Solved by verified expert:here is some course material you may need to complete the assignment. the assignment is attached. please let me know if you have any questions. this is a quantitative method assignment

assignment.pdf

deck.pdf

random_sampling.pdf

2.pdf

Unformatted Attachment Preview

Management 3: Quantitative Methods in Business

Session 9 Assignment (20 points)

This is a simulation exercise in R. Because you’ll be creating simulated data in this exercise, there is no data file to

download/import from TED.

Case Study: The Marquez Family Farm

Abraham Marquez runs a family-owned orange grove in Santa Clarita, California. Recently, he’s been considering establishing a

presence at a farmer’s market; in particular, the Leucadia farmer’s market. Before committing his time, energy, and money to

setting up a booth, he would like to crunch the numbers and estimate how likely it is that the venture would be a profitable one.

The market is held once a week. It’s a pretty long trek from Santa Clarita to Leucadia; Google Maps says the round trip is about

250 miles. Abe considers his truck’s gas mileage and estimates that he would need to buy about 25 gallons of gas each week to

make the round trip. Given how the price of gas at his local gas station fluctuates, Abe estimates that his weekly gas expense

would be normally distributed with a mean of $88.75 and a standard deviation of $3.

As much faith as Abe has in the superior quality of his produce, the reality is that in the eyes of most consumers, oranges are

oranges are oranges. That might change as Abe is able to develop a stable base of loyal customers, but until that time, he isn’t

confident that he’d be able to successfully command a premium price for his product. He decides he’ll need to set his price to be

comparable with those of the other fruit vendors at the market. Because produce vendors typically vary their prices from week to

week, Abe decides on the following strategy: upon arrival at the market each week, he could quickly scan the other booths to

see what other vendors are charging for oranges that day. He would then set his own price accordingly. From his experience as

a customer at similar markets, he knows that the price of oranges tends to vary a lot…their price is typically uniformly distributed

between $1.50 and $3 per pound.

Abe makes a phone call to a friend in Northern California who already sells oranges at a weekly market there. The friend reports

that sales aren’t very predictable…he says that at any given market, he typically sells anywhere from 100 to 400 pounds of

oranges (assume a uniform distribution here, as well). Given this information, Abe decides he’d just bring 400 pounds of oranges

each time, to minimize the risk of selling out. (For the sake of simplicity, we will also assume that any unsold oranges have no

resale value.)

Of course, it also costs money to harvest the fruit in the first place. Between expenses like labor, fertilizer, water, and

processing, he estimates that the total variable cost to produce 400 pounds of oranges is about $150. Last, there is the cost of

the permit required to establish a stand at the market. Fortunately, this cost is fairly small: the Leucadia market only charges

vendors $50/week to set up a booth.

Abe’s task is to sort all this information so that he can decide…is it even worth trying to set up a stand at the farmer’s

market? If he were to do this, what is his expected weekly profit (or loss)?

© Ryan Wagner, 2019. Do not copy or distribute without permission.

1. Classify the values given in the prompt into one of the categories shown below. For any value that is given as a

random variable, state its distribution and parameters. Otherwise, simply list its amount. Note: Remember that

“variable costs” refer only to those expenses that are a direct function of production level. Values that are defined

as random variables are not necessarily “variable costs”, and vice versa.

(½ point each, 2.5 points total)

a. Selling Price:

b. Supply:

c. Demand:

d. Fixed Cost(s):

e. Variable Cost(s):

2. What is the variable cost per pound of oranges? (1 point)

3. Let’s say Abe were to set his selling price at $2.50. At this price, what is the unit contribution margin? (1 point)

4. At this price, what is the minimum quantity of oranges that Abe would have to sell per week to break even? Note:

if your calculation includes a random variable, use the mean value of that variable in your calculation. (2 points)

5. Write the command to generate a set of 10,000 simulations of Abe’s weekly cost of gas, according to the

parameters you defined in Q1. Set the seed as 1, and store the simulations as an object called gas_cost.

(2 points)

6. Write the command to generate a variable called total_cost that is the sum of Abe’s estimated total weekly

costs, using any static costs previously defined, or simulated costs previously generated, as your inputs.

(1.5 points)

7. Write the command to generate a set of 10,000 simulations of the weekly price of oranges, according to the

parameters you defined in Q1. Set the seed as 38, and store the simulations as an object called price. (1 point)

8. Write the command to generate a set of 10,000 simulations of the weekly demand for oranges, according to the

parameters you defined in Q1. Set the seed as 54, and store the simulations as an object called demand.

(1 point)

© Ryan Wagner, 2019. Do not copy or distribute without permission.

9. Revenue: (1 point each, 2 points total)

a. Write the command to create an object called revenue that is calculated as the simulated weekly

revenue, using the previously generated simulations as your inputs.

b. What is the average revenue, according to the simulations? Write the command and result.

10. Profit: (1 point each, 3 points total)

a. Write the command to create an object called profit that is calculated as the simulated weekly profit,

using the previously generated simulations as your inputs.

b. According to these simulations, what is the average weekly profit? Write the command and result.

c. According to these simulations, what is the probability that on any given week, Abe will make a profit at

the market? Write the command and result.

11. Because Abe has other options for selling his produce (e.g., grocery stores, restaurants), he figures this venture is

really only worthwhile if he can reliably make a profit of at least $400 each week.

a. Write the command to generate this probability that this will occur, along with the result. (1 point)

b. Given your answer, what is your recommendation to Abe regarding the farmer’s market? (2 points)

© Ryan Wagner, 2019. Do not copy or distribute without permission.

MGT 3

Quantitative Methods

in Business

R. Wagner

Spring 2019

Session 9

This Week

• Session 8 HW: Diamond Question

• Managerial Economics

• Monte Carlo Simulation

© 2019 Ryan Wagner

Managerial Economics

© 2019 Ryan Wagner

Managerial Economics is the application of economic principles to business

decision-making. Today’s focus will be microeconomic: generally limited to

situations that occur in firms.

• Focused on allocation of resources (e.g., labor, capital, cash flow)

• Profit-driven

Setup

© 2019 Ryan Wagner

You’ve decided to turn your knack for woodworking into a livelihood! Yay!

Some immediate questions:

• What should I charge for my product?

• How much do I need to sell in order to:

• …cover my costs?

• …turn a profit?

To keep it basic, some assumptions going forward:

• One product only: same selling price/manufacturing cost per unit

• Supply = Demand (ignoring shortage/surplus)

Cash Outflows

© 2019 Ryan Wagner

Fixed Costs: any cost that is not a function of sales.

• Rent

• Utilities (e.g., electricity, water, phone, internet, AC, insurance)

• Admin Expenses (e.g., base salaries, advertising expenses, office supplies)

Variable Costs: costs that change with the amount of inventory produced.

• Raw materials (e.g., wood, glue, nails/screws, paint/veneer, shipping costs)

• Direct Labor (e.g., sales commissions, piece rate pay)

𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡 = (𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠 + 𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡𝑠)

Cash Outflows

© 2019 Ryan Wagner

Say our costs are as follows:

Fixed Costs: per time period (e.g., per month)

• Rent – $1,900

• Utilities – $950

Σ𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡 = $4,550

• Admin Expenses – $1,700

Variable Costs: per unit produced

• Raw materials – $60/unit

• Direct Labor – $30/unit

Σ𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡 = $90/𝑢𝑛𝑖𝑡

BUT. It would be not quite right to say we only spend $90 to produce each unit,

considering we still have to pay the fixed costs to keep our business running.

Unit Cost

© 2019 Ryan Wagner

Say you review your books at the end of the month, and find that you:

• spent a total of $15,000. (fixed + variable costs)

• sold a total of Q =120 units.

The total cost per unit =

$15,000

120 𝑢𝑛𝑖𝑡𝑠

= $125

We generalize this as: 𝑈𝑛𝑖𝑡 𝐶𝑜𝑠𝑡 =

𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡

𝑄

=

(𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡 + 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡)

𝑄

As Q increases, Unit Cost decreases (spreading fixed costs over greater output)

Unit Cost

© 2019 Ryan Wagner

Total Var.

Total Cost Unit Cost

Cost

Q

Fixed Cost

1

4,550

90

4,640

4,640

2

4,550

180

4,730

2,365

3

4,550

270

4,820

1,607

4

4,550

360

4,910

1,228

5

4,550

450

5,000

1,000

6

4,550

540

5,090

848

7

4,550

630

5,180

740

8

4,550

720

5,270

659

9

4,550

810

5,360

596

𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡 = $4,550 + (90 ∗ 𝑄)

Cash Inflows

© 2019 Ryan Wagner

Selling Price (P): the selling price per unit. (sticker price; what the customer pays)

Revenue (R): the total cash inflow from sales. (before subtracting costs)

𝑅 = (𝑃 ∗ 𝑄)

Example: if you charge $270/unit, and you sell 80 units: 𝑅 = $270 ∗ 80 = $21,600

Unit Contribution Margin: Selling Price less variable cost per unit.

Example: if you charge $270/unit, and each unit costs $90 to produce,

UCM = $270 − $90 = $180

At this price/cost, each unit contributes $180 towards covering fixed costs.

After recouping FC, you get to pocket the entire UCM for every additional unit produced!

Price vs. Quantity

© 2019 Ryan Wagner

Managers must navigate the trade-off between price and demand.

• The fewer units you sell, the higher the selling price has to be.

(fewer units sold = less opportunity to recoup costs)

• The higher selling price is, the lower your demand will be.

Whatever price point you land on, you’ll have to answer a critical question:

What should your sales target be? (what Q?)

We know at a bare minimum, we need to sell enough units such that our revenue at

least equals our costs. i.e., we need to break even. (cash inflow = cash outflow)

Breakeven Quantity

© 2019 Ryan Wagner

Breakeven occurs at the quantity for which cash inflow = cash outflow.

𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = 𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡

→ 𝑃𝑟𝑖𝑐𝑒 ∗ 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = 𝑇𝑜𝑡𝑎𝑙 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠 + 𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡𝑠

→ 𝑃𝑟𝑖𝑐𝑒 ∗ 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠 + (𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑈𝑛𝑖𝑡 ∗ 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦)

→ 𝑃𝑟𝑖𝑐𝑒 ∗ 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 − 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑈𝑛𝑖𝑡 ∗ 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 = 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠

→ 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 ∗ 𝑃𝑟𝑖𝑐𝑒 − 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑈𝑛𝑖𝑡 = 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠

→ 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 ∗ 𝑈𝑛𝑖𝑡 𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑀𝑎𝑟𝑔𝑖𝑛 = 𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠

𝐵𝑟𝑒𝑎𝑘𝑒𝑣𝑒𝑛 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 =

𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠

(𝑈𝑛𝑖𝑡 𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑀𝑎𝑟𝑔𝑖𝑛)

Breakeven Quantity

© 2019 Ryan Wagner

Using the values from our previous example:

FC

4,550

4,550

BEQ =

=

=

= 25.28

UCM 270 − 90

180

Selling Price = $270

Fixed Costs = $4,550

Var. Cost per Unit = $90

Units are discrete; can’t sell 0.28 of a chair. Look at contribution on either side:

Q = 25: $180 ∗ 25 = $4,500

Q = 26: $180 ∗ 26 = $4,680

just shy of covering fixed costs.

As with sampling and experimental design, any answer with decimals needs to

be rounded up.

Breakeven Quantity

𝐵𝐸𝑄 =

© 2019 Ryan Wagner

𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠

(𝑈𝑛𝑖𝑡 𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑀𝑎𝑟𝑔𝑖𝑛)

Say you start worrying a price point of $270 feels too high…you’re considering

lowering it to $240. All costs stay the same. Solve the BEQ for this price.

1.

Fixed/variable cost per unit stay the same.

2. Solve new UCM = (P – VC/unit)

3. Solve BEQ.

4,550

4,550

𝐵𝐸𝑄 =

=

= 30.33 ≈ 31 𝑢𝑛𝑖𝑡𝑠

240 − 90

150

‘What-If’ Scenarios

© 2019 Ryan Wagner

It would be helpful to quickly see the relationship between selling price and BEQ

for this (and any) price point we might consider.

In R:

fixed_cost <- 4550
vc_per_unit <- 90
price <- 90:500
ucm <- (price – vc_per_unit)
beq <- (fixed_cost / ucm)
seq() Command
© 2019 Ryan Wagner
price <- 90:500 increments by 1…prob. too granular for our needs
(in reality, would likely round price…wouldn’t choose $91 instead of $90).
seq() is a simple, useful function for creating a series that increments by a value
of your choosing. It takes three parameters:
from: starting value
to: ending value
by: value to increment by
ex: seq(from=90, to=500, by=10) (now assign output to price)
Exploring Price/BEQ
© 2019 Ryan Wagner
df <- data.frame(price, beq)
plot(df)
Let’s say we think that in our first months, selling more than 20 units is
unrealistic. We’d like to look at the required prices where BEQ < 20.
Use subset() to return that portion of the data.
subset(df, beq < 20)
Pro tip: add [1,] at the end to return first row where beq < 20
subset(df, beq < 20)[1,]
Some Thoughts on Pricing
© 2019 Ryan Wagner
No deterministic formula for pricing. Pricing strategy involves a nuanced set of
considerations…
Costs
• At a bare minimum, selling price has to be higher than unit cost.
• How many units do I need to sell to keep the lights on? (Breakeven Quantity?)
Value Perception:
• What is the perceived value of my product in the marketplace?
Competition:
• What are my competitors charging for similar products?
• Is my product different enough (in a good way) that I can command a premium?
Elasticity:
• How sensitive are my (existing/potential) customers to changes in prices?
Example: App Development
© 2019 Ryan Wagner
Let’s say you release an app on the Apple store. Identify your fixed costs, variable
costs, and selling price.
•
You keep a developer on retainer for a base monthly salary of $1,000. Fixed cost
•
You also agreed to pay her $0.10 for every download. Variable cost
•
You budget $400/month for targeted social media ads. Fixed cost
•
Users pay a one-time fee of $1.99 to purchase the app. Selling price
•
Apple charges of a fee of 30% of each sale. Variable cost
•
You rented a WeWork studio for you and your developer for $350/month. Fixed cost
Solve the Unit Contribution Margin and BEQ for this scenario.
Example: App Development
© 2019 Ryan Wagner
Unit Contribution Margin = (𝑆𝑒𝑙𝑙𝑖𝑛𝑔 𝑃𝑟𝑖𝑐𝑒 − 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡)
• Selling Price: $1.99
• Variable Costs:
•
•
•
App store: 0.30 ∗ $1.99 ≈ $0.60
Developer commission: $0.10
Σ = $0.70
Unit Contribution Margin: ($1.99 − $0.70) = $1.29
Example: App Development
Breakeven Quantity =
𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠
(𝑈𝑛𝑖𝑡 𝐶𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑀𝑎𝑟𝑔𝑖𝑛)
Fixed Costs:
• Developer base pay: $1,000
• Facebook ad spend: $400
• WeWork rent: $350
• Σ = $1,750
𝐵𝐸𝑄 =
$1,750
$1.29
= 1,356.59 ≈ 1,357 units (per month)
© 2019 Ryan Wagner
Demand
© 2019 Ryan Wagner
Realistically, you don’t have control over demand. Demand is a random variable,
with its own distribution.
Of all the variables we’ve discussed, all you can really control is the selling price.
(ignoring economies of scales, etc.)
So after setting a price, the next critical question is: how likely are you to meet
your sales goal?
Since this question is a function of demand, and demand is a r.v., there’s no
deterministic equation to model this. This is where simulation comes into play.
Monte Carlo Simulation
© 2019 Ryan Wagner
Simulation (often called Monte Carlo simulation) is the process of generating
fake data that obeys one or more specified probability distributions, and using
the results to calculate the profit/loss of a scenario.
This method is repeated a large number of times (say tens of thousands), and the
results are used to estimate the expected net outcome of a situation.
𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑃𝑟𝑖𝑐𝑒 ∗ 𝐷𝑒𝑚𝑎𝑛𝑑 − [𝐹𝑖𝑥𝑒𝑑 𝐶𝑜𝑠𝑡𝑠 + 𝑉𝐶 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 ∗ 𝐷𝑒𝑚𝑎𝑛𝑑 ]
Everything on the RHS is likely to fluctuate in each time period, and could be
modeled as a random variable. No single point estimate for profit.
Monte Carlo Simulation
© 2019 Ryan Wagner
Back to our app developer example:
Demand is the r.v….let’s say you source some info from an online community of
developers, and determine that in any given month, an app is downloaded
~2,000 times, “give or take 1,000 downloads.” (use sd = 1,000)
Begin by decomposing profit into its base components:
𝑃𝑟𝑜𝑓𝑖𝑡 = (𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑣𝑒𝑛𝑢𝑒 − 𝑇𝑜𝑡𝑎𝑙 𝐶𝑜𝑠𝑡𝑠)
𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑃 ∗ 𝑄 − (𝐹𝐶 + 𝑉𝐶 ∗ 𝑄 )
𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑃 ∗ 𝑄 − 𝑉𝐶 ∗ 𝑄 − 𝐹𝐶
Sampling Distributions in R
© 2019 Ryan Wagner
We return to the four probability distribution we’ve covered so far (normal,
binomial, poisson, uniform ), this time the “r” versions of each. …
Each of these distributions has a command that allows you to generate random
numbers that obey that distribution.
rnorm() ← i.e., simulate random numbers that obey a normal distribution.
rbinom()
rpois()
runif()
rnorm()
© 2019 Ryan Wagner
demand <- rnorm(n=100, mean=10, sd=5)
demand
11.34803
6.850073
14.3433
18.63598
10.12094
11.84013
3.453979
13.69311
10.22436
4.758014
18.63926
4.107001
13.26603
8.157168
7.002227
10.27303
18.53839
4.528135
8.553591
mean(demand) = 9.948966
sd(demand) = 5.167387
Note that random number
generators will produce
different results every time.
To reproduce results, you
can manually ‘set the seed’,
by placing the set.seed() command before your run.
set.seed(15); demand <- rnorm(n=100, mean=10, sd=5)
rnorm()
© 2019 Ryan Wagner
Minor tweak: although we are treating demand as a continuous r.v., the demand
figure is realistically limited to integers (can’t have 0.50 of a download). Let’s
round our simulated demand values to the nearest integer.
The round() command takes two parameters:
• x: the vector of numbers you want rounded.
• digits: the number of digits to round to. (0 = round to nearest integer)
demand <- round(x = demand, digits = 0)
Simulation
© 2019 Ryan Wagner
We would like to simulate the estimated distribution of our monthly profit.
𝑃𝑟𝑜𝑓𝑖𝑡 = 𝑃 ∗ 𝑄 − 𝑉𝐶 ∗ 𝑄 − 𝐹𝐶
� ...
Purchase answer to see full
attachment