Download Monopoly and Monopsony Labor Market Behavior

Monopoly and Monopsony Labor Market Behavior
1
Introduction
For the purposes of this handout, let’s assume that firms operate in just two markets: the market for
their product (where they are a seller) and the labor market (where they are a buyer)1 . If the firm
faces a competitive output market, regardless of what quantity they produce they get the constant
market price P for each unit sold. If they face a competitive labor market, regardless of how many
people they hire they have to pay each worker the constant market wage w.
Sometimes firms have ”market power” - the ability to manipulate the price of the market good
based on the quantity they use or produce. In the extreme case, firms may be the only seller or buyer
in a market. If the firm is the only seller, it is called a monopolist, and can change P depending on
how much it produces. If it is the only buyer, it is a monopsonist, and can change w depending on
how much labor it employs.
2
Review of the General Labor Supply Model
Let there be only one input in production - labor. Profit maximizing firms choose labor by setting
the marginal cost of labor equal to the marginal benefit (or marginal revenue product) of labor. In a
competitive model, both P and w are taken as given - the only thing that isn’t determined by market
forces is the marginal product of labor. The first order conditions (FOC) for profit maximization
are2
M RPL = M CL
M PL ∗ M R = M CL
(1)
1
In reality, firms may operate in many markets, selling multiple goods and using multiple inputs in production.
They may face competitive markets in some and have market power in others.
2
The FOC is a result of a generic maximization problem such as max P (Q(L)) ∗ Q(L) − C(L) where the maximization is done over L.
Prepared by Nick Sanders, UC Davis Graduate Department of Economics 2006
where
dOutput
dQ
=
dLabor
dL
dR
dRevenue
=
MR =
dOutput
dQ
dCosts
dC
M CL =
=
dLabor
dL
M PL =
When firms take output price and wage as given, (1) simplifies to
M PL ∗ P = w
(2)
Now let’s consider how things change when P or w is allowed to change with Q and L, respectively.
3
The Monopolist Labor Decision
Since a monopolist is the only producer in a market, they see the entire demand curve when making
their production decision. As such, the more they produce, the lower the price in the market. We
represent this with an inverse demand curve where price is presented as a function of quantity
supplied, or P ≡ P (Q), where
dP
dQ
< 0.
Returning to equation (1), M CL is still w, because the firm takes the price of its inputs as given.
M R, however, is no longer simply P , but is now
d(P (Q) ∗ Q)
dQ
dP
= P (Q) +
Q
dQ
dP Q
=P 1+
dQ P
1
=P 1+
η
M RLmonopolist =
2
(3)
where η is the elasticity of demand (percentage change in quantity demanded divided by percentage
change in price) which is assumed to be negative3 . So for a monopolist
M RLmonopolist = P ∗ (1 − some nonzero number) < P
which reflects that each additional unit sold in the market lowers the price of all other units sold,
and at a given wage monopolists will hire less labor than perfectly competitive firms. As monopolists employ less labor, they have lower production and thus generate a higher market price than
competitive firms.
3.1
Monopolist example
Let the firm face the inverse demand function P (Q) = 50 − 2Q and a cost function C(L) = 50L.
The marginal productivity of labor is constant at 5, i.e. Q(L) = 5L. Find the optimal level of
employment for the firm, the optimal quantity produced, and the price in the market.
3.1.1
Solution
Since the marginal productivity of labor is constant at 5, the only variation in the marginal revenue
product of labor will be marginal revenue. To find marginal revenue, we first need to construct the
revenue function, which is price times quantity, or (50 − 2Q) ∗ Q. The marginal revenue function
is then
d (50 − 2Q) ∗ Q)
d (50Q − 2Q2 )
=
dQ
dQ
= 50 − 4Q
(4)
Equation (4) times the marginal product of labor gives the marginal revenue product of labor. Since
we know the marginal product of labor
dQ
dL
equals 5 (as Q(L) = 5L), M RPL = 250 − 100L.
3
Note that in the case of perfect competition, firms face a perfectly elastic demand curve, which implies that
1
η = ∞. This means that for perfect competition η1 = ∞
= 0 and (3) is back to (2).
3
The marginal cost of labor is the derivative of the cost function with respect to L, or d(50L)
= 50,
dL
which is the same as saying the wage in the labor market is 50. Setting marginal cost equal to
marginal revenue product then gives
250 − 100L = 50
200 = 100L
2=L
Total production is Q(2) = 5 ∗ 2 = 10, and price is given by P (10) = 50 − 2(10) = 30.
4
The Monopsonist Labor Decision
Since a monopsonist is the only buyer in a market, they see the entire supply curve when making
their purchasing decisions. If they are a monopsonist in the labor market, the more people they hire,
the higher the wage. Assuming they are a non-discriminating monopsonist, hiring one additional
worker means they have to pay all workers employed the now higher wage. The marginal cost of
labor is now a function of the quantity hired, or M CL ≡ M CL (L).
Again going back to equation (1), now M R = P because the firm takes the price of its good
as given, but (assuming labor is the only factor of production)
dCosts
dL
d[w(L) ∗ L]
=
dL
dw
= w(L) +
L
dL dw L
=w 1+
dL w
M CLmonopsonist =
where
dw
dL
> 0 (due to an upward sloping labor supply curve4 ) and
4
(5)
L
w
> 0, so
Under perfect competition, the firm’s labor hiring decision cannot affect the wage, so
L
simplifies to w 1 + 0 ∗ w
, and we’re back to the usual competitive result of M CL = w.
4
dw
dL
dw L
dL w
> 0 and
= 0. That means (5)
w 1+
dw L
dL w
> w. Since each worker costs more for a monopsonist to hire than it would for
a perfectly competitive firm, for a given price monopsonists hire less labor than firms in perfect
competition.
4.1
Monopsonist Example
Let a non-discriminating firm face a perfectly elastic demand curve with a price of $5 (note that
saying their demand curve is perfectly elastic is equivalent to saying they face a fixed price that
doesn’t change with production). Their marginal productivity of labor is constant at 10 (Q(L) =
10L), and labor is the only factor of production. They face a labor supply curve equal to Ls =
8w − 32. Find the optimal amount of labor hired and the wage in the market.
4.1.1
Solution
The marginal revenue product of the monopsonist is constant here, since M R = P = 5 and
M PL = 10. But their marginal cost of labor is changing since with each person they hire, they
need to pay all workers a higher wage. To find M CL , we first need to construct the cost curve, and
then take the derivative of that with respect to labor.
The cost curve is the amount of labor hired times the wage, or wL. The wage is given by the
labor supply function, which we can mess with to get w.
L = 8w − 32
8w = L + 32
w=
L
+4
8
5
(6)
which means the cost function C(L) and the marginal cost function M CL are
L
+ 4) ∗ L
8
dC(L))
M CL (L) =
dL
L
= +4
4
C(L) = (
(7)
We can now set marginal revenue product of labor equal to marginal cost of labor (7) and solve for
L.
M RPL = M CL
M PL ∗ M R = M CL
10 ∗ 5 =
L
+4
4
46 ∗ 4 = L → L = 184
Plugging that in to (6) gives w =
184
8
+ 4 = 27. We can also get quantity produced, since
Q(L) = 10L → Q(184) = 1840. We don’t need to calculate price, since that is taken as fixed an
is independent of labor hired/quantity produced.
A-1
Appendix - Proofs Assuming Decreasing M PL
No need to worry about learning and memorizing the stuff in this section . . . this is just for those
people that like to have mathematical intuition with their results. Again, I repeat, do not spend
your time trying to memorize this. That being said, for those of you that are still reading, I’ll
continue.
If we assume decreasing marginal productivity of labor, we can prove mathematically that
monopolists and monopsonists hire less labor than competitive firms (under the same wages and
prices, respectively).
6
A-1.1
Monopolist
Given some wage w, let’s compare the first order conditions for competitive firm labor demand
and the monopolist labor demand. For both, at the optimal L equation (1) holds. But for the
monopolist, the left hand side of (1) is M PL ∗ P 1 + η1 while for the perfectly competitive firm
it is M PL ∗ P . This implies
1
∗P 1+
= M PLcompetitive ∗ P
η
1
monopolist
= M PLcompetitive
M PL
∗ 1+
η
M PLmonopolist
Since 1 + η1 < 1, it must be that M PLmonopolist > M PLcompetitive , which implies that for the
optimal L, Lmonopolist < Lcompetitive (remember marginal product of labor is decreasing in L).
A-1.2
Monopsonist
Given some price P , using the FOC for the monopsonist and the fact that M RPLcompetitive = w
M RPLmonopsonist
dw L
=w 1+
dL w
M PLmonopsonist ∗ P
→
=w
L
1 + dw
dL w
M PLmonopsonist ∗ P
= M PLcompetitive ∗ P by (2)
dw L
1 + dL w
dw L
monopsonist
competitive
M PL
= M PL
1+
dL w
Since 1 +
dw L
dL w
> 1, it must be that at the optimal amount of labor, M PLmonopsonist > M PLcompetitive .
7