Question

#1

Abraham Marquez runs a family-owned orange grove in Santa Clarita, California. Recently, he's been considering estab i shing a presence at a famer'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 consum ers, oranges are oranges are oranges. That might change as Abe is able to develop a stable base of loyal custome rs, but until that time, he isn't confident that he'd be able to successfully co mmand a premium price for his product. He dedides hell 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 we ek to week, Abe decides on the following strategy: upon armival 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 marke ts, he knows that the price of oranges tends to vary a lot...their price is typica ly un ifomly distributed between $1.50 and $3 per pound. Abe makes a phone cal to a friend in Northern California who already sells oranges at a weekly market there. The friend reports that sales aren't very pre dictable.. .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 faily smal: the Leucadia market on ly charges vendors $50/week to set up a booth. Classify the values given in the prompt into one of the categories shown below. For any value that is given as a 1. random variable, state its distribution and parameters. Otherwise, s imply 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): Variable Cost(s): t is the variable cost per pound of oranges? (1 point) 2. W 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 caled total_cost that is the sum of Abe's estimated total weekly costs, using any static costs previously defin ed, 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 de mand for oranges, according to the parameters you defined in Q1. Set the seed as 54, and store the simulations as an object called demand. (1point)

Answer 1

1)

Selling price: Uniform($1.5,$3)

Supply: 400 lb

Demand: Uniform(100 lb,400 lb)

Fixed Cost: Normal($88.75,($3)^2), $50/week

Variable: $150