Pre-Test
This is a draft of the final exam for this course. Each question or cluster of question has an expandable list of skills you will be responsible for acquiring and can expect to be examined on. This pre-test is intended to give us a baseline against which we can measure progress as well as to give you an idea of what is to come.
Answer the following questions to the best of your current ability. Feel free to use the answer "I have no idea." Check your solutions/answers against those given here (for most questions) and submit only a report of how you did on each question. Categories are CORRECT, INCORRECT. If incorrect indicate what you got wrong if close or just indicate "did not know how to do"
Final Exam
When we talk about models we need to mention that a model is a simplified representation of something and the purpose of that representation is to better understand (or illustrate) how that something functions. It is a description of how something happens that can be used to make predictions about things that should be observable if the model is correct, and that leads us to identify things we can manipulate in order to change how things happen in the world.
Equations, diagrams, physical models, verbal descriptions, computer programs, biological systems.
Markov model, agent model, game theory model, stock and flow model
Generic Concepts
1. Name seven "modeling" techniques discussed in this course (not the little details inside of a technique, but the big seven tools).
Difference Equations, Queues, STOCK/FLOW, Markov, Benefit/cost, Discounting, Linear programming, Decision trees
Which of the following represent linear processes?
2. One youth agency is estimated to be able to reduce delinquency by 15%. Two agencies would be able to reduce it by 27%, three by 37%, and five agencies by 50%.
This one is NONlinear: one variable is the number of agencies, the other is the decrease in the rate of delinquency. The information we have suggests that as the number of agencies goes up, the reduction PER agency drops off. Eventually, we can imagine, you'd reach a level where adding one more agency would get you nothing.
3. We predict that 25 people will show up at the protest no matter what. Further, all of our research shows that for each $100 we spend in public relations and networking, we can expect 12 more people to show up.
Here we have a linear process. The problem suggests that every 100 dollars gets you 12 people. We expect that 200 will yield 24 additional people, 300 will yield 36 and so on. And we could write a simple linear equation for the process as described:
protesters=12×dollars spent+25
7. Distinguish between: DESCRIPTIVE/POSITIVE VS PRESCRIPTIVE/NORMATIVE
The goal or purpose of a theory or model can be either to describe what is or to specify what ought to be. When we run a model with the purpose of ascertaining what the consequences of a particular course of action would be we are engaged in descriptive or positive work. This work does not involve a "should" and is not about making decisions. When we are able to state a goal and we want to evaluate which course of action will best move us toward that goal – that is, when we are using an analytical technique to guide our decision making – then we are engaged in prescriptive or normative work.
8. What is an OBJECTIVE FUNCTION and when do we encounter it?
A mathematical expression of what we are trying to optimize, our overall outcome, in techniques like linear programming.
9. What is the difference between a stable and an unstable equilibrium?
Small perturbation from unstable equilibrium leads to movement away from equilibrium. From a stable equilibrium it would lead to a return to the equilibrium point.
9. Label each of the following as representing either stable or unstable equilibria.
- Text vs. number vs. formula in cells
- Cell references and cell range references
- Use formula to calculate values based on other cells.
- standard math functions and operators (including exponentiation and parentheses and order of operations)
- Autofill (values and formulae), use of relative and absolute cell references in formulae for autofill
- Formatting. Alignment, font, borders and fill, number format, decimal places, conditional formatting.
- Workspace control. Row and column width, merging cells, hiding columns, split panes, wrapping text, inserting newline in cell, text
- Rename, move, copy worksheets.
- if() function
- Use sort and find/replace
- Name a cell or cell range
- Create a percentage table from a data count table.
- Absolute vs. relative cell/range references
- Use frequency() to convert list of data values into histogram
- Use formulas, relative/absolute references, and autofill to create a table of values
- rand()
- Pivot tables
- Create one, two and multidimensional.
- Select different value field settings.
- Convert to text appropriate table. Use to clean data.
- Basic descriptive statistics with sum(), average(), mode(), median(), stddev(), frequency(), correl() functions.
- Simple data range to column, bar, pie, scatter. Label and format axes, chart titles, legends, gridlines.
- Add, remove, edit data series. Format data series (change line, point styles, add/customize labels)
- Add/remove/format/label trendlines
- Manipulate strings with text functions (e.g., left(), right(), mid(), find(), substitute(), text())
- De-junkify charts.
- Copy charts to Word. Remove borders, select colors for greyscale reproduction.
- Copy charts to PowerPoint.
- Apply conditional formatting.
- data table
- array functions (e.g., sumproduct, mmult)
- data lookup functions (e.g., vlookup)
- solver
- statistics functions
- Add controls for interaction (sliders, spinners, etc.) to a worksheet
- MSW01.1 Styles
- MSW01.2 Page numbers
- MSW01.3 Sections, headers, footers
- MSW02.1 Captions and crossreferences
- MSW02.2 Drawing
- MSW02.3 Inserting images, charts, etc. and formatting text flow around them.
- MSW02.4 Tables
- MSW02.5 Equations
- MSW02.6 Footnotes/endnotes
- MSW03.1 Sophisticated search and replace
- MSW03.2 Paste options
- MSW03.3 Customizing autocorrect
- MSW04.1 Tables of contents, etc.
- MSW04.2 Track changes, versions, comments
- MSW04.3 Mailmerge
- MSW04.4 Citation and Bibliography tool
Use flowchart to demonstrate stepwise refinement and black-boxing (Edit)
Create and interpret flow chart that includes time and divisions of labor (Edit)
Offer critique and corrections to a poorly formed flowchart (Edit)
Identify and comment on concerns and caveats (Edit)
Draw flow charts in Microsoft Excel, Word or other software (Edit)
1. Sketch a flowchart that represents this bit of logic: “if you are a woman then if you are over 40 you should have this test no matter what but if either parent had diabetes women should have the test no matter what. Men only need to take the test if they are overweight.”
This medical advice protocol has two top level parts: one for women and one for men.
For women, it says those over 40 should have the test. And it says no matter what your age, if either parent had diabetes you should have the test.
For men, it says only take the test if you are overweight.
We might be tempted to distraction by phrases like "no matter what" and "only" but a little bit of parsing and rephrasing helps to clarify.
11. No Question
12. No Question
13. March & Lave describe the "act of modeling" as being composed of the following "steps." Observe a pattern. Describe a process that could produce an observed pattern. Predict what else that process would produce. Put these into this flow chart and write a short paragraph describing their general process of model building as an explanatory endeavor.
- outcomes & events
- mutually exclusive & exhaustive
- independence
- P(A or B)
- P(A and B)
- P(A|B)
- $P(x) \times \mbox{value of outcome X}$
- Expected value of a set of mutually exclusive alternatives (e.g., branches coming out of a chance node) is sum of expected values.
- Read a description, draw the tree
- See a tree, write the description
- Work backwards to calculate values of chance nodes and pick decisions at choice nodes.
- Take an ordinary DT and add a test/no test choice node, calculate value of test
- Expected value of perfect information (EVPI)
- Define false positive and false negative
- "Flip" a tree to calculate expectations of test outcomes
* choice/chance
- exhaustive/mutually exclusive
- folding-back
- imperfect tests
- risk-aversion
- the value of information
- utility theory
14. Identification: CHANCE/CHOICE NODE.
In decision analysis there are two types of "forks in the road." One type represents something the decision maker has control over – whether to go with option A or option B, for example. These are called "choice" nodes and are conventionally represented with a square. The other type represents contingency – points where two or more alternative things can happen – the new machine will work well or it will work poorly, for example – and this depends on factors outside the decision maker's control (or are even subject to pure chance). These are called "chance" nodes and are represented by a circle. At a chance node we calculate the value of the branch by taking the value of each sub-branch and multiplying by the probability of that sub-branch to get the expected value. At a choice node, we calculate the value of the node as the minimum or maximum (depending on the problem) of the sub-branches coming off it.
15. Identification: EXPECTED VALUE.
The long run average value of an uncertain outcome. Calculated by summing the products of the probability of outcome and the value of the outcome for all possible outcomes. Used, for example, in decision analysis to calculate the value of a "chance node."
16. Consider this decision chart about whether or not to use a new training curriculum at the city's firefighter academy. Its developers make claims that it produces better trained firefighters in less time and at less cost. Your experts think there's about a 50:50 chance that it performs as well as claimed. You've estimated that the net benefit of the old, traditional training approach is 5 while the savings and better training promised with the new curriculum would be worth 8. If however, it turned out not to work, then the cost of retraining would bring the total benefit down to 3. Fortunately you have the option of doing a pilot test at cost T to determine whether the new training methods work well or not. This information is laid out in a decision tree. Explain how to read the tree and what we should do.
Cost/Benefit Analysis
17. Define EXTERNALITY.
A cost or benefit borne by an entity that is not party to a transaction. Most commonly, a cost that is not covered by the parties to an exchange. The presence of externalities is one kind of market failure. The most common example is pollution caused by a manufacturing process; it has an impact on people other than those who make or buy the product.
18. Define MARGINAL COST.
The cost of the last item. "Marginal" refers to the rate of change of a measure at a particular point. In general, the cost of the first unit of something is more than the cost of later units – there is an "economy of scale." The marginal cost at any given level of input/output is the cost for increasing the output just a little. The usual criterion for selecting the appropriate level of any activity is to stop increasing the level when the marginal cost is just equal to the marginal benefit.
19. Define PARETO CRITERION.
In decision making and welfare analysis we are often faced with considering alternatives that benefit some and penalize others. The Pareto criterion for decision making says that you can only select from among options that make at least somebody better off without making anyone worse off.
20. An Oakland youth oriented non-profit is considering building a new headquarters. Funding is all lined up. The building being considered will cost 1.75 million. The net savings in rent and utilities and such over the years net out to 1.5 million. Maintenance on the new building is expected to be 750,000 over the period in question. Should they go ahead with the project?
This is a straightforward "type 1" benefit-cost analysis case: deciding whether or not to undertake a single project without a specific budget constraint. We might want to think a bit about laying out the costs and benefits over time and calculating the present value of future savings and costs before we start. It sounds like the main building cost is an up-front cost and the savings and the maintenance come in over a period of years. The problem states, though, that the savings "net out to 1.5 million" and the maintenance is 750,000 so maybe that's already been done.
In any case, we'll line up the costs and the savings and any other benefits and net them all out. If the net benefit is greater than zero, then we go ahead with the project.
21. City officials are considering four different approaches to relieving pressure on the local housing market. Cost estimates for four projects A, B, C, and D have been obtained. For each project experts have also estimated the expected benefits in terms of taxes, savings on services, etc. How would you start to think about this? What else would you want to know?
This time it sounds like we have a benefit cost analysis scenario in which we need to choose between projects subject to a budget and/or select the appropriate level of each project. What we are going to need is some information on the different levels of each project that are possible along with estimates of the cost of each level and how much impact we expect it to have on the housing market. A big challenge, we can expect, is trying to figure out how to quantify the impact on the problem. I'll have to learn a bit about how folks in housing policy think about such things.
Once we get the information on programs A through D we'll calculate the marginal benefit
Discounting
22. What is the basic equation for compound interest. Let $P$ be the principal and $R$ be the interest rate per period and $t$ the time in periods.
(1)
\begin{align} P(t)=P_0 \times (1 + R)^t \end{align}
23. DEFINE internal rate of return.
24. Lots of talk recently about green this and green that. Homeowners, cities, counties, states and the federal government all have to figure out whether to invest in green technologies of various kinds. Typically these things have big up front costs and relatively long pay back periods. How does one best think about which ones make sense?
25. We must choose between two energy saving projects. For each one we have data on how long it will take to install, what the payment schedule will be and how much each will let us save over the years along with projected maintenance costs.
Multiple Attributes Problems
26. Consider the following rankings of several alternative projects under consideration at your agency. Cheaper is better. High political attractiveness is better. Higher effectiveness is better.
| Alternative |
Cost |
Political Attractiveness |
Actual Effectiveness |
| Option A |
100 |
Not |
84 |
| Option B |
300 |
Not |
67 |
| Option C |
300 |
High |
49 |
| Option D |
200 |
Medium |
93 |
a. Is any option strictly dominated by another? D»B. A>=B.
b. Suppose Cost is more important than political attractiveness is more important than effectiveness. Give a lexicrographic ordering of the alternatives. A > D > C > B
c. Suppose we are satisficing and our standards are: it has to cost less than 300 and at least a little bit politically attractive. What are our choices? D
d. Suppose we have an objective function that goes like this:
Rank options on each factor. In the case of ties, split the sum (i.e., a tie for 2nd place would split 2nd and 3rd and each would get 2.5). Add up the point scores.
What option do we choose?
A 1 3.5 2 6.5
B 3.5 3.5 3 10
C 3.5 1 4 8.5
D 2 2 1 5
Randomness as a Tool: Random Experiments and Monte Carlo Simulation
- Apply basic concepts of probability (events and outcomes, and, or, conditional)
- Craft an argument in favor of random trials as gold standard
- Demonstrate understanding of random selection from different shape distributions (uniform, normal, Poisson)
- Use rand() and related functions and data table function in Excel to generate Monte Carlo simulation
- Suggest indications and contraindications, identify and respond to typical concerns and caveats
27. If a group of monkeys took the GRE, what do we expect their average score would be? Make any necessary assumptions and explain your reasoning.
28. When building computer models, why do we sometimes use random numbers that are normally distributed, sometimes uniformly distributed, and sometimes coming from a special distribution such as the Poisson distribution?
Stock & Flow Models
- Translate system description into stock and flow diagram
- Translate diagram into difference equations
- Describe generic system performance possibilities
- Find a good source for refresher on system dynamics
- Concepts and Terms: positive/negative feedback, causal loops, equilibrium
- Suggest indications and contraindications, identify and respond to typical concerns and caveats
29. Suppose you have been put in charge of establishing an "army" of community organizers as a part of a new Obamanization of American Communities. The goal is to have a steady state program size of 800,000 organizers. Our program involves a recruitment phase, a training phase, a placement phase and an active duty phase. Recruitment happens continuously with training academies starting every 6 months. The training itself lasts 6 months and is followed by a 6 month placement rotation, and then a year of active duty volunteer work.
We will assume that logistics and such are taken care of and we are only interested in building a model that will help us to see how recruitment and retention rates are related to the the number of volunteers on the street at any given point.
During training we expect 25% of recruits to drop out. A further 10% will leave during the placement phase. Once organizers get to active duty status we expect 100% retention for one year. At the end of one year, about 30% of organizers sign up for one more year of active duty.
First describe the situation in words. If the program begins with 100,000 recruits on 1 January 2011, what will the first few half-years look like in terms of activity?
Explain what this diagram shows and what sorts of equations we would associate with it (based on the above information).
Project Management
- Describe components of logic model, interpret and criticize logic model, translate project description into conventional logic model
- Demonstrate familiarity with existence of broad range of project management strategies
- Gantt charts, PERT, CPM, stakeholder analysis, ISO 900x Standards
- Explain/describe what a PERT chart and a Gantt chart are.
- Explain/describe related concepts: critical path, scope, deliverables, tasks, work breakdown, activities, sequence, resource requirement, time and cost, schedule, budget, business plan, monitoring, control, outcomes assessment, evaluation
- Identify and respond to typical concerns and caveats
30. Consider this PERT chart illustrating the process of designing an airplane. In each box you will see the number of days we expect the subprocess to take (ET) and the longest amount of time it can be expected to take with delays and problems and such. How long do we expect the process to take from start to finish?
Back of the Envelope Calculations
- Describe general principle
- orders of magnitude
- Apply simple units of analysis assessment to ball park a problem
- Demonstrate awareness of making parallel assumptions (i.e., all optimistic or all pessimistic) in setting up an approximation
- Use BOTEC to come up with high and low estimates for a problem
- Identify and respond to typical concerns and caveats
31. What is the order of magnitude of the number of seconds in a year (show your reasoning).
(2)
\begin{align} = 1 Year \times \frac{365 days}{year} \times \frac{24 hours}{day} \times \frac{60 minutes}{hour} \times \frac {60 seconds}{minute} \end{align}
(3)
\begin{align} \approx 3.65 \times 10^2 \times 2.4 \times 10^1 \times 6 \times 10^1 \times 6 \times 10^1 \end{align}
(4)
\begin{align} \approx 3.65 \times 2.4 \times 6 \times 6 \times 10^5 \end{align}
(5)
\begin{align} \approx 3.65 \times 6 \times 2.4\times 6 \times 10^5 \end{align}
and since
(6)
\begin{equation} \end{equation}
3.65 \times 6 \approx 20
[[/math]]
and
(7)
\begin{align} 2.4\times 6 \approx 15 \end{align}
(8)
\begin{align} \approx 2 \times 10 \times 1.5 \times 10 \times 10^5 \end{align}
(9)
\begin{align} \approx 3.0 \times 10^7 \end{align}
Actual value is 31,536,000
32. If there are about 36 million people in California, about how many school age children are there? Show your reasoning.
{{
Many approaches are possible. Here's a very crude one: assume the population is uniformly distributed across ages from, say, 1 to 72. That would put 1/2 a million in each year. If school age means 5 to 18 (13 years) that would be about 7.5 million.
Next we would want to shift the distribution down (since old people die) and so, perhaps take a first guess at 10 million?
Age pyramids for California.
Here are the 2000 numbers
| Age |
Male |
Female |
| 5-9 |
1,396,480 |
1,329,400 |
| 10-14 |
1,317,135 |
1,253,687 |
| 15-19 |
1,271,626 |
1,179,262 |
Total 7,747,590
}}
Markov Models
- Describe general idea and types of problems that lend themselves to Markov model
- Describe requirements of a system to be eligible and how to solve "something else" problems
- ranslate between description, transition matrix, diagram
- From diagram, set up equations in Excel, chart results, identify equilibrium states
- Terms and concepts: Markov chain vs. Markov process, regular, absorbing, cyclical, state, absorbing state, transient state, eigen vecgtor, transition matrix
- Suggest indications and contraindications, identify and respond to typical concerns and caveats
33. What is the eigen vector of a Markov model?
34. A criminologist and an activist decide to collaborate on a project designed to reduce prison population. In the spirit of starting simple, they identify 4 states in which people can find themselves: never imprisoned; incarcerated; on parole; post-parole. The period of time in their analysis will be one year. Suppose 70% of the population has never been incarcerated. Each year 2% of these people are imprisoned. Of those currently incarcerated, 5% are released each year onto parole. Average parole is 5 years so that a person on parole has a 20% chance of finishing parole. Those on parole have a 10% chance of finding themselves back in prison in any given year. Individuals who are post parole have a 4% chance of returning to prison in any given year.
35. Look at this diagram about the fate of mortgages and tell us what happens to mortgages that are late. Explain what these figures and tables are and interpret what they mean.
36. You have been assigned to assess a social services project meant to address the "continuum of care" around family homelessness. The continuum of care concept is based on the idea that in a problem like homelessness affected persons pass through stages both on the way into and out of the problem. At different stages, different services are needed (e.g., a homeless family needs emergency shelter NOW; a family living in a shelter needs resources and contacts to enable them to move into transitional housing; a family in transitional housing may need help assessing what kind of permanent housing is appropriate to their situation; a family teetering on the edge of losing its home may need mortgage support). Traditionally, agencies specialize in particular stages and may not collaborate at all with agencies concerned with other stages.
Your client has a simple request. It wants to build a picture of "the problem" and the stages that families pass through on the into and out of homelessness. They have categorized some of the stages:
- Pre-at-risk families who are currently doing well but who could themselves at risk with the loss of a job or other unanticipated circumstance. We think of these families as "pre-at-risk" and there are services targeted at "risk prevention" – educating homeowners about re-finance options, renters' legal aid, etc.
- Families at risk of homelessness. Families currently with a home but at risk for losing the home due to foreclosure or because of behavioral problems (drugs, etc.) on the part of adults in family.
- Currently Homeless Families. Families currently living "on the street."
- Emergency Shelter Families. Short term, safe and decent shelter provided as an alternative to the streets. Families currently staying in emergency shelters. Time in shelters varies depending on, among other things, availability of staff to assist in placements in transitional housing and the availability of transitional housing.
- Families currently in transitional housing. Families who have moved out of emergency shelters but have not found permanent housing. The program here consists of training and assessment. Training is about financial management, job skills, employment finding, etc. The assessment is to determine whether on-going support will be needed to help these families succeed.
- Families in permanent housing. Client families who have been placed in permanent (usually rental) housing. Ongoing services such as those mentioned above under "pre-at-risk" are often appropriate on an ongoing basis for these families.
- Families in permanent supportive housing. Families in need of special services to make their continued tenure in permanent housing viable. These include services such as education, employment assistance, health care, substance abuse treatment and mental health care, child care, and transportation.
What kind of a model would you suggest building?
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This sounds a lot like a Markov model. Homeless families pass from one state to another and it sounds like they are only ever in one (states are mutually exclusive) and as long as we include the entry and exit states they are exhaustive of all the possibilities. We might decide on a time period, say a week, and then ask the various service providers how many new clients they get each week and how many they typically "say good bye to" each week and where these clients go. We'd get the data from interviews as well as client records. Some of the stages are probably fixed duration (e.g., a drug treatment program might always take 6 weeks) but others will depend on availability of resources. Once our research has gotten us some numbers we can estimate the transition probabilities and run the model to identify equilibrium state populations and such.
To better understand how the process works, I'd probably want to make a flow chart based on interviews with staff to understand how they understand the flows of clients through the system and the categorization of the clients.
I might also think a little about difference equations and queuing in conjunction with this situation. It is not inconceivable that we'd find some bottlenecks in the system – in fact, the whole notion of a "continuum of care" implies a connected system in which one agency "passes" clients to another and this is almost certainly a recipe for the creation of bottlenecks. It may well be that these do not show up now because families "fall in the cracks" – a better connected system may have more waiting problems.
}}
Queuing Models
- Describe general idea and types of problems that lend themselves to queuing model
- Describe requirements of a system to be amenable to use of QM
- Translate between description, diagram, equations
- Explain why wait time is deadweight loss and nominal fees can be efficient
- Terms and concepts: arrivals, servers, Poisson distribution/process, wait time, deadweight loss
- Suggest indications and contraindications, identify and respond to typical concerns and caveats
37. Explain the basic idea of "waiting time as deadweight loss."
38. WHAT KIND OF MODEL? The DMV is mindful of its poor reputation for service. You are employed as a consultant to help them deal with it. In particular, they are trying to figure out whether some sort of appointment system will be effective both in terms of customer satisfaction and overall productivity.
39. El Oakalo, CA is a small town with a beautiful new public aquatics facility with a great pool and very nice surrounding areas. Since it was built with taxpayer money, it's free to town residents. Unfortunately, there's not a lot to do in town and so everybody shows up — even folks who have no interest in swimming and just want to sit around under beach umbrellas. Regulations restrict the number of people on site to 575 and this means that lots of swimmers face big delays before they can get in the pool. Help!
40. No question
41. No question
Agent Models
- Suggest indications and contraindications, identify and respond to typical concerns and caveats
42. No question.
43. No question.
44. No question.
Linear Programming
- Understand vocabulary and concepts associated with linear programming.
- Translate word problems into inequalities for use in linear programming model.
- Be able to solve linear programming model graphically
- Translate simple linear programming problem into Excel and use solver to find optimum.
- Suggest indications and contraindications, identify and respond to typical concerns and caveats
45. You are working for an agricultural cooperative which is helping local farmers figure out how to optimize the mixture of crops they plant. A typical farmer has 10 acres to plant in wheat and rye. She has to plant at least 7 acres. However, she has only the equivalent of $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover, the farmer has to get the planting done in 12 hours and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the expected profit is $500 per acre of wheat and $300 per acre of rye how many acres of each should be planted to maximize profits?
46. A non-profit supplier of after-school materials has orders for 600 copies from San Francisco and 400 copies from Sacramento. The organization has 700 copies in a warehouse in Novato and 800 copies in a warehouse in Lodi. It costs $5 to ship a text from Novato to San Francisco, but it costs $10 to ship it to Sacramento. It costs $15 to ship from Lodi to San Francisco, but it costs $4 to ship it from Lodi to Sacramento. How many copies should the organization ship from each warehouse to San Francisco and Sacramento to fill the order at the least cost?
47. No question.
Difference Equations
- Understand notation and be able to write out equations for simple difference equations
- Translate a problem into difference equations
- Implement difference equations model in Excel and graphically display outcome
- Convey understanding of relationship between difference equations and related models/techniques.
- Suggest indications and contraindications, identify and respond to typical concerns and caveats
48. Every year 10 percent of the public housing units in Oakfrisco deteriorate to the point where they are uninhabitable and must be demolished. Current plans and budget constraints call for the construction of 800 new units per year. Is there an equilibrium number of housing units? Is it a stable equilibrium?
{{
I'd set this up as a difference equation. In time period 0 I have X total units and each year 0.1X units are taken off line and 800 come online. It would look like this
| Year |
Total Units |
Demolition |
New |
| 0 |
X |
0.1X |
800 |
| 1 |
0.9X+800 |
0.09X+80 |
800 |
As set up this is probably not quite right since the brand new units presumably don't deteriorate at the ten percent per year rate. I'd have to inquire as to how many years down the line do they start to fall apart (and presumably I'd think about the costs and benefits of building them in such a way that there is this 10% per year rate of becoming unusable – my gut would tell me this is not a good use of funds).
}}
49. Based on what we have learned about how the slope affects the type of equilibrium, draw arrows on the black line in this diagram to indicate where the system would go in the four labeled sections. Describe a situation that this diagram might represent.
