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QUESTION 1

Consider a project that has a positive NPV at the firm’s discount rate.

Which of the following statements is true?

.

.

None of these are necessarily true.

.

.

The internal rate of return (IRR) is smaller than

the firm’s discount rate.

.

.

The internal rate of return (IRR) is larger than

the firm’s discount rate.

.

.

The internal rate of return (IRR) is equal to the

firm’s discount rate.

QUESTION 2

Consider a project that requires an upfront payment of $150. Suppose

the project is expected to generate cash flows of $70 one year from

now, and $90 two year from now. If the firm’s discount rate is 7%, which

of the following statements is true?

.

.

This project should not be undertaken.

.

.

The firm should pass on this project only if there

is a project with a higher NPV.

.

.

None of these are true.

.

.

This project has a positive NPV.

QUESTION 3

If an innovation is likely to generate $2,000 three years in the future,

and the interest rate is 5%, the present value of that flow of profits is

best approximated by:

.

.

$2,

315

.

.

$1,

728

.

.

$1,

905

.

.

$1,

700

QUESTION 4

When capital is scarce, which of the following tools is most effective for

determining which competing R&D projects should receive funding?

.

.

The Payback

Period

.

.

The IRR

.

.

Comparison of

NPV

.

.

The BC Ratio

QUESTION 5

This question and the following relate to the following information.

Suppose a jar contains 7 blue jellybeans and 3 red jellybeans. What is

the probability that a random draw of a jellybean is red? (Type your

answer in decimal form, e.g. “0.5”.)

QUESTION 6

Using the information from the problem above, what is the probability

that the first jellybean is blue?

QUESTION 7

A roulette wheel has 38 numbers: 0, 00, and numbers 136. Let X denote the number on which the ball lands. The relationship

between X and the prize money, P, is given by the following table:

X

0

00

Odd

number

Even

number

P

$

5

$

1

0

$

2

$

4

The prize money, P, is a random variable. How much does the casino

need to charge each gambler (per spin of the wheel) to break-even over

time? (Do not include dollar signs, and use three decimal places.)

QUESTION 8

Y 0 1

f( 0 0

y . .

) 8 2

Consider the following distribution of motorcycles per family, :

What is E(Y)?

QUESTION 9

This question and the next relate to the following probability distribution

for random variable X, which is the number of cars a family owns:

X 0 1 2

f( 0 0

x . . ?

) 2 4

What is the (missing) probability of a family owning 2 cars, f(2) Assume

no family owns more than 2 cars.

QUESTION 10

Using the information from the above question, what is E(X)?

QUESTION 11

A firms marginal cost function is

given by:

.

.

q*=9

0

.

.

q*=7

0

.

.

q*=1

00/3

.

.

q*=7

5

If the future price of the firm’s output is uncertain, and the firm expects

that the price equals $35 or $45 each with a probability of 1/4, and $60

with probability 1/2, what is the optimal level of production for this firm

in the presence of the uncertainty?

QUESTION 12

There is a 0.10 probability that you have profits of 30, a 0.30 probability

that you have profits of 40, and your profits are 100 otherwise. What are

your expected profits?

.

.

6

0

.

.

3

4

.

.

1

0

0

.

.

7

5

Homework 4

ECON 431, Fall 2021

Instructions: Show all of your work!!!! Then scan to a pdf, or if necessary take pictures of your work

(scanning to pdf preferred). Upload your complete assignment using the Blackboard link provided by

midnight, Friday 11/7. Please submit pdf, Word, and jpeg files only!

1. Suppose the number of potential adopters of a new technology is N=21, and ?=0.07.

Assuming a Central Source Model, calculate the number of adopters of the new technology

for t=0, 1, 2,
, 10 and fill out the table below. Assume that the central source is one of

the 21 adopters such that D(0)=1.

t

?D/?t

0

D(t)

N?D

1.00

1

2

3

4

5

6

7

8

9

10

2. Now assume an Epidemic Model with N=21, and ?=0.07. Calculate the number of

adopters of the new technology for t=0, 1, 2,
, 10 and fill out the table below. Assume

that D(0)=1.

t

0

1

2

3

4

5

6

7

8

9

10

?D/?t

D(t)

1.00

N?D

3. How does this diffusion curve in problem 1 differ from that in problem 2 above?

Construct a graph of each with t on the x-axis and D(t) on the y-axis to facilitate this

discussion.

4. A model of network externalities. Suppose that there are 50 potential consumers in the

market for a new technology that exhibits network effects. There is a uniform distribution

of consumers with individual valuations, v, ranging from $1, $2,
, $50. Consumer

valuation from consuming the technology is given by vN, where N is the number of

consumers adopting the technology.

Consumers with purchase the product as long as their valuation is greater or equal to the

price, so that the marginal consumer has a valuation such that p=vN. The number of

consumers adopting the technology is given by the number of people with valuation

greater than v, i.e. N = 50 v.

a) Using the information above, derive the relationship between the price of the product and the

number of consumers adopting the product, N. Characterize this relationship does it reflect

a typical market demand curve?

b) If the price for the product is $600, find the three equilibrium number of adopters in the

market.

c) Graph this market, with the price of the product on the vertical axis and the number of

adopters on the horizontal axis.

d) Which of these equilibria are stable, and which are unstable? Explain.

5. Consider the two innovations you identified in Problem 3 on Homework 1. For each of

those innovations, address the following:

a) Consider the firm(s) that is (are) most successful for the innovations you identified. Were they

first- or later-movers in that innovation market? If they werent fist-movers, who was?

b) If the successful firms werent first-movers, why did the first-movers fail? What advantages do

you think the successful later firms had?

c) If the successful firms were first-movers, what advantages did they gain from entering first?

More specifically, did they benefit from any increasing returns generated by information

cascades or network effects?

Economics of Innovation and

Intellectual Property

CSU East Bay

Lecture 13: Decision-Making Under Uncertainty

Lecture Outline

Basics in Probability

Expected Value

Extending the Firms Framework

First
Some Motivation

Why study uncertainty?

It is pervasive

We take actions to address uncertainty and the resulting risk:

o State contingent consumption bundles

o Insurance markets

It forms the basis of many interesting areas of economic inquiry:

o Principal-agent problems, i.e. information economics

o Value of information

o Auction theory

o Environmental outcomes and optimal environmental policy

o Economics of crime

3

Probability

A random variable, ????, takes possible values, ???????? , as outcomes of a random

phenomena.

Can be discrete or continuous

The rolling of a dice
the flipping of a coin
drawing numbers from a

hat
measuring heights or speeds of a car.

The probability of ???? taking a specific value, ???????? , denoted by ???????? ???? = ???????? , or

simply ????????(???????? ), is the chance that the outcome ???????? occurs.

Well define the probability of an event, A, as:

???????? ???? =

???????????????????????? ???????? ???????????????????????????????? ???????? ?????????????? ???? ????????????????????????

???????????????????? ???????????????????????? ???????? ???????????????????????????????? ????????????????????????????????

4

Probability Basics

(i) Probabilities are bounded by 0 and 1: 0 ? ????????(???????? ) ? 1

If an event never occurs, then ???????? ???????? = 0

If an event occurs with certainty, then ???????? ???????? = 1

(ii) If we have a collection of collectively exhaustive and mutually exclusive

possible outcomes of ????, then: ?????????=1 ???????? ???????? = 1

A set is collectively exhaustive if all possible outcomes of the variable ???? are

included.

A set is mutually exclusive if only one outcome occurs for each realization of ????.

In other words, one and only one outcome actually occurs!

5

Example: Dice Rolling

Consider the rolling of a fair, six-sided die.

What is the probability that you roll a 3?

There are 6 possible outcomes, and one of these is a 3, so:

???????? ???? = 3 =

1

6

What is the probability that you roll a number greater than a 3?

There are 6 possible outcomes, and three of them are greater than a 3, so:

3

1

???????? ???? > 3 = 6 = 2

6

Example: Dice Rolling

Consider now, the rolling of a weighted six-sided die!

It rolls a 6 half of the time and other possible values with equal probability.

What is the probability that you roll a 1?

Well, we know the sum of the probabilities has to equal 1.

1

You get a 6 with probability 2 and the probability you get a 1 is the same as

getting a 2, 3, 4, or 5! So:

???????? ???? = 1 = ???????? ???? = 2 = ???????? ???? = 3 = ???????? ???? = 4 = ???????? ???? = 5

? 5 ? ???????? ???? = 1 + ???????? ???? = 6 = 1

1 1

1

? ???????? ???? = 1 = ? =

2 5 10

7

Expected Value

We use the probability distribution the listing of the probabilities of all

possible distinct outcomes of the random variable to calculate the

expected value of the random variable:

The expected value is a weighted average, where the value of each

outcome is weighted by its probability of occurring:

????

???? ???? = ???????? ????1 ????1 + ???????? ????2 ????2 + ? + ???????? ???????? ???????? = ? ???????? ???????? ????????

????=1

8

Example 1: Expected Value

Calculate the expected value of the roll of a fair die.

???? ???? = ???????? 1 ? 1 + ???????? 2 ? 2 + ? + ???????? 6 ? 6

where each of the probabilities is equal to one-sixth

1

1

1

? ???? ???? = ? 1 + ? 2 + ? + ? 6

6

6

6

1

???? ???? =

1 + 2 + 3 + 4 + 5 + 6 = 3.5

6

What is the expected value of the unfair die in the previous slide?

Answer: 4.5

9

Recall Profit-Maximizing Output of the Firm

We can think through the problem using our economic decision-making

rule: compare the marginal benefits (????????) with the marginal costs (????????)!

We are deciding to produce another unit or not, so compare the

marginal benefits and marginal costs of an additional unit of production.

The marginal benefit is the marginal revenue received, which is the

market price (under perfect competition)! ???????? = ???????? = ????

The marginal cost is the cost of additional resources needed to produce

the extra unit!

At the optimum level of production, ???????? ???????? ? = ???? = ???????? ???????? ?

This is our profit-maximization condition!

10

Graphically

E

Q*

11

Expected Profits

When maximizing profit in the presence of uncertainty, we should choose

the action that yields the greatest expected profits!

The random variable here is profits, ????.

???????????????????????? = ???? = ???????????????????? ???????????????????????????? ? ???????????????????? ???????????????? = ???? ? ???? ? ????(????)

???? ???? = ???????? ????1 ????1 + ???????? ????2 ????2

more generally
.

????

???? ???? = ? ???????? ???????? ????????

????=1

12

Expected Profits

Given the information available, in the presence of uncertainty the best we can

do is to maximize your expected profits!

Fundamentally concerned with uncertainty about some parameter we are

facing:

o Uncertain market prices due to uncertain economic conditions

o Uncertain costs

o Uncertain regulations and other political risks

????

max ???? ???? = ? ???????? ???????? ????(????)????

????

????=1

We arrive at similar conditions for profit-maximizing decisions:

???? ???????? = ???? ????????

13

Example: An Illustrative Example

Consider a firm trying to decide how much to produce. However, there is

uncertainty about the market conditions when the product is brought to

market one of two market prices will prevail!

The price is expected to be relatively low, ???? = 4, with probability ¼.

Otherwise, the price is expected to be relatively high, ???? = 20.

The cost function is ???? ???? = 2????2 such that marginal costs (MC) are:

???????? ???? = 4????

What level of output should the firm choose given the uncertainty?

14

Example Continued

???? ???? = ???????? ???? = 4 ? ????4 + ???????? ???? = 20 ? ????20

1

3

2

max ???? ???? = ? 4???? ? 2???? + ? 20???? ? 2????2

????

4

4

1 ? ???? + 15 ? 3???? = 0

or: E(MR)=E(MC)

1 + 15 = 4????

????? = 4

15

Expected Profits

The quantity that maximizes expected profits will always yield profits less

than under certainty youll always wish you had chosen a higher or

lower quantity!

If ????1 maximizes profits when price is ????1 , and ????2 maximizes profits when

price is ????2 :

o ???? ????1 , ???? ? < ???? ????1 , ????1
o ???? ????2 , ???? ? < ???? ????2 , ????2
Decisions will not be perfect in hindsight!
Using expected profits does maximize profits if the decision is made many
times, i.e. it maximizes profits in the long run!
16
Example 3: Maximizing Expected Profits
????
Your perfectly competitive firms cost function is given by ???? ???? = ???????? .
????
You need to decide how much to produce today, but you do not know
for sure the market price that will prevail tomorrow when you go to sell
your output.
With probability 1/3 the price will be $30, otherwise the price is $45.
????
Note that, given this cost function, we have ???????? ???? = ????
????
Given this uncertainty, what is the profit-maximizing level of output for
your firm today? What are the expected profits?
17
Example 2 Continued
???? ???? = ???????? ???? = 30 ? ????30 + ???????? ???? = 45 ? ????45
1
1 2
2
1 2
max ???? ???? = ? 30???? ? ???? + ? 45???? ? ????
????
3
4
3
4
1
2
10 ? ???? + 30 ? ???? = 0
6
6
or: E(MR)=E(MC)
1
10 + 30 = ????
2
???? ? = 80
1
???? ???? = 40 ? 80 ? ? 802 = 1600
4
18
When Should a Firm Innovate?
What should be the most that the firm is willing to pay for an innovation?...
The expected gain in profit!
o May be uncertainty
o May be multiple time periods
From the perspective of marginal analysis, innovate if:
?
?
????=0
????=0
? ???? ???????????????? ???????? ???????????????????????????? ? ? ???? ???????????????????? ???????? ????????????????????????????????????????????
In such a context, we can extend the decision-making over time framework:
???? ???????????? = ???? ???????? ???????? ???????????????????????????????? ? ????(???????? ???????? ????????????????????)
Expected Profits
Decision-making based upon expected profits requires a complete and full
accounting of (a) the possible outcomes of an uncertain scenario and (b)
the probabilities of each possible outcome occurring!
Careful assessment of the probabilities of potential outcomes is important,
and often a key set of assumptions that requires sensitivity analysis.
All too often, we dont see all possible outcomes, and/or are too vague or
optimistic about our assessments of probabilities.
The framework allows for analyzing more complex, layered uncertainties,
by looking at stages of the possible outcomes!
The decision tree and concept of conditional probabilities is incredibly
useful here!
20
Question 4
O out of 1 points
When capital is scarce, which of the following tools is most effective for determining which competing R&D projects should receive funding?
Selected Answer:
The Payback Period
Question 10
O out of 1 points
Using the information from the above question, what is E(X)?
Selected Answer:
0.4
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Explanation & Answer:
20 Questions
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