Standard Approach: Partial Equilibrium Analysis
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Analyze a single market in isolation from the rest of the
economy.
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Underlying assumption: the market is small enough so that
general equilibrium e§ects (i.e. repercussions through
adjustments in other markets) can be ignored.
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Should really be seen as a Örst approximation.
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Examples: Markets for various grocery items such as wine,
milk, bread. Markets for various services: hairdressing,
banking services etc.
Standard Approach: Partial Equilibrium Analysis
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Modeling tool: quasilinear preferences:
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The quantity of the good bought by individual i is denoted by
qi ! 0.
Utility to i from consuming qi units: vi (qi ) .
Outside good y that can be thought of as money or a
composite good reáecting all other consumption.
The good is priced at p > 0 per unit of consumption. The
composite good is priced at 1, and hence p is also the relative
price.
Initial holdings of mi units of the composite good or money.
Quasilinear utility:
ui (yi , qi ) = vi (qi ) + yi .
Standard Approach: Partial Equilibrium Analysis
Consumerís problem:
max vi (qi ) + yi
yi ,q i !0
subject to
yi + pqi = mi .
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Interpretation: y is a composite good that represents the
major part of the consumption outlays for all consumers.
Hence changes in qi have a linear e§ect on the utility from
consumption of y .
If the consumerís total budget is mi ! 0, then only mi " pqi
is left for other goods.
Substituting from the budget constraint into objective
function, we get:
max vi (qi ) + mi " pqi .
qi
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Note that this formulation applies to choosing continuous
units as well as choosing discrete units.
Standard Approach: Partial Equilibrium Analysis
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We will assume that the function vi is increasing and has
decreasing marginal utilities. For discrete units,
vi (qi ) ! vi (qi " 1) ,
vi (qi ) " vi (qi " 1) ! vi (qi + 1) " vi (qi ) ,
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and in the continuous case, vi0 (qi ) ! 0 and vi00 (qi ) $ 0, i.e.
vi (%) is an increasing and concave function.
In the continuous case, the Örst order condition for maximum
is
vi0 (qi ) = p
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We denote the solution to the consumerís problem by
qi (p ) .
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By di§erentiating the Örst-order condition, we get the law of
demand:
qi0 (p ) $ 0,
or in words, the individual demand is downward sloping.
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Standard Approach: Partial Equilibrium Analysis
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Production side: cost function
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Firm j supplies qjs units of the good.
! "
The cost function cj qjs measures the cost of delivering qjs
units on the market in terms of the composite good (or
money).
The produced good is priced at p in the market, and the Örm
chooses qjs to maximizes its Önal wealth:
! "
s
max
m
+
pq
"
c
qjs ,
j
j
j
s
qj
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where mj is the initial holdings of money by Örm j.
This is called the Firmís problem.
The solution to the Örmís problem gives the supply function
qjs (p ) .
Standard Approach: Partial Equilibrium Analysis
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For the most part, we assume that the cost function is
inceasing and convex in the sense that for all j and qjs ,
# $
#
$
cj qjs ! cj qjs " 1 ,
and
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# $
#
$
#
$
# $
cj qjs " cj qjs " 1 $ cj qjs + 1 " cj qjs
! "
! "
with discrete supply levels or cj0 qjs > 0 and cj00 qjs ! 0 in
the case of a continuous qjs .
First order condition:
# $
cj0 qjs = p.
The solution to this equation is called the individual supply of
Örm j and it is denoted by qjs (p ) .
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By di§erentiation:
qjs 0 (p ) =
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cj00
1
! " > 0.
qjs
In words, more is supplied at higher output prices.
Standard Approach: Partial Equilibrium Analysis
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Market place:
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In competitive analysis, Örms and consumers meet in an
anonymous market.
Anonymity means that the price is the same for all participants
and not dependent on the identities i and j.
Both the buyers and the sellers are price takers: the price is
there and does not depend on individual demands qi and
supplies qjs .
Price is linear: the cost of buying qi units is p % qi rather than
a more general function p (qi ) .
All buyers and all sellers know the price in the market.
Standard Approach: Partial Equilibrium Analysis
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We cover only the case with continuous demands and
supplies, but all arguments generalize to the discrete case too.
The market functions as follows
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Market demand function Q is obtained by summing over i all
individual demand functions:
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 qi (p ) ,
Q (p ) =
i =1
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where I is the total number of consumers in the market.
By the individual laws of demand, we get
Q 0 (p ) =
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 qi0 (p ) < 0.
i =1
Market demand curve is thus downward sloping.
Standard Approach: Partial Equilibrium Analysis
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Market supply Q s (p ) is obtained by summing over j all
individual supply functions:
Q s (p ) =
J
 qjs (p ) ,
j =1
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where j is the total number of Örms in the market.
By individual laws of supply, we get
Q s 0 (p ) =
J
 qjs 0 (p ) > 0.
j =1
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An equilibrium in the market is a pair (p & , Q & ) such that
markets clear:
Q & = Q (p & ) = Q s (p & ) .
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Observe that in equilibrium, each Örm j supplies quantity
qjs (p & ) and each consumer i demands qi (p & ) .
Markets, E¢ciency and Welfare
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How to deÖne e¢ciency? This is hard a priori.
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How to deÖne ine¢ciency? An economic allocation is clearly
ine¢cient if it is possible to improve the welfare of some
individuals without hurting others.
An allocation is Pareto-e¢cient if it is not clearly ine¢cient.
Or in other words:
DeÖnition
A feasible allocation is Pareto-e¢cient if there is no other feasible
allocation where at least one of the agents is better o§ and none of
the agents is worse o§.
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Yet another way of phrasing this: Starting from a
Pareto-e¢cient allocation, you cannot help anyone without
hurting someone else.
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What do we mean by agentsí welfare here?
For consumers, it is measured by their utility function
yi + v (qi ).
For Örms, it is measured!by "their proÖt in terms of the
composite good y " c q s .
Markets, E¢ciency and Welfare
Theorem (First Fundamental Welfare Theorem (Invisible
Hand))
Every competitive equilibrium is Pareto-e¢cient
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With quasi-linear preferences, there are no other
Pareto-e¢cient allocations
Theorem
At any Pareto-e¢cient vi0 (qi ) = cj0 (qj ) for all i and j, and
therefore the aggregate quantity produced is given by Q & .
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To see why this must be true, notice that proÖtable trade
exists between buyers i and i 0 if vi0 (qi ) 6= vi00 (qi 0 ) , a
0
proÖtable
! " of production exists between j and j if
! " reallocation
cj0 qjs 6= cj00 qjs0 and a proÖtable trade exists between i
and j if vi0 (qi ) 6= cj0 (qj ) . Hence the allocation is Pareto
e¢cient only if vi0 (qi ) = cj0 (qj ) = p for some p.
Markets, E¢ciency and Welfare
Finally, an allocation is feasible only if at least as much is produced
as consumed. E¢ciency implies that total amount consumed must
be exactly the amount produced. But then
Q (p ) = Q s (p ) ,
and this happens only when p = p & and Q (p ) = Q s (p ) = Q & .
Summary
In competitive anonymous markets:
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Prices adjust to clear the market
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Competitive equilibrium allocation is e¢cient
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A unique price obtains in the market (Law of one price).
Evaluating the predictions
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Predictions seem to work reasonably well in centralized
markets such as commodity exchanges
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What about closer to home: supermarkets?
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Markets do seem to clear reasonable well
Law of one price fails.
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This failure has been documented in numerous markets.
Does not seem to be too sensitive to the deÖnition of market
Source: Kaplan & Menzio (2014): The Morphology of Prices
Evaluating the predictions
How to quantify the price dispersion?
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Kaplan & Menzio (2014) have data on purchases collected by
Nielsen. They have computed:
How to explain this?
Maybe the stores are di§erentiated
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Some stores higher quality
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Some stores at more attractive locations
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But variation by store accounts only for 10% of total price
variation
Maybe supermarkets have di§erent costs
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Maybe they have di§erent wholesale prices
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But wholesale price depends on the chain to which the store
belongs and the chain explains very little of the price variation.
How to explain this?
Maybe supermarkets are special
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shoppers get a basket of goods, not a single good
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But Sorensen (2000) shows similar price dispersion for
prescription drugs in pharmacies in a small town.
In Sorensen (2000), variation in prices seems to be
independent in the sense that some drugs are expensive while
others are cheap at a given pharmacy.
Maybe shoppers are not aware of all the prices in their market
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This is the story line pursued in the next lecture
How to explain this?
Maybe there is a pricing cycle?
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Weíll take this up later in the course on intertemporal price
discrimination
Maybe coupons make the prices di§erent for di§erent individuals
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Covered in the section on price discrimination
Is the price dispersion economically important?
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For a representative shopping basket, K&M compute
dispersion in the price index depending on the deÖnition of
good as follows: