For each problem, please express your answer in a short sentence. Show your work. Unless otherwise stated, assume that fractional units of goods can be produced and sold, and fractional units of inputs can be used (meaning answers do not necessarily need to be whole numbers).

Problem Set # 2
Ben Milner, Econ 281
Due Date: Oct. 31, 2022
For each problem, please express your answer in a short sentence. Show your work.
Unless otherwise stated, assume that fractional units of goods can be produced and
sold, and fractional units of inputs can be used (meaning answers do not necessarily
need to be whole numbers). Round your final answers to three decimal places when
necessary.
Problem 1 (20 Marks)
For each of the following production functions, please do the following:
i) graph the isoquant curve corresponding to Q = 100 (Tip: you don’t need to use
the same scale on all your graphs);
ii) for each of labour and capital, use the corresponding marginal product functions
to describe whether diminishing marginal returns exist;
iii) find the marginal rate of technical substitution of labour for capital;
iv) describe if there are increasing, constant, or decreasing returns to scale.
(a) Q = 3√
L +
√
K, MPL =
3
2
√
L
, MPK =
1
2
√
K
(b) Q = 10√
KL, MPL = 5
√
√
K
L
, MPK = 5
√
√
L
K
(c) Q =
1
3
√
KL, MPL =
1
6
√
√
K
L
, MPK =
1
6
√
√
L
K
(d) Q = 3L + 2K, MPL = 3, MPK = 2
(e) Q = K2
√
L, MPL =
K2
2
√
L
, MPK = 2K
√
L
. . . . . . . . .
Problem 3 2
Problem 2 (30 Marks)
SouthPawShoes makes dog boots for left-handed pets. Its production function is
Q = 100L
0.3K0.2
, where Q is the number of pairs of boots produced, L is the number of hours of work put in by the company’s specially trained workforce, and K
is the number of sewing machines the company uses to make the boots. Note that
MPL =
30K0.2
L0.7
, and MPK =
20L0.3
K0.8
.
(a) There is a huge left-handed dog show coming up, and SouthPawShoes needs
to fill a big order: 3000 pairs of boots. Suppose that the hourly wage rate at SouthPawShoes was $25, and that the rental rate for sewing machines was $10 per hour.
What combination of labour and sewing machines will allow them to fill this order in
the cheapest way? How much will it cost?
(b) The management at SouthPawShoes are trying to anticipate how much labour
they will use next year. They think the rental rate will remain at $10, and they think
they will again produce 3000 pairs of boots, but they are unsure what the going wage
rate will be. Find the company’s demand curve for labour.
(c) Suppose it takes quite a long time to find a skilled boot worker, and that due
to their strong union it is also difficult to fire them. Thus, in the short run, L is fixed
at 100. Assume again that the hourly wage rate is $25 and the rental rate for sewing
machines is $10 per hour. What is the cheapest way for the company to make 3000
pairs of boots now, and what will it cost?
(d) Suppose L is still fixed at 100, but the company has figured out that they
can save a huge amount of time and effort by printing out paper boot-outlines first,
and using these to aid the sewing process. Thus, their new production function is
Q = 100L
0.3K0.2M0.5
, where M is the number of printing machines the company
uses. Now MPL =
30K0.2M0.5
L0.7
, and MPK =
20L0.3M0.5
K0.8
, and MPM =
50L0.3K0.2
M0.5
. If the
rental rate for printing machines is $50 per hour, what is the cheapest way for the
company to make 3000 pairs of boots now, and what will it cost?
. . . . . . . . .
Problem 3 20 Marks
Marley runs a crypto mining warehouse where she mines Bubble Coin. She has already put down a non-refundable deposit of $5000 to rent the warehouse, and also
has to pay an additional $2500 per month she stays there. She has also spent $15000
buying servers, but could get this money back by selling them if she wanted to. Her
variable costs are 5Q + Q2
(meaning MC = 5 + 2Q), where Q is amount of Bubble
Coin mined each month.
Econ 281 PS # 2
Problem 4 3
a) How many Bubble Coins will Marley mine this month if Price = $455? What
will her profit be?
b) At what price of Bubble Coin should Marley shut down the warehouse?
Note: there is some flexibility in how the above question can be interpreted. As
long as you give a logical argument explaining why you treated each cost the way you
did, full marks are available.
. . . . . . . . .
Problem 4 30 Marks
There are 96 producers in the perfectly competitive market of Queen Elizabeth II
commemorative tea cosies. Of those 96, 70 each have the following short-run total
cost curve: ST C(Q) = 22.5 + 0.1Q2 + 20Q, where Q is output (meaning each have
short-run marginal cost curve SMC(Q) = 0.2Q + 20). The remaining 26 each have
the following short-run total cost curve: ST C(Q) = 40 + 0.1Q2
(meaning each have
the short-run marginal cost curve SMC(Q) = 0.2Q).
a) Assuming all fixed costs are sunk in the short run, find each producer’s shortrun supply curve, and also the market supply curve.
b) Assuming all fixed costs are nonsunk in the short-run, find each producer’s
short-run supply curve, and also the market supply curve.
c) Assume again that all fixed costs are sunk in the short run. If the market demand curve is Q = 8990−300P, what is the short-run equilibrium price and quantity?
d) Assume again that all fixed costs are nonsunk in the short run. If the market
demand curve is again Q = 8990 − 300P, what is the short-run equilibrium price and
quantity?
e) Suppose that in the long run, all firms face the following total cost curve:
T C(Q) = 600.625 + 0.1Q2
(meaning each have the marginal cost curve MC(Q) =
0.2Q). As well, firm entry and exit is possible in the long run. Assuming market
demand is Q = 8990 − 300P, how many producers will there be in the market in the
long run?
.