MAT244H1F Final Report

Fall 2024/2025

The Setup

The city of Oronto is concerned about its future. Having watched other cities around the country deal with high rent prices and a fluctuating population, they decided to get ahead of the game. They commissioned a study on the relationship between rent and population in their city from the prestigious Housing Institute of Greater Oronto (HIGO).

HIGO came up with the following model:

P 0 (t) = P(t) · (10 − P(t)) − R(t)

R 0 (t) = 0.5 · R(t)

where

• t is measured in decades from the year 2024.

• P(t) is the population in millions at decade t.

• R(t) is the average price of a rental unit at decade t in units of thousands of dollars (per month).

HIGO based their model on the following assumptions:

(A1) The city land has a natural carrying capacity (i.e., a maximum number of people that can sustainably live within the city limits).

(A2) People are attracted by people and will move to the city if many other people already live in the city.

(A3) People will move away from the city if the rent is too high.

(A4) Rent increases with inflation.

For your information, the current population of Oronto is 5 million people and the average rent is $2500 per month.

The Task

Your job is two-fold. You aim to

• inform. the city council of the consequences/predictions of HIGO’s model for the city of Oronto; and,

• propose legislation (affecting rent) so that the population and rent of Oronto eventually stabilizes.

Rules

1. Make reasonable assumptions. You will be exploring HIGO’s model and proposing your own, modified, model based upon your proposed legislation. As the city of Oronto is fictional, you are allowed a great deal of freedom to make assumptions and proposals. However, your assumptions must be reasonable.

Tips on reasonability:

• Don’t assume the city has an infinite budget. For example, proposing that the city subsidize all rents to $0 is not reasonable (nor would any proposal that results in rents ending up at a steady state of $0).

• The population should not stabilize at 0. If a city starts out at 5 million people, a city council will not be interested in a plan that sees the city disappear or nearly disappear.

• Your forecast won’t work for centuries, so make sure your argument doesn’t depend on a rent-population relationship holding 500 years from now.

You may create assumptions on city budgets or federal assistance to the city, etc. as long as the numbers you propose are feasible (e.g., you shouldn’t require a national budget that is 10× that of the US).

2. Legislation can only affect rent. You are proposing legislation to the city council to stabilize the rent and population of Oronto. Your proposal for legislation can only affect rent. For example, you cannot propose legislation where people are forbidden from moving away from the city, etc..

3. Consider your audience. You are making a proposal to a city council. If your proposal is to compute the arctan of rent multiplied by the cubed root of population and then take the coefficients of the decimal expansion of that number and use them to create a Taylor series which computes the optimal rent at decade t, the city council won’t understand and won’t consider your proposal. Keep things as simple as possible so the council can understand.

The Report

You will be investigating the HIGO model and proposing your own modification of said model. Normally, one would prepare a long technical report and an executive summary. However, you will only be turning in a short technical report and an executive summary.

Your final report will consist of three parts. All parts must be typed, though you may hand-annotate figures or insert equations by hand.

1. Background Assumptions (1 page). In one page, you should outline the most important background assumptions needed to understand your report and execu-tive summary. For example, if you assume that Oronto is a fiscally conservative city, or that the council is divided among those who want the largest city possible, and those who prefer a smaller population, those assumptions go here.

Note: you must still list your assumptions where relevant elsewhere in your exec-utive summary/technical report. The purpose of this page is for you to give the marker the necessary context to understand your writeups.

2. Executive Summary (2 pages). You are to write a 2 page executive summary. Your audience is the Oronto city council. Your executive summary must

• Explain your findings about what the (unmodified) HIGO model predicts

• Explain your proposed legislation and the predicted consequences of your legislation

• Address trade-offs the city may need to make to meet its desired goals

Your executive summary should follow the guidelines of a good summary (e.g., as outlined by USC). It may be single or double spaced, but it should be readable (e.g., do not squish the margins to try to fit in more words).

3. Technical Report (2 pages). You are to write a technical report. Your technical report must be at most 2 pages, excluding figures. Including figures, your technical report may be no more than 5 pages. Your figures may be interspersed among the text of your report or they may be listed at the end of your report (it is preferable that you intersperse your figures).

Your audience is fellow MAT244 students. Your goal with the technical report is to explain how you analyzed the HIGO model, what your proposed legislation is, and how your proposed legislation affects/modifies the HIGO model.

Your technical report must

• Include relevant assumptions

• Explain how you analyzed the HIGO model

• Explain your proposed legislation

• Show a system of equations that models the city rent/population if your legislation were to be adopted

• Include figures/diagrams/equations/etc. to support your conclusions

Your technical report should be an independently readable document. Since the technical report is quite short, it does not need to address issues like the pros/cons of your legislation (that will show up in the executive summary). However, it is not a place for “scratch work”. For example, your technical report would not walk through every equation you solved when finding an affine approximation/finding eigenvalues/etc., but it may include overviews like, “After finding an affine ap-proximation about the equilibrium solution, we found the coefficient matrix has eigenvalues 2 and 7, suggesting that the equilibrium is unstable.”

You are given a lot of freedom on this project. Have fun and be creative! Through this project, we are looking to see you demonstrate that you can analyze a system of differential equations using the tools that we’ve learned this semester and that you can modify a differential-equations based model and analyze the result. If you demonstrate that in a well-written way, that’s exactly what we’re looking for!

Submitting

Submit on Gradescope. Since your reports will be typed, we are not using a template-based submission.