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Roadmap:
Up to now, we’ve focused primarily on markets for
goods and services.
In chapter 18, we look at markets for . . .
Factors of production:
The inputs used to produce goods and services
including . . .
. . . land, labor, raw materials, and . . .
Capital: the equipment and structures used to produce
goods and services.
“Physical capital” vs. “financial capital”
In factor markets (labor markets, for example):
Households are suppliers; firms are demanders.
“Derived demand:” Factors of production are
demanded for their usefulness in producing output
and generating profit.
Let’s consider the (short-run) demand for labor . . .
. . . borrowing a numerical example from chapter 13.
For a factory of given size . . .
Quantity of Marginal product
Quantity of
output
of labor (MPL)
labor
(workers/day) (widgets/day) (widgets/worker)
0
0
1
50
2
90
3
120
4
140
5
150
50
40
30
20
10
Marginal product of labor (MPL): the increase in output
that arises from an additional unit of labor, holding
other input(s) fixed.
Diminishing marginal product of labor: the property of a
production function whereby the marginal product of
labor declines as the quantity of labor increases.
Assume that the firm is a price taker in both its output
(widget) and labor markets.
Value of the marginal product of labor (VMPL): the
marginal product of labor times the price of the
output.
VMPL = MPL x p
(where p is the price of a widget)
Let’s assume that p = $2/widget . . .
. . . and return to the example.
Value of the
Quantity of Marginal product marginal product
Quantity of
of labor (VMPL)
output
of labor (MPL)
labor
($/worker)
(workers/day) (widgets/day) (widgets/worker)
0
0
1
50
2
90
3
120
4
140
5
150
50
100
40
80
30
60
20
40
10
20
Graphing VMPL as a function of employment:
($/worker)
VMPL
(workers/day)
VMPL slopes down . . .
. . . because MPL slopes down . . .
. . . because of diminishing marginal product.
Determining the profit-maximizing employment level:
($/worker)
Suppose that the firm
faces a wage of w.
w
VMPL
L1
At L1, VMPL > w.
(workers/day)
Hiring one more worker adds more to revenue
than to cost.
Increase employment to increase profit.
Determining the profit-maximizing employment level:
($/worker)
Suppose that the firm
faces a wage of w.
w
VMPL
L1
At L2, VMPL < w.
L2
(workers/day)
Lay-off one worker: Cost savings is greater
than lost revenue.
Decrease employment to increase profit.
Determining the profit-maximizing employment level:
($/worker)
Suppose that the firm
faces a wage of w.
w
VMPL
L1
L2
(workers/day)
L*
To maximize profit, hire workers up to the point where
VMPL = w . . . at L*.
Determining the profit-maximizing employment level:
($/worker)
w
w
w
VMPL
L*
(workers/day)
L*
To maximize profit, hire workers up to the point where
VMPL = w . . . at L*.
As wage increases or decreases . . . employment adjusts along VMPL
L*
VMPL gives the firm’s (short-run) demand curve for labor.
Recap:
A firm uses “factory” (fixed input) and “labor”
(variable input) to produce output.
The firm is a price-taker in output and labor mkts.
Back in chapter 14: Profit-maximizing output level
determined by P = MC
Now: Profit-maximizing employment of labor
determined by w = VMPL
But these aren’t two different problems . . .
. . . just two different views of the same problem.
(Deciding on output determines the labor input
required. Deciding on labor employment determines
how much output you get.)
Can we reconcile the two profit-max rules?
Profit-maximizing output level characterized by:
p = MC
=
TC
Q
Consider changes resulting from addition of 1 more worker.
=
=
w
Q from 1 more worker
w
MPL
Multiplying both sides by MPL, we have:
p x MPL = w
VMPL = w
. . . the equation that characterizes the
profit-maximizing labor employment level.
So the two conditions (p = MC and VMPL = w) are
really the same condition.
Markets for capital
One key feature of capital:
It’s “durable.”
That is, it can deliver a stream of productive
services over time.
So really two markets for capital:
Market for the ownership of capital:
Determines “purchase price” -- price of title to a
unit of capital.
Market for services of capital:
Determines “rental price” -- price of the right to
use a unit of capital for a limited time.
Ownership of capital entitles the owner to a stream of
rental payments over time.
So the purchase price is related to WTP (today) for a
stream of (future) payments.
“Present value of a future sum”
(As we’ll see: lots of applications -not just capital markets.)
Suppose I promise to give you $100 1 year from today.
(And assume I’m trustworthy!)
How much would you be willing to pay (today) in
exchange for my promise?
Less than $100! . . .
. . . because you can invest at a positive interest rate.
If you were to invest $ X at interest rate r, at the end of
1 year you would have (in $):
X + r X = X (1 + r).
For example: If X = $1000 and r = 0.05 (5%), at the
end of 1 year you would have $1050.
The “present value” of $100 1 year from now is the
amount you would have to invest today to have $100
in one year.
In other words, it’s the value of X such that
X (1 + r) = 100.
Solving:
X = 100/(1 + r).
Example: With r = 0.05, the present value of $100 1
year from now is $95.24.
Coming back to the original question:
The amount you’d be willing to pay (today) for $100
in 1 year is the present value ($95.24, if r = 5%).
Why?
If you give me $95.24 today . . .
. . . or if you invest $95.24 today at 5% . . .
. . . either way, you come out the same:
$100 in your pocket 1 year from today.
Now suppose I promise to give you $100 2 years from
now. PV of this promise?
If you were to invest $ X at interest rate r, at the end of
2 years you would have . . .
X (1 + r)(1 + r) = X (1 + r)2.
Note: This is NOT(!) the same as X (1 + 2r).
“compound interest”: interest on interest
http://en.wiki . . . /Compound_interest
PV of $100 2 years from now is the value of X such that
X (1 + r)2 = 100.
Solving:
X =
100
(1 + r)2
With r = 0.05, the PV of $100 2 years from now is $90.70.
Now generalize: Let PV denote “present value.”
Let FVt denote a “future value t years from now.”
FVt = PV (1 + r)t
PV =
FVt
(1 + r)t
Important note: Throughout these examples -to keep things simple -- we’re assuming no
intra-year compounding. Interest is paid or is
owed just once, at the end of the year.
(http://www.studyfinance.com/lessons . . .)
A simple capital pricing example:
A machine will generate rental income payments of
$100 -- 1 year from today,
$100 -- 2 years from today, and
$50 -- 3 years from today . . .
. . . then it will break down (no “salvage” value).
My WTP for title to the machine would be PV of future
stream of rental receipts.
Assume r = 0.05 (5%). (This would be determined by
alternative investment opportunities.)
PV =
100
1 + 0.05
+
100
(1 + 0.05)2
+
50
(1 + 0.05)3
= $229.13
Another way of thinking about this result:
If I were to deposit $229.13 in a 5% savings account
today, I could withdraw $100 in 1 year . . .
. . . another $100 in 2 years . . .
. . . and have a remaining balance of $50 in 3 years.
Another PV example: Installment loans.
(like home mortgages, auto loans, etc.)
Borrower gets lump sum loan amount up front . . .
. . . repays principal and interest over time in a
series of scheduled installment payments.
The interest rate associated with an installment loan is
whatever interest rate makes the PV of installment
payments equal to the loan amount.
Example: Suppose I borrow $5,000 . . .
. . . and terms of loan require 10 installment
payments of $1000 each due 1 year
from today, 2 years from today, etc.
This loan has an interest rate of 15.1% because:
1000
(1 + 0.151)
... +
1000
+
+
(1 + 0.151)2
1000
(1 + 0.151)9
+
1000
+ ...
(1 + 0.151)3
1000
(1 + 0.151)10
= 5,000
Another PV example:
Selling structured settlement for (“up-front”) lump-sum
cash payment.
“Structured settlements:” a stream of guaranteed
future cash payments.
Lottery winners; accident victims who are
successful plaintiffs in personal injury
lawsuits.
Sales of structured settlements sometimes called
“factoring transactions.”
“Factoring companies”
The biggest: J. G. Wentworth, Philadelphia
(http://www.jgwfunding.com/)
“It’s my money and I need it now!”
(http://www.jgwentworth . . .)
Factoring transaction is similar to installment loan:
“Borrower” (accident victim) gets money up-front.
Repays “loan” amount plus interest to “lender”
(Wentworth) by signing-over structured
settlement payments.
Some lawmakers believe that factoring companies . . .
. . . take advantage of desperate people and
. . . charge interest rates that are “too high.”
Sound familiar?
Similar concerns raised over “car title loans”:
Small emergency loans secured by a car title (lender
keeps set of keys), often charging high interest rates,
and usually requiring payment within one month.
(http://www.cnn.com/ . . .)
Iowa law (signed by Gov. Culver, March 27, 2007)
caps car title loan rates at 21 – 36%.
The interest rate implicit in a factoring transaction is
. . . the interest rate for which the PV of structured
settlement equals up-front lump-sum payment.
Example (from U.S. News and World Report, “Settling
for Less,” 1/25/99): Chris Hicks (20-year-old
accident victim from Oklahoma).
Structured settlement: $1000/month for 32 months
then, after that: $1500/month for 26 months.
(total = $71,000)
Sold to Wentworth for $37,500.
Wentworth implicitly charged an interest rate equal to
the value of r that satisfies:
1000
(1 + r)
. . . +
+
1000
(1 + r)2
1000
(1 + r)32
. . . +
+
1000
+
(1 + r)3
1500
(1 + r)33
1500
(1 + r)57
+
+
+ . . .
1500
(1 + r)34
1500
(1 + r)58
Solution: r = 0.0224 (2.24%)
+ . . .
= 37,500
Monthly!!
Equivalent to annual interest rate roughly 12 x as big – close to 30%!
(Car title loan rates up to 360%!! http://www.state.ia.us/. . .)