UNH Why Study of Markets with Frictions Is Interesting or Important Reflection

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What did you learn from this class? Discuss why the study of markets with frictions is (or is not) interesting or important.

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Part I: Labor Markets
Introduction
Search theory provide a novel, compared to classical economics,
way to study markets with frictions.
Labor markets are just one application, but a very important one:
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the economic fortunes of most people are largely determined
by labor market outcomes;
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this includes employment-unemployment experiences and
wages
Also, many policy makers focus a lot (perhaps too much) on
unemployment – e.g., the Fed’s “dual mandate.”
Motivation
The classical tools of supply and demand analysis are very useful
for understanding some aspects of labor markets – e.g., the e¤ects
of taxation.
But they are limited:
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not consistent with coexistence of unemployment and
vacancies;
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not consistent with long and variable durations of
unemployment and vacancy spells
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not consistent with wage dispersion for similar workers/jobs.
It’s doubtful classic S and D analysis has anything at all to say
about equilibrium unemployment or vacancies.
Even it we allow w to be too high and call S-D unemployment, or
too low and call D-S vacancies, we cannot get both at once.
And certainly we cannot get wage dispersion.
Big Ideas
Search theory is designed to study random processes of workers
…nding and losing jobs, and …rms …nding and losing employees.
It builds on the notion that it takes time and other resources to
…nd most things, including a desirable house, spouse, car, … and,
of course, a job if you are a worker, or a worker if you are a …rm
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Early search models were concerned with individual activities
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Modern search models integrate individual activities into
general equilibrium settings.
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Importantly, frictions allow deviations from the “law of one
price” in standard GE theory.
More Big Ideas:
Search theory is also one of the few ways in which economists
study agents trading with each other:
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in standard GE theory agents simply trade along their budget
lines at prices they take as given.
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there is no notion of who works for whom – there is only
aggregate S and D of labor,
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prices are determined by the theory – i.e., by us as economists
– to clear the market, not by the individuals in the model
Search is an very natural way to study decentralized markets
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frictions make things interesting!
A Decision Problem
Consider an unemployed individual looking for a job.
Assume time is discrete time his horizon is in…nite, t = 1, 2, …
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this is a good simpli…cation for many purposes.
Assume he wants to maximize the PV of life-time earnings with
discount factor ? = 1/(1 + r ), where r > 0.
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income is w if employed at w (generally, this can include
wages, plus bene…ts, commuting distance, etc.)
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income is b if unemployed (generally, this can include UI
bene…ts, plus home production, value of leisure, etc.)
One interpretation is risk neutrality; there are others, including
complete insurance markets.
A Digression on Probability Theory
Consider a random variable x. It is called a discrete random
variable if it takes values in a set X = fx1 , x2 , …g, called the
support, each with some probability. We write this as x = x1 with
prob ? 1 , x = x2 with prob ? 2 , … and, in general, x = xj with prob
? j . An example is shown in the left panel of the diagram below.
The probabilities are such that ? j
0 for all j, and typically
? j > 0 for all j (if ? j = 0 for some j we can eliminate xj from X ).
The support X may contain a …nite or an in…nite number of
elements, but in any case the law of total probability is ?j ? j = 1.
We are often interested in certain statistics, such as the mean, or
expected value, denoted Ex and de…ned by
Ex = ? ? j xj .
j =1
Figure: A Discrete (left) and a Continuous (right) Random Variable
A continuous random variable has support with a continuum of
values, e.g. X = [x, x ], where x and x > x are the lower and
upper bounds of an interval. We are not so interested in the
probability of each potential realization x, but instead in the
probability density function, or pfd, f (x ). It tells us the probability
of certain events, e.g., the probability that x is in a subinterval
(x1 , x2 ) X is given by the area under the curve between x1 and
x2 , as shown in the right panel. That area is given by the integral,
prob (x1
x
x2 ) =
Z x2
f (x ) dx.
x1
An integral is a generalization of a sum: partition (x1 , x2 ) into
subintervals; take a value of x in each subinterval and compute an
area, f (x ) times the length of the subinterval; sum up these areas;
repeat taking smaller subintervals; then continuing in this way, in
the limit we get the integral.
A special case is an interval [x, x? ], starting at lower bound x and
ending at x? > x, so prob (x x x? ) = prob (x x? ) = F (x? ),
where F ( ) is called the called the cumulative distribution
function, or cdf. It is the area under f ( ) to the left of x?:
Z x?
F (x? ) =
f (x ) dx.
x
So at any x the pdf is the derivative of the cdf, f (x ) = F 0 (x ).The
law of total probability is now
F (x ) =
Z x
f (x ) dx = 1,
Z x
xf (x ) dx.
x
and the mean is given by
Ex =
x
Some random variables are neither discrete nor continuous – e.g.,
x can take values in a set fx1 , x2 , …g with some probability, and
values in an interval [x, x ] with complimentary probability. In any
case, we still have a cdf F (x? ) = prob (x x? ). Indeed, in general a
cdf describes any random variable x, including the discrete case,
the continuous case, and combination cases. In general we write
the expectation as
Z
Ex =
x
xdF (x ) .
x
If F ( ) is di¤erentiable then dF (x ) = F 0 (x ) dx = f (x ) dx, so the
expectation of a continuous random variable is a special case. With
a little more e¤ort, one can see that the expectation of a discrete
random variable is a special case, too.
For present purposes, we do not need all the power of formal
probability theory, but we will use some of the notation and idea,
in particular the idea of the mean. In fact, we are interested not
only in the mean or expected value of x, but in the expected value
of functions of x. For instance, if x is consumption and U (x ) is a
utility function, when x is random expected utility EU (x ), or more
concisely just EU, is given by
EU =
Z x
U (x ) dF (x ) .
x
Note that we sometimes write the above as Ex U to make it clear
that x is the random variable.
Also note that the expectation operator is linear – i.e., for any
constants a and b, E(a + bx ) = a + bEx.
Important Concept: An agent is said to be risk neutral when his
utility function U (x ) is linear, meaning he only cares about the
mean: EU (x ) = U (Ex ). He is strictly risk averse when U (x ) is
strictly concave, in which case EU (x ) < U (Ex ) for any genuinely random x (i.e., it has more than one value in the support, so where there is actually some risk). And he is strictly risk loving when U (x ) is strictly convex, in which case EU (x ) > U (Ex ) for any
genuinely random x.
In economics and …nance it is often assumed that agents are risk
averse, or, sometimes, risk neutral. But even if their direct utility
function is concave, their indirect utility (value) function can be
convex, as we will see below, making them prefer more risk.
From the graph this is clear: EU (x ) < U (Ex ) (i.e., risk aversion) is the same as U (x ) concave. As a limiting case, EU (x ) = U (Ex ) (i.e., risk neutrality) is the same as U (x ) linear. Back to the Decision Problem Assume one job o¤er w arrives each period, which come as i.i.d. random draws from a known cdf F (w ). I if multiple o¤er arrive in a period, simply interpret F (w ) as the cdf of a new random variable, the best o¤er. I of course learning about F (w ) is important, but that is a complication best ignored for now. Also assume: I rejected o¤ers are lost forever (no recall); I accepted o¤ers entail permanent employment (no quits); I in fact, these restrictions are not binding, given our other assumptions. A Dynamic Programming Formulation Let W (w ) be the (present) value of accepting w . Let U the value of rejecting it, which does not depend on w given our assumptions. Due to stationarity, we can write these recursively as follows: W (w ) = w + ?W (w ) U = b + ?Ew? max fW (w? ), U g Here Ew? is the expectation of the value of the next o¤er w? , and max captures the idea that w? can be accepted or rejected. Hint: to remember notation, W and U stand for "working" and "unemployed." Optimal Search Behavior First it is clear that W (w ) = w + ?W (w ) ) W (w ) = w 1 ? , which actually should be obvious, given accepted o¤ers entail permanent employment. As W (w ) is strictly increasing and U is independent of w , there is a unique w , called the reservation wage, where W (w ) = U. Then the optimal search strategy is clearly: w < w ) reject and w w ) accept Hence, while the problem may seem di¢ cult, the solution is conceptually simple! The optimal search strategy has a simple representation in terms of w , although we still have to solve for w . An Aside on Risk Before solving for w , notice something interesting in the above diagram: the value of the next o¤er maxfW (w ), U g is convex. This means that agents like risk! The reason is simple: they have the option to reject low w while accepting high w , and a more disperse o¤er distribution lets them better take advantage of that. Now if they actually get utility U (w ) and U (b ) while employed and while unemployed, this e¤ect is tempered by risk aversion captured by assuming U ( ) is concave. But we can still say they like risk in they dist’n of U (w ) induced by the the dist’n of w . Finding w Notice b + ?Ew? max fW (w? ), U g = w / (1 w = (1 ? ) b + (1 = (1 ? ) b + (1 ?), and so ?) ?Ew? max fW (w? ), U g w? w ?) ?Ew? max , 1 ? 1 ? Cancelling 1 ? in the last term, we have w = T (w ), where T is a nice function – i.e., a contraction – given by T (w ) = (1 ?) b + ?Ew? max fw? , w g . In other words, w is the solution to T (w ) = w . One might ask, is there a solution to T (w ) = w ? If so, is it unique? And, how can we …nd it? Since T (w ) is a contraction, w = T (w ) has a unique solution, and it can be found it by iterating on wn +1 = T (wn ) for any w1 . In fact, if w1 = b then wn is the reservation wage when there are n periods left to search, and wn ! w as n ! ?. Another Version of the Solution Subtracting ?w from BS of w = T (w ) and simplifying, we get w =b+ ? 1 ? Ew? max fw? w , 0g , R w? Using ?/ (1 ?) = 1/r and Ew? g (w? ) = 0 g (w? )dF (w? ) for any function g ( ), we get an eqn often seen in the literature w =b+ 1 r Z w? (w? w )dF (w? ), w where max is taken into account by the lower limit of the integral, and w? denotes the upper bound. Intuitively, the LHS is the opportunity cost of rejecting w , while the RHS is the bene…t, b plus the value of further search, which is the discounted expected gain from seeing another o¤er. Some Simple Experiments From the reservation wage eqn, it is easy (if you know calculus, as discussed below in a Technical Digression) to derive: ?w ?w > 0 and
> 0.
?b
??
Intuitively, when you …nd unemployment less painful (higher b) or
you are more patient (higher ?) you become more selective.
In particular, higher UI bene…ts make agents less inclined to accept
o¤ers, so they have longer unemployment spells on average.
But of course this does not mean they are worse o¤!
Can one even contemplate such questions using S and D analysis?
Generalizations
Now suppose o¤ers arrive with probability ?
unemployed.
1 each period while
And workers lose their job (e.g., get laid o¤) each period with
probability ? 1 while employed.
The previous methods imply there is again a reservation wage w .
If we also assume, mainly to ease notation, that w and b accrue at
the end of each period, the recursive equations are
W (w ) = ? [w + (1
U
= ? [b + (1
?) W (w ) + ?U ]
?) U + ?Ew? max fW (w? ), U g]
assuming that losing a job means you must wait a period for the
next o¤er, just like rejecting a job.
Further Analysis
Subtracting ?W (w ) from BS of the …rst eqn, we get
(1
?) W (w ) = ?w + ?? [U
which simpli…es using r = (1
W (w )]
?) /? to
rW (w ) = w + ? [U
W (w )]
Similarly, from the second eqn, we get
rU = b + ?Ew? max fW (w? )
U, 0g .
These set ‡ow values equal to current reward plus excepted capital
gains or losses.
Conveniently, they also hold in continuous time.
More Analysis
From rW (w ) = w + ? [U
W (w )] and rW (w ) = w , we get
(r + ? ) [W (w )
W (w )] = w
w .
Also, from rW (w ) = rU
w
= b + ?Ew? max fW (w? )
?
= b+
Ew? max fw?
r +?
W (w ) , 0g
w , 0g .
Again inserting the integral for the expectation, we arrive at
w =b+
?
r +?
Z w?
(w?
w )dF (w? ) ,
w
generalizing the baseline model with ? = 1 and ? = 0.
Technical Digression
Here we need some calculus, and in particular Leibniz’rule,
?
?x
Z b (x )
f (x, t )dt =
a (x )
Z b (x )
a (x )
fx (x, t )dx + f [x, b (x )]b 0 (x )
f [x, a (x )]a0 (x
Recall the previous reservation wage eqn,
w =b+
?
r +?
Z w?
(w?
w )dF (w? ) ,
w
Taking the derivative of BS wrt b, we get
?
?w
= 1+
?b
r +?
)
Z w?
w
?w
dF (w? ) = 1
?b
?w
=
?b
1+
?
r +?
? ?w
[1
r + ? ?b
F (w )]
1
2 (0, 1) .
[1 F (w )]
So an increase in b raises w , but by less than the increase in b.
One can similarly derive ?w /?r , ?w /?? and ?w /??.
Durations and Unemployment
An employed workers transits to unemployment at rate ?, implying
an expected duration 1/? by the standard formula.
Similarly, an unemployed workers transits to employment at rate
? = ? prob (w? w ), implying an expected duration 1/?.
If there are many similar workers, the aggregate unemployment
rate u adjusts in continuous time according to the di¤erential eqn
u? = (1
u) ?
u?
? (u ) .
Then u? = 0 implies the the steady state, or natural rate, of
unemployment is
?
u =
.
?+?
Convergence to the natural rate: Clearly u < u ) u? > 0 and
u > u ) u? < 0, so u ! u starting from any u. Dynamic Equilibria The dynamic system moves towards u from any starting point, although "shocks" can move us away. Hence, even if the economy is stochastic, there is a tendency to move toward the natural rate. Even in the long run there will be unemployed workers. Even without stochastic shocks individuals experience random durations of unemployment and employment spells. Moreover, across employed workers there will be a distribution of wages, G (w ) = F (w jw w ). Heterogeneity of workers gives w dispersion, too, but search gives it even with homogeneous workers – it’s due to luck, although that can be expected to average out in the long run. More Experiments The previous results continue to hold, ?w ?w > 0 and
> 0,
?b
??
but now we can additionally say that higher b or ? increase u .
Additionally one can check
?w
?w
> 0 and
< 0, ?? ?? which means agents are more selective when it is easier to get an o¤er, and less selective when is it easier to lose a job. The impact of ? and ? on u are complicated, however, since they directly a¤ect transitions, plus they a¤ect w . Another Extension If one wants to model quits, assume w changes stochastically on job at rate ?, say, to a random draw from F (w? jw ), so thatn rW (w ) = w + ? [U W (w )] + ?Ew? max fW (w? ) W (w ), U W (w )g Note: This can also be interpreted as learning on the job. If higher w implies the conditional distribution F (w? jw ) is better, the employed use the same w for quits as the unemployed use for acceptance. In the simplest case where w? is independent of w , we get ? ? w =b+ r +? Z ? (w? w )dF (w? ) , w which shows that w < b when ? > ?.
Model explains quits negatively correlated with wage and tenure.
Yet Another Extension
Importantly, in the data there are also job to job transitions with
no intervening spell of unemployment.
To capture that we can add search on the job, letting ?0 and ?1 be
arrival rates of o¤ers for the unemployed and employed.
If, e.g., outside o¤ers are drawn from the same F (w ) then
= b + ?0 Ew? max fW (w? ) U, 0g
rW (w ) = w + ? [U W (w )] + ?1 Ew? max fW (w? )
rU
W (w ), 0g .
This is similar to having w change, as in the previous extension,
except now you have the option of keeping the old w .
This extension also admits w < b, in this case when ?0 < ?1 . Explains job changes negatively correlated with wage and tenure. One More Extension One can also model the arrival of o¤ers as a choice. In the case of no search on the job, e.g., the only change is rU = max [b + ?Ew? max fW (w? ) U, 0g k (?)] , which says that the arrival rate ? is a choice, but it entails a cost k (?), where it is natural to impose k 0 > 0 and k 00 > 0.
We still get the equations analyzed above, plus the FOC
Ew? max fW (w? )
U, 0g = k 0 (?) ,
which equates the marginal bene…t and cost of an additional o¤er.
This also helps explain some labor market facts.
Conclusion
The last extension, to the choice of ?, adds some marginal
considerations to an otherwise discrete choice framework.
But it is worth emphasizing that discrete choice theory is, for a
great many applications, the more natural mode of analysis.
Lots of decisions in life involve discrete choice, like either accepting
or rejecting job o¤ers.
This is quite di¤erent from classical economic analysis, which
simply sets hours of work so the marginal cost (loss of time) just
equals the marginal bene…t (higher income).
I think this is why students enjoy learning search theory – to them,
it’s novel, realistic and relevant for labor markets.
And for many other decisions that are naturally understood as
discrete choices – should I marry this person, buy that house, go to
graduate school or join the army, etc.

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