MPV_APPE 2018, block 6
Shadow pricing – deriving prices from the demand curves
(model example)
Region plans to build a new bridge that would lower the travelling costs between two cities by 100
CZK/trip. This costs reduction includes shorter travelling time, lower vehicle depreciation and lower
fuel consumption of 120 CZK/trip in total minus 20 CZK toll per bridge crossing – a Region’s income.
Before the bridge there were total of 1 million trips per year, an estimate after the bridge is built is
1.5 million trips per year. Calculate annual benefits of the new bridge for drivers and for the Region.
Shadow pricing represents a way how to estimate the value of goods and services using a marketbased
approach. In an optimal situation it is possible to find an alternative to the evaluated
goods/service and use this price. If not, we can try to use shadow prices.
We use shadow pricing in situations where the actual price of goods/services does not exist or does
not reflect the true value (often due to the regulations, temporary fluctuations, or some other
market failures). A shadow price then can be perceived as a price that would be achieved in a perfect
competition market. Shadow price represents a proxy value, usually considered as a value of what
one has to sacrifice in order to get an additional unit. An estimation of a shadow price is usually done
by observing subjects’ behavior. The two main approaches to estimated shadow prices are based on
benefits or on costs.
Benefits approach includes estimation methods like change in productivity, or forgone earnings. We
can calculate these prices using a CBA for the specific goods/service. Costs approach includes
methods like replacements/restoration costs, preventive expenditure, damage avoidance costs, etc.
In practice these methods can be used in a form of a natural experiment or pilot projects (small scale
experiments). Impacts of such projects are classified, quantified and monetized. Using an
extrapolation, shadow prices can be estimated for the whole project.
Alternative way of estimating shadow prices is by using pseudo- or quasi-demand curves. We use the
initial state and the expected state after the project’s realization and based on that estimate the
demand curve. We then calculate consumer’s benefit resulting from the change of the state.
Solution steps: In our case we have data about demand before and after the project realization, thus
we derive shadow prices from the relevant demand curve. We can start with benefits to the Region,
which are basically number of trips multiplied by the bridge crossing toll (de facto revenues).
In case of drivers we need to separate the group that would travel regardless of the new bridge –
they would enjoy benefits of the full cost saving, and the remaining group with decreasing benefits
due to the slope of the demand curve. In other words, benefit for the driver no. 1,000,001 is almost
the full costs savings, while the benefit for the driver no. 1,500,000 is basically zero – this is the very
last driver that decided to do the trip, as it was practically on par with the alternative of, for instance,
staying at home or taking an alternative route.
Solution: Benefit for the drivers is 125 mil. CZK/year, benefit for the Region is 30 mil. CZK/year.
MPV_APPE 2018, block 6
WTP – Willingness to Pay
(model example)
Municipality is considering introducing a new waste collection system – instead of central container
(fully paid by the municipality) use bins for the individual households. Estimate the social benefits of
such new system using WTP. This could tell the municipality at what level should it set the fee for the
new system in order to cover the additional expenditures.
Population (200 total) has been sorted based on their willingness to pay for the new system from the
lowest fee to the highest and divided into the quartiles. Assume that the willingness to pay grows
linearly within each quartile. First person is not willing to pay anything, person at the first quartile
can pay 300 CZK, at the second quartile 380 CZK, at the third quartile 440 CZK, and last one 560 CZK.
WTP method represents the second approach how to estimate shadow prices and is based on stated
(declared) preferences. It is a kind of a Contingent Valuation Method (CVM) that uses
questionnaires/surveys. In contrast with the market-approach based on observation of the subjects,
this approach estimates the values based on subjective opinions (theoretical preference), not from
actual behavior, that has been confirmed at the market.
The method itself consists of collecting data from a survey, in which a relevant sample of subjects
state the maximum price they are willing to pay for some goods or service. Survey can start with
lower amounts and continue higher until maximum price is stated. An alternatively is a directly
question about the highest price. Based on collected data we derive a pseudo-demand curve.
The benefits estimated by WTP method is, as with other methods based on preferences, calculated
as the consumer surplus – the area under the derived pseudo-demand curve.
Analogical method is WTA – Willingness to Accept, where you estimate the price, resp. equivalent
that would make subject accept the presence of some negative factor.
Main problem of WTP and WTA methods is in often observed difference between stated and
revealed (real, market) preferences, and the correct use of this method requires taking this into the
consideration. Respondents in case of WTP and WTA often overstate their willingness to pay, resp.
their minimal accepted equivalent for sustaining some negative effect. Another problem of these
methods is acquiring the data from a representative sample during the survey.
Solution steps: Based on the collected data we derive a pseudo-demand curve of the willingness to
pay and calculate the area under it. Total willingness to pay in 4th
quartile is (50*440+(560-
440)*50/2) = 25 000 CZK, in 3rd
quartile is (380*50+(440-380)*50/2) = 20 500 CZK, in 2nd
quartile is
(50*300+(380-300)*50/2) = 17 000 CZK, and in the first quartile is (300*50/2) = 7 500 CZK.
Solution: Social benefit from new waste collection system using bins would be 70 000 CZK.
MPV_APPE 2018, block 6
Hedonic method
(model example)
You are asked to evaluate benefits of an anti-flood dam that would reduce the probability of a flood
in a residential area of 80 houses. Use hedonic method for the evaluation if you know that the price
of houses in given area can be estimated using linear regression model:
𝑝𝑖 = 𝛼 + 𝛽(𝑝𝑟𝑜𝑏) + 𝛾(𝑥𝑖) + 𝜀𝑖
where 𝑝𝑖 means estimated price of the house, 𝑝𝑟𝑜𝑏 probability of a flood occurrence in some time
horizon, and 𝑥𝑖 some other parameters of house that we do not need to consider in this case.
Based on the survey of house prices the coefficient have values 𝛼 = 75, 𝛽 = −200 and 𝛾 = 1. Dam
construction is estimated to reduce the risk of flood in some time horizon from 0.4 to 0.05.
Hedonic method represents an approach used for estimating the values of certain measures, good or
services that directly affect the market prices of some other goods. Usual application area is the
realities market, which reflects the impacts of changes of local environmental attributes.
Hedonic method assumes that the price of a certain market good is directly influenced by its
characteristics and our considered measure (resp. good, service). Steps of this method consist of
collecting sufficient data regarding prices and characteristics of considered market goods and their
subsequent statistical analysis. The result is then a price function of a market good consisting of
effects of individual characteristics. Based on the impact of changing individual parameters we
estimate the social benefits of introducing evaluated measure (good, service). In contrast with the
stated preferences, here we use the real data, revealed preferences.
When applying this method in a project evaluation we examine the impact on the welfare (e.g.
change in the prices of realities) when changing selected variables that we can influence (deciding
whether to undergo the project). Final monetized value of the project realization (resp. measure
introduction) is then calculated as a difference between the original state and the new state after the
project realization.
Solution steps: In our case we calculate the difference the value of houses before and after the
construction of the anti-flood dam. Specifically we compare the total value of houses using
regression model using the original and new parameters. Numerically this is a difference between
𝑝𝑖 = 75 − 200 ∗ (0.4) = −5, and 𝑝𝑖 = 75 − 200 ∗ (0.05) = 65, resp. simplified into ∆𝑝𝑖 = −200 ∗
(0.05 − 0.4) = 70 for a single house. Other parameters than flood occurrence probability in this
case do not change, therefore we do not need to consider them in the calculation.
Change in the value of a single house in this case is a value increase by 70.
Solution: Total benefit of building an anti-flood dam for the residential area has a value of 5600.
MPV_APPE 2018, block 6
TCM – Travel Cost Method
(model example)
A municipality owns a lookout tower. Annual maintenance costs including personnel are 300,000 CZK
(material, repairs, etc.). Municipality wants to know the value of the social benefits of the tower.
Calculate them using TCM. Consider travel costs of 2 CZK/km, travel pace of 30 km/hour, average
hourly wage 120 CZK, and an entrance fee of 10 CZK/person. Further parameters are in the table.
Assume discontinuous changes in demand for visits (each zone has same demand level), and
irrelevant visits from additional zones. Also consider that visits are not cumulative (only the tower).
Zone Distance Population
Probability of visit
per year
0 1 200 50%
1 10 10,000 5%
2 30 100,000 2%
TCM represents a method of evaluating nonmarket goods based on observing the market behavior of
subject (revealed preferences). It is used mostly when appraising attractive locations (points of
interest, POI) or ecosystems, where it is naturally difficult to set a direct market monetary value.
This method assumes that the value of evaluated goods is equal to the loss (costs), that subjects are
willing to accept by travelling to the destination with the POI, specifically the costs for travelling (fuel,
public transportation), value of the lost time due to the travelling, and the entrance fees, if there are
any. These costs are different for each of the considered zones based on the distance.
Initially we identify the demand zones in the form of amounts of visits and related total costs per visit
for each considered zone. From these zones we derive a pseudo-demand curve for visiting
considered POIs. The value of POI is then represented by the area under the pseudo-demand curve.
Disadvantage of this method is, for instance, the necessity of sufficient data, as well as inaccuracies
due to the combined visits to the multiple POIs (costs are then shared between the visits), or the
perceived benefits from the travelling itself.
Solution steps: We have data about three zones, we do not consider more. We also do not consider
standard demand curve shape, but only three homogenous groups of visitors based on the zones.
We calculate costs per visit for each zone: 0) 100*(2CZK*2*1km+120CZK*2*1km/30+10CZK);
1) 500*(2CZK*2*10km+120CZK*2*10km/30+10CZK); 2) 2000*(2CZK*2*30km+120CZK*2*30km/30
+10CZK).
Solution: Benefits of the lookout tower are 2200 CZK + 65000 CZK + 740000 CZK = 807 200 CZK/year.
MPV_APPE 2018, block 6
Dominated and non-dominated variants (dominance analysis)
(model example)
Local police department wants to purchase new cars. Following table contains data about several
considered models. Decide which models are dominated, determine basal and ideal variants and full
solution set (the models that should be considered for the further choice).
Model Acceleration Top speed Fuel consum. Trunk size Price
Forman 17 141 8.1 450 220
Felicia combi 16 148 7.6 450 250
Lada 1500 15 153 7.6 480 210
Trabant 30 110 8.1 380 180
With increasing amount of available options and evaluated criteria in a multicriteria evaluation it can
become very difficult to stay oriented. One way to simplify it is to reduce the available set of options,
let’s call it a full solution, in which we do not consider further those options that are not relevant
(dominated ones) – the options that practically cannot be chosen over some other available option.
Irrelevant variant is in this case the one to which there exists at least one other option that is not
worse in any of the considered criteria while being better in at least one criterion. Such variant is
then considered as a dominated variant, and is being dominated by all other variants that fulfill the
condition of being better in at least one criterion while not being worse in any other.
Available variants in most cases do not dominate each other, meaning that one is better in some
criteria, while worse in other. Then they are considered to be mutually non-dominated.
Sometimes it might help to determine theoretically worst and best variant. The worst variant is the
one with the worst available values from the set, it is called a basal variant (B), and contains basal
values. On contrary, with the best available values from the set we get the ideal variant (I) with the
ideal values of criteria from the evaluated set of options.
Solution: Basal variant has an acceleration of 30, top speed of 110, fuel consumption 8.1, trunk size
380 and costs 250. Ideal variant has an acceleration of 15, top speed of 153, fuel consumption 7.6,
trunk size 480 and costs 180. Forman and Felicia are dominated (by Lada), Lada and Trabant are not
dominated. For further evaluation we would consider Lada and Trabant (full solution).
*In case of more complex problems you can use available tools, like SANNA from PSE.
MPV_APPE 2018, block 6
Transformation of minimizing criteria to maximizing ones
(model example)
Police department wants again to purchase new cars. Following table contains data about considered
models. Some of the criteria are minimizing. Transform all such to maximizing criteria.
Model Acceleration Top speed Fuel consum. Trunk size Price
Forman 17 141 8.1 450 220
Felicia combi 16 148 7.6 450 250
Lada 1500 15 153 7.6 480 210
Trabant 30 110 8.1 380 180
In practice of multicriteria evaluation we often encounter situations where some parameters are
better if maximized (like output level), while other minimized (like price). Transformation to the one
type can reduce possibility of making a mistake due to such difference and can be also useful later.
We can transform the values of minimizing criteria to maximizing using the following transformation:
𝑦𝑖𝑗(𝑚𝑎𝑥) = 𝐵(𝑚𝑖𝑛) − 𝑦𝑖𝑗(𝑚𝑖𝑛)
Where y(max) means transformed value from a min criterion to a max criterion, B(min) means basal
value of given min criterion (in such case the highest value of such criterion), and y(min) means
original value of min criterion.
*if we have a fixed available interval of values, like using grades 1-5, we use 5 (worst) as a basal value
independently from the fact that none of the evaluated variants actually got grade 5 in the criterion.
Solution:
Model T-Acceleration Top speed T-Fuel consum. Trunk size T-Price
Forman 13 141 0 450 30
Felicia combi 14 148 0,5 450 0
Lada 1500 15 153 0,5 480 40
Trabant 0 110 0 380 70
MPV_APPE 2018, block 6
WSA – weight sum approach
(model example)
Police department still wants to purchase new cars. Following table contains data about considered
models. Use WSA to select the best variant (weights of criteria are 30%, 10%, 30%, 30%).
Model Acceleration Top speed Fuel consum. Price
Octavia 9 200 6.8 410
Rapid 10 190 6.5 360
Fabia 11 180 6.3 330
WSA means that individual evaluated criteria are assigned with certain weights that represent their
level of importance in the final evaluation. Significantly worse parameters in one less important
criterion therefore do not mean that the variant will automatically not be selected, as long as it has
competitive parameters in other more important criteria. For correct use os WSA we need to
transform original values to the appropriate form. We transform values to the same type and then
normalize them, so we can use type-comparable values. Final score for each variant is then a scalar
product of normalized values of criteria and their weights.
Solution steps: Transformation formula for normalizing the maximizing criteria:
𝑦𝑖𝑗(𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑) =
𝑦𝑖𝑗 𝑚𝑎𝑥 − 𝐵𝑗
𝐼𝑗 − 𝐵𝑗
Resp. transformation formula for normalizing the minimizing criteria:
𝑦𝑖𝑗(𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑) =
𝐵𝑗 − 𝑦𝑖𝑗 𝑚𝑖𝑛
𝐵𝑗−𝐼𝑗
Using these transformations we get normalized matrix of values between 0 and 1 and then we
multiply the values with the weights. Normalized matrix looks like this:
Model N-Acceleration N-Top speed N-Fuel consum. N-Price
Octavia 1 1 0 0
Rapid 0.5 0.5 0.6 0.625
Fabia 0 0 1 1
Solution: Octavia gets 40%, Rapid 56.75% and Fabia 60%. Best model is Fabia.
MPV_APPE 2018, block 6
Scales and ranges – assigning points within a scale
(model example)
Region is deciding between projects of several water plants on different rivers. Three projects were
submitted, with criteria of building costs, running costs, output and safety level (range 0-10). You as
an expert should evaluate criteria of projects on a scale of 0-100 and choose the best.
River Building costs Running costs Output Safety (0-10)
Bobrava 170 73 67 9
Ponávka 132 38 45 7
Želetavka 99 41 33 5
Method of scales requires ability of the quantitative evaluation of given parameters within evaluated
criteria, meaning that the evaluator assigns values based on his expert opinion. Unlike with strictly
mathematical methods this allows the consideration of other factors as well, like the experience of
the evaluator, preferences or other aspects. The better the value of a parameter, the more points
should be assigned. Thanks to that we do not have any more issues with min/max criteria and their
transformations or normalization.
On the other hand, the disadvantage of this method is the dependency on the subjective evaluation
of parameters. For reducing the risk of making an incorrect decision, multiple independent expert
evaluations are often used. Final score is then a result of sum of individual evaluations, or their
weighted sum, if opinions of different experts are weighted differently.
Analogically we can assign different weights to different criteria based on their importance. Assigned
points for each parameter are then weighted and summed together afterwards. Such method is then
called scoring method.
Example of a possible solution, a subjective assignation of points to the parameters (0-100 scale):
River Building costs Running costs Output Safety Sum
Bobrava 40 50 100 90 280
Ponávka 61 100 77 70 308
Želetavka 80 95 50 50 275
MPV_APPE 2018, block 6
Lexicographic method
(model example)
Region wants to build a bridge over the river Jevišovka. Estimated parameters of variants are in the
following table. For the decision use lexicographic method with criteria preferences C→B→D→A, and
requirements for the criteria being A ≥ 440, B ≥ 6, C ≥ 7 a D ≤ 50.
Bridge A) Capacity B) Looks C) Place D) Costs
of victory (1) 406 7 8 50
Red (2) 444 8 8 39
of friendship (3) 505 4 9 44
of labor (4) 568 5 10 32
of proletariat (5) 541 8 6 52
Lexicographic method evaluates available variants sequentially based on the importance of individual
criteria and limiting requirements. In the first step we take the most important criterion and discard
the variants that do not meet the requirements for this criterion. In the second step we continue
analogically with the reduced set of remaining variants. The evaluation process is finished when only
one variant remains, and this variant is chosen as the best. It is not necessary to evaluate options
using all available criteria, if we get to the point when there is only one option left. In case we are left
with multiple variants even after the last criterion, we need to use some additional method for
choosing a compromise variant (for instance selecting one with the best parameter in the last
evaluated criterion).
The disadvantage of this method is that we are practically taking into account only the last evaluated
criterion and do not consider previous as long as the minimal requirements were met. Moreover, the
result can be notably biased by the criteria preferences selection. In cases with no non-dominated
variants, with “appropriate” preference order and minimal requirements it is sometimes possible to
secure any of the variants as the winning one.
Solution steps: In the first step we evaluate according to the C criterion – we discard variant 5,
remaining set is {1, 2, 3, 4}. In the second step we evaluate according to the B criterion – we discard
variants 3 and 4, remaining set is {1, 2}. In the third step we evaluate according to the D criterion –
we do not discard any variants, the set remains {1, 2}. In the fourth step we evaluate according to the
A criterion – we discard variant 1, remaining set is {2}.
Solution: Based on the lexicographic method we choose the Red bridge (2).
MPV_APPE 2018, block 6
Public procurement evaluation
(model example)
You are a member of the evaluation committee choosing the company constructing a migration
corridor over the highway. Assume that all applicants met all requirements like delivering the
complete documentation, references, qualifications, etc. and submitted bids on time. Evaluate bids.
Bid Price (mil. CZK) Delivery (months)
Guarantee
(months)
Technical aspects
(0-10 scale)
A 15.2 25 43 9
B 16 23 69 6
C 18.9 19 86 7
weight 50% 20% 15% 15%
Process of public procurement evaluation is close to the WSA (weighted sum approach). Values of
each variant are transformed into the normalized form and multiplied by given weights. The
difference is in the different transformation formula. Normalized values in this case do not
necessarily acquire value 0 in case of the worst parameter – they acquire a proportional value
compared to the best parameter (maximizing criterion), or value calculated as an inverse
proportional (minimizing criterion). Normalized values are finally multiplied by weights and summed.
In case of maximizing criteria we calculate transformed values as a proportion of the best value. In
case of minimizing criteria we calculate the value as an inverse proportional (using flipped fractions).
Solution steps: Normalized matrix of parameters (below) is multiplied by given weights:
Bid Criterion 1 Criterion 2 Criterion 3 Criterion 4
A 1.00 0.76 0.50 0.90
B 0.95 0.83 0.80 0.60
C 0.80 1.00 1.00 0.70
Solution: Bid A acquired score of 86.20%, bid B acquired 85.06%, and bid C acquired 85.71%. Bid A is
victorious and this applicant should be asked to deliver the realization of the public procurement.