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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

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