Pareto Efficiency and Market
Lam, Wing Tung
ECON2210 Fall 2023
ˆ Positive economics is about predictions and testable implications.
ˆ Welfare economics concerns about whether an exogenous change in the environment
leaves an entity better off or worse off. Its ultimate goal is to design policies to
make certain groups, and possibly everyone, better off. The distinction between the
positive and welfare economics is not meant to be a dichotomy, but a reflection of
the differences in interests.
ˆ In the first part, we introduce the notion of Pareto efficiency. It is a general welfare
criterion which applies to any institution.
ˆ For expositional simplicity, we consider an exchange economy where the total quantities
of the goods are exogenously given. We focus on how to efficiently allocate existing
goods among households with different preferences.
ˆ In the second part, we define the notion of competitive equilibrium in an exchange
economy. It is the equilibrium concept in the general equilibrium framework whereby
prices in all markets are determined simultaneously. Compared to partial equilibrium,
we turn from studying a single market in isolation to studying the entire economy.
ˆ We conclude with the welfare property of an equilibrium allocation. We introduce the
celebrated First Welfare Theorem, which states that a perfectly competitive market
always achieves Pareto efficiency.
ˆ The bigger picture is that we move from studying the equilibrium allocation in a
competitive market to evaluating the market as an institution for allocating resources
1
1
Pareto Efficiency
ˆ In the partial equilibrium framework, we have learnt that the equilibrium allocation
in a perfectly competitive market is efficient in the sense that it maximizes the total
(economic) surplus and ensures no dead weight loss. In this notes, we extend welfare
economics beyond the partial equilibrium framework.
ˆ We now introduce the notion of Pareto efficiency, the most general criterion of
a “socially desirable” allocation. Compared to the measures of total surplus and
dead weight loss, Pareto efficiency is much more general because it applies to any
institution, not limited to markets.
Exercise (Robinson and Friday). Suppose there are only two households,
Robinson and Friday, and two goods, bread and rum, on an island. Initially,
Robinson is endowed with 4 units of bread while Friday is endowed with 2 units
of bread and 2 bottles of rum. The two households cannot produce bread and
rum themselves.
Robinson prefers more bread but is indifferent about the quantity of rum. Friday
prefers more rum but is indifferent about the quantity of bread.
Suppose, initially, Robinson and Friday do not exchange their goods but simply
consume their endowments. Do you find such an allocation “socially desirable”?
why?
ˆ Suppose there are J households in the economy. We index households by j =
1, 2, . . . , J.
– In setting with only two or a few households, we index households by letters
j = A, B, C, . . . instead.1
ˆ X j = (xj1 , xj2 , . . . , xjN ) denotes household j’s consumption bundle.
– Superscript denotes household
1
You should not literally interpret the setting with two households as there are only two humans in the
economy. The correct interpretation is that there are two types of representative households. Each type of
household represents a large number of households who are homogeneous, or very similar, in their preferences
and endowments. E.g. Different age group, workers with various levels of educational attainment.
2
– Subscript denotes good
– Power is placed outside bracket e.g., (xjn )3 denotes cubic of xjn
ˆ (X 1 , X 2 , . . . , X J ) is called an allocation of consumption goods. It describes the
consumption bundles for every household.
ˆ To achieve generality, we directly consider the set of FEASIBLE allocations of consumption
goods.
– By changing the allocation of inputs, the economy will produce different quantities
of consumption goods. We can incorporate the production functions and the
endowments all into the set of feasible allocations.2
– In the canonical setting, the households have preferences over the consumption
bundles, but not the production decision or firms’ profit per se.3 Ultimately,
only the allocation of consumption goods matters.
ˆ For simplicity, we consider an exchange economy where the only source of the goods
is the endowments. The households cannot produce new goods but exchange their
endowed goods with each other.
ˆ Household j’s endowment bundle is denoted as E j = (ej1 , ej2 , . . . , ejN ), where ejn is the
endowed quantity of good n
ˆ We then look at how the endowed goods can be reallocated among the households.
An allocation is feasible if for each good, the total quantities consumed equal to the
total quantities endowed
x1n + x2n + . . . + xJn = e1n + e2n + . . . + eJn , n = 1, 2, . . . , N
– There are N equality constraints in total.4
– In subsequent discussion, we consider only feasible allocations and omit the word
“feasible” if no confusion arises.
2
This is formally known as the production possibilities curve.
This setting implicitly assumes no externalities caused by the production.
4
In principle, we may allow the households to discard endowments and replace the equality with an
inequality constraint that total quantity consumed cannot exceed the total quantity endowed. The results
are the same as we assume monotonic preferences.
3
3
ˆ We focus on the setting with only two households and two goods. In this setting, we
can represent the set of feasible allocations using the Edgeworth box.
– Household A is bottom left corner and household B is top right corner
– Horizontal axis is good 1 while vertical axis is good 2
– The length of an axis is the total quantity of that good endowed
– Each point inside the Edgeworth box is a feasible allocation, satisfying
B
A
B
xA
1 + x1 = e 1 + e 1
B
A
B
xA
2 + x2 = e 2 + e 2
The distances from the borders are quantities consumed by the corresponding
household
ˆ It is always feasible everyone to simply consume their endowment, X A = E A and
X B = E B . Hence, the endowments (E A , E B ) is also a point in the Edgeworth box
4
ˆ We now turn to comparing different allocations.
Exercise. Suppose a planner reallocates the goods among the two households.
The new allocation is labelled as (X A , X B )
We can deduce the direction of reallocation from the Edgeworth box
– Is good 1 reallocated from household A to B, or the other way around?
– Is good 2 reallocated from household A to B, or the other way around?
– In an exchange economy, household A consumes more of good n if and only if
household B consumes less of good n. When comparing quantities of goods, it
suffices to look at household A’s consumption bundle in the Edgeworth box.
ˆ Each household j ranks the bundles X j and hence the allocations based on her
preference.
We can represent the households’ preferences using the indifference
curves.
5
Household A’s preference over allocations
Household B’s preference over allocations
– The arrows in the colored graphs indicate the direction of increasing utility
ˆ Now we are going to compare feasible allocations from a normative standpoint. That
is, are some allocations more “socially desirable” than others?
Definition (Pareto Dominance). A feasible allocation (X 1 , X 2 , . . . , X J )
Pareto dominates another feasible allocation (Y 1 , Y 2 , . . . , Y J ) if every household
j weakly prefers X j to Y j , and some household strictly prefers X j to Y j .
– Suppose (X 1 , X 2 , . . . , X J ) Pareto dominates (Y 1 , Y 2 , . . . , Y J ). When we switch
the allocation from latter to former, no one is worse off and opposes the switch,
but someone is strictly better off.5
– We also call (X 1 , X 2 , . . . , X J ) a Pareto improvement over (Y 1 , Y 2 , . . . , Y J )
ˆ When comparing two allocations, it is possible that neither one Pareto dominates
the other.6 This happens if some household strictly prefers one allocation while some
strictly prefers the other allocation.
5
Suppose all households strictly prefer (X 1 , X 2 , . . . , X J ) to (Y 1 , Y 2 , . . . , Y J ). It is logically correct to
say that all households weakly prefer the former allocation, and some household strictly prefers the former
allocation to the latter. Therefore, (X 1 , X 2 , . . . , X J ) Pareto dominates (Y 1 , Y 2 , . . . , Y J ).
6
In mathematics terminology, Pareto dominance is said to be a partial order of allocations, not a complete
order.
6
ˆ Pareto dominance is an ordinal ranking of allocations. Unlike the measure of dead
weight loss, the concept of Pareto dominance does not quantify how much an allocation
is “better” than another allocation.
ˆ It is uncontroversial that a Pareto dominated allocation cannot be “socially desirable.”
Put it differently, a “socially desirable” allocation must not be Pareto dominated.
Definition (Pareto Efficiency). A feasible allocation is Pareto efficient if no
other feasible allocations Pareto dominates it.
– In words, an allocation is Pareto efficient if it is impossible to make someone
strictly better off without making some other household strictly worse off. Pareto
efficiency is also called Pareto optimality.
– Typically, there can be many Pareto efficient allocations. Pareto efficiency can
be thought of a minimal or necessary requirement for an allocation to be “socially
desirable”.
– The set of all Pareto efficient allocations is called the Pareto set.
Exercise (Robinson and Friday). Consider again Robinson and Friday on the
island. Recall that Robinson prefers more bread but is indifferent about the
quantity of rum. Friday prefers more rum but is indifferent about the quantity
of bread.
We denote Robinson’s and Friday’s by X R and X F respectively. Let bread and
rum be good 1 and 2. Initially, they consume their own endowment. We represent
the initial allocation as X R = E R = (4, 0) and X F = E F = (2, 2)
1. Graph the initial allocation in an Edgeworth box
2. For each household j, draw the set of his indifference curves over X j and
indicate the direction of increasing utility.
3. Propose an allocation which is Pareto dominated by the initial allocation
4. Propose an allocation which neither Pareto dominates nor is Pareto
dominated by the initial allocation.
7
5. Propose an allocation which Pareto dominates the initial allocation
6. Find out the unique Pareto efficient allocation.
ˆ Notice that the Pareto set depends only on the total quantities of the goods but not
the initial distribution of the endowments. It is because the former determines the
set of feasible allocations.
ˆ With only two households and two goods, we can find out Pareto improvements over
an allocation graphically using the Edgeworth box. We illustrate it using the colored
diagram below.
– Represent an allocation (X A , X B ) in the Edgeworth box
– Depict household A’s indifference curve passing through X A . The shaded area
is the feasible allocations which make household A better off
– Depict household B’s indifference curve passing through X B . The shaded area
is the feasible allocations which make household B better off
– The overlapped area (including the boundary) is the allocations which Pareto
dominates (X A , X B )
ˆ We now apply the principle of marginalism to characterize the condition for Pareto
efficient allocations.
– First, we look at how to construct a Pareto improvement by reallocating incremental
amount of goods among households.
8
– If the given allocation is Pareto efficient, no incremental adjustments in the
allocation can be a Pareto improvement. This observation allows us to derive a
set of conditions for Pareto efficient allocations
A (X A ) > M RS B (X B ). The following graph depicts the situation
ˆ Suppose M RS12
12
– Suppose we reallocate an incremental amount ∆x2 of good 2 from household A
to household B and an incremental amount ∆x1 of good 1 from household B
to household A. Both parties will be strictly better off if
∆x2
B (X B )
> M RS12
∆x1
– Graphically, the new allocation will be inside the overlapped area.
A (X A ) >
M RS12
– In words, the above inequality ensures that household A is willing to forgo more
than ∆x2 units of good 2 in exchange for ∆x1 units of good 1 while household
B is willing to forgo more than ∆x1 units of good 1 in exchange for ∆x2 units
of good 2. Thus the reallocation benefits both of them.
A (X A ) < M RS B (X B ).
ˆ The opposite happens for the case M RS12
12
9
– In this case, household A is willing to forgo more units of good 2 in exchange
for an additional unit of good 1 than household B.
– We can construct a Pareto improvement by reallocating an incremental amount
∆x1 of good 1 from household A to household B and an incremental amount
∆x2 of good 2 from household B to household A. The reallocation makes both
parties strictly better off if
A (X A ) <
M RS12
∆x2
B (X B )
< M RS12
∆x1
– The above graph depicts the reallocation of goods
Exercise. In each of the following graphs, shade the allocations which are Pareto
improvements and describe how the goods are reallocated.
A (X A ) > M RS B (X B )
M RS12
12
A (X A ) < M RS B (X B )
M RS12
12
Example. Suppose there are two households j = A, B and two goods n = 1, 2.
B
The distribution of endowments is that E A = (2, 2)
qand E = (0, 1). Household
A
A’s preference is represented by U A (xA
1 , x2 ) =
A
xA
1 x2 while Household B’s
B
B
B
preference is U B (xB
1 , x2 ) = x1 + x2 .
b A = (2, 1) and X
b B = (0, 2). It is not Pareto
Consider the feasible allocation X
efficient. Explain how to reallocate the goods to make both parties better off.
10
b A = (2, 1) and X
b B = (0, 2)
Let us compare the MRS for the two households at X
A
M RS12
(2, 1) =
1
B
< 1 = M RS12
(0, 2)
2
It indicates that household A is willing to forgo more units of good 1 in exchange
for one more unit of good 2 than household B. We can make both parties
better off by reallocating some incremental amount of good 1 from household A
to household B and some incremental amount of good 2 from household B to
household A
Exercise (Cobb-Douglas preferences). Suppose there are two households j = A, B
and two goods n = 1, 2. The distribution of endowments is that E A = (1, 2) and
E B = (1, 1). Two
same preferences, which are represented
q households have the q
B
B
A
A
A
A
B
by U (X ) = x1 x2 and U (X ) = xB
1 x2 .
The following allocations are feasible but not Pareto efficient. For each allocation,
explain how you may reallocate the goods to make both parties better off.
Represent your answer in the Edgeworth box.
b A = (1, 1); X
b B = (1, 2).
1. X
b A = (0, 1); X
b B = (2, 2).
2. X
ˆ Consider an allocation (X A , X B ) in the INTERIOR of the Edgeworth box, where all
households are consuming positive quantities of all goods. If the indifference curves
of the two households are not tangent to each other at (X A , X B ), we can always
find a Pareto improvement. This allows us to characterize the condition for Pareto
efficient allocations in the INTERIOR of the Edgeworth box.
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Remark. Suppose the two households have well-behaved preferences which can
be represented by continuously differentiable utility functions.
(X A , X B )
An allocation
in the interior of Edgeworth box is Pareto efficient if and only if the
tangency condition holds,
A
B
M RS12
(X A ) = M RS12
(X B )
– The tangency condition is not only a necessary but also a sufficient condition,
under the assumption of well-behaved preferences.
Example (Cobb-Douglas preferences). Suppose there are two households j =
A, B and two goods n = 1, 2. The distribution of endowments is that E A = (1, 2)
and E B = (1, 1).
Two q
households have the same
q preferences, which are
B
B
A
B
represented by U A (X A ) = xA
xB
1 x2 and U (X ) =
1 x2 .
B
A
B
The feasible allocations are those satisfying xA
1 + x1 = 2 and x2 + x2 = 3.
We first consider the allocations in the interior of the Edgeworth box. The
tangency condition can be written as
xB
xA
A
B
2
2
= M RS12
(X A ) = M RS12
(X B ) = B
=η
A
x1
x1
12
where η denote the slope of the two indifference curves at the tangent point.
A
B
B
A
B
Substitute xA
2 = ηx1 and x2 = ηx1 into the condition x2 + x2 = 3, we obtain
B
3 = (xA
1 + x1 )η. We then solve out η =
3
2
B
from the condition xA
1 + x1 = 2.
Therefore, a feasible allocation satisfies the tangency condition if and only if it
A
B
B
satisfies 3xA
1 = 2x2 and 3x1 = 2x2
Second, we consider the allocations in the boundaries of the Edgeworth box.
Both households are indifferent between all bundles with x1 = 0 or x2 = 0, which
yield zero utility level. In this case, it is efficient to allocate all the goods to the
other household. The Pareto efficient allocations are X A = (2, 3), X B = (0, 0)
and X A = (0, 0), X B = (2, 3). Both allocations satisfy the listed conditions.
To summarize, a feasible allocation is Pareto efficient if and only if it satisfies
both the conditions
A
B
B
3xA
1 = 2x2 ; 3x1 = 2x2
Graphically, the set of Pareto efficient allocations is the diagonal of the Edgeworth
box, which slope is 32 .
ˆ The tangency condition does not apply to the allocations on the boundaries of
Edgeworth box, where a household does not consume one of the goods.
13
Remark. Suppose the two households have well-behaved preferences which can
be represented by continuously differentiable utility functions.
B
– An allocation (X A , X B ) with xA
2 = 0 or x1 = 0 is Pareto efficient if and
only if
A
B
M RS12
(X A ) ≥ M RS12
(X B )
B
– An allocation (X A , X B ) with xA
1 = 0 or x2 = 0 is Pareto efficient if and
only if
A
B
M RS12
(X A ) ≤ M RS12
(X B )
B
– A Pareto efficient allocations on the boundaries with xA
2 = 0 and x1 = 0 can
A (X A ) > M RS B (X B ). Even though household A is willing to
feature M RS12
12
forgo more units of good 2 in exchange for an additional unit of good 1 than
household B, we cannot reallocate good 2 from household A to household B or
reallocate good 1 from household B to household A. The previous construction
of a Pareto improvement no longer applies.
14
– By a symmetric argument, a Pareto efficient allocations on the boundaries with
B
A
A
B
B
xA
1 = 0 and x2 = 0 can feature M RS12 (X ) < M RS12 (X ).
Exercise (Perfect substitutes). Suppose there are two households j = A, B and
A
two goods n = 1, 2. Household A’s preference is represented by U A (xA
1 , x2 ) =
A
B B
B
B
B
2xA
1 +x2 and household B’s preference is represented by U (x1 , x2 ) = x1 +2x2 .
The initial distribution of endowments is that E A = (2, 2) and E B = (0, 1).
1. For each household, draw the set of indifference curves and indicate the
direction of increasing utility.
b A = (1, 2) and X
b B = (1, 1). Graph the set of
2. Consider the allocation X
b A, X
b B ) in an Edgeworth box.
allocations which Pareto dominates (X
3. Shade the Pareto set in the Edgeworth box. Explain why the allocations
B
with xA
2 = 0 or x1 = 0 are Pareto efficient.
Intuitively, we have a Pareto improvement whenever household A exchanges one
unit of good 2 for one unit of good 1 with household B. Household A is willing
to forgo two units of good 2 for that unit of good 1 while household B is willing
to forgo two units of good 1 for that unit of good 2. Such reallocation always
makes both households better off. Pareto improvement is not feasible only when
household A runs out of good 2 or household B does not have any good 1.
Example (continued). The distribution of endowments is that E A = (2,q
2) and
B
A
A
A
A
E = (0, 1). Household A’s preference is represented by U (x1 , x2 ) = xA
1 x2
B
B
B
while household B’s preference is U B (xB
1 , x2 ) = x1 + x2 .
We now find out all Pareto efficient allocations
1. Consider the interior of the Edgeworth box, show that an allocation is Pareto
A
efficient if and only if it satisfies xA
1 = x2 .
15
The tangency condition is given by
xA
A
A
B
2
= M RS12
(xA
1 , x2 ) = M RS12 = 1
xA
1
A
Only the allocations with xA
1 = x2 satisfies the tangency condition.
A
2. Consider the boundaries of of the Edgeworth box with xA
1 = 0 or x2 = 0.
What is the Pareto efficient allocation?
Notice that the household A is indifferent between all such allocations
A
A
because U A (xA
1 , 0) = U (0, x2 ) = 0. Yet we can reallocate goods from
household A to household B to make household B better off. A Pareto
improvement is no longer feasible when household B already consumes all
the goods. Therefore, the only Pareto efficient allocation is X A = (0, 0) and
X B = (2, 3). Graphically, this is the bottom left corner of the Edgeworth
box.
3. Consider the boundary of of the Edgeworth box with xB
2 = 0. It follows that
A
B
xA
2 = e2 + e2 = 3. Show that none of these allocations is Pareto efficient.
A
B
For any feasible allocation, xA
1 ≤ e1 + e1 = 2. Hence,
A
M RS12
(xA
1 , 3) =
3
3
≥
A
2
x1
All such allocations are not Pareto efficient because we can construct Pareto
improvements
A
M RS12
(xA
1 , 3) ≥
3
B
> 1 = M RS12
(xB
1 , 0)
2
4. Consider the boundary of of the Edgeworth box with xB
1 = 0. It follows
A
B
that xA
1 = e1 + e1 = 2. Find out the Pareto efficient allocations on this
boundary.
The Pareto efficient allocations are those satisfying the condition
xA
2
2
A (2, xA ) ≥ M RS B (0, xB ) = 1
= M RS12
2
12
2
16
B
Hence, any allocation on this boundary with xA
2 ≥ 2 and x2 ≤ 1 is Pareto
efficient
Putting all the results together, we shade the Pareto set as below
ˆ We can apply the principle of marginalism to generalize the conditions for Pareto
efficiency to settings with multiple goods and types of households.
– When there are more (types of) households, we have to consider a larger set
of adjustments in the allocations. The task of characterizing the set of Pareto
efficient allocation becomes much more complicated. The appendix contains
examples of three households.
ˆ For simplicity, we focus on the setting with two households and two goods from now
on.
ˆ The criterion of Pareto efficiency has nothing to do with Egalitarianism or equality.
In particular, allocating all goods to one household is Pareto efficient.
– Suppose household A has all the goods while household B consumes no goods.
The preferences are monotonic. If we reallocate some goods from household A
to household B to make the latter better off, the former will be made worse off.
17
– Graphically, the bottom left and top right corners of the Edgeworth box are
always Pareto efficient allocations.
ˆ Household A’s and B’s welfare vary under different Pareto efficient allocations.In
general, when there are two households, household A and B rank the Pareto efficient
allocations in opposite ways.
– Suppose (X A , X B ) and (Y A , Y B ) are two different Pareto efficient allocations.
If household A strictly prefers X A to Y A , then household B must strictly prefer
Y B to X B . The reason is that (X A , X B ) cannot Pareto dominate (Y A , Y B ).
ˆ Let us visualize this observation in the colored graph below.
– Suppose there are infinitely many Pareto efficient allocations in an exchange
economy. The Pareto set forms a curve in an Edgeworth box.
– When we move along the curve from the bottom left corner to the top right
corner, household A will be on a higher indifference curve while household B
will be on a lower indifference curve.
ˆ The concept of Pareto efficiency completely detaches from any market structure, and
can be applied to any application. Therefore, Pareto efficiency serves as a criterion
to evaluate whether an institution allocates resources efficiently.
ˆ In subsequent sections, we study the allocation efficiency in a perfectly competitive
market
– For examples, we assume no problems of information asymmetry, externalities
and non-excludable public good.
18
2
Competitive Equilibrium
ˆ In a competitive market, households take the prices as given. Household j can afford
a bundle (xj1 , xj2 ) as long as
p1 xj1 + p2 xj2 ≤ p1 ej1 + p2 ej2
ˆ When there are two households, we can graphically represent budget sets in the
Edgeworth box
Household A’s budget set
Household B’s budget set
– In a household’s perspective, her budget line passes through her endowment
bundle.
– The households face the same prices, so the slopes of budget lines are the same.
ˆ In household A’s perspective, she may choose any goods bundle in her budget set.
In general, some bundles such as X̂ A in a household’s budget set are outside the
19
Edgeworth box. If household A attempts to purchase the bundle X̂ A , there will be
excess demand for good 2. The relative price of good 2 will increase, making the
households to adjust their consumption bundles. The prices will adjust until the
markets clear.
ˆ The notion of competitive equilibrium captures how prices and allocation are determined
in a competitive market. It is also known as Walrasian equilibrium.
Definition (Competitive equilibrium). A competitive equilibrium in an
e A, X
eB)
exchange economy consists of prices (e
p1 , pe2 ) and an allocation (X
satisfying the following two sets of conditions:
e j is her optimal consumption bundle on the
– For any household j = A, B, X
budget line
pe1 xj1 + pe2 xj2 = pe1 ej1 + pe2 ej2
– The markets for goods all clear. That is,
A
B
x
eA
eB
1 +x
1 = e1 + e1
A
B
x
eA
eB
2 +x
2 = e2 + e2
ˆ In words, the equilibrium prices simultaneously clear all markets and no household
will adjust their consumption bundles.
ˆ For simplicity, we define a competitive equilibrium where there are only two households
and two goods. In appendix, we extend the definition to any setting with J ≥ 2
households and N ≥ 2 goods.
ˆ Graphically, a competitive equilibrium can be represented in Edgeworth box
20
– Market clearing: equilibrium allocation is inside the Edgeworth box, including
the boundaries
– Optimal bundle: for each household, the bundle is on the budget line and the
highest indifference curve attainable
e A, X
e B ) is a competitive equilibrium, so is (te
Remark. If (e
p1 , pe2 ) and (X
p1 , te
p2 )
A
B
e
e
and (X , X ), where t is any positive number
– The budget lines remain the same, so are optimal bundles. The market clearing
conditions are unaffected by the prices.
ˆ The implication of the remark is that the absolute levels of prices are indeterminate.
Only relative prices are determinate.
– When finding a competitive equilibrium, we may normalize price of one good,
say good 1, to unity. We effectively express all prices in unit of good 1.
ˆ Let us revisit an example in the previous notes “Market equilibrium”
– We now apply the concept of competitive equilibrium to study the effects of a
drop in endowment.
21
Example (Cobb-Douglas preferences). Suppose there are two households j =
A, B and two goods n = 1, 2. The distribution of endowments is that E A = (1, 2)
and E B = (1, 1).
represented by
Two q
households have the same
q preferences, which are
B
B
A
B
A
= x1 x2 and U (X ) = xB
1 x2 . Fix prices p1 and
U A (X A )
p2 , the optimal consumption bundles are given by
p1 + 2p2 A
;x
e2 =
p1
B
1 p1 + p2
;x
e2 =
x
eB
1 = 2
p1
x
eA
1 =
1
2
1
2
1
2
p1 + 2p2 p2
p1 + p2 p2
We normalize pe1 to one and solve for pe2 . Market for good 1 clears when
x
eA
eB
1 +x
1 =
1 + 2p2 1 + p2
+
=1+1
2
2
We solve out pe2 = 23 . Substituting pe2 =
2
3
into the demand functions, we then
obtained the equilibrium allocation
eA =
X
7 7
6, 4
5 5
6, 4
eB =
;X
Let us compare the households’ endowment bundles and their consumption
bundles in equilibrium.
x
eA
1 =
7
6
> 1 = eA
eA
1; x
2 =
7
4
< 2 = eA
2
That is, household A is exchanging good 2 for more good 1 in the market. As
the market clears, household B must be on the other side of trade. She is selling
good 1 for more good 2.
x
eB
1 =
5
6
< 1 = eB
eB
1; x
2 =
5
4
> 2 = eB
2
We then represent the competitive equilibrium in an Edgeworth box
22
Exercise (Cobb-Douglas preferences). Two
households
have
the
same
q
B
B
A
A
=
x1 x2 and U (X ) =
U A (X A )
preferences, which are represented by
q
A = (1, 1).
B
xB
1 x2 . Now suppose the household A’s endowment drops to E
Show that the new equilibrium prices are pe1 = pe2 = 1 and new equilibrium
eA = X
e B = (1, 1).
allocation is X
Notice that when household A’s endowment of good 2 declines, the equilibrium
allocation of both goods changes.
– The concept of competitive equilibrium accounts for the linkage across markets,
exemplifying general equilibrium analysis. In particular, a change in one market,
e.g., a reduction in endowment, affects the prices and allocation of all goods.
Exercise (continued). Suppose there are two households j = A, B and two goods
B
n = 1, 2. The distribution of endowments is that E A = (1.6,
q 3) and E = (0.4, 0).
A
Household A’s preference is represented by U A (X A ) = xA
1 x2 while Household
B
B’s preference is U B (X B ) = xB
1 + x2 .
23
Even though the distribution of endowments is different, we already worked out
the Pareto set in the previous example.
1. Is the initial allocation E A = (1.6, 3) and E B = (0.4, 0) Pareto efficient?
2. We now verify that the following prices and allocation is a competitive
equilibrium
pe1 = 1; pe2 = 0.8
e A = (2, 2.5); X
e B = (0, 0.5)
X
e A = (2, 2.5) is household A’s optimal bundle given the
(a) Verify that X
prices pe1 and pe2
e B = (0, 0.5) is household B’s optimal bundle given the
(b) Verify that X
prices pe1 and pe2
(c) Verify the markets for the two goods both clear
In this equilibrium, household B exchanges all her endowment of good 1
with household A for 0.5 units of good 2. We now represent the competitive
equilibrium in the Edgeworth box.
24
ˆ Since a household always has the option not to trade and consume her endowment
e j ) ≥ U j (E j ).
bundle, any trade occurred must be mutually beneficial, U j (X
ˆ An implication is that if the initial distribution of goods is Pareto efficient, exchange
of goods cannot make some party strictly better off. The households have no reasons
to trade as there are no gains from trade.7
Remark. Suppose the households’ preferences are all well-behaved. If the initial
distribution (E A , E B ) is Pareto efficient, then households do not trade in a
competitive equilibrium.
Example (continued). Suppose there are two households j = A, B and two goods
B
n = 1, 2. The distribution of endowments is that E A =
q(2, 2) and E = (0, 1).
A
Household A’s preference is represented by U A (X A ) = xA
1 x2 while Household
B
B’s preference is U B (X B ) = xB
1 + x2 .
Verify that there is a competitive equilibrium in which the households simply
consume their endowments.
7
Results of this kind is generally known as no-trade theorem.
25
In such an equilibrium, the allocation is given by
e A = E A = (2, 2); X
e B = E B = (0, 1)
X
Hence the two markets clear. The first step is to find out the equilibrium prices
e A = (2, 2) to be optimal for
that clear the markets. For the interior solution X
household A, the tangency condition must hold
p1
2
A
= M RS12
(2, 2) =
2
p2
Hence, pe1 = pe2 . We again normalize the price of good 1 to be unity, thus
pe1 = pe2 = 1.
The remaining step is to verify that the consumption bundle (0, 1) is optimal for
household B. Notice that household B takes the two goods as perfect substitutes,
B
M RS12
(x1 , x1 ) = 1 =
pe1
pe2
e B = (0, 1)
She is indifferent between all bundles on her budget line, and finds X
optimal.
From the preceding discussion, the initial distribution of endowments E A = (2, 2)
and E B = (0, 1) is a Pareto efficient allocation. No trade can benefit both parties.
Consequently, the two households do not trade and consume their endowments in
equilibrium. In fact, the above equilibrium is the unique competitive equilibrium
given that E A = (2, 2) and E B = (0, 1).
3
Allocation efficiency of competitive market
ˆ Our baseline formulation of the household’s utility maximization problem and the
competitive firm’s profit maximization problem implicitly assumes a perfectly competitive
market
26
ˆ We now evaluate the allocation efficiency of a perfectly competitive market.
Theorem (First Welfare Theorem). Suppose the households’ preferences are
all strictly monotonic.
In a perfectly competitive market, the equilibrium
allocation of any competitive equilibrium is Pareto efficient.
– In partial equilibrium framework, we learnt that the equilibrium allocation in
a perfectly competitive market maximizes total economic surplus, eliminating
any dead weight loss. First Welfare Theorem is the generalization in the general
equilibrium framework.
– In words, when the market is in equilibrium, all gain from trade must have been
exhausted and no further exchange of goods can be mutually beneficial. This
outcome is precisely captured by the concept of Pareto efficiency.
– The conclusion of First Welfare Theorem applies to not only an exchange economy
but also an economy with production.
– First Welfare Theorem requires virtually no assumptions on preferences and
production functions. The general proof for an exchange economy is provided
in Appendix.
Example (Cobb-Douglas preferences). Suppose there are two households j =
A, B and two goods n = 1, 2. The distribution of endowments is that E A = (1, 2)
and E B = (1, 1).
Two households
have the sameqpreferences, which are
q
B
B
A
B
represented by U A (X A ) =
xA
xB
1 x2 and U (X ) =
1 x2 . We solved out
the equilibrium prices pe1 = 1 and pe2 =
eA =
X
7 7
6, 4
2
3
and the equilibrium allocation
eB =
;X
5 5
6, 4
Since the consumption bundles are optimal, they satisfy the tangency conditions
7
4
7
6
5
4
5
6
pe1
3
A 7 7
= M RS12
( , )=
=
6 4
pe2
2
pe1
3
B 5 5
= M RS12
( , )=
=
6 4
pe2
2
27
Hence, the equilibrium allocation is in the interior of the Edgeworth box and
satisfies the tangency condition for the two households
A eA
B eB
M RS12
(X ) = M RS12
(X )
We conclude that the equilibrium allocation is Pareto efficient.
We then represent the equilibrium allocation in the Edgeworth box.
Recall that the Pareto efficient allocations in this case are those on the diagonal
A
B
B
satisfying 3xA
1 = 2x2 and 3x1 = 2x2 . If we draw the budget line passing through
the endowment bundle, the intersection point with the diagonal is the equilibrium
allocation.
ˆ Unlike the set of Pareto efficient allocations, competitive equilibria depend on the
distribution of endowments, not only the total quantities of goods endowed. This is
because the endowment bundle affects a household’s budget line.
ˆ Typically, there are many Pareto efficient allocations in an exchange economy. First
Welfare Theorem only states that an equilibrium allocation is one of them.8
8
In light of First welfare theorem, Pareto set is also called contract curve. The name captures the idea
28
Exercise (continued). Suppose there are two households j = A, B and two goods
B
n = 1, 2. The distribution of endowments is that E A = (1.6,
q 3) and E = (0.4, 0).
A
Household A’s preference is represented by U A (X A ) = xA
1 x2 while Household
B
B’s preference is U B (X B ) = xB
1 + x2 .
We already solved out the competitive equilibrium
e A = (2, 2.5); X
e B = (0, 0.5)
pe1 = 1; pe2 = 0.8 X
Notice that the equilibrium allocation is on the boundary of the Edgeworth box.
Verify that the equilibrium allocation is Pareto efficient. Since x
eB
1 = 0, it suffices
to show that
A eA
M RS12
(X ) =
pe1
B eB
> M RS12
(X )
pe2
ˆ One may argue that if a Pareto improvement is possible, the households should have
incentive to coordinate themselves to achieve it.
– We first need information on production functions and endowments to find out
the set of feasible allocations
– We then need information on everyone’s preference to find out Pareto efficient
allocations
– It may be too costly to acquire or verify the information. In this case, we need
to incentivize everyone to truthfully report her preference, production function
and endowment.
– Coordination among a large number of parties is easy said than done.
ˆ In this perspective, a competitive market solves the problems of information acquisition
and transmission while different participants are separately acting in their own interests.
– Households and competitive firms observe the market prices and take them as
given.
that the equilibrium allocation specified in the contracts between households must lie on the Pareto set.
29
– A household chooses her most preferred bundle in the budget set without any
knowledge of other households’ preferences and the production functions
– A competitive firm chooses the input bundle to maximize its profit without any
knowledge of households’ preferences and the production functions of other firms
ˆ In the classical essay “The use of knowledge in society”, Friedrich Hayek provides a
vivid description
“Fundamentally, in a system where the knowledge of the relevant facts
is dispersed among many people, prices can act to coordinate the separate
actions of different people in the same way as subjective values help the
individual to coordinate the parts of his plan. . . . Assume that somewhere
in the world a new opportunity for the use of some raw material, say tin,
has risen, or that one of the sources of supply of tin has been eliminated.
It does not matter for our purpose–and it is very significant that it does
not matter –which of these two causes has made tin more scarce. All that
the users of tin needs to know is that some of the tin they used to consume
is now more profitably employed elsewhere and that in consequence they
must economize tin. There is no need for the great majority of them
even to know where the more urgent need has risen, or in favor of what
other needs they ought to husband the supply. If only some of them know
directly of the new demand, and switch resources over to it, and if the
people who are aware of the new gap thus created in turn fill it from still
other sources, the effect will rapidly spread throughout whole economic
system and influence not only all the uses of tin, but also those of its
substitutes, and so on; and all this without the great majority of those
instrumental in bringing about these substitutions knowing anything at all
about the original cause of theses changes.The whole acts as one market,
not because any of its members survey the whole field, but because their
limited individual fields of vision sufficiently overlap so that through many
intermediaries the relevant information is communicated to all. The mere
fact that there is one price for any commodity . . . brings about the solution
which . . . might have been arrived at by one single mind possessing all the
information which is in fact dispersed among all the people involved in the
process.”
30
ˆ A key insight from First Welfare Theorem is that we can allocate scare resources
efficiently by artificially creating a perfectly competitive market
– In appendix, we discuss how the NGO Feeding America allocates donated food
to over two hundreds food banks across U.S. everyday using a market-based
mechanism.
ˆ It is a common misconception that First Welfare Theorem suggests to eliminate
regulations or other sources of inefficiency e.g. market power and externalities, as
much as possible.
– First Welfare Theorem only states that if the regulations and sources of inefficiency
are ALL eliminated, the equilibrium allocation in the now perfectly competitive
market will be Pareto efficient.
– First Welfare Theorem never states that if SOME sources of inefficiency is
eliminated or partially mitigated, the new equilibrium allocation must be a
Pareto improvement over the initial one. In fact, the opposite can happen.
ˆ In this light, economists study how non-market institutions can improve the equilibrium
allocation in various applications where some sources of inefficiency or market frictions
are always present.
4
Key learning outcomes
ˆ Concept: Pareto dominance and Pareto efficiency
ˆ Skill: Connecting graphical representation and mathematical conditions for Pareto
efficient allocations
ˆ Concept: Competitive equilibrium
ˆ Concept: First Welfare Theorem
ˆ Methodology: Evaluating institutions using Pareto efficiency
31
5
Appendix
5.1
Additional examples on Pareto efficiency
Example. Suppose Anthony, Barney and Cathy are trapped inside a elevator one
evening, waiting for the rescue in the morning. Anthony have 7 Toblerone chocolate
in his backpack while Barney have 5 mint candies in his pocket while. Cathy has no
food with her.
They are going to share the confectionery and make it through the night. Let mint
candies and chocolate be good 1 and good 2 respectively. Anthony’s, Barney’s and
Cathy’s preferences can be respectively represented by
A
A A
U A (xA
1 , x2 ) = min(x1 , x2 )
B
B
B
U B (xB
1 , x2 ) = 2x1 + x2
C
C C
U C (xC
1 , x2 ) = x1 x2
A
For example, xB
1 denotes the quantity of mint candies consumed by Barney while x2
denotes the quantity of chocolate consumed by Anthony.
B
C
A
An allocation (X A , X B , X C ) is feasible if it satisfies xA
1 + x1 + x1 = 5 and x2 +
C
xB
2 + x2 = 7.
1. Explain whether one of the following allocations Pareto dominates the other
A
B
B
C
C
(xA
1 , x2 ) = (4, 0); (x1 , x2 ) = (1, 1); (x1 , x2 ) = (0, 6)
(y1A , y2A ) = (3, 6); (y1B , y2B ) = (2, 1); (y1C , y2C ) = (0, 0)
U A (4, 0) < U A (3, 6); U B (1, 1) < U B (2, 1); U C (0, 6) = U C (0, 0).
The first
allocation is Pareto dominated by the second allocation.
2. Explain whether the following allocation is Pareto efficient
A
B
B
C
C
(xA
1 , x2 ) = (2, 2); (x1 , x2 ) = (0, 2); (x1 , x2 ) = (3, 3)
B (0, 2) = 2 > 1 =
No, notice that M RS12
3
3
C (3, 3). We can make them
= M RS12
both better off by reallocating an increment amount of good 1 from Cathy to
32
Barney and reallocating good 2 from Barney to Cathy. For example, the given
allocation is Pareto dominated by the following feasible allocation
B ) = (2, 2); (y B , y B ) = (1, 0); (y C , y C ) = (2, 5)
(y1A , yA
1
2
1
2
Anthony is indifferent as he consumes the same bundle. U B (1, 0) = 2 = U B (0, 2),
so Barney is indifferent too. Cathy is strictly better off as U C (2, 5) = 10 > 9 =
U C (3, 3).
3. Explain why the following allocation is Pareto efficient.
A
B
B
C
C
(xA
1 , x2 ) = (4, 4); (x1 , x2 ) = (1, 3); (x1 , x2 ) = (0, 0)
We now argue that a Pareto improvement is impossible.
First consider an allocation (Y A , Y B , Y C ) that makes Anthony strictly better
off. Since the two goods are perfect complements, he must have more of both
goods y1A > 4 and y2A > 4. Cathy is consuming nothing initially. The goods she
consumed can only come from Barney. Barney is left with less of both goods
and worse off! We conclude that a Pareto improvement cannot make Anthony
strictly better off.
We then argue a Pareto improvement cannot make Barney strictly better off.
Anthony cannot be made worse off either. That means, we cannot take any
goods away from Anthony and the new allocation must satisfy min(y1A , y2A ) = 4.
The remaining feasible adjustments all involve taking goods away from Barney.
So Barney cannot be made better off.
Now consider a reallocation that makes Cathy strictly better off. She must
consume more of both goods, y1C > 0 and y2C > 0. Again, the additional goods
can only come from Barney or Anthony, leaving either of them worse off.
Therefore, we conclude that if we reallocate the goods to make any one of them
strictly better off, another household must be worse off. No Pareto improvement
is possible.
33
Exercise (continued). Consider the setting in the previous example.
A
1. Explain why every Pareto efficient allocation must satisfy xA
1 = x2
2. Explain why the following allocation is Pareto dominated.
A
B
B
C
C
(xA
1 , x2 ) = (0, 0); (x1 , x2 ) = (2, 6); (x1 , x2 ) = (3, 1)
Example (Grass is always greener on the other side of the fence). There
are
three
households j = A, B, C and three goods n = 1, 2, 3. Household A is endowed one unit
of good 1. Household B is endowed one unit of good 2. Household C is endowed one
unit of good 3. An allocation is feasible if the total quantities consumed is one for
every good.
B
C
xA
n + xn + xn = 1, n = 1, 2, 3
A
B
B
B
B
The households’ preferences are represented by U A (X A ) = xA
2 +x3 , U (X ) = x1 +x3
C
and U C (X C ) = xC
1 + x2 . We going to show that the set of Pareto efficient allocation
B
C
is exactly the feasible allocations satisfying xA
1 = x2 = x3 = 0.
B
C
1. Show that a Pareto efficient allocation must satisfy xA
1 = x2 = x3 = 0.
A
Consider an allocation with xA
1 > 0. If x1 is taken from Household A and added
to X B . Then household B is strictly better off while the other two households
are indifferent about the change. So the new allocation is a Pareto improvement.
C
A similar argument applies if xB
2 > 0 or x3 > 0.
B
C
2. Show that any feasible allocation satisfying xA
1 = x2 = x3 = 0 is Pareto efficient.
B
C
Let (X A , X B , X C ) be a feasible allocation satisfying xA
1 = x2 = x3 =
0. Suppose, to the contrary, that a feasible allocation (Y A , Y B , Y C ) Pareto
dominates (X A , X B , X C ). Under (Y A , Y B , Y C ), no one is worse off, and say
34
household A is strictly better off. From the utility functions,
A
y2A + y3A > xA
2 + x3
B
y1B + y3B ≥ xB
1 + x3
C
y1C + y2C ≥ xC
1 + x2
Summing up inequalities across all households, we obtain a contradiction!
1 + 1 + 1 = (y1A + y1B + y1C ) + (y2A + y2B + y2C ) + (y3A + y3B + y3C )
≥ (y2A + y3A ) + (y1B + y3B ) + (y1C + y2C )
A
B
B
C
C
> (xA
2 + x3 ) + (x1 + x3 ) + (x1 + x2 )
C
A
C
A
B
= (xB
1 + x1 ) + (x2 + x2 ) + (x3 + x3 ) = 1 + 1 + 1!
The first and last equalities hold because the total quantities of the goods must
be the same as the quantities endowed. The first weak inequality holds because
y1A , y2B and y3C are non-negative.
Since all households are symmetric in our setting, we relabel other households as
A. Thus the above argument covers all other cases.
35
5.2
Additional examples on competitive equilibria
ˆ Let us first extend the definition of a competitive equilibrium to any exchange
economy.
Definition (Competitive equilibrium). A competitive equilibrium consists
e J ) satisfying the
e 1, X
e 2, . . . , X
of prices (e
p1 , pe2 , . . . , peN ) and an allocation (X
following two sets of conditions:
– The markets for goods all clear. That is, for every good n = 1, 2, . . . , N
x
e1n + x
e2n + · · · + x
eJn = e1n + e2n + · · · + eJn
(Market clearing)
e j is her optional bundle on the budget line
– For any household j, X
pe1 xj1 + pe2 xj2 + · · · + peN xjN = pe1 ej1 + pe2 ej2 + · · · + peN ejN
Exercise (Grass is always greener on the other side of the fence). There
are
three households j = A, B, C and three goods n = 1, 2, 3. Household A is
endowed one unit of good 1. Household B is endowed one unit of good 2.
Household C is endowed one unit of good 3. An allocation is feasible if the total
quantities consumed is one for every good.
B
C
xA
n + xn + xn = 1, n = 1, 2, 3
A
B
B
The households’ preferences are represented by U A (X A ) = xA
2 + x3 , U (X ) =
B
C
C
C
C
xB
1 + x3 and U (X ) = x1 + x2 . There are many competitive equilibria.
1. Verify that the following is a competitive equilibrium
e A = (0, 1, 0); X
e B = (0, 0, 1); X
e C = (1, 0, 0)
(e
p1 , pe2 , pe3 ) = (1, 1, 1); X
Since all three goods are symmetric, we can relabel the goods so that p1 ≥ p2 ≥
p3 . In all equilibria, the equilibrium prices must be pe1 = pe2 = pe3 . It suffices to
rule out the following two cases:
36
2. Suppose p1 ≥ p2 > p3 . Show that the market for good 3 cannot clear.
3. Suppose p1 > p2 ≥ p3 . Show that either the market for good 2 or the market
for good 3 cannot clear.
Example (continued). Suppose there are two households j = A, B and two goods
A
n = 1, 2. Household A’s preference is represented by U A (X A ) = 2xA
1 + x2 and
B
household B’s preference is represented by U B (X B ) = xB
1 + 2x2 . The initial
distribution of endowments is that E A = (2, 2) and E B = (0, 1).
Let us normalize price of good 2 to unity to simplify notation.
1. Argue that given the households’ preferences, the relative price in a
A and M RS B , 2 ≥
competitive equilibrium must lie between M RS12
12
pe1
pe2
≥ 12 ,
irrespective of the households’ endowments
Fix p2 = 1. If p1 > 2, then everyone buys only good 2 and the market for
good 1 cannot clear. If p1 <
1
2,
then everyone buys only good 1 and the
market for good 2 cannot clear.
2. Show that the unique competitive equilibrium is given by (e
p1 , pe2 ) = (2, 1),
A
B
e
e
X = (2, 2) and X = (0, 1).
Fix p2 = 1. Suppose 2 > p1 ≥ 21 , then it is optimal for household A to buy
only good 1. From the household A’s budget constraint,
p2
A
B
x
eA
1 = 2 + 2 p1 > 3 > e1 + e1
The market for good 1 cannot clear! Hence, we conclude that (e
p1 , pe2 ) =
(2, 1).
At (e
p1 , pe2 ) = (2, 1), it is optimal for household B to consume only good 2.
e B = (0, 1) from her budget constraint. Household A is
We deduce that X
indifferent about the any bundle on his budget line
A
2xA
1 + x2 = 2 × 2 + 2
eA =
Market clearing conditions for the two goods are met if, and only if X
(2, 2). We also verify all the equilibrium conditions in the above analysis,
37
e A = (2, 2) and X
e B = (0, 1) is the only competitive
so (e
p1 , pe2 ) = (2, 1), X
equilibrium.
3. Now suppose household B discovers a mine of good 1 so that the new
b B = (1, 1). Show that
distribution of endowments is that E A = (2, 2) and E
e A = (3, 0)
the unique competitive equilibrium is given by (e
p1 , pe2 ) = (2, 1), X
e B = (0, 3). How does the relative price of good 2 change in response
and X
to the increase in endowment?
Let us tackle this type of question in another direction. At (e
p1 , pe2 ) = (2, 1), it
e B = (0, 3)
is optimal for household B to consume only good 2. We deduce X
from her budget constraint. Household A is indifferent about the any bundle,
e A = (3, 0), on his budget line
including X
A
2xA
1 + x2 = 2 × 2 + 2
e A = (3, 0) and X
e B = (0, 3) satisfy the market clearing conditions for the
X
e A = (3, 0) and
two goods. Therefore, we verify that (e
p1 , pe2 ) = (2, 1), X
e B = (0, 3) is a competitive equilibrium.
X
We now rule out the possibility of other equilibria. At (e
p1 , pe2 ) = (2, 1),
B
e
e B = (0, 3),
X = (0, 3) is unique optimal bundle for household B. Given X
e A = (3, 0) can satisfy market clearing conditions for the two goods.
only X
If 2 >
p1
p2
≥ 12 , then it is optimal for household A to buy only good 1. From
the household A’s budget constraint,
p2
A
x
eA
bB
1 = 2 + 2 p1 > 3 = e1 + e
1
The market for good 1 cannot clear!
Observe that relative price of the goods remain unchanged amid an increase
in the supply of good 1! This qualitatively differs from the prediction a
partial equilibrium analysis.
4. Now suppose a communist government steps in and redistributes the
A
B
= ( 32 , 32 ). Show that the unique
e A = (3, 0) and
competitive equilibrium is given by (e
p1 , pe2 ) = (1, 1), X
e B = (0, 3). Is household A worse off after the redistribution?
X
endowments evenly, so that E
38
= E
Fix p2 = 1. If 1 > p1 ≥ 12 , then it is optimal for household A to buy only
good 1. From household A’s budget constraint,
x
eA
1 =
3
2
3 p2
2 p1
+
> 3 = ebA
bB
1 +e
1
The market for good 1 cannot clear!
If 2 ≥ p1 > 1, then it is optimal for household B to buy only good 2. From
household B’s budget constraint,
x
eB
2 =
3 p1
2 p2
+
3
2
> 3 = ebA
bB
2 +e
2
The market for good 2 cannot clear!
We conclude that equilibrium prices must be (e
p1 , pe2 ) = (1, 1). At such
A
B
e
e
prices, X = (3, 0) and X = (0, 3) are the unique optimal bundle for the
two households, and satisfy market clearing conditions.
Notice that half units of both goods have been confiscated from household A.
However, the relative price of good 1 decreases in equilibrium, household A is
still consuming the same bundle as before! It turns out that the households
are all indifferent about the redistribution.
39
5.3
Proof of First Welfare Theorem
e 1, X
e 2, . . . , X
e J ) be a competitive equilibrium.
ˆ Let prices (e
p1 , pe2 , . . . , peN ) and allocation (X
– Since preferences are strictly monotonic, the equilibrium prices are all positive.
Otherwise, households will consume infinite amount of goods which pen = 0.
ˆ I now argue that the equilibrium allocation is Pareto efficient.
e 2, . . . , X
e J ) is Pareto dominated by another
e 1, X
ˆ Suppose, to the contrary, that (X
feasible allocation (Y 1 , Y 2 , . . . , Y J ).
ˆ The allocation (Y 1 , Y 2 , . . . , Y J ) is feasible only if for each good n = 1, 2, . . . , N ,
J
X
ejn
=
j=1
J
X
ynj
j=1
Multiply both sides of the equality with pen and sum across all goods, we obtain
N
X

pejn
n=1
J
X

ejn  =
N
X

pejn
n=1
j=1
J
X

ynj 
j=1
By changing the order of summation, the equality can be further rearranged as
J X
N
X
pejn ejn
j=1 n=1
=
J X
N
X
pen ynj
(1)
j=1 n=1
– Equality (1) means that total market value of consumption bundles (Y 1 , Y 2 , . . . , Y J )
and total market value of endowments are equal.
e 1, X
e 2, . . . , X
e J ).
ˆ We now exploit the fact that (Y 1 , Y 2 , . . . , Y J ) Pareto dominates (X
e j ).
It requires that for all households, U j (Y j ) ≥ U j (X
ˆ Fix a household j. If
ejn ynj
n=1 p
PN
<
ejn ejn ,
n=1 p
PN
then strict monotonicity implies that
the household can find a even better bundle on the budget line. This is impossible
e j is the most preferred bundle in the budget set PN pejn xjn ≤ PN pejn ejn .
because X
n=1
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n=1
Therefore, we can conclude that for every household j
N
X
pejn ynj ≥
n=1
N
X
pejn ejn
n=1
e j ). Again from
ˆ Pareto dominance requires that for some households U j (Y j ) > U j (X
e j is the most preferable bundle in the budget set, we can deduce that
the fact that X
for these households, the consumption bundle Y j is not affordable
N
X
pejn ynj >
n=1
N
X
pejn ejn
n=1
ˆ Sum the inequalities across all households,
J X
N
X
pejn ynj
j=1 n=1
>
J X
N
X
pejn ejn
(2)
j=1 n=1
e 1, X
e 2, . . . , X
e J ) only if its total
– In words, (Y 1 , Y 2 , . . . , Y J ) Pareto dominates (X
market value of (Y 1 , Y 2 , . . . , Y J ) is strictly above that of endowments.
ˆ Inequality (2) contradicts the previous equality (1)! Therefore, we conclude that
e 1, X
e 2, . . . , X
e J ) is Pareto efficient.
(X
ˆ Bare in mind that we deduce equality (1) merely using the fact that (Y 1 , Y 2 , . . . , Y J )
is a feasible allocation. We deduce the second inequality (2) from the fact that
e 1, X
e 2, . . . , X
e J ) are optimal for the respective households given the market prices
(X
e 1, X
e 2, . . . , X
e J ).
(e
p1 , pe2 , . . . , peN ), and the requirements that (Y 1 , Y 2 , . . . , Y J ) Pareto dominates (X
e 1, X
e 2, . . . , X
e J ) is completely driven by the
ˆ In this perspective, Pareto efficiency of (X
fact every household picks her most preferable bundle in her budget set!
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5.4
Feeding America’s allocation of donated food
ˆ Feeding America is one of the largest NGO in U.S. It has to allocate, on average,
around 50 truckloads of foods daily to 210 food banks across U.S.
ˆ Feeding America decides the quantities of different food to be given to local food
banks. The following factors complicates its decision
– The quantities and the composition of the food donation Feeding America
received varies from day to day.
– The allocated food accounts for around a quarter of the total amount received
by the food banks. Food banks also receive food donation themselves from other
sources. Depending on what other donation they receive, their “preferences” or
need for different categories of food are constantly changing.
– Food banks are liable for the transportation. The cost depends on the locations
of the food bank and the food.
ˆ Prendergast (2017) “How Food Banks Use Markets to Feed the Poor” describes how
Feeding America allocates donated food using a market-based mechanism.
ˆ Before 2005, Feeding America adopted a central assignment system.
– Feeding America assigned each food bank a permanent “goal pound,” the average
weight it was supposed to receive. Food banks were ranked on their goal pounds
relative to the pounds they recently received. Feeding American assigned food
lots to the food banks based on the ranking.
– The food bank cannot choose among different lots. It may only accept or reject
the assigned lot. It Even if a food bank refused a lot, the lot will be counted
against their need measure as if it had been accepted. The rejected lot would
then be assigned to the next in line.
– The assignment system only counted the weight of a food lot. It treated all
categories of food equally, be it fresh produce or cereal.
ˆ In 2005, Feeding America switched from a central assignment system to a market-
based mechanism for food allocation.
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– Feeding America creates a token currency called ”shares” and assigns each food
bank a permanent “goal factor,” the portion of token currency it receives daily.
It starts with distributing the token currency to the food banks.
– The food banks then submit bids for the food twice a day. The highest bid for
a food is its market price.
– The food banks pay for the food using the token currency in their accounts.
– Feeding America then redistributes its revenue, the token currency received, to
the food banks at the end of every day.
– Food banks can access credit. They pay off those debts with at least half of
their daily redistributed tokens until the debt is paid off.
– Negative prices are allowed for food with high transportation cost.
– Food banks may even put up their own food for sale, termed as “maroon
pounds.”
ˆ Since its implementation, the new market-based mechanism receives support from
the food banks. Prendergast (2017) discusses the institutional details and analyzes
the sources of efficiency gains of the market-based mechanism. In particular,
– There is a large variation in prices across different food categories.
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For examples, average prices for staple food and diapers are more than sixtyfold
of the average prices of fresh produce and beverages. In contrast, the origin
central assignment system does not differentiate between different food.
– The traded prices also inform Feeding America about what food are in high
demand and allow it to better target donors.
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