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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 . . .

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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.


Quantity of

output

(widgets/day)

0

50

90

120

140

150

Quantity of

labor

(workers/day)

0

1

2

3

4

5

For a factory of given size . . .

Marginal product

of labor (MPL)

(widgets/worker)

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.


Marginal product

of labor (MPL)

(widgets/worker)

50

40

30

20

10

Quantity of

output

(widgets/day)

0

50

90

120

140

150

Quantity of

labor

(workers/day)

0

1

2

3

4

5

Value of the

marginal product

of labor (VMPL)

($/worker)

100

80

60

40

20


($/worker)

VMPL

(workers/day)

Graphing VMPL as a function of employment:

VMPL slopes down . . .

. . . because MPL slopes down . . .

. . . because of diminishing marginal product.


($/worker)

w

VMPL

L1

(workers/day)

Determining the profit-maximizing employment level:

Suppose that the firm

faces a wage of w.

Hiring one more worker adds more to revenue

than to cost.

At L1, VMPL > w.

Increase employment to increase profit.


($/worker)

VMPL

L2

(workers/day)

Determining the profit-maximizing employment level:

Suppose that the firm

faces a wage of w.

w

L1

Lay-off one worker: Cost savings is greater

than lost revenue.

At L2, VMPL < w.

Decrease employment to increase profit.


($/worker)

VMPL

L2

L*

(workers/day)

Determining the profit-maximizing employment level:

Suppose that the firm

faces a wage of w.

w

L1

To maximize profit, hire workers up to the point where

VMPL = w . . . at L*.


($/worker)

w

w

w

VMPL

L*

L*

L*

(workers/day)

employment adjusts along VMPL

As wage increases or decreases . . .

Determining the profit-maximizing employment level:

To maximize profit, hire workers up to the point where

VMPL = w . . . at 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 of this promise?

X (1 + r)2 = 100.

Solving:

100

X =

(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.”


FV of this promise?t = PV (1 + r)t

FVt

PV =

(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: of this promise?

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).


100 of this promise?

100

50

+

+

1 + 0.05

(1 + 0.05)2

(1 + 0.05)3

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.)

= $229.13

PV =

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. of this promise?

(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.


1000 of this promise?

1000

1000

+ . . .

+

+

(1 + 0.151)2

(1 + 0.151)3

(1 + 0.151)

1000

1000

. . . +

+

= 5,000

(1 + 0.151)9

(1 + 0.151)10

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:


Another PV example: of this promise?

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” of this promise?

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 . . . of this promise?

. . . 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 of this promise?

. . . 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.


1000 of this promise?

1000

1000

+

+

+ . . .

(1 + r)

(1 + r)2

(1 + r)3

1000

1500

1500

. . . +

+

+

+ . . .

(1 + r)32

(1 + r)33

(1 + r)34

1500

1500

. . . +

+

(1 + r)57

(1 + r)58

Wentworth implicitly charged an interest rate equal to

the value of r that satisfies:

= 37,500

Solution: r = 0.0224 (2.24%)

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/. . .)


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