This document provides an introduction to multiattribute decision making and decision theories. It discusses several key aspects of multiattribute choice models, including:
1) The number and nature of attributes that are used to differentiate decision alternatives.
2) The structure of the feasible set of alternatives.
3) The basis of evaluation, such as preference relations or criterion functions.
4) Independence and separability assumptions that are required to obtain additive representations of preferences.
The document outlines some classic evaluation theories under certainty that do not involve probabilities, and discusses the concept of separability, which reduces complexity by allowing decentralized preferences across attribute groups.
Lecture slides on Decision Theory. The contents in large part come from the following excellent textbook.
Rubinstein, A. (2012). Lecture notes in microeconomic theory: the
economic agent, 2nd.
http://www.amazon.co.jp/dp/B0073X0J7Q/
Absolute and Relative Clustering
4th MultiClust Workshop on Multiple Clusterings, Multi-view Data, and Multi-source Knowledge-driven Clustering (Multiclust 2013)
Aug. 11, 2013 @ Chicago, U.S.A, in conjunction with KDD2013
Article @ Official Site: http://dx.doi.org/10.1145/2501006.2501013
Article @ Personal Site: http://www.kamishima.net/archive/2013-ws-kdd-print.pdf
Handnote: http://www.kamishima.net/archive/2013-ws-kdd-HN.pdf
Workshop Homepage: http://cs.au.dk/research/research-areas/data-intensive-systems/projects/multiclust2013/
Abstract:
Research into (semi-)supervised clustering has been increasing. Supervised clustering aims to group similar data that are partially guided by the user's supervision. In this supervised clustering, there are many choices for formalization. For example, as a type of supervision, one can adopt labels of data points, must/cannot links, and so on. Given a real clustering task, such as grouping documents or image segmentation, users must confront the question ``How should we mathematically formalize our task?''To help answer this question, we propose the classification of real clusterings into absolute and relative clusterings, which are defined based on the relationship between the resultant partition and the data set to be clustered. This categorization can be exploited to choose a type of task formalization.
Statistical Inference Part II: Types of Sampling DistributionDexlab Analytics
This is an in-depth analysis of the way different types of sampling distribution works focusing on their specific functions and interrelations as part of the discussion on the theory of sampling.
Discuss and apply comprehensively the concepts, properties and theorems of functions, limits, continuity and the derivatives in determining the derivatives of algebraic functions
Talk at the modcov19 CNRS workshop, en France, to present our article COVID-19 pandemic control: balancing detection policy and lockdown intervention under ICU sustainability
BYD SWOT Analysis and In-Depth Insights 2024.pptxmikemetalprod
Indepth analysis of the BYD 2024
BYD (Build Your Dreams) is a Chinese automaker and battery manufacturer that has snowballed over the past two decades to become a significant player in electric vehicles and global clean energy technology.
This SWOT analysis examines BYD's strengths, weaknesses, opportunities, and threats as it competes in the fast-changing automotive and energy storage industries.
Founded in 1995 and headquartered in Shenzhen, BYD started as a battery company before expanding into automobiles in the early 2000s.
Initially manufacturing gasoline-powered vehicles, BYD focused on plug-in hybrid and fully electric vehicles, leveraging its expertise in battery technology.
Today, BYD is the world’s largest electric vehicle manufacturer, delivering over 1.2 million electric cars globally. The company also produces electric buses, trucks, forklifts, and rail transit.
On the energy side, BYD is a major supplier of rechargeable batteries for cell phones, laptops, electric vehicles, and energy storage systems.
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Empowering the Unbanked: The Vital Role of NBFCs in Promoting Financial Inclu...Vighnesh Shashtri
In India, financial inclusion remains a critical challenge, with a significant portion of the population still unbanked. Non-Banking Financial Companies (NBFCs) have emerged as key players in bridging this gap by providing financial services to those often overlooked by traditional banking institutions. This article delves into how NBFCs are fostering financial inclusion and empowering the unbanked.
what is the future of Pi Network currency.DOT TECH
The future of the Pi cryptocurrency is uncertain, and its success will depend on several factors. Pi is a relatively new cryptocurrency that aims to be user-friendly and accessible to a wide audience. Here are a few key considerations for its future:
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1. Mainnet Launch: As of my last knowledge update in January 2022, Pi was still in the testnet phase. Its success will depend on a successful transition to a mainnet, where actual transactions can take place.
2. User Adoption: Pi's success will be closely tied to user adoption. The more users who join the network and actively participate, the stronger the ecosystem can become.
3. Utility and Use Cases: For a cryptocurrency to thrive, it must offer utility and practical use cases. The Pi team has talked about various applications, including peer-to-peer transactions, smart contracts, and more. The development and implementation of these features will be essential.
4. Regulatory Environment: The regulatory environment for cryptocurrencies is evolving globally. How Pi navigates and complies with regulations in various jurisdictions will significantly impact its future.
5. Technology Development: The Pi network must continue to develop and improve its technology, security, and scalability to compete with established cryptocurrencies.
6. Community Engagement: The Pi community plays a critical role in its future. Engaged users can help build trust and grow the network.
7. Monetization and Sustainability: The Pi team's monetization strategy, such as fees, partnerships, or other revenue sources, will affect its long-term sustainability.
It's essential to approach Pi or any new cryptocurrency with caution and conduct due diligence. Cryptocurrency investments involve risks, and potential rewards can be uncertain. The success and future of Pi will depend on the collective efforts of its team, community, and the broader cryptocurrency market dynamics. It's advisable to stay updated on Pi's development and follow any updates from the official Pi Network website or announcements from the team.
Poonawalla Fincorp and IndusInd Bank Introduce New Co-Branded Credit Cardnickysharmasucks
The unveiling of the IndusInd Bank Poonawalla Fincorp eLITE RuPay Platinum Credit Card marks a notable milestone in the Indian financial landscape, showcasing a successful partnership between two leading institutions, Poonawalla Fincorp and IndusInd Bank. This co-branded credit card not only offers users a plethora of benefits but also reflects a commitment to innovation and adaptation. With a focus on providing value-driven and customer-centric solutions, this launch represents more than just a new product—it signifies a step towards redefining the banking experience for millions. Promising convenience, rewards, and a touch of luxury in everyday financial transactions, this collaboration aims to cater to the evolving needs of customers and set new standards in the industry.
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
how to sell pi coins in all Africa Countries.DOT TECH
Yes. You can sell your pi network for other cryptocurrencies like Bitcoin, usdt , Ethereum and other currencies And this is done easily with the help from a pi merchant.
What is a pi merchant ?
Since pi is not launched yet in any exchange. The only way you can sell right now is through merchants.
A verified Pi merchant is someone who buys pi network coins from miners and resell them to investors looking forward to hold massive quantities of pi coins before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
Financial Assets: Debit vs Equity Securities.pptxWrito-Finance
financial assets represent claim for future benefit or cash. Financial assets are formed by establishing contracts between participants. These financial assets are used for collection of huge amounts of money for business purposes.
Two major Types: Debt Securities and Equity Securities.
Debt Securities are Also known as fixed-income securities or instruments. The type of assets is formed by establishing contracts between investor and issuer of the asset.
• The first type of Debit securities is BONDS. Bonds are issued by corporations and government (both local and national government).
• The second important type of Debit security is NOTES. Apart from similarities associated with notes and bonds, notes have shorter term maturity.
• The 3rd important type of Debit security is TRESURY BILLS. These securities have short-term ranging from three months, six months, and one year. Issuer of such securities are governments.
• Above discussed debit securities are mostly issued by governments and corporations. CERTIFICATE OF DEPOSITS CDs are issued by Banks and Financial Institutions. Risk factor associated with CDs gets reduced when issued by reputable institutions or Banks.
Following are the risk attached with debt securities: Credit risk, interest rate risk and currency risk
There are no fixed maturity dates in such securities, and asset’s value is determined by company’s performance. There are two major types of equity securities: common stock and preferred stock.
Common Stock: These are simple equity securities and bear no complexities which the preferred stock bears. Holders of such securities or instrument have the voting rights when it comes to select the company’s board of director or the business decisions to be made.
Preferred Stock: Preferred stocks are sometime referred to as hybrid securities, because it contains elements of both debit security and equity security. Preferred stock confers ownership rights to security holder that is why it is equity instrument
<a href="https://www.writofinance.com/equity-securities-features-types-risk/" >Equity securities </a> as a whole is used for capital funding for companies. Companies have multiple expenses to cover. Potential growth of company is required in competitive market. So, these securities are used for capital generation, and then uses it for company’s growth.
Concluding remarks
Both are employed in business. Businesses are often established through debit securities, then what is the need for equity securities. Companies have to cover multiple expenses and expansion of business. They can also use equity instruments for repayment of debits. So, there are multiple uses for securities. As an investor, you need tools for analysis. Investment decisions are made by carefully analyzing the market. For better analysis of the stock market, investors often employ financial analysis of companies.
what is the best method to sell pi coins in 2024DOT TECH
The best way to sell your pi coins safely is trading with an exchange..but since pi is not launched in any exchange, and second option is through a VERIFIED pi merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and pioneers and resell them to Investors looking forward to hold massive amounts before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade pi coins with.
@Pi_vendor_247
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
how to swap pi coins to foreign currency withdrawable.DOT TECH
As of my last update, Pi is still in the testing phase and is not tradable on any exchanges.
However, Pi Network has announced plans to launch its Testnet and Mainnet in the future, which may include listing Pi on exchanges.
The current method for selling pi coins involves exchanging them with a pi vendor who purchases pi coins for investment reasons.
If you want to sell your pi coins, reach out to a pi vendor and sell them to anyone looking to sell pi coins from any country around the globe.
Below is the contact information for my personal pi vendor.
Telegram: @Pi_vendor_247
How to get verified on Coinbase Account?_.docxBuy bitget
t's important to note that buying verified Coinbase accounts is not recommended and may violate Coinbase's terms of service. Instead of searching to "buy verified Coinbase accounts," follow the proper steps to verify your own account to ensure compliance and security.
USDA Loans in California: A Comprehensive Overview.pptxmarketing367770
USDA Loans in California: A Comprehensive Overview
If you're dreaming of owning a home in California's rural or suburban areas, a USDA loan might be the perfect solution. The U.S. Department of Agriculture (USDA) offers these loans to help low-to-moderate-income individuals and families achieve homeownership.
Key Features of USDA Loans:
Zero Down Payment: USDA loans require no down payment, making homeownership more accessible.
Competitive Interest Rates: These loans often come with lower interest rates compared to conventional loans.
Flexible Credit Requirements: USDA loans have more lenient credit score requirements, helping those with less-than-perfect credit.
Guaranteed Loan Program: The USDA guarantees a portion of the loan, reducing risk for lenders and expanding borrowing options.
Eligibility Criteria:
Location: The property must be located in a USDA-designated rural or suburban area. Many areas in California qualify.
Income Limits: Applicants must meet income guidelines, which vary by region and household size.
Primary Residence: The home must be used as the borrower's primary residence.
Application Process:
Find a USDA-Approved Lender: Not all lenders offer USDA loans, so it's essential to choose one approved by the USDA.
Pre-Qualification: Determine your eligibility and the amount you can borrow.
Property Search: Look for properties in eligible rural or suburban areas.
Loan Application: Submit your application, including financial and personal information.
Processing and Approval: The lender and USDA will review your application. If approved, you can proceed to closing.
USDA loans are an excellent option for those looking to buy a home in California's rural and suburban areas. With no down payment and flexible requirements, these loans make homeownership more attainable for many families. Explore your eligibility today and take the first step toward owning your dream home.
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
This is because pi network is not doing any pre-sale or ico offerings, the only way to get my coins is from buying from miners. So a merchant facilitates the transactions between the miners and these exchanges holding pi.
I and my friends has sold more than 6000 pi coins successfully with this method. I will be happy to share the contact of my personal pi merchant. The one i trade with, if you have your own merchant you can trade with them. For those who are new.
Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
#kyc #mainnet #picoins #pi #sellpi #piwallet
#pinetwork
3. Attribute mapping: Settings
X the set of decision alternatives
n ≥ 2 attributes used to differentiate among the objects in X
∀i ∈ 1...n, there is a set Xi whose elements are values/level of
attribute i
∀i ∈ 1...n, there is an attribute mapping fi : X → Xi
each fi assigns to each object in X a specific level of the ith
attribute.
for each x ∈ X, the fi functions produce a n-tuple
(f1(x), f2(x), ..., fn(x)) in X1 × ... × Xn.
4. Attribute mapping: Settings
The fi functions describes x in terms of its values on the n
attributes.
The fi functions are descriptive or identification functions.
x!
X!
f1!
x1!
X1!
fn!
xn!
Xn!
f2!
x2!
X2!
. . .!
x → (f1(x), f2(x), ..., fn(x))
5. Attribute mapping: Examples
f1(x), f2(x), ..., fn(x)
X is a set of simple probability measures on R and fi (x) is the
ith central moment of x.
X is the set of commodity bundles. In standard
microeconomics fi are linear functions.
X is the set of commodity bundles. In Lancaster (1966)
decision needs to map first each x ∈ X into a vector of
characteristics (f1(x), f2(x), ..., fn(x)).
X is the set of people. In literature on inequality, decision is
often based on a vector of wealths (f1(x), f2(x), ..., fn(x)).
6. Attribute mapping: Remarks
x → (x1, x2, ..., xn)
In many cases, fi (x) is abbreviated as xi and elements in X are
n-tuples (x1, ..., xn) in the product set X1 × ... × Xn.
X is a proper subset of X1 × .... × Xn: X ⊆ X1 × .... × Xn.
elements of X1 × .... × Xn that are not in X are unrealizable or
infeasable combinations of attributes.
many axiomatic preference theories assume that
X = X1 × .... × Xn.
implicit assumption of attribute mapping (non neutral): two
elements in X that map in the same n-tuple (x1, ..., xn) are
indifferent/ have equal value.
7. Criterion functions: Definition
criterion function: real value function on X reflecting the value
of the elements in X.
various names: objective functions, goal functions, scoring
functions, ranking functions, utility functions.
difference between:
attribute mapping: objective characteristics of
alternatives/consequences
criterion functions: subjective values on a scale.
with m criteria and criteria functions gj : X → R, each x is
mapped into:
(g1(x), ..., gm(x)) (1)
convention: preference monotonically increases in each gj .
9. Classification of theories: basic classification
evaluation theories under certainty (Debreu, 1959)
evaluation theories under risk (Keeney and Raiffa, 1976),
based on expected utility
evaluation theories under uncertainty (Savage,1954)
10. Classification of theories: 7 key aspects (Fishburn, 1978)
1. the number and nature of attributes and criterion functions
2. the structure of the feasible set of alternatives
3. the basis of evaluation
4. the ordering assumptions
5. the independence assumptions
6. the degree of compensatoriness
7. the subjectives judgements of the decision-maker
11. Aspect 1: the number and nature of attributes
The number of attributes:
n finite, e.g n = 2, n = 3, n > 3...
n = 2 and n ≥ 3 can be very different
n infinite, e.g delayed outcomes
The nature of attributes:
Xi ∈ {0, 1} (qualitative)
Xi ∈ R (quantitative)
12. Aspect 2: the structure of the feasible set
Some theories assume that X or {(g1(x), ..., gm(x)) : x ∈ X}
is
(a) a Cartesian Product
X1!
X2!
Product set!
Subset!
(b) a Subset of product set
feasible sets are convex/connected/compact separable
topological or Euclidean spaces
less structured feasible sets
Xi dense is much simpler, most tractable results for cartesian
products
13. Aspect 3: the basis of evaluation
many theories are based on holistic binary preferences relation
theories based on quaternary preference-intensity comparison
relation
revealed-preference theory
stochastic preference theory
most theories are based (or can be interpreted) on binary
comparisons
14. Aspect 4: ordering assumptions
asymmetric preference relation: x y
complete preference relation: x y or y x (or both)
transitivity x y and y z ⇒ x z
negative transitivity x z ⇒ either x y or y z
asymmetric + transitive = strict partial order
asymmetric + negatively transitive = strict weak order
the indifference relation ∼ is an equivalence relation (reflexive,
symmetric, transitive)
the relation is a weak order (reflexive, complete, transitive)
without completness, an incomparability relation is assumed
for and ∼
transitivity is key
15. Aspect 5: independence assumptions
Crucial element to obtain an additive representation of preferences
(Gossen, 1854)
U(x) =
i
ui (xi )
separability assumptions (strong, weak)
independence assumptions
widely used in (applied) economics
functional forms (Cobb-Douglas, CES, Bergson...)
applied and normative economics: EU, time preferences
decentralization and aggregation
utility trees, price/quantities indexes...
16. Remarks on additivity
U(x) =
i
ui (xi )
additivity implies independence of marginal utility of attribute
i from the level of any other attribute.
second order cross-partial derivatives equal to zero (e.g broad
aggregates of goods: food, clothing, housing).
nice property: take into account independence of certain
attributes
unfortunate property: defined in a cardinal way: public enemy
number one for ordinalists
solution:
reject the definition of additivity in terms of second order cross
partial derivatives
define additivity in terms of the particular restrictions which it
implies.
17. Remarks on additivity in consumer theory: cross-price
derivatives
cross-price derivatives are proportional to income derivatives
( ∂xi
∂pk
= µ∂xi
∂w , for i = k)
the change in the demand for food (clothing) induced by a
change in the price of housing is proportional to the change in
the demand for food (clothing) induced by a change in income.
the factor of proportionality µ is invariant to the quantity of
good i (but not to price pk ).
∂xi
∂pj
+ xj
∂xi
∂w
= −
λ
∂λ/∂w
∂xi
∂w
∂xj
w
, i = j
18. Remarks on additivity in consumer theory: substitution
effects
substitution analyzed in terms of "competition for the
consumer’s dollars"
substitution effect proportional to income derivative and
Lagrange multiplier λ
Slutsky equation invariant to monotonic increasing
transformations
→ all utility functions belonging to the same class as a given
additive utility function have the same substitution effects
but Lagrange multiplier is not invariant to any monotonic
increasing transformations, only to linear one.
well, I’m not totally sure of what happens here → needs
deeper (basic) analysis
all attributes are substitutes
independent goods in cardinal sense are indeed substitutes in
an ordinal sense
19. Remarks on additivity in consumer theory (indirect utility)
Houthakker (1960): additive indirect utility function
cross-price derivatives are proportional to the quantities
affected.
all cross-price elasticities with respect to each price are the
same
if both direct and indirect additive utilities are the same, all
elasticities with respect to total expenditure are equal and
unitary (straight Engel curves).
20. Aspect 6: degree of compensatoriness
example: local changes that preserve indifference can be made
around one point in the space. Small deviations from x or y
does not reverse the ranking.
typical noncompensatory preferences: lexicographic preferences
(preference ranking are reversed at the limit point of the
sequence and indifference sets are singletons).
key axiom: continuity or Archimedean axiom.
variety of intermediary cases.
X1!
X2!
x!
y!
Bx!
By!
21. Aspect 7: decision maker’s subjective judgements
some evaluative models impose the same evaluate realization
to all decision makers.
monotonicity, efficient frontiers
first-order stochastic dominance
vector dominance relations if each criterion function is ordinaly
equivalent across decision makers.
most compensatory preference models presume different
decision makers have different tradeoff structures.
23. Basis references
Debreu (1960), Gorman (1968): quantity aggregation.
Blackorby, Primont, and Russell (1978): direct and indirect
utility function, duality and expenditure function.
Strotz (1957), Gorman (1959): price aggregation and
two-stage budgeting.
24. Building blocks
no probabilities, no uncertain events
set of objects to be evaluated is a subset X of a product set
X1 × X2 × ....Xn
evaluation with a global preference/utility function or one of
the criterion functions gj defined over X
asymmetric preference relation on X "better than": x y and
y x connot both hold, ∀x, y ∈ X.
25. Separability in economics
separability : reduces the complexity of the allocation decision
decentralization of preferences:
first allocation to broad classes of commodities (e.g Food and
Clothes "budgets")
then allocation within each class knowing only intra-class
prices (e.g food prices)
allows to group items in close substitutes or complements
Leontief(1947)-Sono(1945,1961) definition of separability:
∂
∂xk
∂U(x)/∂xi
∂U(x)/∂xj
= 0
MRS between pairs of goods in the first group (e.g food,
including i, j) is independent of quantities in the second group
(e.g clothes, including k)
an aggregator function exists for the first group
26. Separability (Blackorby, Primont and Russel, 1978)
binary partition of the set of the variables indices:
( x1, ..., xi
group 1: x1
, xi+1, ..., xn
group 2: x2
)
conditional preference ordering over group 2: 2
(x1):
x2 2
(x1) ˆx2
⇔ (x1
, x2
) (x1
, ˆx2
)
commodities 2 are separable from commodities 1 if 2
(x1) is
independent of the quantities in group 1: this defines 2
equivalent to the characterization:
U(x1
, x2
) ≥ U(x1
, ˆx2
) ⇔ U(ˆx1
, x2
) ≥ U(ˆx1
, ˆx2
)
27. Separability and functional structure (Debreu, 1959, Gorman, 1968)
complete, continuous, transitive ⇒ same for 2
utility function for commodities in group 2: U2
U2 is independent of x1: u2 = U(x2) group-2 utility
U2 is an aggregator function, for a given a1 in group 1:
U2(x2) = U(a1, x2)
U2 aggregate quantities for group-2 commodities
allows to define a macro function U of x1 and the aggregator:
U(x1
, U2
(x2
)) := U(x)
not necessarly a symmetric concept, not necessarly unique
as long as the allocation in group 2 in optimal, within-group
(i.e group 2) allocation will be optimal
28. Price aggregation (Gorman, 1959)
drawback of separability: the consumer needs the optimal
expenditure in each sector before proceeding to within-sector
allocations
allocation depends on all prices and income → decentralization
is disappointing.
idea: use sectoral prices and total expenditure as information
rather than n commodity prices.
homogeneity of degree zero of the indirect utility function
ensure the existence of price aggregation (no separability here)
means that demand for commodity i in sector r depends on
income, vector of price indexes and prices.
optimal allocation in sector r based on income and price
indexes, demand still depends on all prices (because of no
separability)
29. Two-stage budgeting (Strotz, 1957, Blackorby and Russel, 1997)
separability (quantity aggregation) alone is disappointing, price
aggregation alone is a bit useless...
Strotz (1957): use both!
question: under what conditions can the consumer
determine optimal group expenditure using only group price
indices
spend the allocated budget using only sector-specific prices
Preferences are separable in R groups and the conditional
indirect utility function is twice-differentiable.
indirect utility function of the generalized Gorman polar form
aggregator function are generalization of hometheticity plus
homothetic sectoral utility functions.
Econometrics and identification: Blundell and Robin’s (2000)
latent separability
30. Two-stage budgeting (Strotz, 1957, Blackorby and Russel, 1997)
separability (quantity aggregation) alone is disappointing, price
aggregation alone is a bit useless...
Strotz (1957): use both!
question: under what conditions can the consumer
determine optimal group expenditure using only group price
indices
spend the allocated budget using only sector-specific prices
Preferences are separable in R groups and the conditional
indirect utility function is twice-differentiable.
indirect utility function of the generalized Gorman polar form
aggregator function are generalization of hometheticity plus
homothetic sectoral utility functions.
Econometrics and identification: Blundell and Robin’s (2000)
latent separability
31. Additive structures
Basically, solve the dimensionality problem by dividing the products
into smaller groups and allow for a flexible functional form within
each group.
U(x) = U(U1
(x1
) + U2
(x2
) + ... + Un
(xn)) :=
n
i
ui (xi )
one of oldest idea in the representation of preferences (Gossen,
1854)
usually judged as too restrictive for economics, but....
Bergson-Samuelson, with ui including different weights
Discounted utility model, with ui including different time
weights (constant discounting, hyperbolic discouting...).
Cobb-Douglas, quasi-linear, CES functions (e.g
Spence-Dixit-Stiglitz)
Stone-Geary, almost ideal demand system, Pigou’s law
(Deaton, 1974)
decentralization and aggregation, price indexes...
32. Additive structures and separability
separability on subsets {1, ..., R} allows for macro functions:
U(x) = U(u1(x1), u2(x2), .., uR(xR), x1...Rc
)
to go one step further: two versions of separability:
weak separability (on all coordinates)
strong separability (on all possible partitions)
weak separability corresponds to
U(x) = U(u1(x1), u2(x2), .., un(xn)) for n ≥ 3.
strong separability corresponds to the additive representation
for n ≥ 3:
U(x) =
n
i
ui (xi )
for n = 2 further conditions are necessary (e.g the Thomsen
condition).
33. Separability over coordinates
Xi is separable from the other attributes/criteria if and only if:
(a1, ..., xi , ..., an) (a1, ..., yi , ..., an)
⇒ (b1, ..., xi , ..., bn) (b1, ..., yi , ..., bn)
for all cases where the four n-tuples are in X.
allows to define an asymmetric relation i on Xi from on X
by:
xi i yi ⇔ (a1, ..., xi , ..., an) (a1, ..., yi , ..., an)
for some (a1, ..., xi , ..., an), (a1, ..., yi , ..., an) ∈ X
X is supposed to be a large subset of X1 × ... × Xn to avoid triviality:
cases where i is empty, i.e there are never two n-tuples that have the
same values of Xi for all but one i and different values of Xi for the other
i.
35. Strong separability
a subset I : {Xi : i ∈ I} is strongly separable from its
complement Ic : {1, ..., n}I if and only if:
(xi for i ∈ I, ai for i ∈ Ic
) (yi for i ∈ I, ai for i ∈ Ic)
⇒ (xi for i ∈ I, bi for i ∈ Ic
) (yi for i ∈ I, bi for i ∈ Ic)
for all cases where the four n-tuples are in X.
typical exemple (Kreps): I are food items, Ic are clothing,
housing... Separability means:
that combinations of food does not depend whether the
consumers are wearing skirts or t-shirts
37. Independence
Coordinate independence:
(x1, ..., ai , ..., xn) (y1, ..., ai , ..., yn)
⇔ (x1, ..., bi , ..., xn) (y1, ..., bi , ..., yn)
for all icases where the four n-tuples are in X.
Coordinate independence implies weak separability
Coordinate independence allows to replace any single pair of
equal coordinates
Weak separability allows to replace n − 1 t-uples of pairs of
equal coordinates
Notation ai x ai y ⇔ bi x bi y
if all the Xi ’s are the same, the condition is called the
"sure-thing principle".
39. Independence and additive utility (Krantz et al. 1971)
outcome set X = n
i=1 Xi with n ≥ 3
binary relation on X × X satisfying
essentiality (each attribute matters)
solvability wrt. every attribute
weak order + independence and Archimedean axiom for
every attribute
40. Limitations
not general n = 2 needs additional assumptions
X is a Cartesian product of the Xi , if not additional- highly
technical - assumption needed (e.g Chateauneuf and Wakker,
1993)
restricted solvability has only meaning for continuums (time,
money) but not for discrete sets (qualitative)
41. Interdependent preferences
Debreu (1960): independence for some subsets of goods may
fail when other subsets are independent of their complements.
Multiplicative form u(x1, ..., xn) = u1(x1) × ... × un(xn)
without constant sign: ui (xi ) > 0 and ui (yi ) < 0.
ideal points x∗: independence vs. interdependence:
x*!
xi! yi!
a!
b!
(c) spherical isoutility
x*!a!
b!
xi! yi!
(d) elliptical isoutility
42. Next time...
interpretation of the axiomatic in terms of indifference curves
elicitation of utility functions for multiattribute choice
standard sequence and tradeoff consistency
risk and uncertainty