Calculus and Linear Algebra for Commerce



MAT133: Calculus and Linear Algebra for CommerceInstructions BookletIntroductionThis project is based on course content discussed in Modules E – J. There are three main pieces to it: the individualproblems, the pod project, and the reflective questions. This document outlines the instructions for accessing andsubmitting each part.Each part of the project is to be submitted on gradescope.ca. You should have received an invite to log intoGradescope to your official UofT email address; please use the link contained within in this invite to log in for thefirst time. Do not submit from a non-UofT email address. Note that you may re-submit or change your submissionas many times as you like before the submission deadline. No late submissions will be accepted.Caution: Please be sure to always use gradescope.ca for this class and not gradescope.com. If you type“gradescope” into Google, the first result will be gradescope.com and not gradescope.ca.You will also create a presentation based on your pod project. Instructions for the presentation will be posted in aseparate document.Contents1 Individual Submission 21.1 The Code and the OK List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Your Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Submission Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Pod Project: Modelling a Restaurant during COVID 42.1 Modelling the Restaurant’s Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Description of Project Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Submission Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Reflection Questions 74 Appendix 711 Individual SubmissionAs part of your first submission, you’ll need to include an academic integrity statement along with your own solutionsto the individual problems.1.1 The Code and the OK ListUofT upholds high standards of academic integrity. It is your responsibility to read and understand the Code ofBehaviour on Academic Matters and to adhere to the list of “OK” resources below.The OK ListThis OK list is a closed list of allowed resources, not just a list of examples. If you are unsure of what you areallowed to use, do not hesitate to ask on Piazza.OK: Collaboration with other members of your pod and hive – this is encouraged for all parts of this projectand required for the Pod Project portion.OK: Anything that can be found on the MAT133 Quercus pageOK: Your own and your hive member’s previous MAT133 work, including any projects, tests, or homeworkOK: Your own and your hive members’ notes from this course and your other coursesOK: Any other textbook (online or physical) you have access toOK: Online learning videos (e.g. Khan Academy)OK: Data sources including, but not limited to, those listed in Appendix AOK: General advice on the MAT133 Piazza page (e.g. “Can someone please help me understand RREF?”)OK: wolframalpha.com, desmos.com, any calculatorOK: Anything else declared as OK in a written announcement by the course coordinatorExamples of Not OK ThingsHere are some examples of things that are not OK. These are just examples. Unless something is on the OKlist, it is not OK.NO: Communicating about the project with anyone not in your hive. This means, for example, that you mustnot use group chats to share project content if these group chats involve anyone who is not in your hive.NO: Including your name on a submission to which you did not contribute. For example, if you neglect to doyour part of the pod project but include your name on the report, this is not only unfair to your pod,but it is also academically dishonest to claim credit for their work.NO: Asking for answers on Ed (e.g. ”how do you solve Question 2…”)NO: Accessing or posting on so-called “tutoring websites” like chegg.com or Easy EducationNO: Using online forums like stackexchangeNO: Entering the question text into a search engineAn important note on group chats: If you have administration privileges for any online chat that involvesanyone not on your team and in which assignment content is shared, you must delete any non-authorized contentfrom that group chat as soon as you see it (if this is technologically possible on that platform). Otherwise, you areconsidered to have helped someone cheat and therefore committed an academic offence.21.2 Your StatementAs part of your individual project submission, you’ll include an abbreviated statement to Gradescope that impliesthe following.In submitting this assessment … In short… I confirm that my conduct during this take-home exam adheresto the Code of Behaviour on Academic Matters.I know the Code.… I confirm that I have not acted in such a way that would constitute cheating, misrepresentation, or unfairness, including but notlimited to, using unauthorized aids and assistance, impersonatinganother person, and committing plagiarism.I didn’t cheat.… I confirm that the work I am submitting in my name is myown work. The work submitted for the pod project is the work ofmyself and my pod.This is my work.… I confirm that I have only used the aids marked as approvedin the “OK List”.I only used “OK” aids.On the front page of your individual submission, you’ll be asked handwrite and sign the abbreviated academicintegrity statement, as follows:“I have read the project instructions. I know the Code. I didn’t cheat. This is my work. I only used “OK” aids. Ipledge upon my honour that I have not violated the Code during this assessment.”Include your student ID number, the date, and your signature at the bottom. Here is an example:1.3 Submission InstructionsThe individual problems are to be completed by each student individually. You are allowed and encouraged todiscuss these problems with other members of your hive, but you must write your own solution for submission.Please note the following general policies for the individual problems:• You must handwrite directly into the problem template, using either a tablet or printer and scanner, justas you did for Test 1.• Your solutions should written clearly and concisely in a linear fashion. Do not submit messy scrapwork.• Explain all your steps. Your solution should be easy to understand by any other student in the class.• When applicable, please state your final answer in the form of a sentence, including units.The individual problems will be accessed and submitted on gradescope.ca. (They will be posted shortly.)32 Pod Project: Modelling a Restaurant during COVIDThis part of the project is to be completed in your pods. You should collaborate as a group on all parts of thisproject and you may organize the work however you’d like. However, we recommend everyone work through ALLcalculations together, then designate one person to be primarily responsible for the write-up on each part.• If your pod has 3 members, then we recommend assigning one pod member to be primarily responsible forthe write-up of each of the parts (a-c), then all 3 members can check each other’s work, then discuss and writeup part (d) together.• If your pod has 4 members, then we recommend assigning the write-up of one part to each member, but thatthe pod member for (d) also offer assistance to all of the other 3 pod members on their parts, while overseeingand coordinating the project as a whole.Important: Please be sure to also add all pod members’ names to your Gradescope submission. While it is yourresponsibility to get into contact with all of your pod members and to establish a method of communication foryour pod, you might still have difficulties getting into contact with a member. In this scenario, please send an emailto admin133@math.utoronto.ca with your concern, as soon as possible. If you write to us within 2 days of thedeadline, we may not have enough time to provide helpful advice!Please read the section below on Modelling the Restaurant’s Profit carefully before reading your pod’s tasks below.2.1 Modelling the Restaurant’s ProfitDuring the pandemic, many restaurants experienced a significant drop in in-person dining and an increase indemand for takeout. As a result, some restaurants have opened up additional kitchens (sometimes known as “ghostkitchens”) where they prepare meals for take-out and delivery. In this project, your pod will provide consultationto a restaurant who has opened up an additional kitchen during the pandemic.As the world recovers from the pandemic, restaurants are still struggling to keep afloat, due to capacity restrictions orpublic hesitancy to return to in-person dining. Some governments are providing incentives to encourage consumersto safely return to in-person dining, along with subsidies for restaurants. Considering new subsidies from thegovernment for in-person dining, your restaurant is now seeking to strike the right balance between the two services(dine-in vs. takeout) in order to maximize its profits.You will be using the Cobb-Douglas production function to model the quantity of meals produced by your restaurant.The Cobb-Douglas production function is a popular economic model used to relate different inputs, such ascapital and labour, to the output of production, Q over a given time period. We will consider the following inputs:• Capital, denoted by K, is the business’s capital expenditures over the time period. This could include moneyspent on space or equipment rental, maintenance, supplies, etc.• Labour, denoted by L, is the number of worker-hours used over the time period.In this case, the Cobb-Douglas production function is:Q(K, L) = AKαLβ,for positive constants A, α and β. We assume that α + β = 1 so that α = 1 − β. The individual portion of theproject is designed to help develop your understanding of this model.We will utilize the Cobb-Douglas production function to determine quantities of dine-in and takeout meals, basedon the labour and capital dedicated toward each. To calculate the profit function, we firstly make some assumptionsabout our restaurant. We assume that:A1 Everything produced by the restaurant is sold to a customer.A2 The capital costs of operating the restaurant’s two kitchens are the same.A3 The parameters A, α, and β are the same for the restaurant’s two kitchens.4Using these assumptions and the Cobb-Douglas production function, the monthly profit can be modelled byProfit = p1 × AK1−βLβ1| {z }quantity ofdine-in meals+ p2 × AK1−βLβ2| {z }quantity oftakeout meals− (w1L1 + w2L2 + 2K)| {z }costs of operation, (1)where:• K is the capital costs per month on each kitchen.• L1 is the number of worker-hours towards preparing and serving dine-in meals per month, in thedine-in kitchen.• L2 is the number of worker-hours towards preparing and serving takeout meals per month, in theghost kitchen.• p1 is the price of a dine-in meal.• p2 is the price of a takeout meal.• w1 is the hourly wage given to a dine-in worker.• w2 is the hourly wage given to a takeout worker.• A is the scaling factor from the Cobb-Douglas production function.The owner of the restaurant pays all employees the same wage, regardless of their task. Hence, w1 = w2 = w.She also intends to keep all employees working the same hours throughout these challenging times. No one will belaid off, but no new workers will be hired either. In other words, L = L1 + L2 is fixed. She is reaching out to youfor guidance on what proportion r of hours would be best devoted to the main kitchen preparing dine-in meals. IfL1 = rL, then L2 = (1 − r)L.2.2 Description of Project Tasks(a) Determining parameter values and optimizing the profit without subsidies.In this section, your pod will carry out research to determine reasonable values of the constants p1, p2, w, K, L, A,and β. Because these constants largely depend on the type of restaurant, you are able (and expected) to makeadditional assumptions in order to make it easier to determine constants. For the constant β in particular, afew possible resources you could refer to in order to choose a value are listed in the appendix in Section 4 ofthis document.i. What’s the restaurant’s story? Based on your findings during your pod’s research, you will likelydevelop an idea of the type of restaurant you want to focus on. Different types of restaurants havedifferent roles in the “restaurant universe” and so find themselves in different circumstances as a result. Ina paragraph, give your restaurant a name and location, describe it, and explain its circumstances. (Note:Please choose a city that is still experiencing reduction in seated diners relative to 2019, according to theOpen Table dataset linked here: https://www.opentable.com/state-of-industry.)• In the context of your restaurant’s circumstances, please clearly indicate and explain any additionalassumptions you make in order to arrive at your chosen values for the constants.• If you use data to determine your constants, indicate the data source and how it was used in order toarrive at the constant value.ii. Simplify the profit function. Using Equation (1) and your values for the parameters (p1, p2, w, K, L, A,and β), express the profit as a function of the proportion r of labour hours to be to be devoted to thedine-in kitchen.iii. Optimizing the profit. Find the optimal proportion of worker-hours dedicated to in-person dining, r,in order to maximize the restaurant’s profit. Be sure to justify your answer mathematically and expressyour final answer in a form that will be clear for the restaurant owner to understand.(b) Optimizing the profit with subsidies.Suppose now that as part of a new economic recovery plan, the government offers a subsidy on dine-in meals.For each meal sold, the government will give an additional s × 100% of the price of the meal to the restaurant.For example, if s = 0.08, the restaurant receives an additional 8% from the government on the price of eachdine-in meal. In other words, the restaurant would receive (1 + s)p1 = 1.08p1 for each dine-in meal sold.i. Simplify the profit function including subsidies. The new subsidies will lead to larger profits for therestaurant. Using the parameter values found in (a), express the new profit as a function of both s and r.5ii. Optimize. Consider s to be a fixed, but currently unknown parameter value. Determine the optimal valueof the ratio r of labour hours, which leads to maximum profit for the restaurant, including governmentsubsidies. (Your answer will depend on the subsidy s.) Call this optimal ratio r∗ = f(s).iii. Sketch a graph of r∗ = f(s). Your main goal here is to use calculus to sketch a graph of the optimalratio r∗ as a function of s on the domain [−1, ∞). Be sure to:• Find the values of f(0) and lims→∞ f(s). Explain why these values intuitively make sense in thissituation.• Find any critical points and determine the sign(s) of dr∗ds everywhere in the domain. Explain why youranswer intuitively makes sense in this situation.• Find any inflection points.• Use the above information to sketch a graph of r∗ = f(s) on the domain [−1, ∞), labelling any notablepoints or values found above.• Use Desmos to confirm that your sketch is correct.iv. Write a short paragraph to communicate your findings to the owner of the restaurant.(c) Closing the ghost kitchen and returning to “normal”.i. Closing the ghost kitchen: If the subsidy s were large enough, the restaurant should consider closingtheir ghost (takeout) kitchen and saving on the associated capital costs. How large would s need to bein order for the restaurant to close the ghost kitchen and offer dine-in meals only, to maximize theirprofit? In other words, at what value of s does the profit obtained from operating only the dine-in kitchen(with the subsidy) exceed the maximum profit from running both the dine-in and the ghost kitchen (withthe subsidy)? Refer to the results from part (b). You may use Desmos to help you solve this problemnumerically. Be sure to include any graphs you use in this report and explain your reasoning. Write ashort paragraph about your findings to the owner of the restaurant.ii. OpenTable is an app that has kept track of the number of reservations at the restaurants it works withfrom 2019-2021. The first table available at https://www.opentable.com/state-of-industry keepstrack of the percentage increase or decrease in number of seated diners in restaurants in 2021 (whereCOVID-related restrictions may be in effect) compared with the same day on 2019 (where there were norestrictions). You may download the full data set from their website.• Give a brief outline of key dates with regards to any lockdowns or relevant restrictions (being placed orlifted) during the time period. Can any trends in your data be possibly linked to these dates, periods,or resurges in COVID-19?• Focus on the data over the most recent month. Roughly how large would s need to be in order torestore the revenue for a typical dine-in restaurant in this city to pre-pandemic levels? Assume herethat the price per meal is held fixed throughout the pandemic. State your result in plain language forthe restaurant owner.(d) Implications and limitations of the modelIn this section, you will explore the implications and limitations of the restaurant model you have been using.In your submission, please discuss the following questions.• Recommendations: In plain language and with reference to the results in parts (a-c), summarize themain recommendations you’d make to the owner of the restaurant.• Limitations: In developing our mathematical model, we made some assumptions and simplificationsabout how the restaurant operates. While a simplified model can still give us useful results, it may alsoneglect other factors affecting the profit in real life. Describe several aspects of the model that maynot accurately reflect how a restaurant or take-out service operates. For at least one of these aspects,investigate how might we change the modelling process to address this limitation.2.3 Submission Instructions• Please include a cover page that lists the names, student IDs and UofT email addresses of all group members.• Be sure to include ALL pod members on your submission in Gradescope. All group members that appear onthe cover page must appear in Gradescope. Penalties may apply to pods who do not follow these instructions.6• As usual, your submission must be in the form of a single pdf.• NEW: Please make sure to pay special attention to your pod’s writing to ensure its quality and clarity.This includes integration of mathematics and figures into your paragraphs, ensuring all figures and graphs arewell-labeled, and being mindful of the overall appearance of your submission.3 Reflection QuestionsReflection is key to learning from our experiences. These questions are intended as a venue for you to reflect onyour experiences working on this project.Your grade will be based on your efforts to thoughtfully reflect on each question, according to the instructionsprovided. While we do ask that you write in full English sentences, we will not judge your grammar or syntax.Please don’t worry too much about writing perfectly! Think of it as being like a journal entry or an email to a friend.The estimated time to complete these questions is approximately 30 minutes.The reflection questions will be posted to Quercus shortly after your presentations are complete. Your will besubmitted directly to Quercus.4 AppendixReferences for βBelow are a list of references that might help with choosing a value of β in part (a). Note that you do not have tounderstand them completely, but rather are only expected to extract the information which is useful to your podfor the purpose of choosing β.• C.W. Cobb, P.H. Douglas. A Theory of Production. 1928. URL https://www.jstor.org/stable/1811556• Office of the Parliamentary Budget Officer. PBO’s Approach to Measuring Potential GDP. 2018. URL https://www.pbo-dpb.gc.ca/web/default/files/Documents/Reports/2018/Potential%20GDP/Measuring_Potential_GDP_EN.pdf• J. Barlow, I. Vodenska. Socio-Economic Impact of COVID-19 Pandemic. 2020. URL https://papers.ssrn.com/sol3/papers.cfm?abstract_id=37207857

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