Chapter 3: Optimization in Microeconomics – Choosing the Best Feasible Option

Optimization in Microeconomics

Introduction to Optimization

Optimization is a fundamental concept in microeconomics, referring to the process by which economic agents—such as individuals, households, businesses, and governments—make choices that maximize their net benefit given constraints. The best feasible option is known as the optimum or optimal choice.

• Optimization: The process of selecting the best feasible option from a set of alternatives.

• Economic Agent: Any individual or entity making choices in an economic context.

• Optimum: The best feasible option, yielding the highest net benefit.

Example: A consumer choosing between different apartments based on rent and commuting costs.

Challenges in Making Optimal Choices

While optimization is the goal, several factors can make it difficult to always choose the best option:

• Limited Information: Economic agents may not have access to all relevant data.

• Complexity: Sorting through information and alternatives can be complicated.

• Inexperience: Lack of experience with a situation can hinder optimal decision-making.

Techniques of Optimization

Optimization Using Total Value

This technique involves calculating the total value (or net benefit) of each feasible option and selecting the one with the highest value. All costs and benefits are translated into common units, such as dollars per month.

• Total Value: The sum of all benefits minus the sum of all costs for each alternative.

• Net Benefit: $\text{Net Benefit} = \text{Total Benefit} - \text{Total Cost}$

• Choose the alternative with the highest net benefit.

Example: Comparing apartments by adding rent and commuting costs to find the lowest total cost.

Optimization Using Marginal Analysis

Marginal analysis focuses on the change in net benefit when moving from one alternative to another. It is often faster than total value analysis because it only considers differences between options.

• Marginal Cost (MC): The change in cost when switching between alternatives. $MC = \Delta \text{Cost}$

• Marginal Benefit (MB): The change in benefit when switching between alternatives. $MB = \Delta \text{Benefit}$

• Marginal Net Benefit (MNB): The change in net benefit. $MNB = MB - MC$

• The optimal choice is where moving away from it would decrease net benefit.

Example: Deciding whether to move to a different apartment by comparing the additional commuting cost to the benefit of a better view.

Application: Renting the Optimal Apartment

Translating Costs and Benefits

When comparing apartments, all costs (rent, commuting time, transportation, parking, wear and tear, opportunity cost of time) are converted into a common unit, typically dollars per month.

• Opportunity Cost of Time: The value of time spent commuting, calculated as hours per month multiplied by the value per hour.

• Total Cost: $\text{Total Cost} = \text{Rent} + \text{Commuting Cost}$

Example Table: Apartment Comparison

Apartment	Commuting Time (hours/month)	Rent ($/month)	Commuting Cost ($/month)	Total Cost ($/month)
Very Close	5	1180	50	1230
Close	10	1150	100	1250
Far	15	1100	150	1250
Very Far	20	1030	200	1230

Additional info: Table values inferred from context and typical textbook examples.

Effect of Opportunity Cost of Time

Changing the value of time (e.g., from $10/hour to $15/hour) affects the total cost calculation and can change which apartment is optimal.

• Higher opportunity cost of time increases the commuting cost, making closer apartments more attractive.

Example Table: Total Cost with Different Opportunity Costs

Apartment	Commuting Time (hours/month)	Rent ($/month)	Commuting Cost ($/month, $15/hr)	Total Cost ($/month)
Very Close	5	1180	75	1255
Close	10	1150	150	1300
Far	15	1100	225	1325
Very Far	20	1030	300	1330

Additional info: Table values inferred from context and typical textbook examples.

Marginal Analysis in Practice

Principle of Optimization at the Margin

The optimal feasible alternative is the one where moving away from it would make you worse off, and moving toward it would make you better off. Marginal analysis is especially useful when only a few aspects differ between alternatives.

• Focuses on incremental changes rather than total values.

• Helps identify the point where net benefit is maximized.

Formulas for Marginal Analysis

• $MC = \Delta \text{Cost}$

• $MB = \Delta \text{Benefit}$

• $MNB = MB - MC$

Evidence-Based Economics: Housing Location and Rent

How Location Affects Rental Cost

Empirical data shows that rental costs typically decrease as distance from the city center increases. This reflects the trade-off between commuting time and rent.

• Central locations have higher rents due to convenience and reduced commuting costs.

• Peripheral locations have lower rents but higher commuting costs.

Example Table: Rent vs. Distance from City Center

Distance from City Center (miles)	Rent ($/month)
0	1600
5	1300
10	1100
15	950

Additional info: Table values inferred from context and typical textbook examples.

Worked Problems: Apartment Choice Using Marginal and Total Value Analysis

Example Problem: Choosing Between Two Apartments

Suppose you are choosing between Apartment East and Apartment West, both costing $950/month. East has a better view valued at $25/month, but West is closer to the airport, saving you commuting time valued at $20/hour.

• Total Value Analysis: East yields $25 more benefit due to the view.

• Marginal Analysis: If West saves you 2 hours/month in commuting, the cost savings is $40. Net benefit of moving East to West is $25 (view) - $40 (commuting) = -$15. West is optimal.

Formula: $MNB = MB - MC$

Adjusting for Changes in Opportunity Cost

If your value of time increases (e.g., after a raise), the marginal cost of commuting increases, which can change the optimal choice.

• Recalculate $MC$ using the new value of time.

• Compare $MB$ and $MC$ to determine the optimal apartment.

Summary Table: Optimization Techniques

Technique	Method	Key Formula	Application
Total Value	Calculate net benefit for each option	$\text{Net Benefit} = \text{Total Benefit} - \text{Total Cost}$	Choose option with highest net benefit
Marginal Analysis	Compare incremental changes between options	$MNB = MB - MC$	Choose option where moving away decreases net benefit

Additional info: All tables and some explanations inferred and expanded for completeness and clarity.