Final review nopause

ECON W4850 Final Exam Review
Ambuj Dewan
May 8, 2015
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Three Perspectives on Theories of Human Behavior
Milton Friedman
Daniel Kahneman
Daniel McFadden
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Milton Friedman
It doesn’t matter if a theory isn’t “realistic” as long as it
generates correct predictions.
Even if people aren’t literally making the computations
necessary for ﬁnding an optimal solution in their heads, what
matters is that they behave as if they were.
e.g. An expert billiards player doesn’t perform complex
kinematic calculations when deciding how to make a shot.
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Daniel Kahneman
People are capable of reasoning, but many judgments are
made quickly and intuitively.
Deliberative judgment is slow and costly.
Intuitive judgments display systematic biases.
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Daniel McFadden
Identiﬁes three diﬀerent kinds of rationality:
1. Perception rationality: Beliefs about one’s situation
constructed from rational inference from available evidence
2. Preference rationality: Choices made based on preferences
that are coherent and consistent across situations
3. Process rationality: Choices made to maximize utility in a
given situation
Most behavioral biases are perceptual.
Proposes “new economic analysis” where people want to
rationally optimize but are prone to errors caused by
perceptual biases
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Perceptual Errors
The perceived magnitude of a stimulus is a random function
of its true magnitude.
Fechner proposes:
perceived magnitude = true magnitude + random error
Results in a probability distribution of perceived magnitudes
for each true magnitude
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The Psychometric Function
Comes from two-alternative forced choice experiments
Given two stimuli, asked which one is bigger, more intense,
brighter, etc.
Can hold one fixed while the other changes
On the horizontal axis: the ratio (or difference) of the two
stimuli
On the vertical axis: the proportion of responses where the
variable stimulus is chosen over the fixed one
Can also come from signal detection theory. (Here the “fixed
stimulus” is no stimulus.)
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Point of subjective equality (PSE): point where variable
and fixed are chosen equally often (yes and no for SDT)
50th percentile of psychometric function
Just noticeable difference (JND): increase in ratio required
to go from choosing variable stimulus 50% of the time to
choosing it 75% of the time
Difference between 75th and 50th percentiles of psychometric
functions.
Weber’s Law: The increase in a stimulus needed to produce
a JND is proportional to the stimulus intensity.
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Imperfect Discrimination and Signal Detection Theory
Suppose there are two stimuli, i = 1, 2. µ1 < µ2 are their true
magnitudes. Their perceived magnitudes are distributed as
ˆθi ∼ N(µi , σ2).
Consistent with Fechnerian view that true magnitudes
generate a probability distribution of perceived magnitudes.
Degree of discriminability (d ):
d =
µ2 − µ1
σ
Easier to tell apart two distributions when the diﬀerence in
means increases or when their standard errors decrease.
Decreases “overlap” between distributions.
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Signal Detection Theory
Any “rule” for detecting the presence of a stimulus will
generate misses (not detecting a stimulus) and false alarms
(incorrectly saying there’s a stimulus when there is none).
In class, we showed (using the Neyman-Pearson lemma)
that a test that minimizes the rate of misses subject to a given
false alarm rate takes the form of a likelihood ratio test: say
“yes” when the likelihood ratio exceeds some cutoff β.
If perceived magnitudes are normally distributed, this is
equivalent to saying “yes” when the perceived magnitude
exceeds some threshold c.
c depends on the false alarm rate we’re willing to accept.
There’s a trade-off between reducing misses and reducing false
alarms.
This trade-off is given by the receiver operating
characteristic (ROC) curve.
The larger d is, the less severe this trade-off is.
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The Ideal Observer Model
Idea: People optimally choose the cutoff β∗ so as to maximize
their expected rewards.
Problem: We observe conservative cutoff placement:
people choose cutoffs between β∗ and 1.
One proposed explanation is that people choose cutoff to have
the same overall frequency of misses and false alarms. (See
Problem Set 1.)
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Endogenous Discriminability
Our ability to discriminate between stimuli might get better
over time, because we can examine more evidence.
σ shrinks, so d increases.
There is a trade-oﬀ between speed and accuracy in decision
making.
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Sequential Evidence Accumulation (SEA)
Suppose you have access to any number of pieces of evidence rk,
but that each one is costly. Then, an eﬃcient decision rule (that
minimizes the number of tests subject to no higher a rate of false
alarms and misses) is a sequential probability ratio test:
Say “yes” if LR(r) ≥ βs.
Say “no” if LR(r) ≤ βn.
Gather another piece of evidence if LR(r) ∈ (βs, βn).
for some thresholds βn < 1 < βs.
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Drift-Diffusion Model (DDM)
Basically the continuous-time version of SEA.
Let r(t) be a Brownian motion process with drift parameter µ
and instantaneous variance σ2.
Say “yes” if upper boundary reached first.
Say “no” if lower boundary reached first.
Continue gathering evidence as long as process remains
between boundaries.
See Problem Set 2.
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Drift-Diﬀusion Model (DDM)
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Estimation
Example: Experiments of Kaufman et al. (1949)
Perception of numerosity
Guess the number of dots on a screen
For small numbers of dots (5 or 6), use a process called
subitizing, which is nearly 100% accurate; basically just
counting the dots
For larger numbers of dots, use a less accurate method that
formulates a best guess based on what is perceived —
estimation
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Estimation
Let x be the true value of what we’re trying to estimate. An
estimator ˆx is unbiased if E[ˆx|x] = x.
The bias of an estimator is b(x) ≡ E[ˆx|x] − x.
Unbiasedness seems like a desirable property, but is it really
what we want?
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Estimation
Minimum-MSE Estimation
The mean squared error (MSE) of an estimator is:
E[(ˆx − x)2|x].
It can be shown that MSE = E[(b(x))2 + v(x)], where
v(x) = var[ˆx|x].
There’s a trade-oﬀ between lowering the bias and lowering the
variance.
The estimator that minimizes the MSE is ˆx(r) = E[x|r];
estimates are equal to the posterior mean, given the data.
The average bias of this estimator is zero.
The bias of this estimator covaries negatively with the true
value of x.
Thus, minimum-MSE estimation necessitates conservatism;
estimates are biased to be less extreme than the true values.
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Estimation
Over-confidence
Over-confidence: True values are less extreme than
estimates.
It is possible for both conservatism and over-confidence to be
present simultaneously.
Conservatism is a property of E[ˆx|x].
Over-confidence is a property of E[x|ˆx].
However, over-confidence is inconsistent with minimum-MSE
estimation; minimum-MSE estimation requires that estimates
be equal to the posterior means, not more extreme than them.
See Problem Set 3.
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The Limits of Information Processing
There’s a limit to how much information about a stimulus we
can process.
There’s a limit to the number of diﬀerent things we can tell
apart (e.g. Pollack, 1952).
What we’re interested in is how much uncertainty about a
stimulus is reduced by observing the subject’s response.
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Mutual Information
A measure of the reduction in uncertainty is Shannon’s
mutual information:
I ≡
i,j
p(si , rj ) log2
p(si , rj )
p(si )p(rj )
This measures the average reduction in entropy
(H ≡ − i pi log2 pi ) from observing the subject’s response.
If stimulus s and response r are independent, then I = 0; we
learn nothing about the stimulus by observing r.
If responses are perfectly accurate, then I = H; we learn the
stimulus perfectly by observing r.
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Comparative and Absolute Judgment
Comparative judgment is a simple matter of comparing two
stimuli in quick succession. We’re quite good at this.
Absolute judgment involves comparing a stimulus to a set of
mental categories. We’re quite bad at this when the number
of mental categories is not small. There’s a limit to the
number of mental categories we can have.
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Multiple Dimensions and Set Size Effects
If options differ on multiple dimensions, it becomes very
difficult to categorize them on a single scale; it’s not
immediately clear which options are better than others.
The larger the set of options, the harder it is to learn about
each option. This reduces the accuracy of categorization.
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Choice Overload
If there are lots of options that differ on multiple dimensions,
this increases the bias in the estimation of their values.
Recall that minimum-MSE makes conservatism an optimal
response.
The estimated values of high-value options are attenuated,
which increases the probability that their perceived value will
fall below the reservation value.
This explains why choice can be demotivating.
However, there’s a countervailing force: when the set size
increases, this increases the probability that there will be
high-value options.
This explains why experts may benefit from lots of choice;
they have high accuracy in their judgments, so the second
force outweighs the first.
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Environmental Statistics and Perceptual Coding
Our brains allocate more neural resources to detecting,
discriminating, and processing more common stimuli.
Properties of allocation of neural resources:
width of tuning curve ∼ 1/environmental frequency
cell density ∼ environmental frequency
discrimination threshold ∼ 1/environmental frequency
Example: more neural resources devoted to dealing with
vertical angles than oblique ones
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Eﬃcient Allocation of Attention
Suppose we have a set of channels to which to allocate
attention, i = 1, 2, . . . , I. We want to minimize the overall
error rate i πi ei , but we have an attention budget
i c(ei ) ≤ M.
Cost decreases in error rate; less costly to pay less attention.
Result: pay more attention to more probable channels; for
each channel, optimal ei is decreasing in πi .
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Decision Utility vs. Experienced Utility
Decision utility: Prospective evaluations on the basis of
which we make decisions
Experienced utility: Actual degree of enjoyment from
outcome
Focusing illusion: Placing disproportionate weight on some
attributes of the decision problem over others
Implication: Decision utility is a biased estimate of experienced
utility.
e.g. Schkade and Kahneman (1998): People over-emphasize
climatic conditions when thinking about where to live.
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Predicting Attention Allocation
Theory of Koszegi and Szeidl (2012)
Each option has attributes {xa}.
Consumption utility: a xa
Decisions based on focus-weighted utility: a gaxa
ga increases in the range of variation of attribute a; pay more
attention to attributes that vary more.
These weights are consistent with the theory of rational
inattention.
Intuition: Reducing the posterior variance helps us make better
decisions, but what’s costly is reducing the posterior variance
relative to the prior variance. Hence, we can get the cheapest
improvements by paying more attention to the attributes that
account for more of the prior uncertainty.
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Real Effects of Monetary Policy
Rational inattention can explain real effects of monetary
policy, why prices don’t adjust immediately in response to
changes in monetary policy.
Firms pay more attention to sector-specific conditions than
aggregate conditions when making pricing decisions.
This makes sense; sector-specific conditions are more highly
variable.
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Discrimination
Many disparities in economic outcomes along lines of race,
gender, ethnicity, etc.
Disparities persist despite decline in declared prejudice.
Could be due to reduced willingness to reveal prejudice, or
difference between conscious attitudes and implicit ones.
Bertrand and Mullainathan (2004): sent fictitious résumés to
help-wanted ads
50% more callbacks for white names than black ones
Not just level effects — quality had less of an impact for black
names
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Statistical Discrimination
In theory, race, gender, ethnicity, etc. should be irrelevant in
determining ability. Does that mean discrimination is
irrational?
Suppose there are two groups, B and W , and that signals of
ability from B are noisier. Then, minimum-MSE estimation
dictates a greater degree of conservatism in the estimation of
the ability levels of members of group B.
⇒ High-ability members of B are more undervalued.
⇒ Lower incentives for members of B to accumulate human
capital, which can lead to self-fulﬁlling expectations of low
productivity (Coate & Loury, 1993).
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Attention Allocation and Discrimination
Why are signals of minority group’s ability noisier?
Suppose people have ﬁnite capacity for attention. Then they
allocate less attention to processing information from
members of groups encountered less frequently (i.e. minority
groups).
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Expected Utility Theory
Evaluate prospects by their expected utility, i.e. the sum of
the utilities generated by each outcome, weighted by their
probabilities.
i
pi ui
u is usually a concave function of wealth ⇒ risk aversion
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Problems with Expected Utility Theory
Aversion to small gambles — implies unrealistic rejection of
larger gambles (Rabin paradox)
Isolation effect — EUT evaluates final outcomes, not the
paths taken to reach them
Both risk seeking and risk aversion — People are
risk-averse for some gains and risk-seeking for others; difficult
to explain without weirdly-shaped utility functions
Allais Paradox — The common components of two gambles
are not necessarily evaluated in the same way
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Prospect Theory
Prospect — deﬁned by a set of net gains {xi } and
associated probabilities {pi }.
Evaluate prospects by computing:
i
π(pi )v(xi )
v(x) — value function
π(x) — probability weighting function
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Prospect Theory
Value Function
Strictly increasing
v(0) = 0
Strictly concave for x > 0
Strictly convex for x < 0
Kinked at x = 0, i.e. lim
x→0−
v (x) > lim
x→0+
v (x)
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Prospect Theory
Probability Weighting Function
Strictly increasing, continuous
π(0) = 0, π(1) = 1
π(p) > p for small p, π(p) < p for large p
Subcertainty — π(p) + π(1 − p) < 1 for p ∈ (0, 1)
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Prospect Theory
Addressing the Problems with Expected Utility Theory
Aversion to small gambles — Kink in value function implies
loss aversion
Isolation eﬀect — PT evaluates net gains
Both risk seeking and risk aversion — Underweighting of
small probabilities explains why people become risk-seeking
(averse) for small probability of a large gain (loss)
Allais Paradox — Subcertainty implies that common
components of gambles can’t just be “subtracted away”
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Prospect Theory
Prospect Theory as Perceptual Bias
Concavity in gains, convexity in losses is an example of
diminishing marginal sensitivity in the coding of magnitudes
(cf. Weber’s Law).
If more processing capacity allocated to losses than gains,
then assessments of magnitudes of losses are more precise.
Assessments of losses are less conservative.
Implies kink of value function at zero.
Underweighting small probabilities and overweighting large
ones is an example of conservatism.
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Intertemporal Choice
Consumption plans are evaluated using a discounted utility
function:
U(ct, ct+1, . . .) =
j≥0
Dj u(ct+j )
u(c) is the per-period utility function.
Dj are discount factors.
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Exponential Discounting
Dj = δj for some δ ∈ (0, 1]
Implies time consistency:
U(ct, ct+1, . . .) = u(ct) + δU(ct+1, ct+2, . . .)
Plan that is thought to be optimal at t will still be thought of
as optimal at t + 1.
Discount rate ρj ≡ −
ln Dj
j is constant.
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Hyperbolic Discounting
In reality, people are time-inconsistent — often want to
change their previously determined consumption plans.
Hyperbolic discounting — discount rate ρj ≡ −
ln Dj
j is a
(weakly) decreasing function of time.
Common formulation: present bias
D0 = 1, Dj = βδj
for j ≥ 1.
The future is discounted sharply relative to the present, but
successive future periods are discounted at a constant rate
relative to each other.
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Na¨ıvet´e and Sophistication (O’Donohue & Rabin, 1999)
Na¨ıve hyperbolic discounter: At each date, chooses a plan
that is optimal given current preferences, not taking into
account that future self might want to change plan.
Sophisticated hyperbolic discounter: At each date, chooses
a plan that takes into account how future selves will behave.
Makes decisions as if playing a game against future selves —
these decision problems are solved by backwards induction.
Data on gym memberships seem to indicate people act more
like na¨ıve hyperbolic discounters (DellaVigna & Malmendier,
2006).
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Poverty and Present Bias
Endogenous increased discounting of future consequences
caused by reduced processing capacity (Mullainathan &
Shaﬁr, 2013).
Poverty itself results in myopic behavior.
Dire need causes the poor to focus their attention on the
short term.
Result: “tunnel vision” that ignores future matters (e.g.
consequences of borrowing at high interest rates).
Policy implications: may not be enough to try to change the
behavior of the poor simply by providing incentives (because
they may not be well perceived) or providing information (that
may not be paid attention to).
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Asymmetric Dominance Eﬀect
Regularity: Adding an option to a choice set shouldn’t
increase purchases of the previous options.
But adding a decoy option can lead to violations of regularity.
Asymmetric Dominance Eﬀect: Suppose there are two
options A and B, neither of which dominates the other. C
(the decoy) is introduced, and is dominated by A (the target
good), but neither dominates nor is dominated by B. This can
increase purchases of A.
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Comparisons
What goods decoys increase purchases of depend on what
consumers pay attention to.
Heath and Chatterjee (1995) ﬁnd that asymmetrically
dominated decoys increase purchases of
high-quality/high-price goods among MBA students at an
urban research university (probably pay more attention to
quality), but increase purchases of low-quality/low price goods
among undergraduates at a rural state university (probably
pay more attention to price).
A favorable comparison can increased perceived value along a
dimension to which one pays little attention.
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Contrast Effects
Many judgments/decisions made on the basis of comparisons.
Can lead to contrast effects.
E.g. rats’ willingness to lick sucrose solution influenced by
contrast in concentration (Flaherty et al., 1983).
Also create visual illusions.
Simultaneous illusions: caused by other stimuli that are
simultaneously present.
After-effects: caused by other stimuli that were previously
present.
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Flaherty et al. (1983)
Example of Contrast Eﬀects
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Zollner Illusion
Example of a Simultaneous Illusion
The diagonal lines are actually parallel.
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Land-Horn Computational Model
An ordered sequence of detectors (left to right in the visual
ﬁeld).
Detectors are edge detectors — respond to local contrast.
Signal from detector n is:
sn = xn − 1
2(xn−1 + xn+1)
where xn is luminance at location n.
Edges ignored below a certain contrast level. sn replaced
yn = f (sn), where:
f (s) =



s + δ, s ≤ −δ
0, −δ < s ≤ δ
s − δ, s > δ
Brain attempts to reconstruct xn from yn.
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Sensitivity to Differences
Encoding of differences implies contrast effects.
Why are sensory systems more sensitive to differences than
absolute magnitudes? Two proposed explanations:
1. Signal-Extraction Hypothesis: Brain tries to estimate
something that can’t be directly observed. If changes in the
observable quantity are highly likely to reflect to changes in the
quantity of interest, then it makes sense to encode differences.
2. Recalibration Hypothesis: Sensory circuits can encode only a
limited number of distinct states, so the mapping from sensory
magnitudes to subjective representation is recalibrated when
the environment changes. (e.g. Eyes adjusting to light in a
dimly or brightly lit room.)
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Salience
Theory of Bordalo, Gennaioli, & Shleifer (2013).
Salience of an attribute for a good depends on comparison of
that attribute with its average.
Example of a salience function:
σ(ak, ¯a) =
|ak − ¯a|
ak + ¯a
Quality is salient for good k if σ(qk, ¯q) > σ(pk, ¯p); price is
salient for good k if reverse is true.
Utility weights: (δ ∈ (0, 1])
gsal = 2
1+δ gnot = 2δ
1+δ
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Salience and Decoy Effects
Suppose there are two goods, neither of which dominates the
other. (One is higher-Q and higher-P; the other is lower-Q
and lower-P.) Then higher Q or lower P is salient iff qk
pk
> ¯q
¯p .
If P is salient for high-quality good h, adding a decoy d with a
low enough value per dollar (qd
pd
) can bring ¯q
¯p below qh
ph
and
possibly flip preferences.
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Before Introducing Decoy
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After Introducing Decoy
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Context Dependence of Willingness to Pay
Willingness to pay (WTP) for a good with price p and quality
q depends on choice context {(0, 0), (q, p), (q, pe)}, where pe
is the expected price.
pe depends on where you are — e.g. willing to pay more for a
beverage at a resort than at a convenience store.
Raising pe raises ¯p, which can ﬂip salience from P to Q and
increase WTP.
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Standard Asset Pricing Theory and the Equity Premium
Puzzle
According to standard asset pricing theory, asset prices and
expected returns should be such that:
E[Rk] = Rrf −
cov(Rk, Λ)
E[Λ]
So if returns on risky asset higher than risk-free rate of return,
it must be because risky returns and the marginal utility of
income are negatively correlated.
Problem: growth in aggregate consumption expenditure
exhibits low covariance with stock market returns — equity
premium puzzle (Mehra & Prescott, 1985).
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Prospect Theory and the Equity Premium Puzzle
PT can explain the EPP (Benartzi & Thaler, 1995)
People care about gains or losses of each investment decision,
not ﬁnal outcomes.
People care about variability of returns, rather than
considering how randomness “cancels out” across investments.
People are loss-averse.
Barberis (2013) also notes that distribution of stock market
returns is negatively skewed ⇒ PT predicts an equity premium
also because people overweight small probability of large loss.
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Pricing Long-Lived Assets
Price should not only depend on dividends, but also what one
might expect to sell asset for.
Rational expectations: Price assets so that expected returns
are equal across assets, assuming that future traders also price
assets in this manner.
Fundamental value determined according to:
E[Rt] =
E[dt + Pt]
Pt−1
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Asset Bubbles
Price temporarily much higher than fundamental value, then
crashes.
Possible explanation: Mistaken valuations are self-conﬁrming.
Past overvaluation of stock becomes reason to expect it will
continue to be overvalued.
It may be fully rational to purchase an overvalued stock if you
expect it to continue to be overvalued.
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Public Policy
If consumer choices are the result of cognitive biases, not true
preferences, then there is a strong argument for intervention.
Problem: How do we determine what true preferences are?
Try to measure experienced utility.
E.g. “happiness” surveys.
Can conduct welfare analysis incorporating biases if true
valuations are known (e.g. Allcott and Taubinsky, 2015).
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