Now in the long run I is no longer a fixed variable so the firm will optimize (minimize) its costs with
respect to I and R.
c) (tR)(tI) = t2RI ---> Increasing returns to scale
LATC = 20/Q1/2 which is a decreasing function of Q. ---> The technology and costs shows Economies of
Scale. Yes both are consistent.
If Price of R is 121.
Short Run Variable Cost = VC = wR = 100*Q/I = 121*Q/1000 = 0.121Q
Short Run Total Cost = FC + VC = 1*1000 + 0.121Q = 1000 + 0.121Q
Long Run Total cost function
MRTS = w/r ---> I = 121R then Q = 121R2 or Q = I2/121
121R + I = LTC
Plugging value of R and I in terms of Q we get
LTC = 22Q1/2
So both the long run and short run costs go up and the price goes up.
3)
This expansion path shows an
extreme case. Really, it is likely to
curve up to the left, so the ratio of K
to L increases as output goes up
The expansion path might look like this one where the L is not changing even when the auto mall has
more cars in its inventory. Also looking at the green line, we can see that the output increases more
than proportionate to the increase in input. So if the costs doubled, the output increased more than the
costs. That is why we are talking about economies of scale.
4) Under the first strategy, the structure of the firms is not really changing. Only the collective input and
output is different than the input and output usage of these small firms in isolation. Hence, a returns to
scale analysis can give more insights as to how good this strategy is. By not combining physical plants,
this strategy really shows scale changes as it is not showing any substitution between capital and labor.
Under the second strategy, the structure of the organization is changing and that would prompt a
change in the long run average cost of the firm. This can be studied relatively easily by studying
economies of scale. By restructuring the mix of inputs, the venture capitalists are attempting to
increase efficiency through input substitution.
It is easy to study economies of scale and returns to scale when the expansion paths are straight lines
but in real world when the expansion paths are nonlinear, it is difficult to compare these criteria.
Therefore, we can't really say which strategy is better.
5)The first tax will have no effect on variable cost or marginal cost in the short run. This is because it
does not affect wage or labor in short run and marginal cost when derived from total cost will drop the
addition of the tax because it will be seen. The distance between ATC with tax and AVC should be equal
to AFC with tax. In the long run the LAC curve will shift upwards and become steeper on the left side and
asymptote with the old LAC curve on the right side. This is because as the firm produces more output
the license fee will become relatively cheaper. Once again the MC curve will remain the same for the
same reason as before.
The second tax will cause a shift in AVC, ATC, and MC curves in the short run. This is because MC is a
function of Quantity and now that quantity is more expensive MC must be higher. This causes a shift up
in the ATC because it is a function of quantity as well and quantity now costs more. AVC have gone up
because Average Product of Labor has now gotten more expensive because the firm will now produce
less quantity. In the long run both MC and LRAC will increase for the same reasoning as before.
The third tax will cause AVC, MC, and ATC all to increase while AFC will remain the same. This is because
labor is a variable cost and it has now become more expensive which means that ATC must go up. Now
every unit of output is more expensive because the labor to produce it is more expensive. In the long
run costs have to increase because labor has increased in price. What will happen is the ATC curve will
shift up and become steeper on the left hand side and asymptote with the old ATC curve. This is because
as the firm produces more and more output they can substitute more labor for capital. MC shifts up as
well in this situation for the same reason as in the short run.
Extra Credit Question
a) Q = (KL)1/2 Q=161/2L1/2=4L1/2 in the SR. Hence L=Q2/16 in the SR. STC=16+ Q2/16
SAC=16/Q+Q/16. SAVC=Q/16 SMC=Q/8
b) Long run cost curve
First we find out the optimal combination i.e. MRTS = w/r ---> 0.5K-.5L.5/0.5K.5L-.5 = 1 --> K = L
Putting this in our production function we get Q = K = L
wK + rL = LTC
Q + Q = LTC
LTC = 2Q
LAC = 2 and LMC = 2
c) Since the AC is constant, this is constant economies of scale.
Also (tK)1/2(tL)1/2 = t(KL)1/2 ---> Satisfies Constant Returns to Scale