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14.01 Problem Set 3
Due at 5pm on October 13th, 2023
Late problem sets are not accepted.
1 Long-Run Supply Curve (20 Points)
Consider a firm with the following production function
F (K, L) = L
α K
β
The firm faces a wage level w and rental rate of capital r.
- (5 Points) Show that the long run supply curve for this production function is
β α
(^) α+β w α+β r 1
p = q
α+β
− 1
α β
Solution: To determine the optimal choice of labor and capital, we equate the M RT S with −w/r:
M PL w
M P
K
r
αL
α− 1 K
β w
βL
α K
β− 1 r
αK w
=.
βL r
Substituting this into the production function, we get:
β
q = L
α
βw
L
αr
αr
β
α+β
1
α+β L = q ,
βw
which leads to α
βw
α+β 1
α+β K = q.
αr
Plugging this into the cost function, we find:
β (^) α β
α+β α+β
α+β
α
αr 1 βw 1 w α+β r 1 α+β α+β α+β C(q) = wL + rK = w q + r q = (α + β) q.
βw αr α β
For profit maximization (where M R = M C), we have:
β α
α+β w α+β r 1
p = q
α+β
− 1
.
α β
Now suppose the firm has the following production function
F (K, L) = K + L
- (5 Points) Find the long run supply as a function of r, w, and q.
Solution: The MRTS is constant and equal to M RT S = −1. Depending on the value of −w/r, we can
categorize into three cases:
Case 1 : −w/r > M RT S =⇒ r > w If the firm aims to minimize costs, it will rely solely on labor for
production in this case. Thus, the cost function is C(q) = wL + rK = wq, leading to a marginal cost of
M C = w. The long-run supply curve, derived by equating M R = M C, will then be p = w.
Case 2 : −w/r < M RT S =⇒ r < w If the firm’s goal is cost minimization, it will exclusively use capital for
production in this scenario. This results in a cost function C(q) = wL + rK = rq, with a marginal cost of
M C = r. The long-run supply curve, derived from M R = M C, is p = r.
Case 3 : −w/r = M RT S =⇒ r = w In this situation, the firm is indifferent between labor and capital; any
combination of K and L will minimize costs. The cost function becomes C(q) = wL + rK = rq, and the
marginal cost is M C = r. The long-run supply curve, derived by equating M R = M C, can be p = r or p = w.
Hence, the long-run supply curve is: ⎧
⎪w r > w ⎨
p = r r < w
⎪ ⎩
r or w r = w
We differentiate into these three cases because we’re comparing the cost of substituting labor for capital
against a constant MRTS. For instance, a higher substitution cost rate implies that we wouldn’t want to
replace any labor with capital.
- (5 Points) True of False? Justify your answer: An increase in wages always decreases supply in the long run.
Solution: False. If r < w then the long run supply curve does not depend on wages.
Let L(w, r, q) denote the labor demand when wages are w, the rental rate of capital is r, and the quantity produced
is q. Define the elasticity of labor demand as
∂L(w, r, q) w
ε
L
w ∂w L(w, r, q)
- (5 Points) True or False? Justify your answer: The elasticity of labor demand is always larger (in absolute
value) in the long run than in the short run.
Solution: True. In the short run the demand for output does not depend on prices. If capital is fixed then
the only way to increase production is to increase labor, regardless of prices. Then, the short run elasticity
is equal to zero. In the long run, capital is a variable input so there will be substitution between labor and
capital when wages change.
q
quantity produced by firm i, which we denote by qi. In equilibrium we have qi =. Then the individual 6
supply curve is
1 1 2 p = 2r 2 q i
For aggregation we need a relationship between q i
and p. After some algebra
2 1 p
qi =
r 2
Then
2 6 p
q = =⇒
r 2
p = r q
3
1
2
1
2
1
2
q
p
p
∗
S
D
S
0
- (6 Points) Suppose demand is still Q d
= 10 − p. How does the equilibrium price and quantity compare when
w = 1 vs when w =
q ? You don’t need to provide an algebraic expression, it is sufficient toshow in agraph 6
how they compare. Provide an intuition.
∗∗ ∗∗ ∗ Solution: At the new equilibrium, we have q < q
∗ and p > p. As the supply curve shifts from a horizontal
position to an upward-sloping position from its previous constant state, the new equilibrium quantity decreases
given the same demand. Additionally, as costs rise, the equilibrium price should also increase since the firm
has an incentive to sell at a higher price. Graphically
p
p
∗
S
D
S
0
p
∗∗
∗∗ ∗ (^) q q q
3 Aggregate Supply (22 Points)
In downtown Boston there is a vibrant farmers’ market. In this market, there are three apple orchards in the area
who all specialize in producing the same variety of apples. The total short-run cost functions for these producers
are
c 1
(q) =
q
2
c 2
(q) =
q
2
c 3
(q) =
q
2
- (5 Points) Derive each firm’s short-run supply curves. Do firms choose to produce at any price?
1 Solution: We observe that c i
(q) = q
2
- 5 − i. Firm i maximizes its profit when the marginal revenue (p) 5 −i
equals the marginal cost:
p = q.
5 − i
Thus, each firm has supply curves of q 1
= 2p, q 2
= 1. 5 p, and q 3
= p.
Given that firms produce according to profit maximization, and there are decreasing returns to scale, we know
that p = M C(q) > AV C(q) and therefore the firm will never choose to shut down.
- (5 Points) Let Qs denote the aggregate supply of apples in the farmer’s market. Derive the aggregate supply
of apples in the farmers’ market as a function of the price of apples p.
Solution: The aggregate supply of apples in the farmers’ market is:
QS (p) = 2p + 1. 5 p + p = 4. 5 p.
Suppose the demand for apples is given by Qd = 1 − p.
- (5 Points) What is the market equilibrium quantity Q
∗ and price p
∗ ? How much does each firm produce in
equilibrium?
Solution: Equilibrium is attained when supply equals demand. Given:
∗ ∗
- 5 p = 1 − p ,
∗ 2 ∗ 9 we find p =. Therefore, the equilibrium quantity is Q
∗ = 1 − p =. At this equilibrium, each firm 11 11
∗ ∗ 4 ∗ ∗ 3 ∗ ∗ 2 produces: q = 2p = , q = 1. 5 p = , and q = p =. (^1 11 2 11 3 )
- (7 Points) In the long run, would you expect the number of apple orchards to increase or decrease over time?
Justify your answer. Your answer needs to have an intuitive explanation, as well as a mathematical result to
back your intuition.
Solution: In the long run, firms (or apple orchards, in this case) typically look for positive profits to sustain
their operations. If apple orchards are consistently making a negative profit as
2 5 − i 1 (5 − i)
2 120 ∗ ∗ ∗ πi = p q − ci(q i
) = − − 5 + i = − (5 − i), i 11 11 5 − i 121 121
it’s not sustainable for them to continue operating indefinitely. Then, the number of apple orchards will
decrease over time because some producers will exit the market.
- (7 Points) How much does each firm produce in equilibrium as a function of J and N? Derive an expression
for the firm’s profits as a function of J and N.
Solution: Each firm will produce the equilibrium quantity divided by the number of firms. Thus, each firm
N i produces quantity qi(p) = (^) N. J+ 50
To find the profits of firm i, we solve the following:
N N N 1
π i
(N, J) = ∗ − [( )
2 ∗ + 2] N N N J + J + J + 2 50 50 50
πi(N, J) =
N
2 − 2
2 J +
N
50
- (8 Points) Suppose now that there is free entry. Derive the number of suppliers, J, that will produce in the
LR long run as well as the aggregate quantity, Q
LR and price p in the long run. How does it depend on the
size of the town N? Provide an intuition.
Solution: If there is free entry, then the long-run profits must be zero, which implies that the price is equal
to the marginal cost for each firm p = M C = qi. To get J, we set the profit equation from the previous
question to 0.
1 N
2 0 = ( − 2 N (^2) J
50
N
2
J +
N
50
N
2
N J
50
N
N J
50
N
2 J
+ = N
24 N
2 J
12 N
J
Using the equations from part 4, we can calculate the following:
N
LR p = 12 N N
25 50
N
LR p = 25 N
50
LR
p = 2
Q
LR
12 N
Q
LR
24 N
The number of suppliers increases as the size of the town N increases. This follows intuition because if there
are more people that have the same demand of milk, then it would make sense to have more milk suppliers.
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14.01 Principles of Microeconomics
Fall 2023
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