Transportation and Decision-Making: Quick Notes
Price per unit and usage thresholds
A core concept in economic decision-making is understanding the price per unit of any good, service, or resource. This metric allows for a direct comparison of costs between different options or over varying levels of consumption. The price per unit is generally defined as:
Priceperunit=\frac{TotalCost}{TotalUnitsQuantity}
Understanding the price per unit is crucial when considering opportunity cost because it enables you to accurately assess the true cost of using a resource or choosing an option. For example, when deciding between two alternatives, calculating the price per unit for each allows you to compare them on an equal basis. If you choose option A, the opportunity cost isn't just the nominal price of option B, but the value of the 'units' you forego from option B for the same 'cost' per unit. This helps in identifying the most efficient use of resources and ensures that decisions are based on the effective value and cost, rather than just the total expenditure.
Opportunity cost and sunk costs
Key concepts:
Opportunity cost: the value of the next best alternative you give up when you choose one option over another. It is what you forego by not choosing that other option.
Sunk costs: past expenditures that cannot be recovered. Do not let sunk costs bias current decisions; only future costs and benefits matter.
1. Resources, Alternatives, and Costs vs. Value
When choosing between two alternatives (say, option 1 and option 2), the opportunity cost of choosing option 1 is the value you forgo from option 2 plus the price/cost of option 2, compared to the cost of option 1.
Value (V) = what you gain from the choice (utility, satisfaction, or payoff).
Price/Cost (P) = what you spend in resources.
So, the foregone net gain of not picking option 2 is:
Formula:
OC(1)=(V2−P2)−(V1−P1)⟹V2−V1+P1−P2OC(1)=(V2−P2)−(V1−P1)⟹V2−V1+P1−P2
Explanation: We take the net value of option 2 (V₂ – P₂), subtract the net value of option 1 (V₁ – P₁). Rearranging gives the compact form V₂ – V₁ + P₁ – P₂, which is exactly your structure.
2. Marginal Costs and Marginal Benefits (Quantity Choice)
When deciding “how much” of a good to consume or produce, the opportunity cost is the sacrifice of resources at the margin. The rule is that we keep consuming/producing until the extra value (MB) equals the extra price/cost (MC).
Formula:
Choose Q where MB(Q)=MC(Q)Choose Q where MB(Q)=MC(Q)
Explanation: If MB > MC, we are missing out on value by stopping too early; if MB < MC, we’re wasting resources. The equality ensures no alternative use of resources would give a better payoff.
3. Total Benefits (Value) vs. Total Costs (Price)
Once a quantity is chosen, the opportunity cost of that decision is shown by comparing total value and total price. The decision is efficient only if total value outweighs total costs.
Formula:
Net Value(Q)=V(Q)−P(Q)Net Value(Q)=V(Q)−P(Q)
Explanation: If Net Value is positive, the chosen option creates gains. If negative, the resources could have been better used elsewhere (the opportunity cost is too high).
✅ Summary of the Three Principles with Formulas:
Alternatives (trade-offs): OC(1)=V2−V1+P1−P2OC(1)=V2−V1+P1−P2
Marginal decision rule: MB(Q)=MC(Q)MB(Q)=MC(Q)
Total evaluation: Net Value(Q)=V(Q)−P(Q)Net Value(Q)=V(Q)−P(Q)
Worked example: event ticket decision
Context numbers (for illustration):
Benefit of attending the event: B = 500
Price of attending: P = 200
Refundable portion of the price if you do not attend: R = 50
Other costs associated with attending (e.g., snacks, transit): S = 20
Value of the next best alternative (e.g., a D&D event): V_{alt} = 50
To evaluate whether to attend, compute the net benefit of attending over the alternative:
\text{Net benefit} = B - (P - R + S) - V_{alt}
Plugging in the numbers:
\text{Net benefit} = 500 - (200 - 50 + 20) - 50 = 500 - 170 - 50 = 280
Since the net benefit is positive, attending is preferable to the alternative under these values.
To find the thresholdRefund amount needed to make not attending as good as attending, set the net benefit to zero and solve for the refundable amount $R$:
0 = B - (P - R + S) - V{alt} \quad\Rightarrow\quad R{\text{threshold}} = P + S + V{alt} - B Using the numbers: R{\text{threshold}} = 200 + 20 + 50 - 500 = -230
A negative threshold means that, with these values, any positive refundable amount already makes attending preferable; not attending would only occur if the event’s benefit dropped below the combined cost and alt-value. In other words, under these numbers, attendance is clearly favored.
Assumptions to state explicitly (helps with quick review):
The value of the next-best alternative is given (here, V_{alt}=50).
The refundability structure is known (here, R=50 of the ticket price is refundable).
The benefit of attending is a fixed value (here, B=500).
Other costs (e.g., snacks, transit) are included in S.
The comparison is against the next best alternative, not against all possible options.
Quick takeaways
Use price per unit to decide on passes versus per-ride payments.
Distinguish between substitutes and unique goods to understand pricing flexibility.
Always compare benefits to the sum of (adjusted) costs plus the value of the next-best alternative, not just the sticker price.
Ignore sunk costs; focus on future costs and future benefits when making a choice.
Implicit Costs
These are the opportunity costs of using resources already owned by the firm, for which no direct money payment is made. They represent the value of the next best alternative use of a resource. For example, the implicit cost of using a building owned by a business is the rent the business could have earned by leasing it out.
Explicit Costs
These are direct payments made to others in the course of running a business, such as wages, rent, and material costs. Explicit costs are tangible out-of-pocket expenses.
Sunk Costs
Sunk costs are past expenditures that cannot be recovered. These costs should be ignored when making present and future economic decisions because they are unchangeable and irrelevant to the current trade-offs. Only future costs and benefits matter.
Marginal Costs
Marginal cost is the additional cost incurred from producing one more unit of a good or service. It is calculated as the change in total cost divided by the change in quantity.
\text{Marginal Cost} = \frac{\Delta \text{Total Cost}}{\Delta \text{Quantity}}
Marginal Benefit
Marginal benefit is the additional satisfaction or utility received from consuming one more unit of a good or service. Decision-makers often compare marginal cost with marginal benefit to optimize outcomes.
Total Benefit
Total benefit is the sum of all benefits received from an activity or from consuming a certain quantity of a good or service. It represents the overall value or satisfaction obtained.
Total Cost
Total cost is the sum of all explicit and implicit costs associated with an activity or production process. It includes all expenditures (explicit) and the value of forgone opportunities (implicit).
Net Benefits
Net benefits are calculated as the total benefits minus the total costs of a particular action or project. A positive net benefit indicates that the benefits outweigh the costs, making the action economically desirable.
\text{Net Benefits} = \text{Total Benefits} - \text{Total Costs}
Understanding Opportunity Cost Calculation
Opportunity cost is the value of the next best alternative that was not taken. To calculate it, identify all the benefits and costs of each option, focusing on what you must give up to pursue one over another. Remember to exclude sunk costs, as they do not affect current decisions. Economic decisions are made by comparing the net benefits of one choice against the net benefits (or implicit costs) of the best alternative. The true cost of any decision is the value of what is sacrificed.
Worked Example: Mei's Movie Ticket Decision
Problem: Mei paid 15 for a non-refundable ticket for a 2-hour long movie. She could sell this ticket for 10. If she doesn’t go to the movie, she could work for 2 hours and earn 50 (total). Based on this information, if Mei chooses work over the movie, her value from watching the movie is at most ______. A. 25.99 B. 39.99 C. 49.99 D. 59.99 E. None of the above.
Solution:
To determine Mei's decision, we apply the concept of opportunity cost. The 15 she initially paid for the ticket is a sunk cost because it's non-refundable and cannot be recovered regardless of her decision to go or not. Therefore, it is irrelevant for the current decision.
Instead, we focus on the opportunity cost of going to the movie. If Mei goes to the movie, she gives up:
The income from working: 50
The revenue from selling the ticket: 10
Thus, the total opportunity cost of going to the movie is 50 + 10 = 60
Now, let's consider the net benefits for each option:
Option 1: Go to the movie
Let V{movie} be the value Mei gets from watching the movie. Net Benefit of Movie = V{movie} - \text{Opportunity Cost} = V_{movie} - 60Option 2: Work (and sell the ticket)
Benefit from working = 50
Benefit from selling the ticket = 10
Net Benefit of Working = 50 + 10 = 60
Mei chooses work over the movie, which implies that her net benefit from working is greater than or equal to her net benefit from watching the movie:
\text{Net Benefit of Working} \ge \text{Net Benefit of Movie}
60 \ge V_{movie} - 60
This inequality directly compares her choice, however, the question asks for the maximum value from watching the movie that would still lead her to choose work. This means the value of the movie must be less than the opportunity cost of attending it, for her to prefer the alternative.
If Mei chooses work, it must be because the benefit she gets from the movie (V_{movie}) is less than the 60 she foregoes by attending the movie. In other words, if her personal value of watching the movie is less than or equal to the total lost income/revenue (her opportunity cost), she will choose to work.
Therefore, her value from watching the movie (V{movie}) must be at most 60. Among the given options, 59.99 is the highest value that satisfies V{movie} < 60, meaning if the movie's value is 59.99, she would strictly prefer to work (because 60 > 59.99).
Answer: D. 59.99
Worked Example: Event Ticket Decision
Context numbers (for illustration):
Benefit of attending the event: B = 500
Price of attending: P = 200
Refundable portion of the price if you do not attend: R = 50
Other costs associated with attending (e.g., snacks, transit): S = 20
Value of the next best alternative (e.g., a D&D event): V_{alt} = 50
To evaluate whether to attend, compute the net benefit of attending over the alternative:
\text{Net benefit} = B - (P - R + S) - V_{alt}
Plugging in the numbers:
\text{Net benefit} = 500 - (200 - 50 + 20) - 50 = 500 - 170 - 50 = 280
Since the net benefit is positive, attending is preferable to the alternative under these values.
To find the threshold refundable amount (R_{\text{threshold}}) needed to make not attending as good as attending, set the net benefit to zero and solve for the refundable amount R:
0 = B - (P - R + S) - V{alt} \quad\Rightarrow\quad R{\text{threshold}} = P + S + V_{alt} - B
Using the numbers:
R_{\text{threshold}} = 200 + 20 + 50 - 500 = -230
A negative threshold means that, with these values, any positive refundable amount already makes attending preferable; not attending would only occur if the event’s benefit dropped below the combined cost and alt-value. In other words, under these numbers, attendance is clearly favored.
Assumptions to state explicitly (helps with quick review):
The value of the next-best alternative is given (here, V_{alt}=50).
The refundability structure is known (here, R=50 of the ticket price is refundable).
The benefit of attending is a fixed value (here, B=500).
Other costs (e.g., snacks, transit) are included in S.
The comparison is against the next best alternative, not against all possible options.
Quick Takeaways
Always identify both explicit and implicit costs, especially the opportunity cost of your next best alternative.
Ignore sunk costs when making current decisions; focus solely on future costs and future benefits.
When comparing options, choose the one that offers the highest net benefit, which means the total benefits minus total costs (including opportunity costs).
Marginal analysis (comparing marginal costs and marginal benefits) is crucial for making decisions about small, incremental changes.