Optimization Lecture Notes

Chapter 3: Optimization

Optimization: Doing the Best You Can

• Optimization is about making the best choice given the available options.

Comparing Apartments

• Factors to consider when comparing apartments:

  • Rent cost

  • Commute time

• Examples of apartment locations and their rent and distance from downtown:

  • Downtown Anchorage: 0 miles, $1,621

  • Hillside Anchorage: 5 miles, $1,811

  • Wasilla: 43 miles, $1,373

  • Palmer: 43 miles, $1,345

  • Eagle River: 20 miles, $1,626

  • Downtown LA: 0 miles, $2,785

  • Santa Clarita: 34 miles, $2,564

  • Fontana: 50 miles, $1,905

Do You Always Make the Best Choice?

• Reasons why people may not always make the best choice:

  • Uncertainty/risk: Unforeseen changes that are beyond control.

  • Limited information: Not having all the necessary details.

  • Complexity: Difficulty in processing available information.

  • Inexperience: Lack of knowledge in a particular situation; experience leads to better outcomes.

Domain-Specific Risk Preferences

• People exhibit different risk preferences depending on the domain (e.g., financial risk vs. health/safety risk).

• Refer to related work by Elke Weber.

• Behavioral Economics (Econ 123) studies why people optimize in some situations but not in others.

• Examples of situations where people may not optimize:

  • Self-control problems (procrastination, addiction, etc.)

  • Inexperience or being new to a task

Optimization Application: Renting the Optimal Apartment

• Consider only time and rent cost.

• How do you value time (opportunity cost)?

• Example work locations and distances:

  • City center

  • Very close: 5 miles

  • Close: 10 miles

  • Far

  • Very far

Calculating the Dollar Cost of Commute Time

• If the round-trip commute takes 20 hours/month (very far) and the opportunity cost of time is $10/hour, then the dollar cost of that commute is:
  20 \frac{hours}{month} \times $10 \frac{$}{hour} = $200 \frac{$}{month}

Example: Commuting Time, Rent, and Opportunity Cost

• Suppose your opportunity cost is $10 per hour:

  Apartment	Commuting Time (hours/month)	Rent ($/month)
  Very Close	5 hours	$1,180
  Close	10 hours	$1,090
  Far	15 hours	$1,030
  Very Far	20 hours	$1,000

Total Cost Calculation

• $10 per hour opportunity cost:

  Apartment	Commuting Time (hours/month)	Commuting Cost ($/month)	Rent ($/month)	Total Cost: Rent + Commuting ($/month)
  Very Close	5 hours	$50	$1,180	$1,230
  Close	10 hours	$100	$1,090	$1,190
  Far	15 hours	$150	$1,030	$1,180
  Very Far	20 hours	$200	$1,000	$1,200

Total Cost Analysis

• Total Cost = Rent + Commuting & Opportunity Cost

• Example with $10 per hour opportunity cost shows the 'Far' apartment as the optimum.

Impact of Higher Opportunity Cost

• $15 per hour opportunity cost:

  Apartment	Commuting Time (hours/month)	Commuting Cost ($/month)	Rent ($/month)	Total Cost: Rent + Commuting ($/month)
  Very Close	5 hours	$75	$1,180	$1,255
  Close	10 hours	$150	$1,090	$1,240
  Far	15 hours	$225	$1,030	$1,255
  Very Far	20 hours	$300	$1,000	$1,300

Comparing Total Cost Curves

• Illustration of total cost curves for employees with $10/hour and $15/hour opportunity costs of time.

• The optimum apartment choice varies based on the opportunity cost.

Marginal Analysis

• An alternate way of analyzing: Marginal Analysis.

• Instead of looking at total costs and total benefits, we can look at incremental changes.

• Marginal Effect: The change in the total as quantity changes by one.

  • Examples: Grades, weight, etc.

• Optimization using marginal analysis is often faster to implement than optimization using total value because optimization using marginal analysis focuses only on the ways that alternatives differ.

• Marginal analysis forces us to focus on what is changing when we compare alternatives.

Optimization: Marginal Analysis Examples

• Dense urban to semi-urban?

  • Costs …

  • Benefits …

• 4 vs. a 5-day trip?

  • Costs of the 5th day …

  • Benefits of the 5th day …

  • If Benefits > Costs, then take the 5th day.

Marginal Analysis: Process

1. Translate all costs and benefits into common units, like dollars per month.

2. Calculate the marginal consequences of moving between alternatives.

3. Choose the best alternative with the property that moving to it makes you better off and moving away from it makes you worse off.

Marginal Analysis: Apartment Example

• We’ll solve the problem and the optimum won’t change—but the way that you think about the problem will.

Marginal Cost

Completed Marginal Analysis Table

Interpreting Marginal Analysis

• Moving from Very Close to Close is $40 less per month: MOVE.

• Moving from Close to Far is $10 less per month: MOVE.

• Moving from Far to Very Far is $20 more per month: DO NOT MOVE.

Principle of Optimization at the Margin

• An optimal feasible alternative has the property that moving to it makes you better off and moving away from it makes you worse off.

Total Cost Curve and Marginal Cost

• Illustration of Total Cost Curve with Marginal Costs highlighted between apartment options.

Evidence-Based Economics: Location and Rental Cost

• Q: How does location affect the rental cost of housing?

Portland Study

• Wilson and Frew (2002)

• Ring Road system around Portland.

• Rent around the Portland area for comparable apartments.

• Commute time vs. Rental Prices.

Findings: Apartment Rent vs. Distance

• Apartment Rent in Portland, Oregon, Depends on Distance from the City Center

• Approximate location of highways that ring the city center.

• Graph showing rent decreasing as distance from the city center increases.

Journal of Real Estate Practice and Education

• Estimating the Connection between Location and Property Value

• James Frew & Beth Wilson

• Pages 17-25 | Published online: 14 Jun 2020

• Abstract: A hedonic regression model of rent values that is used to assess property values in Portland, Oregon, an urban area that has multi-centric rent gradients, is presented.

Distance and Rental Prices

• “In most cities, though not all, the farther you are from the city center, the more rental costs fall (holding apartment quality fixed). For example, in Portland, Oregon, rents fall by 33 percent as you move from the city center to otherwise identical apartments 6 miles out of town.”

Anchorage, AK Rental Market

• The median rent for 2 bedroom apartments in Anchorage, AK is $1,531.

• The price range for 2 bedroom apartments is $945 to $2,995.

• In the last year, rent has increased by $46 compared to the previous year.

Eagle River, AK Rental Market

• The median rent for 2 bedroom apartments in Eagle River, AK is $1,292.

• The price range for 2 bedroom apartments is $1,200 to $1,700.

• In the last year, rent has increased by $316 compared to the previous year.

Wasilla, AK Rental Market

• The median rent for 2 bedroom apartments in Wasilla, AK is $1,450.

• The price range for 2 bedroom apartments is $1,200 to $1,800.

• In the last year, rent has increased by $600 compared to the previous year.

Evidence-Based Economics: Boston's

Evidence-Based Economics: Boston's Mass Pike Tolls

• Q: How much would drivers benefit if tolls were eliminated on the Massachusetts Turnpike?

Massachusetts Turnpike

• A major highway (I-90) that runs through Massachusetts.

• A portion of the Pike, west of Boston, is tolled.

Tolls on the Mass Pike

• The tolls raise about $150 million per year (2016).

• The tolls are used to pay for highway maintenance and other transportation projects.

Toll Removal

• Removal of the tolls would benefit drivers by saving them time and money.

• However, it would also reduce the amount of money available for transportation projects.

Benefits of Removing Tolls

• The Massachusetts Department of Transportation (MassDOT) studied the effects of removing the tolls (2016).

• MassDOT estimated that removing the tolls would:

  • Save drivers about $27 million per year in toll payments.

  • Save drivers about $25 million per year in time costs (due to reduced congestion).

Costs of Removing Tolls

• The costs of removing the tolls would include:

  • A reduction in the amount of money available for transportation projects.

  • An increase in congestion on the Pike.

Conclusion

• MassDOT concluded that the benefits of removing the tolls would outweigh the costs.

• However, the decision to remove the tolls is ultimately a political one.

Evidence-Based Economics: Boston's Mass Pike Tolls

• Q: How much would drivers benefit if tolls were eliminated on the Massachusetts Turnpike?

Massachusetts Turnpike

• A major highway (I-90) that runs through Massachusetts.

• A portion of the Pike, west of Boston, is tolled.

Tolls on the Mass Pike

• The tolls raise about $150 million per year (2016).

• The tolls are used to pay for highway maintenance and other transportation projects.

Toll Removal

• Removal of the tolls would benefit drivers by saving them time and money.

• However, it would also reduce the amount of money available for transportation projects.

Benefits of Removing Tolls

• The Massachusetts Department of Transportation (MassDOT) studied the effects of removing the tolls (2016).

• MassDOT estimated that removing the tolls would:

  • Save drivers about $27 million per year in toll payments.

  • Save drivers about $25 million per year in time costs (due to reduced congestion).

Costs of Removing Tolls

• The costs of removing the tolls would include:

  • A reduction in the amount of money available for transportation projects.

  • An increase in congestion on the Pike.

Conclusion

• MassDOT concluded that the benefits of removing the tolls would outweigh the costs.

• However, the decision to remove the tolls is ultimately a political one.