Valuation & Benefit-Cost Analysis

Lecture

Dr. James Doss-Gollin

Monday, February 9, 2026

A Decision Problem

Today

1. A Decision Problem

2. Utility Theory

3. Dealing with Time

4. Handling Uncertainty

5. Worked Example: House Elevation

6. Critiques & Defense

7. Wrap-Up

Target: 1:00. Open with the hook — get students thinking about a concrete decision before introducing formalism.

How Much Would You Pay?

Imagine you own a home in Houston’s Meyerland neighborhood.

• Your house flooded in 2015, 2016, and 2017
• A contractor offers to elevate your house by 5 feet for $80,000
• Should you do it?

Target: 1:02. Pause for a few hands. Cold call 1-2 students for their gut reaction. Key point: students will have intuitions, but we need a systematic framework.

What Do We Need to Know?

• What does elevation cost? (up-front + disruption)
• What are the benefits? (avoided flood damage, insurance savings, peace of mind)
• How likely is future flooding?
• How long do we plan to live there?
• How do we compare costs now vs. benefits later?

Target: 1:05. This list motivates each piece of the BCA framework we’ll develop.

Utility Theory

Today

1. A Decision Problem

2. Utility Theory

3. Dealing with Time

4. Handling Uncertainty

5. Worked Example: House Elevation

6. Critiques & Defense

7. Wrap-Up

Utility Functions

We need a way to score how “good” a decision is. Define a utility function:

\[ u(a, \mathbf{s}): \mathcal{A} \times \mathcal{S} \to \mathbb{R} \]

• \(a \in \mathcal{A}\): a decision (or action)
• \(\mathbf{s} \in \mathcal{S}\): the state of the world (everything uncertain)
• \(u\): maps (decision, state) to a real number

Target: 1:08. Emphasize that “utility” is just a general term — could be called objective function, loss function, etc. The key idea is a function that takes a decision and an uncertain state and returns a number.

Choosing Your Metric

You are free to define “goodness” however you want.

• Monetary cost
• Lives saved
• Acres of wetland preserved
• Some weighted combination

This is a feature, not a bug — but it means every BCA embeds value judgments.

Target: 1:10. The freedom to choose metrics is powerful but also means BCA is never “objective.” Every choice of what to measure is a choice about what matters.

Money as a Common Unit

Utility is often measured in dollars.

Not because money is all that matters, but because it provides a common unit for comparing different things.

Tip

For house elevation, monetary costs and benefits include:

• Up-front construction cost
• Change in flood insurance premiums
• Change in property value
• Expected future flood damages avoided

Target: 1:12. Transition to the harder question: what about things that are hard to monetize?

Opportunity Costs

How do we put a dollar value on something that isn’t traded in a market?

Opportunity cost: the value of the next best alternative.

• If we can protect one square mile of pristine wetland for $1 million, that’s the cost of not protecting a comparable square mile elsewhere
• If a household spends 10 hours per flood event cleaning up, the value of their time is an opportunity cost
• But: are these really substitutes?

Target: 1:14. Opportunity cost is the economist’s main tool for monetizing non-market goods. It’s useful but imperfect — the “but are they really substitutes?” question is genuine and unresolved.

Decreasing Marginal Utility

A key insight from economics: the more you have of something, the less each additional unit is worth.

using CairoMakie

let
    fig = Figure(; size=(600, 350))
    ax = Axis(
        fig[1, 1];
        xlabel="Consumption (\$)",
        ylabel="Utility",
        xticklabelsvisible=false,
        yticklabelsvisible=false,
        xgridvisible=false,
        ygridvisible=false,
    )
    x = range(0.1, 10; length=200)
    lines!(ax, x, log.(x); linewidth=3)
    fig
end

Each additional dollar of consumption provides less additional utility.

Target: 1:16. Key example: $1,000 means a lot more to someone earning $30K/year than to someone earning $300K/year. This matters because BCA typically ignores who bears the costs and benefits.

Implication: Distribution and Social Welfare

If marginal utility decreases with wealth:

• A dollar of damage hurts a low-income household more than a wealthy one
• Standard BCA treats a dollar the same regardless of who receives it
• And there is no single “right” way to aggregate individual welfare into social welfare

We will revisit equity and values later in the semester.

Target: 1:18. Don’t try to resolve this — just make sure students see that “total benefits > total costs” is not the whole story. We’ll return to values and trade-offs in Week 9.

Dealing with Time

Today

1. A Decision Problem

2. Utility Theory

3. Dealing with Time

4. Handling Uncertainty

5. Worked Example: House Elevation

6. Critiques & Defense

7. Wrap-Up

Costs Now, Benefits Later

Climate adaptation costs and benefits are spread over decades.

How should we weigh a dollar of benefit 30 years from now against a dollar of cost today?

Target: 1:21. Rhetorical question — transition to discounting.

Net Present Value

Discount future values to the present using discount rate \(r\):

\[ NPV(a, \mathbf{s}) = \sum_{t=0}^{T} \frac{B_t(a, \mathbf{s}) - C_t(a, \mathbf{s})}{(1 + r)^t} \]

• \(B_t - C_t\): net benefits (benefits minus costs) in year \(t\)
• \(r\): discount rate (e.g., 2% per year)
• \(\frac{1}{(1+r)^t}\): how much we value a dollar \(t\) years from now
• \(T\): time horizon

Target: 1:23. Write the formula on the board if possible. Emphasize that this is a choice — the discount rate is not a law of nature.

Alternative Notation

You will also see NPV written as:

\[ NPV(a, \mathbf{s}) = \sum_{t=0}^{T} (1 - \gamma)^t \left[ B_t(a, \mathbf{s}) - C_t(a, \mathbf{s}) \right] \]

• These are mathematically equivalent — just different parameterizations
• Our formula: discount rate \(r\), discount factor \(\frac{1}{(1+r)^t}\)
• Alternative: discount rate \(\gamma\), discount factor \((1-\gamma)^t\)
• Relationship: \((1 - \gamma) = \frac{1}{1+r}\), so \(\gamma = \frac{r}{1+r}\)

Important

Always check which convention a paper or report uses — a “2% discount rate” gives different numbers depending on the formula!

Target: 1:25. Important to flag because students will encounter both conventions. Don’t spend too long here — just make sure they know to check.

Worked Example

At \(r = 2\%\), what is $1 in 10 years worth today?

\[ \frac{1}{(1.02)^{10}} \approx \$0.82 \]

What about $1 in 50 years?

\[ \frac{1}{(1.02)^{50}} \approx \$0.37 \]

At \(r = 7\%\): $1 in 50 years \(= \frac{1}{(1.07)^{50}} \approx \$0.03\)

Target: 1:27. The jump from 2% to 7% is dramatic — this foreshadows why the discount rate debate matters so much. Ask: “Which discount rate should we use for climate policy?”

Handling Uncertainty

Today

1. A Decision Problem

2. Utility Theory

3. Dealing with Time

4. Handling Uncertainty

5. Worked Example: House Elevation

6. Critiques & Defense

7. Wrap-Up

How BCA Is Commonly Practiced

Most real-world analyses pick a single “best guess” scenario.

• A Houston water utility picks one assumption for each uncertain input (population, climate, regulations, costs)
• Calculate NPV under this single scenario
• If NPV > 0, the project “passes”

Target: 1:29. This is how most BCA is actually done, especially outside of academic settings. Set up the critique.

Why Single-Scenario BCA Is Fragile

Decisions that look great under one scenario can fail badly under another.

• Build a reservoir for future demand → but what if future is drier than expected? (Reservoir doesn’t fill)
• Build a reservoir for future demand → but what if future is wetter than expected? (Reservoir was unnecessary — funds should have gone to flood protection)
• The “best guess” is almost certainly wrong in some dimension

Target: 1:31. This connects to a core theme of the course: single-scenario analysis is brittle. Students will encounter this again in robustness and exploratory modeling.

Expected Utility

A better approach: pick the decision with the best expected NPV:

\[ a^* = \arg \max_a \; \mathbb{E}_{\mathbf{s}} \left[ NPV(a, \mathbf{s}) \right] \]

Recall:

\[ \mathbb{E}_{\mathbf{s}} \left[ NPV(a, \mathbf{s}) \right] = \int p(\mathbf{s}) \, NPV(a, \mathbf{s}) \, d\mathbf{s} \]

This requires a probability distribution over states of the world.

Target: 1:33. Connect to Monte Carlo from Week 4: “You already know how to compute expectations with samples!” NPV already sums over time; the expectation integrates over uncertainty.

Connecting to Monte Carlo

You already know how to do this!

1. Sample many states of the world \(\mathbf{s}_1, \mathbf{s}_2, \ldots, \mathbf{s}_N\) from \(p(\mathbf{s})\)
2. Compute \(NPV(a, \mathbf{s}_i)\) for each sample
3. Average: \(\hat{\mathbb{E}}[NPV(a,\mathbf{s})] \approx \frac{1}{N}\sum_{i=1}^N NPV(a, \mathbf{s}_i)\)
4. Compare across actions \(a \in \mathcal{A}\)

Target: 1:35. This makes the abstract formalism concrete. Students did exactly this in Lab 4. Emphasize: BCA + Monte Carlo = expected utility analysis.

The Standard BCA Recipe

1. Define the set of actions \(\mathcal{A}\) (e.g., elevate by 0, 3, 5, or 8 feet)
2. Define costs and benefits \(B_t(a, \mathbf{s})\), \(C_t(a, \mathbf{s})\)
3. Choose a discount rate \(r\)
4. Estimate \(p(\mathbf{s})\) — probability distribution over states of the world
5. Pick \(a^* = \arg \max_a \; \mathbb{E}[NPV(a, \mathbf{s})]\)

Target: 1:36. Pause here. Now let’s apply it.

Worked Example: House Elevation

Today

1. A Decision Problem

2. Utility Theory

3. Dealing with Time

4. Handling Uncertainty

5. Worked Example: House Elevation

6. Critiques & Defense

7. Wrap-Up

Setting Up the Problem

Return to our Meyerland homeowner. Two actions: do nothing (\(a_0\)) or elevate 5 ft (\(a_1\)).

	Do Nothing (\(a_0\))	Elevate 5 ft (\(a_1\))
Up-front cost	$0	$80,000
Annual flood probability	5%	0.5%
Expected damage per flood	$50,000	$5,000
Expected annual damage	$2,500	$25

Target: 1:37. Walk through: EAD = probability \(\times\) damage. These are simplified numbers — real analysis would integrate over all flood depths. Ask students to compute the EAD values themselves before revealing.

Computing NPV

Annual benefit of elevation: \(\$2{,}500 - \$25 = \$2{,}475\) per year in avoided damages.

Over \(T = 30\) years at discount rate \(r\):

\[ NPV = -\$80{,}000 + \sum_{t=1}^{30} \frac{\$2{,}475}{(1+r)^t} \]

Discount rate \(r\)	PV of benefits	NPV	Pass?
2%	$55,500	-$24,500	No
5%	$38,000	-$42,000	No
7%	$30,700	-$49,300	No

At these numbers, elevation doesn’t pass BCA at any discount rate. What would need to change?

Target: 1:40. Pause and ask the class. Possible answers: higher flood probability, longer time horizon, include insurance savings, include non-monetary benefits. The point is that BCA forces you to articulate why you think a project is worthwhile.

Sensitivity: What If Flood Risk Is Higher?

Suppose climate change increases annual flood probability to 10% (EAD: $5,000/yr, benefit: $4,975/yr):

Discount rate \(r\)	PV of benefits	NPV	Pass?
2%	$111,500	+$31,500	Yes
5%	$76,500	-$3,500	Barely no
7%	$61,700	-$18,300	No

The same project passes or fails depending on two assumptions: flood probability and discount rate.

Target: 1:43. This is the key pedagogical moment. BCA is not giving us “the answer” — it’s showing us which assumptions drive the answer. This is why sensitivity analysis matters (Week 7).

Critiques & Defense

Today

1. A Decision Problem

2. Utility Theory

3. Dealing with Time

4. Handling Uncertainty

5. Worked Example: House Elevation

6. Critiques & Defense

7. Wrap-Up

White House Circular A-94

The 2023 revision of federal BCA guidance (U.S. Office of Management and Budget, 2023):

• Intended to govern how federal agencies conduct BCA
• Lowered the default discount rate from 7% to ~2%
• Expanded guidance on equity and distributional effects
• Its current status is uncertain under the new administration

Target: 1:44. The 2023 update was a significant shift — reflects decades of academic argument (including Arrow et al.) finally reaching policy. A shift from 7% to 2% can flip a project from “fails” to “passes,” as we just saw in our worked example.

The Discount Rate Problem

Two rationales for discounting (Arrow et al., 2013):

1. Consumption-based (~3%): Future generations will be richer, so an extra dollar matters less to them
2. Investment-based (~7%): A dollar invested today yields more than a dollar in the future

Both assume future generations are better off — what if climate change makes them worse off?

Target: 1:46. Preview for Wednesday’s reading (Arrow et al.).

High Discount Rates Favor Inaction

At \(r = 7\%\), damages 50 years out are worth $0.03 today.

• Infrastructure with 50-100 year lifetimes
• Sea-level rise over centuries
• Species extinction is permanent

The discount rate can determine whether a project “passes” BCA — more than any other single parameter.

Target: 1:48. This is why the A-94 update mattered so much. A shift from 7% to 2% can flip a project from “fails” to “passes.”

Decision-Support, Not Decision-Making

Important

BCA is a tool for informing decisions, not making them.

• It forces us to be explicit about our assumptions
• Applied well, it’s iterative refinement of understanding; applied poorly, it’s a black box that justifies a decision already made
• The number matters less than what you learn by computing it

Target: 1:50. This is the most important conceptual point of the lecture.

Wrap-Up

Today

1. A Decision Problem

2. Utility Theory

3. Dealing with Time

4. Handling Uncertainty

5. Worked Example: House Elevation

6. Critiques & Defense

7. Wrap-Up

Key Takeaways

1. BCA provides a systematic framework for comparing decisions — but every analysis embeds value judgments
2. Utility theory gives us the math; opportunity costs, discounting, and expected value give us the tools
3. The discount rate is a choice, not a fact — and it can flip a project from “fails” to “passes”
4. BCA is decision-support: the assumptions matter more than the final number

Wednesday Preview

Paper Discussion: Discounting & BCA

• Read Arrow et al. (2013)
• Optional: skim U.S. Office of Management and Budget (2023)
• Before class: look up how one federal agency (USACE, EPA, FEMA, or TWDB) conducts BCA — what discount rate do they use? What counts as a benefit?
• Come ready to discuss: which rationale for discounting do you find most convincing?

Target: 1:53. Remind students of the reading assignment and discussion questions posted on the course website.

References

Arrow, K., Cropper, M., Gollier, C., Groom, B., Heal, G., Newell, R., et al. (2013). Determining benefits and costs for future generations. Science, 341(6144), 349–350. https://doi.org/10.1126/science.1235665

U.S. Office of Management and Budget. (2023). Circular No. A-94: Guidelines and Discount Rates for Benefit-Cost Analysis of Federal Programs (Circular). Executive Office of the President. Retrieved from https://www.whitehouse.gov/wp-content/uploads/2023/11/CircularA-94.pdf