11. Measuring willingness to pay for climate change mitigation

What 1,512 Germans said they would pay, and how much the question format changed the answer

A German government-sponsored internet survey, 1,512 respondents, asking what people would pay each year for climate mitigation. Half were asked a dichotomous choice question — would you pay €X, yes or no — and half a two-way payment ladder, naming the most they would certainly pay and the least they would certainly refuse.

The point of the design is that both groups were asked about the same policy. Any difference in the answers is a property of the question, not of what people want.

New concepts: contingent valuation, Likert scales and reverse-coding, index construction, and Cronbach’s alpha for whether several questions measure one thing. Book pages: project, R walk-through, solutions.

using XLSX, DataFrames
using Statistics
using HypothesisTests, Distributions
using CairoMakie
using DoingEconomics

CairoMakie.activate!(type = "svg")
use_doingecon_theme!()
ImportantThis workbook needs a different reader

XLSX.readtable fails on this file:

KeyError: key "xl/worksheets/sheet2.xml" not found

The file is strict OOXML — the ISO variant of the xlsx format — and XLSX.jl’s strict-to-transitional conversion path cannot handle it, even though the sheet it names is present in the archive. XLSX.readxlsx opens it fine, so reading the sheet as a matrix and building the frame by hand works:

"""Read a worksheet as a DataFrame, taking row 1 as the header.

`XLSX.readtable` cannot open this workbook (strict OOXML), but `readxlsx` can, and
indexing a sheet with `[:]` gives the full cell matrix.
"""
function sheet_frame(path, name)
    cells = XLSX.readxlsx(path)[name][:]
    return DataFrame([cells[2:end, j] for j in axes(cells, 2)],
                     Symbol.(string.(cells[1, :])); makeunique = true)
end

path = rawpath("11", "climate-wtp.xlsx")
survey = sheet_frame(path, "Data")
(rows = nrow(survey), columns = ncol(survey))
(rows = 1512, columns = 79)

One consequence to watch: empty cells come back as nothing rather than missing on this path, so the numeric coercion below has to handle both.

dictionary = sheet_frame(path, "Data dictionary")
show_all(dictionary)
79×2 DataFrame
Row Variable Description
Any Any
1 id Participant ID number
2 sex Gender
3 age Age (given in bands)
4 education Highest educational attainment (1 = in school, 2 = without school degree, 3 = secondary general school, 4 = intermediate general school, 5 = polytechnic school, 6 = university preparatory school)
5 voc_training Highest vocational degree (1 = in training, 2 = no voc degree and not in training, 3 = apprenticeship/vocational school degree, 4 = technical college of the GDR, 5 = technical college, 6 = university of applied sciences, 7 = university)
6 prob_a Most important problem: unemployment
7 prob_b Most important problem: rising prices
8 prob_c most important problem: gov. debt
9 prob_d most important problem: taxes
10 prob_e most important problem: educational system
11 prob_f most important problem: public health system
12 prob_g most important problem: public pension system
13 prob_h most important problem: immigration
14 prob_i Most important problem: environment
15 prob_j most important problem: energy system
16 worry are you worried about global warming (1 = very worried, 5 = not at all worried)
17 know subjective knowledge about global warming
18 infoall How attentively did you read the info text?
19 costs DC format: costs in Euro per year
20 abst_format Treatment question format (ref = DC format, ladder = payment ladder (TWPL/PL))
21 DC_ref_outcome DC format: response
22 dcothers_dk DC format: don't know how others vote
23 dcothers_perc DC format: how many percent do you think will vote in favor?
24 PL_zerowtp PL format: Zero WTP
25 PL_novote PL format: I would not vote
26 WTP_plmin PL format: Price that individual would definitely vote in favour at
27 WTP_plmax PL format: Price that individual would no longer vote in favour at
28 PLothers_dk PL format: don't know how others vote
29 PLothers_perc PL format: how much on average do people say they would pay?
30 protest_1 I can not afford additional expenses.
31 protest_2 It's not fair that the costs of climate protection are to be borne by the households
32 protest_3 I already pay enough for climate protection.
33 protest_4 It does not help if Germany reduced its emissions, if not all, other countries do not
34 protest_5 The industry is doing far too little to protect the climate.
35 protest_6 I am generally opposed to price and tax increases.
36 protest_7 The government is not able to take the right measures to reduce emissions.
37 protest_8 There are many other areas in which the money can be put to better use.
38 protest_9 The specified reduction target is not reachable anyway.
39 protest_10 The money for climate protection should be taken from existing government revenue.
40 protest_11 Wealthier households should pay more for climate protection pay than me.
41 protest_12 I do not have enough information on future action on climate change.
42 protest_13 First other countries need to do more for climate protection before Germany does.
43 scepticism_1 It is too early to say whether climate change is a real problem.
44 scepticism_2 The dangers of climate change are generally exaggerated.
45 scepticism_3 The human impact on climate is generally greatly exaggerated.
46 scepticism_4 The Earth's climate system is very sensitive and can easily get out of balance.
47 scepticism_5 The climate system is too complex so scientists cannot make useful predictions.
48 scepticism_6 Climate change is mainly caused by human activity.
49 scepticism_7 There is clear evidence that global warming is taking place.
50 scepticism_8 Global warming occurs mainly due to natural fluctuations in the Earth's temperature.
51 AR_1 The German energy policy constitutes a substantial contribution to climate protection.
52 AR_2 Germany is responsible for high global CO2 emissions.
53 AR_3 Germany should reduce CO2 emissions, regardless of what other countries do.
54 AR_4 As long as other countries do not reduce their greenhouse gas emissions, climate change will still be a problem.
55 AR_5 The CO2 emissions of Germany have a significant effect on climate change.
56 AR_6 Germany's pioneering role will bring other countries to reduce CO2 emissions.
57 AR_7 Spending more money for climate protection will mitigate climate change.
58 AR_8 Germany's action against climate change are effective.
59 PN_1 I do not buy fruit and vegetables from distant lands, to reduce emissions.
60 PN_2 I feel obliged to take the impact of my daily activities on climate into account.
61 PN_3 I feel better when I'm reducing emissions.
62 PN_4 I have a bad conscience when I drive a car rather than use public transport.
63 PN_5 It is useless to reduce emissions unless everyone else is doing the same thing.
64 PN_6 I feel uncomfortable when I consume energy, thus causing emissions.
65 PN_7 In my daily activities I try to reduce as many emissions as I can.
66 PN_8 My social environment is of the opinion that I should do more for reducing CO2 emissions.
67 cog_1 The government interferes too much with the daily lives of people.
68 cog_2 Sometimes, the government must pass laws so that people act to their own advantage.
69 cog_3 The state should intervene as little as possible in economic matters.
70 cog_4 The government should stop dictating to people how they should live.
71 cog_5 The government should do more to achieve social goals, even if it restricts the freedom of the individual.
72 cog_6 The government is trying to do too much for too many people. We should let the individual have more personal responsibility.
73 liv_sit What best describes your living situation?
74 region In which region are you living?
75 occ_sit Which of the occupational situations from the list applies best to you?
76 kids_nr Number in children in househod
77 member Membership in environmental organisation
78 party Which political party would you vote?
79 hhnetinc Household net income (monthly, in Euros) 
"""Numeric value, or `missing` for blanks and anything non-numeric.

`tryparse` returns `nothing` on failure, which would silently overwrite text columns
with `nothing` if returned directly.
"""
function num(x)
    (x === missing || x === nothing) && return missing
    x isa Number && return Float64(x)
    parsed = tryparse(Float64, string(x))
    return parsed === nothing ? missing : parsed
end

const SCALE_ITEMS = ["scepticism_2", "scepticism_6", "scepticism_7",
                     "cog_1", "cog_2", "cog_3", "cog_4", "cog_5", "cog_6",
                     "PN_1", "PN_2", "PN_3", "PN_4", "PN_6", "PN_7"]
for col in vcat(SCALE_ITEMS, ["WTP_plmin", "WTP_plmax", "costs", "education"])
    survey[!, col] = num.(survey[!, col])
end

(format_values = unique(string.(survey.abst_format)),
 sex_values = unique(string.(survey.sex)),
 dc_outcomes = unique(string.(survey.DC_ref_outcome)))
(format_values = ["ref", "ladder"], sex_values = ["female", "male"], dc_outcomes = ["support referendum and pay", "do not support referendum and no pay", "missing", "would not vote"])

abst_format splits the sample: "ref" is the dichotomous-choice group and "ladder" the payment ladder.

Part 11.1 — Summarizing the data

Q1. What contingent valuation can and cannot measure

Contingent valuation asks people what they would pay for something not traded in a market. Nothing here is revealed behaviour — every number is a statement about a hypothetical. Four limitations bite particularly hard on this survey.

Hypothetical bias. Nobody was charged anything, so there is no cost to overstating. This is the central problem with stated preference and it is not symmetric: the incentive runs toward generosity on a socially approved cause. Revealed-preference methods avoid it by construction, which is why the comparison table in the valuation literature puts stated preference below revealed preference on exactly this axis.

Social desirability. Climate protection is a norm-laden topic in Germany, and a respondent who thinks the “right” answer is a high number can give one at no cost. This is what makes the DC/TWPL comparison in Part 11.2 interesting rather than merely methodological: the two formats offer different opportunities to express approval, and if the answers differ, some of what is being measured is the expression rather than the valuation.

Scope and embedding insensitivity. Stated WTP tends not to scale with the size of the thing being valued — people name similar amounts for saving 2,000 birds and 200,000. A single mitigation target, as here, gives no way to check whether the amounts track the quantity of abatement at all.

Payment vehicle and protest responses. The survey asks about an annual payment, and the protest_1 to protest_13 items exist precisely because some respondents reject the premise rather than valuing the good at zero: “the industry is doing far too little”, “wealthier households should pay more”, “the money should come from existing government revenue”. A zero from someone who objects to the mechanism is not a zero valuation, and treating the two alike biases the mean downward.

The one limitation that matters less here is sample representativeness. It is a serious problem for estimating a national WTP figure, but the DC-versus-TWPL comparison is internal to the sample, and both groups were drawn the same way.

Q2. Recoding

(a) Reverse-coding. Attitudes came on a 1–5 Likert scale, but the statements do not all point the same way. scepticism_6 is “climate change is mainly caused by human activity” — agreeing means less scepticism, the opposite of scepticism_2 (“the dangers are generally exaggerated”). Averaging them as they stand would cancel the signal rather than accumulate it.

# Reversing on a 1-5 scale maps 1<->5 and 2<->4, which is 6 - x.
const REVERSED = ["cog_2", "cog_5", "scepticism_6", "scepticism_7"]
for col in REVERSED
    survey[!, col] = [v === missing ? missing : 6.0 - v for v in survey[!, col]]
end

DataFrame(item = REVERSED,
          statement = [only(dictionary[dictionary[!, 1] .== c, 2]) for c in REVERSED],
          new_min = [minimum(skipmissing(survey[!, c])) for c in REVERSED],
          new_max = [maximum(skipmissing(survey[!, c])) for c in REVERSED])
4×4 DataFrame
Row item statement new_min new_max
String String Float64 Float64
1 cog_2 Sometimes, the government must pass laws so that people act to their own advantage. 1.0 5.0
2 cog_5 The government should do more to achieve social goals, even if it restricts the freedom of the individual. 1.0 5.0
3 scepticism_6 Climate change is mainly caused by human activity. 1.0 5.0
4 scepticism_7 There is clear evidence that global warming is taking place. 1.0 5.0

(b) Ladder categories into euros. WTP_plmin and WTP_plmax hold category numbers 1–14, not amounts. The survey’s actual figures:

const EURO_AMOUNT = Dict(1 => 48.0, 2 => 72.0, 3 => 84.0, 4 => 108.0, 5 => 156.0,
                         6 => 192.0, 7 => 252.0, 8 => 324.0, 9 => 432.0, 10 => 540.0,
                         11 => 720.0, 12 => 960.0, 13 => 1200.0, 14 => 1440.0)

to_euro(v) = v === missing ? missing : get(EURO_AMOUNT, Int(v), missing)
survey.min_euro = to_euro.(survey.WTP_plmin)
survey.max_euro = to_euro.(survey.WTP_plmax)

DataFrame(category = 1:14, euro_per_year = [EURO_AMOUNT[k] for k in 1:14],
          step_ratio = [k == 1 ? missing : round(EURO_AMOUNT[k] / EURO_AMOUNT[k-1], digits = 3)
                        for k in 1:14])
14×3 DataFrame
Row category euro_per_year step_ratio
Int64 Float64 Float64?
1 1 48.0 missing
2 2 72.0 1.5
3 3 84.0 1.167
4 4 108.0 1.286
5 5 156.0 1.444
6 6 192.0 1.231
7 7 252.0 1.312
8 8 324.0 1.286
9 9 432.0 1.333
10 10 540.0 1.25
11 11 720.0 1.333
12 12 960.0 1.333
13 13 1200.0 1.25
14 14 1440.0 1.2

The categories are roughly geometric, not linear. Each step is about 1.2 to 1.5 times the one below, so the ladder spans €48 to €1,440 — a factor of 30 — in fourteen steps. That matters for Part 11.2: a mean taken over these amounts is sensitive to the top categories in a way a mean over evenly spaced options would not be, because the top rungs are far apart.

Q3. Three indices

Each index is the row mean of its items, using the reverse-coded versions where relevant.

const INDEX_ITEMS = [
    "climate" => ["scepticism_2", "scepticism_6", "scepticism_7"],
    "gov_intervention" => ["cog_1", "cog_2", "cog_3", "cog_4", "cog_5", "cog_6"],
    "pro_environment" => ["PN_1", "PN_2", "PN_3", "PN_4", "PN_6", "PN_7"],
]

for (name, items) in INDEX_ITEMS
    survey[!, name] = [all(!ismissing(row[c]) for c in items) ?
                       mean(Float64[row[c] for c in items]) : missing
                       for row in eachrow(survey)]
end

DataFrame(index = first.(INDEX_ITEMS),
          items = [length(i) for (_, i) in INDEX_ITEMS],
          scored = [count(!ismissing, survey[!, n]) for (n, _) in INDEX_ITEMS],
          mean = [round(mean(skipmissing(survey[!, n])), digits = 3) for (n, _) in INDEX_ITEMS],
          min = [minimum(skipmissing(survey[!, n])) for (n, _) in INDEX_ITEMS],
          max = [maximum(skipmissing(survey[!, n])) for (n, _) in INDEX_ITEMS])
3×6 DataFrame
Row index items scored mean min max
String Int64 Int64 Float64 Float64 Float64
1 climate 3 1512 2.344 1.0 5.0
2 gov_intervention 6 1512 3.173 1.0 5.0
3 pro_environment 6 1512 3.019 1.0 5.0

A naming caution that affects how every later result reads. After reverse-coding, all three scepticism items point the same way — higher means more sceptical. The book calls this index “belief that climate change is a real phenomenon”, but as computed a high value is scepticism and a low value is belief. Its mean of about 2.3 on a 1–5 scale therefore says this sample is, on average, not very sceptical. The sign of its correlation with WTP in Part 11.2 only makes sense read that way.

Q4. Do the items in each index measure one thing?

(a) Correlations within each index.

const ITEM_LABELS = Dict(
    "scepticism_2" => "exaggeration", "scepticism_6" => "not.human.activity",
    "scepticism_7" => "no.evidence",
    "cog_1" => "too.much", "cog_2" => "not.pass.laws", "cog_3" => "minimal.intervention",
    "cog_4" => "not.dictate", "cog_5" => "indiv.freedom",
    "cog_6" => "personal.responsibility",
    "PN_1" => "buy.local", "PN_2" => "indiv.impact", "PN_3" => "feel.better",
    "PN_4" => "public.transport", "PN_6" => "conserve.energy",
    "PN_7" => "reduce.emissions")

pairs_table = DataFrame(index = String[], item_a = String[], item_b = String[],
                        correlation = Float64[])
for (name, items) in INDEX_ITEMS, i in eachindex(items), j in 1:i-1
    a = Float64.(collect(skipmissing(survey[!, items[i]])))
    b = Float64.(collect(skipmissing(survey[!, items[j]])))
    push!(pairs_table, (name, ITEM_LABELS[items[i]], ITEM_LABELS[items[j]],
                        round(cor(a, b), digits = 2)))
end
show_all(pairs_table)
33×4 DataFrame
Row index item_a item_b correlation
String String String Float64
1 climate not.human.activity exaggeration 0.39
2 climate no.evidence exaggeration 0.42
3 climate no.evidence not.human.activity 0.46
4 gov_intervention not.pass.laws too.much 0.25
5 gov_intervention minimal.intervention too.much 0.32
6 gov_intervention minimal.intervention not.pass.laws 0.12
7 gov_intervention not.dictate too.much 0.68
8 gov_intervention not.dictate not.pass.laws 0.28
9 gov_intervention not.dictate minimal.intervention 0.33
10 gov_intervention indiv.freedom too.much 0.29
11 gov_intervention indiv.freedom not.pass.laws 0.41
12 gov_intervention indiv.freedom minimal.intervention 0.02
13 gov_intervention indiv.freedom not.dictate 0.27
14 gov_intervention personal.responsibility too.much 0.41
15 gov_intervention personal.responsibility not.pass.laws 0.08
16 gov_intervention personal.responsibility minimal.intervention 0.31
17 gov_intervention personal.responsibility not.dictate 0.46
18 gov_intervention personal.responsibility indiv.freedom 0.1
19 pro_environment indiv.impact buy.local 0.48
20 pro_environment feel.better buy.local 0.43
21 pro_environment feel.better indiv.impact 0.63
22 pro_environment public.transport buy.local 0.42
23 pro_environment public.transport indiv.impact 0.44
24 pro_environment public.transport feel.better 0.46
25 pro_environment conserve.energy buy.local 0.41
26 pro_environment conserve.energy indiv.impact 0.5
27 pro_environment conserve.energy feel.better 0.52
28 pro_environment conserve.energy public.transport 0.57
29 pro_environment reduce.emissions buy.local 0.46
30 pro_environment reduce.emissions indiv.impact 0.65
31 pro_environment reduce.emissions feel.better 0.59
32 pro_environment reduce.emissions public.transport 0.39
33 pro_environment reduce.emissions conserve.energy 0.46
Table 1: Correlation between the items making up each index. Reproduces the book’s Solution figures 11.1, 11.2 and 11.3.

Every pair matches the published tables. All twenty-four correlations are positive, which is the first thing to check after reverse-coding — a negative entry would mean an item still points the wrong way.

(b) Cronbach’s alpha.

DataFrame(index = first.(INDEX_ITEMS),
          items = [length(i) for (_, i) in INDEX_ITEMS],
          alpha = [round(cronbach_alpha(reduce(hcat, [Float64.(collect(skipmissing(survey[!, c])))
                                                      for c in items])), digits = 4)
                   for (_, items) in INDEX_ITEMS])
3×3 DataFrame
Row index items alpha
String Int64 Float64
1 climate 3 0.6848
2 gov_intervention 6 0.7133
3 pro_environment 6 0.8521
Table 2: Cronbach’s alpha for each index. Computed by cronbach_alpha from the shared helpers, which is tested in test/runtests.jl.

Alpha asks whether items behave like repeated measurements of one quantity:

\[\alpha = \frac{k}{k-1}\left(1 - \frac{\sum_i s_i^2}{s_T^2}\right)\]

where \(s_i^2\) is item \(i\)’s variance and \(s_T^2\) the variance of the row totals. If the items were independent, the total variance would just be the sum of the item variances and the bracket would collapse to zero. The more the items covary, the larger \(s_T^2\) grows relative to that sum.

Two of the three clear the conventional 0.7 threshold, and the climate index does not — 0.685, just short. The book describes all the values as “high”; on the usual reading one of them is borderline.

The ordering is the informative part, and it tracks item count exactly: the six-item pro_environment index scores 0.852, the six-item gov_intervention index 0.713, and the three-item climate index 0.685. Three points follow:

  • Alpha rises mechanically with the number of items, because adding items inflates the total variance faster than the sum of item variances. So the climate index is not necessarily built from worse items — it is built from fewer. Its three items correlate 0.39 to 0.46 with each other, which is comparable to several pairs inside the six-item government index.
  • The 0.7 line is a convention, not a test. Nothing changes at 0.699. Treating 0.685 as a failure and 0.713 as a success reads far more into the number than it carries, which is why the item correlations above are the more useful diagnostic.
  • Alpha measures consistency, not validity. Six questions that all mis-measure the same thing score beautifully. It says the items hang together, not that they capture the concept the index is named after — and Q3 already showed this index is named for the opposite of what it counts.

Q5. Are the two groups comparable on demographics?

The whole design rests on this. If the DC and ladder groups differ in who they are, a difference in their answers cannot be attributed to the question format.

const FORMATS = ["ref" => "DC", "ladder" => "TWPL"]
group_rows(code) = survey[coalesce.(survey.abst_format .== code, false), :]

function share_table(column, label)
    out = DataFrame()
    out[!, label] = String[]
    for (_, name) in FORMATS
        out[!, "$name (%)"] = Float64[]
    end
    values = sort(unique(string(v) for v in survey[!, column]
                         if v !== missing && v !== nothing))
    for v in values
        row = Any[v]
        for (code, _) in FORMATS
            g = group_rows(code)
            present = [string(x) for x in g[!, column] if x !== missing && x !== nothing]
            push!(row, round(100 * count(==(v), present) / length(present), digits = 2))
        end
        push!(out, row)
    end
    return out
end

vcat(share_table(:sex, "category"), share_table(:age, "category"))
8×3 DataFrame
Row category DC (%) TWPL (%)
String Float64 Float64
1 female 52.29 51.78
2 male 47.71 48.22
3 18 - 24 9.64 9.49
4 25 - 29 8.65 8.3
5 30 - 39 17.2 17.79
6 40 - 49 22.56 22.33
7 50 - 59 23.86 24.11
8 60 - 69 18.09 17.98
Table 3: Composition of the two question-format groups. Reproduces the book’s Solution figures 11.4 to 11.9. 
DataFrame(format = last.(FORMATS), code = first.(FORMATS),
          respondents = [nrow(group_rows(c)) for (c, _) in FORMATS])
2×3 DataFrame
Row format code respondents
String String Int64
1 DC ref 1006
2 TWPL ladder 506

The two groups are close to identical in composition. Women are 52.29% of the DC group and 51.78% of the ladder group — a gap of half a percentage point. Every age band matches within about half a point too.

Note the group sizes: 1,006 in DC against 506 in the ladder, so this was a 2:1 split rather than an even one. That does not threaten comparability, but it means the ladder estimates rest on half as many people and will carry wider intervals.

Q6. Are the two groups comparable on attitudes?

Demographics being balanced does not guarantee attitudes are, and attitudes are the more direct threat: a group that happens to care more about climate would pay more regardless of format.

balance = DataFrame(index = String[], format = String[], mean = Float64[], sd = Float64[],
                    min = Float64[], max = Float64[], n = Int[])
for (name, _) in INDEX_ITEMS, (code, label) in FORMATS
    v = Float64.(collect(skipmissing(group_rows(code)[!, name])))
    push!(balance, (name, label, round(mean(v), digits = 2), round(std(v), digits = 2),
                    minimum(v), maximum(v), length(v)))
end
balance
6×7 DataFrame
Row index format mean sd min max n
String String Float64 Float64 Float64 Float64 Int64
1 climate DC 2.37 0.85 1.0 5.0 1006
2 climate TWPL 2.29 0.84 1.0 5.0 506
3 gov_intervention DC 3.19 0.66 1.0 5.0 1006
4 gov_intervention TWPL 3.15 0.7 1.0 5.0 506
5 pro_environment DC 3.01 0.82 1.0 5.0 1006
6 pro_environment TWPL 3.03 0.79 1.0 5.0 506
Table 4: The three attitude indices by question format. Reproduces the book’s Solution figures 11.10, 11.11 and 11.12.

Every mean and standard deviation matches the published tables.

The largest gap between the two groups on any index is 0.08 points on a 1–5 scale, against within-group standard deviations of 0.66 to 0.85. The groups are, for this purpose, interchangeable: climate scepticism 2.37 against 2.29, government intervention 3.19 against 3.15, personal responsibility 3.01 against 3.03.

Taken with Q5, that licenses the whole of Part 11.2. Any difference in stated WTP between these two groups is a property of the question, because the people are the same. It is worth being clear that this is what randomisation buys — and that the check was worth running rather than assuming, since the split turned out to be 2:1 rather than even.

Part 11.2 — Comparing willingness to pay across methods

Q1. The ladder responses

(a) The distribution of the two ladder answers.

ladder = group_rows("ladder")
counts(col) = [count(v -> v !== missing && Int(v) == k, ladder[!, col]) for k in 1:14]
min_counts, max_counts = counts(:WTP_plmin), counts(:WTP_plmax)

fig = Figure(size = (920, 500))
ax = Axis(fig[1, 1];
          title = "Both answers pile up at the bottom of the ladder",
          xlabel = "Ladder category (1 = €48 per year, 14 = €1,440)",
          ylabel = "Respondents",
          xticks = 1:14)
for (j, (c, lab)) in enumerate([(min_counts, "Would certainly pay (WTP_plmin)"),
                                (max_counts, "Would certainly refuse (WTP_plmax)")])
    barplot!(ax, (1:14) .+ (j == 1 ? -0.19 : 0.19), c;
             width = 0.34, color = series_color(j), label = lab)
end
Legend(fig[0, 1], ax; orientation = :horizontal, framevisible = false)
fig
Figure 1: Distribution of the lowest amount respondents would certainly pay and the lowest they would certainly refuse, over the fourteen ladder categories. Two series, so the question carries the colour; bars are dodged with a surface gap. The horizontal axis is the category number, which is what respondents actually chose — the euro amounts behind it are not evenly spaced. 
DataFrame(category = 1:14, euros = [EURO_AMOUNT[k] for k in 1:14],
          would_pay = min_counts, would_refuse = max_counts)
14×4 DataFrame
Row category euros would_pay would_refuse
Int64 Float64 Int64 Int64
1 1 48.0 154 8
2 2 72.0 37 42
3 3 84.0 15 26
4 4 108.0 57 52
5 5 156.0 20 39
6 6 192.0 12 24
7 7 252.0 23 33
8 8 324.0 7 17
9 9 432.0 4 19
10 10 540.0 6 12
11 11 720.0 5 18
12 12 960.0 4 9
13 13 1200.0 2 18
14 14 1440.0 2 31
Table 5: Frequency of each ladder category for the two answers.

Both distributions are heavily skewed toward the low categories, and the two are shaped differently in a revealing way. The “would certainly pay” answers concentrate hardest at the very bottom, while the “would certainly refuse” answers sit higher and are more spread out — which is mechanically necessary, since each respondent’s refuse-point is above their pay-point, but the size of the gap is what the design is trying to elicit.

There is visible clustering on particular rungs rather than a smooth decline, which is the signature of a categorical instrument: respondents anchor on round-looking options.

(b) and (c) Average WTP.

survey.WTP_average = [(a === missing || b === missing) ? missing : (a + b) / 2
                      for (a, b) in zip(survey.min_euro, survey.max_euro)]
twpl_wtp = Float64.(collect(skipmissing(survey.WTP_average)))

(respondents = length(twpl_wtp),
 mean = round(mean(twpl_wtp), digits = 2),
 median = round(median(twpl_wtp), digits = 2),
 sd = round(std(twpl_wtp), digits = 2),
 range = extrema(twpl_wtp))
(respondents = 348, mean = 268.53, median = 132.0, sd = 287.71, range = (48.0, 1440.0))

Mean €268.53 and median €132.00, both matching the book. The mean is more than double the median, which is the whole story of this variable: a minority naming amounts near the top of the ladder drags the average far above the typical respondent. With €48 to €1,440 available and a median of €132, most people sat on the bottom few rungs.

Of the 506 ladder respondents, 348 produced a usable average — the rest either named no amount, said they would pay zero, or said they would not vote.

(d) What predicts willingness to pay.

survey.female = [(s === missing || s === nothing) ? missing :
                 Float64(lowercase(string(s)) == "female") for s in survey.sex]

wtp_corr = DataFrame(variable = String[], correlation = Float64[], n = Int[])
for col in [:education, :female, :climate, :gov_intervention, :pro_environment]
    ok = .!ismissing.(survey.WTP_average) .& .!ismissing.(survey[!, col])
    push!(wtp_corr, (string(col),
                     round(cor(Float64.(survey[ok, :WTP_average]),
                               Float64.(survey[ok, col])), digits = 8), count(ok)))
end
wtp_corr
5×3 DataFrame
Row variable correlation n
String Float64 Int64
1 education 0.138174 348
2 female -0.0369497 348
3 climate -0.144621 348
4 gov_intervention -0.188452 348
5 pro_environment 0.187503 348
Table 6: Correlation between average ladder WTP and respondent characteristics. Reproduces the book’s R walk-through 11.6 output.

All five reproduce the published values to eight decimals, with one sign difference explained below.

Every correlation is small — the largest is under 0.19 — so respondent characteristics explain very little of what people said they would pay. Within that, the signs are all interpretable and all point the same way:

  • Climate scepticism −0.145. More sceptical respondents would pay less. Recall from Q3 that a high value on this index is scepticism, so the negative sign is the expected direction, not a puzzle.
  • Preference for government intervention −0.188, the strongest of the five. The index is coded so that high means less appetite for state action, so people who want government to intervene more would pay more.
  • Personal responsibility +0.188. Respondents who feel obliged to act pro-environmentally would pay more.
  • Education +0.138. More educated respondents would pay more, which is the usual finding and confounded with income.
  • Sex −0.037. Effectively zero.
NoteThe sex correlation’s sign depends on an arbitrary coding choice

The book prints +0.03694972; this gives −0.03694972 — the same magnitude, opposite sign. The variable is text, "female" or "male", so turning it into a number requires picking which one is 1. Coding female as 1 gives the negative value; coding male as 1 gives the book’s.

Neither is wrong, but neither is self-explanatory either, which is the point: a dummy’s sign is meaningless without stating its coding. This page codes female = 1, so the negative correlation reads “women stated slightly lower WTP”.

Q2. The dichotomous-choice demand curve

The DC group each saw one amount and voted. Aggregating across the amounts traces out something that looks like a demand curve.

(a) Votes by amount.

const YES = "support referendum and pay"
const NO = "do not support referendum and no pay"
const ABSTAIN = "would not vote"

dc = group_rows("ref")
amounts = sort(unique(skipmissing(dc.costs)))
votes = DataFrame(euros = Float64[], yes = Int[], no = Int[], abstain = Int[], total = Int[],
                  pct_yes_abstain_as_no = Float64[], pct_yes_abstain_dropped = Float64[])
for a in amounts
    rows = dc[coalesce.(dc.costs .== a, false), :]
    outcome = [string(v) for v in rows.DC_ref_outcome]
    y, n, ab = count(==(YES), outcome), count(==(NO), outcome), count(==(ABSTAIN), outcome)
    push!(votes, (a, y, n, ab, y + n + ab,
                  round(100 * y / (y + n + ab), digits = 2),
                  round(100 * y / (y + n), digits = 2)))
end
votes
14×7 DataFrame
Row euros yes no abstain total pct_yes_abstain_as_no pct_yes_abstain_dropped
Float64 Int64 Int64 Int64 Int64 Float64 Float64
1 48.0 32 21 12 65 49.23 60.38
2 72.0 40 30 11 81 49.38 57.14
3 84.0 45 24 12 81 55.56 65.22
4 108.0 31 35 7 73 42.47 46.97
5 156.0 40 31 13 84 47.62 56.34
6 192.0 25 25 11 61 40.98 50.0
7 252.0 28 32 9 69 40.58 46.67
8 324.0 27 41 16 84 32.14 39.71
9 432.0 29 35 11 75 38.67 45.31
10 540.0 22 31 9 62 35.48 41.51
11 720.0 13 39 12 64 20.31 25.0
12 960.0 15 28 14 57 26.32 34.88
13 1200.0 21 42 11 74 28.38 33.33
14 1440.0 15 42 19 76 19.74 26.32
Table 7: Dichotomous-choice votes by the amount offered, as counts and as row percentages. Reproduces the book’s Solution figure 11.17.

(b), (c) and (d) The demand curve, both ways of handling abstentions.

fig = Figure(size = (900, 500))
ax = Axis(fig[1, 1];
          title = "Demand falls with price, and how you count abstentions shifts the level",
          xlabel = "Annual amount offered (€, log scale)",
          ylabel = "Voting yes (%)",
          xscale = log10, xticks = (amounts, string.(Int.(amounts))),
          limits = (nothing, (0, 100)))
for (j, (col, lab)) in enumerate([(:pct_yes_abstain_as_no, "Abstain counted as no"),
                                  (:pct_yes_abstain_dropped, "Abstain excluded")])
    lines!(ax, votes.euros, votes[!, col]; color = series_color(j), linewidth = 2.5)
    scatter!(ax, votes.euros, votes[!, col]; color = series_color(j))
    text!(ax, votes.euros[end], votes[!, col][end]; text = lab, fontsize = 10,
          align = (:right, :top), offset = (-6, -8), color = INK_SECONDARY)
end
fig
Figure 2: Share voting yes against the annual amount offered, under the two treatments of abstentions. Two series, so the treatment carries the colour, and both are direct-labelled. The horizontal axis is on a log scale because the offered amounts are roughly geometric — on a linear axis the four lowest amounts would be indistinguishable.

The curve slopes down, which is the basic sanity check — a higher price brings fewer yes votes — and it is worth appreciating that this is not guaranteed in a contingent-valuation survey. Respondents seeing only one amount cannot anchor on the others, so the downward slope is evidence that people responded to the price rather than to the topic.

Two features stand out. The curve is far from a straight line even on a log axis, and the share saying yes does not approach zero at the top amount — a floor of respondents will approve any amount, which is what “protest in the other direction” looks like.

How abstentions are treated shifts the level but not the shape. Counting them as no is the conservative reading: someone who declines to vote has not agreed to pay. Excluding them assumes abstainers would have split like those who voted. The two curves stay roughly parallel, so the qualitative conclusion does not depend on the choice — but the level difference is large enough that a WTP estimate read off this curve would.

The conservative treatment is the defensible default here, because the alternative assumes abstention is unrelated to price, and the table shows abstention is not evenly spread across amounts.

Q3. Does the question format change the answer?

(a) The two formats side by side. For DC respondents, willingness to pay is the amount they were offered and accepted; for ladder respondents it is the average of their two figures.

dc_wtp = Float64[row.costs for row in eachrow(dc)
                 if row.costs !== missing && string(row.DC_ref_outcome) == YES]

comparison = DataFrame(
    format = ["DC", "TWPL"],
    n = [length(dc_wtp), length(twpl_wtp)],
    mean = round.([mean(dc_wtp), mean(twpl_wtp)], digits = 2),
    sd = round.([std(dc_wtp), std(twpl_wtp)], digits = 2),
    median = [median(dc_wtp), median(twpl_wtp)])
comparison
2×5 DataFrame
Row format n mean sd median
String Int64 Float64 Float64 Float64
1 DC 383 348.19 378.65 192.0
2 TWPL 348 268.53 287.71 132.0
Table 8: Willingness to pay by question format. DC uses the offered amount for respondents who voted yes. Reproduces the book’s Solution figure 11.21.

Every value matches the published table: DC mean €348.19 on 383 respondents, ladder mean €268.53 on 348, with medians of €192 and €132.

(b) A confidence interval for the difference.

test = UnequalVarianceTTest(dc_wtp, twpl_wtp)
lower, upper = confint(test)

DataFrame(quantity = ["Difference in means (DC − TWPL)", "Standard error of the difference",
                      "95% CI lower", "95% CI upper", "95% CI half-width", "p-value"],
          value = round.([mean(dc_wtp) - mean(twpl_wtp),
                          sqrt(var(dc_wtp) / length(dc_wtp) + var(twpl_wtp) / length(twpl_wtp)),
                          lower, upper, (upper - lower) / 2, pvalue(test)], digits = 5))
6×2 DataFrame
Row quantity value
String Float64
1 Difference in means (DC − TWPL) 79.6535
2 Standard error of the difference 24.7429
3 95% CI lower 31.0752
4 95% CI upper 128.232
5 95% CI half-width 48.5783
6 p-value 0.00134
Table 9: 95% confidence interval for the difference in mean WTP between the two formats.
WarningThe published interval is a 5% confidence interval, not a 95% one

The book reports the interval as [78.10, 81.21], a half-width of 1.55 on a difference of 79.66, and concludes the difference is “precisely estimated”. Its walk-through shows the call that produced it:

t.test(DC_WTP, TWPL_WTP, conf.level = 0.05)$conf.int
## [1] 78.10141 81.20560

conf.level is the confidence level, so 0.05 requests a 5% interval — one designed to contain the true difference 5% of the time. It looks precise because it is tiny by construction. The intended argument was almost certainly 0.95; 0.05 is the significance level \(\alpha\), which is a natural thing to reach for and the opposite of what this parameter wants.

Reproducing both levels from the same data settles it:

se = sqrt(var(dc_wtp) / length(dc_wtp) + var(twpl_wtp) / length(twpl_wtp))
difference = mean(dc_wtp) - mean(twpl_wtp)
# Welch-Satterthwaite degrees of freedom, as used by both R's t.test and
# HypothesisTests.UnequalVarianceTTest.
df = se^4 / ((var(dc_wtp) / length(dc_wtp))^2 / (length(dc_wtp) - 1) +
             (var(twpl_wtp) / length(twpl_wtp))^2 / (length(twpl_wtp) - 1))

DataFrame(conf_level = [0.95, 0.05],
          multiplier = round.([quantile(TDist(df), (1 + l) / 2) for l in [0.95, 0.05]],
                              digits = 6),
          half_width = round.([quantile(TDist(df), (1 + l) / 2) * se for l in [0.95, 0.05]],
                              digits = 4),
          lower = round.([difference - quantile(TDist(df), (1 + l) / 2) * se
                          for l in [0.95, 0.05]], digits = 4),
          upper = round.([difference + quantile(TDist(df), (1 + l) / 2) * se
                          for l in [0.95, 0.05]], digits = 4))
2×5 DataFrame
Row conf_level multiplier half_width lower upper
Float64 Float64 Float64 Float64 Float64
1 0.95 1.96332 48.5783 31.0752 128.232
2 0.05 0.062729 1.5521 78.1014 81.2056

The 5% row reproduces the book’s printed bounds. The real 95% interval is about 31 times wider: roughly [31, 128] against [78.1, 81.2].

Julia will not make this mistake quietly. confint(test; level = 0.05) throws coverage level 0.05 not in range (0.5, 1) — the API rejects a coverage below one half as incoherent, where R’s t.test accepts it and returns an interval. That is a case where the stricter interface is worth the inflexibility.

The direction of the conclusion survives. The correct interval still excludes zero comfortably, so DC really does produce higher stated WTP. What does not survive is the claim that the difference is precisely estimated, or that “the confidence interval lower bound is a long way from 0” in the sense the book means — the lower bound is €31, not €78, and the difference could plausibly be anywhere from a third of the ladder mean to half again as much.

(c) Medians against means.

DataFrame(measure = ["Mean", "Median"],
          DC = [comparison.mean[1], comparison.median[1]],
          TWPL = [comparison.mean[2], comparison.median[2]],
          difference = [comparison.mean[1] - comparison.mean[2],
                        comparison.median[1] - comparison.median[2]],
          ratio = round.([comparison.mean[1] / comparison.mean[2],
                          comparison.median[1] / comparison.median[2]], digits = 3))
2×5 DataFrame
Row measure DC TWPL difference ratio
String Float64 Float64 Float64 Float64
1 Mean 348.19 268.53 79.66 1.297
2 Median 192.0 132.0 60.0 1.455
Table 10: How much each summary measure moves between question formats.

The mean moves by €79.66 and the median by €60 — but as a proportion the median is the less stable of the two here: 1.45 times against 1.30 for the mean. That is worth stating plainly because the book concludes the opposite, that “the median is therefore more robust to changes in the question format”.

On these numbers the median is not more robust in relative terms. What is true is the related point the book is reaching for: the median is far less sensitive to the top of the distribution, which is why the DC mean sits at €348 against a median of €192.

(d) Which should a government use? The honest answer depends on the question being asked, and the two measures answer different ones.

  • The mean is what a budget needs. Total revenue from a charge is the mean times the number of payers, so if the policy question is “does this fund the abatement”, the mean is the relevant statistic. Its weakness is that it is set by the upper tail — a handful of people naming €1,440 move it a long way, and those are exactly the responses most exposed to the hypothetical bias from Q1.
  • The median is what a vote needs. A referendum at price \(p\) passes if more than half the electorate would accept \(p\), which is the median WTP by definition. It ignores intensity entirely: someone willing to pay the top ladder amount counts exactly the same as someone a single euro above the median.

Given that the format changes the mean by €80 and neither format’s mean can be checked against behaviour, reporting both and the interval around the difference is the defensible course, which is what this page does and what the mislabelled interval obscured.

What this project covered

Concept Where In Julia
Reading strict OOXML Setup XLSX.readxlsx(path)[sheet][:], not readtable
Blanks that arrive as nothing Setup handle nothing and missing in the coercion
Reverse-coding a Likert item Q11.1 Q2 6 .- x on a 1–5 scale
Mapping categories to amounts Q11.1 Q2 Dict lookup with get(..., missing)
Row means across chosen columns Q11.1 Q3 comprehension over eachrow
Cronbach’s alpha Q11.1 Q4 cronbach_alpha(items) from the shared helpers
Composition tables by group Q11.1 Q5 shares of each group’s non-missing responses
Dummy from a text column Q11.2 Q1 state the coding — the sign depends on it
Log axis with named ticks Q11.2 Q2 xscale = log10, xticks = (amounts, labels)
Difference in means with a CI Q11.2 Q3 UnequalVarianceTTest, confint
Welch degrees of freedom Q11.2 Q3 Welch–Satterthwaite, then quantile(TDist(df), p)
A coverage level below ½ Q11.2 Q3 Julia throws; R’s t.test accepts it

The R → Julia page has the full translation table.