Lecture 3 Welfare Economics by bgc15733

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									                                                                               Welfare economics



                                         Lecture 3
                                     Welfare Economics


Welfare economics is essentially about judging the desirability of social outcomes. In this
lecture we will introduce the normative concept of Pareto efficiency and two positive
concepts; the core and general competitive equilibrium. Much of the lecture will be
devoted to analyzing the relationships among these concepts.


General equilibrium in pure exchange
To begin our discussion of welfare economics we will consider a general equilibrium
model of a pure (barter) exchange economy. In such an economy we will analyze n
consumers trading in m markets. For now, there is no production, no prices, and no
money.

   Define      N = (1, 2, ... , n) -- the set of traders (consumers) in the economy.
               M = (1, 2, ... , m) -- the set of goods available in the economy.

Assume that each good is of homogeneous quality. Two goods that are the same except
for quality should be thought of as different goods. Furthermore, assume that each good
is perfectly divisible. This last is not really necessary, but it makes the analysis easier.


Endowments
Instead of consumers purchasing a bundle of goods out of money income, consumers are
endowed with a bundle of the goods in the economy.

   Let,        wij -- i's endowment of good j.
               wi = (wi1, wi2, ... , wim) -- i's endowment bundle.
               W = (w1, w2, ... , wn) -- the economy's endowment.


Allocations -- consumers take their endowments to market and trade with others to obtain
another bundle of goods which they consume.

   Let,        xij -- i's final demand (consumption ) of good j.
               xi = (xi1, xi2, ... , xim) -- i's consumption bundle.
               X = (x1, x2, ... , xn) -- an allocation.


Feasibility
Note that,             ∑i∈N wij =    w1 j + w2 j + . .. + wnj ,

is the aggregate amount of good j available in the economy. Furthermore,



                                              3.1
                                                                                  Welfare economics



                        ∑i∈N xij =   x1 j + x 2 j + . .. + x nj

is the aggregate consumption of good j.


Definition An allocation X is a feasible allocation if and only if

                        ∑i∈N wij ≥ ∑i∈N xij ,         ∀j ∈ M .

Note that a feasible allocation is one in which the final consumption of each good does
not exceed the amount available at the start of trade.


Preferences
Each individual is fully described by their preferences and endowments. Assume that
each individual trader's preferences are fully described by a utility function. For person i,
ui(xi) = ui(xi1, xi2, ... , xim). Most of the time we will assume that each utility function is
strongly monotonic and strictly quasi-concave.


Edgeworth Box
Let's restrict our economy to two individuals (A, B) trading two goods (1, 2). To
illustrate barter trade we construct what is known as an Edgeworth Box. To construct
this box first consider our description (preferences and endowment points) of the two
individuals. We do not need to draw budget lines since there are no prices in this
economy.


       2                                                2


                                                wB2



                                                                                        1
                                       u1
                                        A                                              uB
 wA2
                                   0                                          0
                                  uA                                         uB

  Α                  wA1                    1     Β         wB1                             1



Now, take B's indifference map, rotate it 180 degree and place it on top of A's
indifference map so the endowment points coincide.



                                                3.2
                                                                                         Welfare economics



                   2                      *
                                        x B1         wB1                 Β
             1

                        0
                       uB

                                    P
             x*                                                           *
                                                                         xB2          wA2 + wB2
              A2

             wA2                                                         wB2
                                                W          u0
                                                            A

                                                                             1
             Α                                      wA1
                                        x*
                                         A1                              2

                                    wA1 + wB1



Remarks
   1) The economy's endowment is W = [wA, wB] = [(wA1, wA2), (wB1, wB2)]. It is also
       called the no-trade allocation. The indifference curves that pass through W gives
                                                                        0
       us the utility levels in the absence of trade ( u0 , uB ) . If these individuals are to
                                                                   A
       trade with each other they have to do at least as well as this.
   2) The height of the box is the amount of good 2 available in the economy, while the
       width of the box is the amount of good 1 available.
   3) Any point in the box or on the boundary represents a feasible allocation. To see
                                     ∗                         ∗      ∗
       this take point P = [ x ∗ , x B ] = [( x ∗ , x ∗ ), ( x B1 , x B2 )], and note that
                               A                A1    A2


                           ∗
                   x ∗ + x B1 = wA1 + wB1
                     A1                             (supply is equal to demand for good 1)
       and                 ∗
                   x ∗ + x B2 = wA2 + wB2.
                     A2                             (supply is equal to demand for good 2)


Trade in the Edgeworth Box Consider the following:

                                2
                                                                             Β
                        1




                                                Q



                                                                                 u2
                                                                                  A
                                                           P                     u1
                                                                     W            A
                            Α                                                    u0
                                                                                  A
                                                                                 1
                                                           0
                                                          uB     B
                                                                     2
                                                                u1 u B       2



                                                3.3
                                                                                   Welfare economics



W is the endowment point again. Trade between the two individuals will take place when
both are made better off. For example, they might trade to allocation P. However, they
won't stop trading at P because there are still gains from trade to be had. At allocation Q,
all gains from trade are exhausted.

   Notes about Q
      1) It is a feasible allocation.
      2) There is no other feasible allocation that will make one of them better off
          without harming the other.
      3) An indifference curve of A's is tangent to an indifference curve of B's.
      4) Both individuals prefer allocation Q to allocation W.


Pareto Efficient Allocations
   Definition 1: A Pareto efficient allocation is a feasible allocation from which there
   is no way to make at least one individual better off without harming another.

                                               (   0        0
                                                                )
   Definition 2: A feasible allocation X0 = x10 , x2 ,..., xn is Pareto efficient if and only
                                                  1
                                                    (   1          1
                                                                    )
   if there is no other feasible allocation X1 = x1 , x 2 , ..., x n such that
               ui(X1) ≥ ui(X0) ∀ i ∈ N                   [no one is harmed by moving to X1]
   and         ui(X1) > ui(X0) for at least one i ∈ N.         [at least one person is better off]

Remarks
   1) The two definitions are equivalent if one accepts the assumption that preferences
      can be represented by a monotonic utility function.
   2) In the previous graph, allocation Q is Pareto efficient, while P and the endowment W
      are not.
   3) These definitions require nothing about tangent indifference curves. Thus, tangent
      indifference curves is sufficient for Pareto efficiency, but not necessary. More on
      this later.
   4) Notice that there is nothing about fairness or equity in these definitions. More
      later.


Pareto Efficiency as a Constrained Optimization Problem
Consider a pure-exchange economy with two people (A, B) and two goods (1, 2), and the
constrained maximization problem of choosing an allocation X = [xA, xB] = [(xA1, xA2),
(xB1, xB2)] to solve

               max     uA(xA1, xA2)
                                       0
               s.t.    uB(xB1, xB2) = uB                i)
                       xA1 + xB1 = wA1 + wB1            ii)
                       xA2 + xB2 = wA2 + wB2.           iii)                               1)


                                              3.4
                                                                              Welfare economics



What we are doing here is trying to find a feasible allocation [constraints ii) and iii)] to
make A as well off as possible (the objective) without harming B [constraint i)]. The
solution to this problem will be a Pareto efficient allocation. The Lagrange equation for
1) is
                L = uA(xA1, xA2) + µ[uB(xB1, xB2) - uB ]
                                                     0



                       + λ1[wA1 + wB1 - xA1 - xB1] + λ2[wA2 + wB2 - xA2 - xB2].

The first-order conditions for an interior solution to 1) are

                                 ∂L      ∂ uA
                                       =        − λ1 = 0                              2)
                                ∂ x A1   ∂ x A1
                                 ∂L       ∂ uA
                                       =        − λ2 = 0                              3)
                                ∂ x A2   ∂ x A2
                                 ∂L        ∂ uB
                                       = µ        − λ1 = 0                            4)
                                ∂ x B1     ∂ x B1
                                 ∂L       ∂ uB
                                      = µ        − λ2 = 0                             5)
                                ∂ xB2     ∂ x B2
                                ∂L   ∂L      ∂L
                                   =      =      = 0                                  6)
                                ∂µ   ∂ λ1   ∂ λ2

                                               ∂ u A ∂ x A1   λ
The first-order conditions 2) and 3) imply                  = 1.                      7)
                                               ∂ u A ∂ x A2   λ2

                                               ∂ uB ∂ x B1    λ
The first-order conditions 4) and 5) imply                  = 1.                      8)
                                               ∂ uB ∂ x B 2   λ2

                               ∂ u A ∂ x A1   ∂ uB ∂ x B1    λ
7) and 8) imply                             =              = 1.                       9)
                               ∂ u A ∂ x A2   ∂ uB ∂ x B 2   λ2

              ∂ ui ∂ xij
Recall that              is the marginal rate of substitution between goods j and k for
              ∂ ui ∂ xik
person i. It is the slope of an indifference curve of i's. In terms of valuation: The
marginal rate of substitution is the subjective marginal value i places on consumption of
good j in terms of good k.

Equation 9) states that at an interior solution to the optimization problem 1), the slope of
an indifference curve for A is equal to the slope of an indifference curve for B. In terms
of value 9) says that at an interior solution, A's subjective marginal valuation of good 1 in
terms of good 2 must be equal to B's subjective marginal valuation of good 1 in terms of
good 2.


                                             3.5
                                                                                         Welfare economics



"Equal slopes" and feasibility require a tangency. One way to think about the solution to
                                                                                    0
1) is to choose the highest indifference curve for A that satisfies the constraint uB .


                      2
                                                            B
                 1
                                                                (x   *
                                                                     A1 ,   x* 2 , x*1, x* 2
                                                                             A      B    B     )


                                                                            u2
                                                                             A
                                                                            u1
                                                                             A

                  A                                                    u0
                                                                        A
                                                                 1
                                                        0   2
                                                       uB




The Set of Efficient Allocations
   Definition: A contract curve is the set of Pareto efficient allocations.


Actually, the "contract curve" may be a space or a point or something other than a
continuous function. In the optimization problem above, the contract curve is found by
             0
varying the uB constraint.


                2
                                                                                 B
           1




                                                                                         u2
                                                                                          A


                                                                                         u1
                                                                                          A

                                                                                         u0
                                                                                          A
            A                                                                        1
                                                   2
                                                  uB   u1
                                                        B
                                                             0
                                                            uB                   2




                                            3.6
                                                                              Welfare economics



Core Allocations
   Definition (blocking): A coalition S ⊆ N can block an allocation X0 if there is some
   other allocation X1 such that


                       ∑  i ∈S
                                 xij =
                                  1
                                         ∑
                                         i ∈S
                                                wij , ∀j ∈ M         i)

                       ui(X1) ≥ ui(X0) ∀ i ∈ S                       ii)
                       ui(X1) > ui(X0) for at least one i ∈ S.       iii)



The first requirement for blocking i) is that the allocation X1 is feasible for the coalition
S. That is, the members of S can 'afford' xi1     ( )
                                                  . The requirements ii) and iii) are that at
                                                        i ∈S

least one member of S is better off and no member is harmed.


   Definition (core): A feasible allocation X is a core allocation if it cannot be blocked
   by any coalition S ⊆ N.


In a core allocation no subgroup can break away from the rest of the economy and trade
only among themselves and be better off.

Notice that the definition for blocking is like the definition for a Pareto efficient
allocation. In fact, the similarity is very real.


   Proposition: Every core allocation is also a Pareto efficient allocation.
   Proof: Toward a contradiction assume that an allocation X0 is a core allocation but is
   not Pareto efficient. If X0 is not Pareto efficient there exists another feasible
   allocation X1 such that

                       ui(X1) ≥ ui(X0) ∀ i ∈ N
   and                 ui(X1) > ui(X0) for at least one i ∈ N.

   But then, X0 can be blocked by S = N. Hence, X0 could not have been a core
   allocation. This contradiction proves the proposition.



The following graph illustrates the set of core allocations in our two-person, two-good
economy.




                                                    3.7
                                                                                Welfare economics




                  2
                                                                       Β
            1

                                                         P                 contract curve

                                          Q


                                                                           core allocations


                                                             W               u0
                                                                              A

             Α                                                              1
                                                                  0
                                                                 uB    2



Notice that the allocation P is efficient but is not in the core because it can be blocked by
person B (he can choose to consume his own endowment). Thus, the reverse of the
proposition is not true. That is, not every Pareto efficient allocation is a core allocation.
Allocation Q is a core allocation because neither A or B, or both together can block it. A
would be worse off consuming her own endowment as would B. Together they cannot
block Q because they can't move from it without harming one or the other or both.

Notice that core allocations depend on the initial endowments, while Pareto efficient
allocations do not.


Positive and normative concepts
The concept of Pareto efficiency is a normative concept. That is, it is a notion of 'what
ought to be'. This definition of efficiency is not derivable from some objective theory of
economic behavior, so, like all normative concepts, there is a subjective value system
underlying its use. Be aware of this.

The core on the other hand is an equilibrium concept. It is reasonable to require that an
economic equilibrium be stable in the sense that no coalition can block it using their own
resources. As an equilibrium concept, the definition of the core is also not derivable
from objective economic theorizing. However, the notion itself is devoid of value
statements.




                                              3.8
                                                                            Welfare economics



Competitive exchange
So far we have considered barter exchange to define efficient and core allocations. Now
we want to look at trade governed by competitive pricing. Once we have characterized
the outcomes of competitive trading (i.e., competitive general equilibrium) we will
analyze these outcomes in terms of efficiency and the core.

Assume:
   1) A general equilibrium model as before
   2) Again there is no production, no storage, and no money.
   3) Each good has a price per unit which traders take as given (the basic assumption of
       competitive behavior).
   4) Each trader has a strongly monotonic and strictly quasi-concave utility function.

Let:
    N = (1, 2, ... , n) -- the set of traders (consumers) in the economy.
    M = (1, 2, ... , m) -- the set of goods available in the economy.
    pj -- the competitive price of the jth good


The budget constraint for any trader i is

       p1xi1 + p2xi2 + ... + pmxim ≤ p1wi1 + p2wi2 + ... + pmwim

or
                        ∑p x
                        j ∈M
                                j   ij   ≤   ∑p w
                                             j ∈M
                                                      j   ij   .



Remarks
   1) pjxij is the market value of i's consumption of good j. Therefore, ∑Mpjxij is the
       market value of i's consumption bundle, or i's expenditure on consumption.
   2) pjwij is the market value of i's endowment of good j. Therefore, ∑Mpjwij is the
       market value of i's endowment bundle.
   3) Since we assume that utility functions are strongly monotonic we can replace '≤'
       with '='. We will do this from now on.


The assumption of competitive behavior also requires that traders maximize their utility
subject to their budget constraint. Thus, each i ∈ N chooses a consumption bundle (xi1,
xi2, ... , xim) to solve

                       max          ui(xi1, xi2, ... , xim).

                         s.t.        ∑p x
                                     j ∈M
                                             j   ij   =   ∑p w
                                                          j ∈M
                                                                   j   ij           10)




                                                           3.9
                                                                                   Welfare economics



The Lagrange equation for 10) is

                       Li = ui(xi1, xi2, ... , xim) + λi[∑Mpjwij - ∑Mpjxij].               11)

The first-order conditions for an interior optimum are

                        ∂ Li    ∂ ui
                              =       − λi pj = 0, ∀ j ∈ M                                 12)
                        ∂ xij   ∂ xij

                        ∂ Li
                             = 0.                                                          13)
                        ∂ λi

12) implies that for any two goods h and k from M,

                        ∂ ui ∂ xih   p
                                   = h , ∀ h and k ∈ M, h ≠ k.                             14)
                        ∂ ui ∂ xik   pk


This is the result that at an individual, interior optimum, the marginal rate of substitution
between any two goods must be equal to the price ratio. Note that since i was chosen
arbitrarily, 14) must be true for each i ∈ N. The optimal consumption bundle for i is
illustrated for the case M = (1, 2) in the first graph below. The second graph illustrates
the comparative static of an increase in the price of good 1.




                                                          2                        p1 '
        2                                                              slope = −        , p1′ > p1
                                                                                   p2
                     ∂ ui ∂ xi1 p1
                                =
                     ∂ ui ∂ xi 2 p2
                                                                    xi∗∗

                          (
                   xi∗ = xi∗1 , xi∗2   )
                                                              xi∗
 wi2                             ui0             wi2                               ui1


    i               wi1                    1          i                wi1                    1




                                               3.10
                                                                                                   Welfare economics



How are prices determined? -- The Walrasian Auctioneer
To mimic actual market operations we add a player whose role is as follows: It calls out
a set of prices. Each trader tells the auctioneer its optimal consumption bundle at those
prices. If quantity demanded is not equal to the amount available for each good, the
auctioneer adjusts prices until all markets clear. When all the markets clear, the traders
consume their final consumption bundles. That all markets clear is another requirement
for a competitive equilibrium. That is, we require that final consumption in a competitive
equilibrium be feasible:

                            ∑x
                            i ∈N
                                   ij   =   ∑w
                                            i ∈N
                                                   ij   , ∀ j ∈ M.                                         15)


An illustration
Consider the graph of a two-person (A, B), two-good (1, 2) economy below. Suppose
                                                                        0
that the auctioneer calls out initial prices ( p10 , p2 ) which results in the allocation
                        0      0                    0
[( x 0 1 , x 0 2 ), ( x B1 , x B 2 )] . At ( p10 , p2 ) neither market clears. In fact,
     A       A


                            x 0 1 + x B1 < w A1 + w B1
                              A
                                      0
                                                                            (excess supply of good 1)
                 and        x 0 2 + x B 2 > w A2 + w B 2
                              A
                                      0
                                                                            (excess demand for good 2).

In this situation, to move toward market clearing, the auctioneer should decrease the
relative price of good 1 and increase the relative price of good 2. That is, in the next
round the auctioneer should call out ( p1 , p2 ) such that p1 p2 < p10 p2 .
                                        1    1              1  1        0



At prices (p1*, p2*) and allocation X*, each consumer is optimizing on their budget sets
and both markets clear. This set of prices and allocation is a competitive equilibrium.

             2         u0
                        A                          u1
                                                    A                  0
                                                                      xB1
                                                                                            Β
      1

       x0
        A2




                                                                                             0
                                                                                            xB 2
                                    X*
                                                                                   W
                                       ∗  ∗
                            slope = − p1 p2                                                                0  0
                                                                                                slope = − p1 p2
       Α
                                                        x01
                                                         A
                                                                                                   1
                                                                     u1
                                                                      B
                                                                               0
                                                                              uB
                                                                                            2

                                                              3.11
                                                                                                         Welfare economics



Definition: A competitive (Walrasian) equilibrium in a pure exchange economy is a set
of prices p = (p1, p2, ... , pm) and an allocation X* = (x1*, x2*, ... , xn*) such that

    A) For each i ∈ N, xi* = (xi1*, xi2*, ... , xim*) is the solution to

        max      ui(xi1, xi2, ... , xim).
          s.t.   ∑p x
                 j ∈M
                         j ij     =       ∑p w
                                          j ∈M
                                                      j   ij    (utility maximization on a budget set)           10)


    B) For each j ∈ M,

                 ∑x
                 i ∈N
                        ∗
                        ij   =     ∑w
                                   i ∈N
                                             ij   .             (feasibility)                                    15)


In the graph above, you noticed that the competitive equilibrium allocation X* is also a
Pareto efficient allocation. It is also a core allocation. It turns out that these are general
results.



The First Theorem of Welfare Economics
       ∗     ∗            ∗
If [( x1 , x 2 , . .. , x n ), ( p1 , p2 , . .. , pm )] ≡ [X*, p] is a competitive equilibrium, then X*
is a core allocation. By implication it is also a Pareto efficient allocation.

Proof: To prove the theorem we use the following facts:

    Fact 1 Let [X*, p] be a competitive equilibrium. If ui(xi') > ui(xi*) for some xi', then

                                 ∑ p j xij '              >    ∑ p j wij   =    ∑ p j xij .
                                                                                       ∗

                                j ∈M                           j ∈M             j ∈M


        That is, if xi' is strictly preferred to xi* by i, it must not be affordable for i. To
        show this assume that i can afford xi'. Then, since he prefers xi' to xi*, he would
        have chosen xi' instead of xi*. But then, X* could not have been a competitive
        equilibrium allocation. This contradiction establishes the result.

    Fact 2 Let [X*, p] be a competitive equilibrium. If ui(xi') ≥ ui(xi*) for some xi', then

                                 ∑ p j xij ' ≥ ∑ p j wij                   =    ∑ p j xij
                                                                                       ∗
                                                                                            .
                                j ∈M                           j ∈M             j ∈M


        That is, if xi' is weakly preferred to xi* by i, it cannot cost less than xi*.




                                                                      3.12
                                                                                                                            Welfare economics



Toward a contradiction of the welfare theorem, suppose that [X*, p] is a competitive
equilibrium but X* is not a core allocation. From the definition of the core, if X* is not a
core allocation there exists another allocation X1 and a blocking coalition S ⊆ N for
which

    i)    ∑      i ∈S
                        xij =
                         1
                                       ∑      i ∈S
                                                     wij , ∀j ∈ M . -- the members of S must be able to achieve their
                                                                                            part of X1 with their own resources.

    ii) ui(xi1) ≥ ui(xi*), ∀ i ∈ S. -- no member of S strictly prefers X* to X1.

    iii) ui(xi1) > ui(xi*), for at least one i ∈ S. -- at least one person in S strictly prefers
                                                        X1 to X*.

Fact 2 and ii) imply that

    iv)   ∑p x
          j ∈M
                        j
                            1
                            ij     ≥        ∑p w
                                            j ∈M
                                                          j    ij       , ∀ i ∈ S.

Fact 1 and iii)

    v)    ∑p x
          j ∈M
                          1
                        j ij       >        ∑p w
                                            j ∈M
                                                      j       ij    , for at least one i ∈ S.


Summing iv) and v) over the members of S yields

          ∑ ∑p x
          i ∈S      j ∈M
                                   j
                                       1
                                       ij     >       ∑ ∑p w
                                                      i ∈S              j ∈M
                                                                               j   ij   ,


which can be written as

          p1 ∑ x i11 + p2 ∑ x i12 +...+ pm ∑ x im > p1 ∑ wi1 + p 2 ∑ wi 2 + ...+ p m ∑ wim .
                                               1

                 i ∈S                          i ∈S                                i ∈S            i ∈S        i ∈S           i ∈S


This can rewritten again as

                                                                       
          ∑ p ∑ x − ∑ w
          j ∈M
                        j
                                i ∈S
                                       1
                                       ij
                                                   i ∈S
                                                                   ij    > 0.
                                                                        

But, since prices are positive,

                                ∑ xij − ∑ wij
                                   1
                                                                          ≠ 0 for some j ∈ M.
                                i ∈S                 i ∈S


This implies that X1 cannot be a feasible allocation for S -- it violates i). We conclude
that an allocation like X1 cannot exist. But this contradicts our assertion that X* is not a



                                                                                            3.13
                                                                               Welfare economics



core allocation. Therefore, X* must be a core allocation. Furthermore, since all core
allocations are Pareto efficient, X* must be Pareto efficient. Q.E.D.

Remarks
   a) The theorem implies that competitive behavior will lead to a desirable (in the
       Pareto sense) social outcome.
   b) Unfortunately the theorem does not hold if the assumptions of competitive trading
       are not met (i.e, no government intervention, no externalities, no market power,
       etc.).
   c) Still we haven't said anything about fairness. There should be no presumption that
       competitive trading will lead to a fair allocation. However, the Second Welfare
       Theorem reveals that we can induce a fair (by some criteria) and efficient
       allocation.



The Second Welfare Theorem
Suppose that all traders have strongly monotonic and strictly quasi-concave utility
functions. Let X* be an efficient allocation such that xij* > 0, ∀ i ∈ N and ∀ j ∈ M. Then
there exists a set of prices p = (p1, p2, ... , pm) and an assignment of endowments W = (w1,
w2, ... , wn) such that (X*, p) is a competitive equilibrium.

Notes
   a) The assumption that xij* > 0, ∀ i ∈ N and ∀ j ∈ M can be relaxed.
   b) In a sense, the theorem says that if you let me choose prices and endowments I can
       guarantee that any efficient allocation of your choice will be a competitive
       equilibrium allocation.

To illustrate the Second Welfare Theorem, consider trade in an Edgeworth box.
                  2
                                                                      Β
             1
                                         u1
                                          A
                            u1
                             B

                                          X*                W0            u0
                                                                           A
                                                                  0
                                                                 uB




                      slope = − p1 p 2                 W1
              Α
                                                                           1
                                                                      2


                                              3.14
                                                                            Welfare economics



Suppose that W0 is the initial allocation of endowments. Suppose we think that trade
between the two individuals will lead to an unfair allocation, and we prefer to see them
trade to the 'fair' and efficient allocation X*. The theorem guarantees that if we pick the
appropriate prices and rearrangement of endowments, X* will result from competitive
trading. The appropriate prices here are (p1, p2) so that p1/p2 = MRSA(X*) = MRSB(X*).
Now pick an endowment W1 so that the budget line in the Edgeworth box is the common
tangent line at X*. Now if A and B start at W1 and trade at prices (p1, p2) they will trade
to X*.

Remarks
   a) The welfare theorems are important because they let us conclude that if we believe
       that people trade in competitive situations, any complaints about the price system
       can be reduced to issues of equity. Furthermore, issues of equity can be
       addressed by rearranging endowments.
   b) In the real-world, we can rearrange endowments by what are called lump-sum
       transfers. Lump-sum transfers are tax/subsidy policies that don't distort
       competitive prices. Unfortunately, there aren't many types of transfers that don't
       distort prices.
   c) Though the welfare theorems are quite powerful, they do depend heavily on the
       assumptions of competitive behavior.
   d) There is another problem that we can't address. What criteria will we use to
       determine what is and what is not fair? Furthermore, what rule do we use to
       choose among fairness criteria?



Shadow prices and competitive prices
The purpose of this section is to show that the Lagrange multipliers from the constrained
optimization problem that characterizes efficient allocations coincide with competitive
market prices.

Proposition 1: Suppose all traders have strongly monotonic and strictly quasi-concave
   utility functions. Then, if (X*, p) is a competitive equilibrium with xij* > 0, ∀ i ∈ N
   and ∀ j ∈ M,

                       ∂ ui ∂ xih   p
                                  = h , ∀ h and k ∈ M, and ∀ i ∈ N.
                       ∂ ui ∂ xik   pk

Proof: Recall that if (X*, p) is a competitive equilibrium each i ∈ N will choose a
consumption bundle (xi1, xi2, ... , xim) to solve

                      max      ui(xi1, xi2, ... , xim).

                        s.t.   ∑p x
                               j ∈M
                                      j   ij   =   ∑p w
                                                   j ∈M
                                                          j   ij                    10)




                                                   3.15
                                                                                                                 Welfare economics



Recall that the necessary conditions for an interior solution to this problem include

                              ∂ ui ∂ xih   p
                                         = h , ∀ h and j ∈ M.                                                                14)
                              ∂ ui ∂ xik   pk

Note that 5) must be true for each i ∈ N. Q.E.D.


Proposition 2: Continue to assume that all traders have strongly monotonic and strictly
quasi-concave utility functions. Then, if X* is a Pareto efficient allocation with xij* > 0
∀ i ∈ N and ∀ j ∈ M,

                              ∂ ui ∂ xih   λ
                                         = h , ∀ h and j ∈ M, and ∀ i ∈ N,
                              ∂ ui ∂ xij   λj

   where λk is the Lagrange multiplier for the feasibility constraint on the kth good.

Proof: If X* is an efficient allocation, it solves the following optimization problem for
each i ∈ N:

                          max               ui(xi1, xi2, ... , xim)
                  ( xij , ∀j ∈M , ∀i ∈N )

                         s.t.               uk(xk1, xk2, ... , xkm) = uk , ∀ k ∈ N, k ≠ i
                                                                       0




                                            ∑x
                                            i ∈N
                                                   ij   =   ∑w
                                                            i ∈N
                                                                     ij   , ∀ j ∈ M.


For an arbitrary i ∈ N, the Lagrange equation is

                                                                                                                            
       Li = ui(xi1, xi2, ... , xim) +                   ∑
                                                   k ∈N , k ≠ i
                                                                    [
                                                                  µ k uk ( xk ) − uk +
                                                                                   0
                                                                                       ] ∑ λ ∑ w − ∑ x  .
                                                                                             
                                                                                          j ∈M
                                                                                                 j
                                                                                                        
                                                                                                     i ∈N
                                                                                                            ij
                                                                                                                 i ∈N
                                                                                                                        ij



The first-order conditions are

                                ∂ Li    ∂ ui
                  i)                  =       − λ j = 0 , ∀ j ∈ M.
                                ∂ xij   ∂ xij

                                ∂ Li        ∂ uk
                  ii)                  = µk        − λ j = 0 , ∀ j ∈ M, ∀ k ∈ N, k ≠ i.
                                ∂ x kj      ∂ x kj

                                ∂ Li   ∂ Li
                  iii)               =      = 0 , ∀ j ∈ M, ∀ k ∈ N, k ≠ i.
                                ∂λj    ∂ µk

From i) and ii)


                                                                   3.16
                                                                                 Welfare economics




                ∂ ui ∂ xih   λ
                           = h , ∀ h and j ∈ M and ∀ i ∈ N. Q.E.D.                        16)
                ∂ ui ∂ xij   λj


Now, Propositions 1 and 2 imply that

                        ph   λ
                           = h , ∀ h and j ∈ M.
                        pj   λj

Thus, we can interpret the Lagrange multipliers from the problem of finding efficient
allocations as the market prices that would emerge from competitive trading.



Welfare maximization
Assume the existence of a social welfare function. This is a mapping U: Rn→R such
that U(u1, u2, ... , un) gives us the collective welfare of N = (1, 2, ..., n) for any
distribution of private utility levels (u1, u2, ... , un). Typically we assume that the social
welfare function is increasing in each private utility, That is, ∂ U/∂ ui > 0, for all i ∈ N.

Now suppose we have the worthwhile goal of maximizing social welfare. How does the
solution to this optimization relate to Pareto efficiency?


Proposition: If an allocation X* maximizes U, X* is efficient.

Proof: Toward a contradiction of the proposition, suppose that X* maximizes social
welfare but is not efficient. If X* is not efficient, there exists a feasible allocation X0,
such that

               i)      ui(xi0) ≥ ui(xi*), ∀ i ∈ N
       and     ii)     ui(xi0) > ui(xi*), for at least one i ∈ N.

But, since U is monotonically increasing in each ui, i) and ii) imply

               U[u1(x10), u2(x20), ... , un(xn0)] > U[u1(x1*), u2(x2*), ... , un(xn*)].

Therefore, X* could not have maximized U. This contradiction proves the proposition.
Q.E.D.

Now, consider the problem of maximizing social welfare subject to the feasibility
constraints:



                                             3.17
                                                                                                                       Welfare economics



                        max               U[u1(x1), u2(x2), ... , un(xn)]
                ( xij , ∀j ∈M , ∀i ∈N )

                       s.t.               ∑x
                                          i ∈N
                                                 ij   =   ∑w
                                                          i ∈N
                                                                  ij   , ∀ j ∈ M.


The Lagrange equation for this problem is

                                                                                                                 
               L = U[u1(x1), u2(x2), ... , un(xn)] +                       ∑ λ ∑ w − ∑ x
                                                                           j ∈M
                                                                                  j
                                                                                          i ∈N
                                                                                                 ij
                                                                                                      i ∈N
                                                                                                             ij   .
                                                                                                                  

Assuming an interior solution, the first-order conditions are

                ∂L      ∂ U ∂ ui
       i)             =     ∗      − λ j = 0, ∀ j ∈ M, and ∀ i ∈ N.
                ∂ xij   ∂ ui ∂ xij

                 ∂L
       ii)            = 0 , ∀ j ∈ M.
                 ∂ λj

From i) we have

                ∂ ui ∂ xih   λ
                           = h , ∀ h and j ∈ M, and ∀ i ∈ N.
                ∂ ui ∂ xij   λj

Since this holds for every i,

                ∂ ui ∂ xih   ∂ uk ∂ xkh
                           =            , ∀ h and j ∈ M, and ∀ i and k ∈ N.
                ∂ ui ∂ xij   ∂ uk ∂ xkj

These marginal conditions are the same as those for Pareto efficient allocations.


Remarks
   a) Though an allocation that maximizes social welfare is efficient, an efficient
       allocation does not necessarily maximize a particular social welfare function.
       This implies that though we may have an efficient allocation, there might be
       another that gives us greater social welfare. In such a case, we would be able to
       make society better off in aggregate, but doing so would harm someone.
   b) However, under certain conditions, it can be shown that an efficient allocation
       always maximizes some social welfare function.
   c) There are real problems with assuming that a social welfare function exists. But, at
       times they are convenient to use.




                                                                 3.18
                                                                                Welfare economics



General equilibrium and the welfare theorems with production
We have examined the relationships among Pareto efficient allocations, core allocations,
and competitive equilibria in pure exchange economies. Now we introduce production
into the economy.

Let
      H = (1, 2, ... , h) -- the set of firms in the economy.
      M = (1, 2, ... , m) -- the set of goods available in the economy.
      p = (p1, p2 , ... , pm) -- constant (competitive) prices.


A production plan for the kth firm is

                 yk = (yk1, yk2 , ... , ykm).

If
      ykj > 0, firm k produces good j as an output
      ykj < 0, firm k uses good j as an input

Note that the goods set M includes outputs for consumption and inputs to production.


An aggregate production plan for the entire economy is

                 y = (y1, y2 , ... , yh).

A production possibilities set for the kth firm is a collection of all production plans that
are technically feasible. Denote the production possibilities set of the kth firm as Yk.


Assume that firms are competitive, and that they choose a production plan (yk) to
maximize profit (πk), taking the vector of prices (p) and the production possibilities set
(Yk) as given. Here,

                 πk = p1yk1 + p2yk2 + ... + pmykm =            ∑p y
                                                               j ∈M
                                                                      j   kj




Note that if good j is an input pjykj < 0 (a cost to the firm), and if good j is an output pjykj
> 0 (a source of revenue for the firm).

The kth firm's optimization problem is to choose a feasible production plan yk to solve

                          max      πk =     ∑p y
                                            j ∈M
                                                   j   kj


                           s.t.    yk ∈ Yk                                              17)



                                                        3.19
                                                                                                 Welfare economics



The solution to 17) is a production plan

               (y   ∗
                    k1                           )
                         , y k 2 , . .. , y km = ( y k 1 ( p), y k 2 ( p), . .. , y km ( p)) .
                             ∗              ∗



                      ∗
In vector notation yk = yk(p). Note that if ykj(p) < 0, it is an input demand function for
good j, and if ykj(p) > 0, it is a supply function for good j. An aggregate production plan
in which each firm chooses inputs and outputs to maximize profit is

               y(p) = [y1(p), y2(p) , ... , yh(p)].


Proposition: An aggregate production plan y(p) maximizes aggregate profit ∑k∈Hπk if
and only if each firm's production plan yk(p) maximizes its individual profit πk. [For this
proposition and its proof see Varian, pg. 339].

Note: For a competitive equilibrium we are going to require that each firm maximizes
profit. Sometimes it is more convenient to maximize aggregate profit. The proposition
tells us that there is no difference between the two operations.


Consumers
Recall that in a competitive exchange economy we required that an equilibrium
allocation     X* = (x1*, x2*, ... , xn*) satisfy

               max           ui(xi*)
                s.t.          ∑ p j xij =
                                     ∗
                                                     ∑ p j wij , ∀ i ∈ N.
                              j ∈M                   j ∈M


In an economy with production there is a complication. What do we do with profit?

Assume that each firm is owned by consumers (not necessarily all consumers). Suppose
that if i is an owner of firm k, she is entitled to a share sik of its profit.


Assume
   1) Each firm is completely owned by individuals so that

                             ∑s
                             i ∈N
                                     ik   = 1.


   2) The shares sik are fixed, and hence, are not traded. In this model there is no stock
       market although we could have included one.




                                                            3.20
                                                                                                                                         Welfare economics



Individual i's share of the profit from firm k is

                                      sikπk = sik ∑ p j y kj ( p) .
                                                           j ∈M



Individual i's income from owning shares in a number of firms is

                                      ∑s
                                      k ∈H
                                              ik   πk =             ∑s ∑ p y
                                                                    k ∈H
                                                                              ik
                                                                                   j ∈M
                                                                                               j    kj    ( p) .


Thus, i's budget constraint in this economy with production is

                     ∑p
                     j ∈M
                                 j   x ij =        ∑p w
                                                   j ∈M
                                                               j    ij    +        ∑s ∑ p
                                                                                   k ∈H
                                                                                          ik
                                                                                                   j ∈M
                                                                                                            j   y kj ( p) .                      18)


We will require that in a competitive equilibrium with production, each i maximizes
utility subject to 18).



Efficient allocations and competitive equilibria
Recall:
        X denotes a consumption allocation.
        y denotes an aggregate production plan.

The pair (X, y) will now be called an allocation. An allocation (X, y) is feasible if and
only if

                     ∑x
                     i ∈N
                            ij       =    ∑w
                                          i ∈N
                                                    ij    +        ∑y
                                                                   k ∈H
                                                                           kj   , ∀ j ∈ M.



An allocation (X, y) is Pareto efficient if and only if there is no other allocation (X0, y0)
such that

    i)   ∑x
         i ∈N
                 0
                ij   =   ∑w
                         i ∈N
                                     ij   +      ∑y
                                                 k ∈H
                                                          0
                                                          ij   ,, ∀ j ∈ M.                                      [(X0, y0) is feasible]


    ii) ui(xi0) ≥ ui(xi), ∀ i ∈ N.                                                        [no one is harmed by moving to (X0, y0)]

    iii) ui(xi0) > ui(xi), for some i ∈ N.                                                [at least one person is better off at (X0, y0)]




                                                                                      3.21
                                                                                                                          Welfare economics



A competitive equilibrium is a triple (X, y, p) such that

   i) Each production plan yk ∈ y = (y1, y2 , ... , yh) is the solution to

                             max               πk =     ∑p y
                                                        j ∈M
                                                                     j       kj


                                 s.t.          yk ∈ Yk,

   ii) Each consumption bundle xi* ∈ X is the solution to

               max           ui(xi*)

                 s.t.            ∑p
                                 j ∈M
                                          j   x ij =    ∑p w
                                                         j ∈M
                                                                         j    ij   +   ∑s ∑ p
                                                                                       k ∈H
                                                                                              ik
                                                                                                   j ∈M
                                                                                                          j   y kj ( p)


   iii) The consumption allocation X is feasible, so that

                ∑x
                i ∈N
                        ij   =     ∑w
                                   i ∈N
                                              ij   +   ∑y
                                                       k ∈H
                                                                kj                             ∀ j ∈ M.




The First Welfare Theorem
If (X, y, p) is a competitive equilibrium, then (X, y) is a core allocation. It is also a Pareto
efficient allocation. [For the proof, see Varian, pp. 345-346].


The Second Welfare Theorem
Suppose that (X, y) is a Pareto efficient allocation with xij > 0, ∀ i ∈ N and ∀ j ∈ M.
Assume further that each consumer has a strongly monotonic and strictly quasi-concave
utility function, and each firm has a closed and convex production possibilities set. Then
with an appropriate choice of endowments and profit shares, there exists a set of prices p
such that (X, y, p) is a competitive equilibrium.


Note: For the second theorem to hold we need each firm's production possibilities set Yk
to be closed and convex. A set is convex if every point on a line segment joining two
points in the set is also in the set. A set is closed if the boundaries of the set are included
in the set. A concave production function will imply a closed and convex production
possibilities set. However, a quasi-concave production function may not. If there is a
region of increasing returns to scale, the production possibilities set will not be convex.




                                                                              3.22
                                                                             Welfare economics



General equilibrium and efficiency with production: The calculus approach
Now we are going to characterize efficient allocations and competitive equilibria with the
marginal conditions from a series of optimization problems. We will derive the marginal
conditions for 1) technical efficiency, 2) Pareto efficiency, and 3) competitive equilibria.
In order to keep things simple we will not fully specify the economy in as much detail as
we did above

Assume
   Two consumers, A and B.
   Two consumption goods, 1 and 2.
   Two inputs into production, L and K.

Technical efficiency
Assume production functions that are strictly concave:

                                x1 = f(L1, K1),
                                x2 = g(L2, K2).

The resource constraints are
                                L = L1 + L2,
                                K = K1 + K2,

where L and K are the aggregate amounts available in the economy. We will ignore the
question of where they come from and who owns them.

To characterize technical efficiency we choose (L1, L2, K1, K2) to solve the following:

       max     x1 = f(L1, K1)
                0
        s.t.   x2 = g(L2, K2)
               L = L1 + L2
               K = K1 + K2.

The Lagrange equation is

       Φ = f(L1, K1) + λ[g(L2, K2) - x2 ] + λL[L - L1 - L2] + λK[K - K1 - K2].
                                      0



The necessary conditions are

               ∂ Φ ∂ L1 = f L − λ L = 0                                     19)
               ∂ Φ ∂ K1 = f K − λ K = 0                                     20)
               ∂ Φ ∂ L2 = λg L − λ L = 0                                    21)
               ∂ Φ ∂ K 2 = λg K − λ K = 0                                   22)



                                             3.23
                                                                              Welfare economics



               ∂ Φ ∂ λ = ∂ Φ ∂ λL = ∂ Φ ∂ λK = 0                                      23)


The first-order conditions 19) through 22) imply

                       fL   g   λ
                          = L = L.                                                    24)
                       fK   gK  λK

That is, the ratio of the marginal products must be equal for all goods. Recall that the
ratio of marginal products is called the marginal rate of technical substitution. Any
production plan (x1, x2, L1, L2, K1, K2) that satisfies 24) and the resource constraints L =
L1 + L2 and K = K1 + K2, is technically efficient.



Production possibilities frontier
Note that the first-order conditions 19) through 23) implicitly define

               L1 = L∗ (x2, L, K)
                     1                 L2 = L∗ (x2, L, K)
                                             2


               K1 = K1∗ (x2, L, K)           ∗
                                       K2 = K2 (x2, L, K)

               λ = λ∗ (x2, L, K)       λL = λ∗L (x2, L, K)     and λK = λ∗K (x2, L, K).


The indirect objective function is

                                                     ∗
                               x1 = f( L∗ , K1∗ ) = x1 (x2, L, K).
                                        1


This is the production possibilities frontier (PPF). It gives us the maximum possible
production of x1 for each level of x2, given the availability of L and K.

From the Envelope Theorem, we have the slope of the PPF

               ∂ x1∗   ∂ Φ∗
                     =      = − λ∗ .
               ∂ x2    ∂ x2

And from the first-order conditions we have

                          fL     f
               − λ∗ = −      = − K < 0.
                          gL    gK




                                            3.24
                                                                                   Welfare economics



Since ∂ x1 ∂ x 2 = − λ∗ < 0, the PPF is downward sloping. The slope of the PPF is
           ∗


sometimes called the marginal rate of transformation. If you check the second derivative
     ∗
of x1 (x2, L, K).you will realize that it is strictly concave if f(L1, K1) and g(L2, K2) are both
strictly concave.


                   x1
                                                                          ∗
                                                                       ∂ x1     f     f
                                                                            = − L = − K
                                                                       ∂ x2    gL    gK




                                                                  PPF
                                                                              x2


Remarks
   1) The production possibilities frontier collects all the combinations of the production
       of the two goods that are technically efficient.
   2) Technical efficiency is a necessary condition for Pareto efficiency


Pareto efficiency
As before, to find the set of Pareto efficient allocations we choose an allocation (xA1, xA2,
xB1, xB2) to solve
                       max uA(xA1, xA2)
                                                0
                        s.t.    uB(xB1, xB2) = uB               i)
                                xA1 + xB1 = x1                  ii)
                                xA2 + xB2 = x2                  iii)
                                 ∗
                                x1 (x2, L, K) = x1              iv)


Constraints ii), iii) and iv) are 'feasibility' constraints. Constraints ii) and iii) state the
supply of each good must be equal to aggregate demand, and iv) states that a production
plan (x1, x2, L1, L2, K1, K2) must be technically efficient.

Combine the last three constraints into one to make the problem a little simpler:

                                     ∗
                        xA1 + xB1 = x1 (xA2 + xB2, L, K)        v)



                                              3.25
                                                                                 Welfare economics



The Lagrange equation is then

                                                       ∗
       L = uA(xA1, xA2) + µ[uB(xB1, xB2) - uB ] + θ [ x1 (xA2 + xB2, L, K) - xA1 - xB1]
                                            0




The first-order conditions are

            ∂L      ∂ uA                       
                  =        − θ = 0             
           ∂ x A1   ∂ x A1
                                               
                                                  ∂ u A ∂ x A2    ∂ x∗
                                                ⇒              = − 1 ,                   25)
                                        ∗         ∂ u A ∂ x A1    ∂ x2
            ∂L      ∂ uA       ∂       x1
                  =        + θ              = 0
           ∂ x A2   ∂ x A2     ∂ x2            
                                               

        ∂L         ∂ uB                        
              = −µ        − θ = 0              
       ∂ x B1      ∂ x B1
                                               
                                                  ∂ u B ∂ x B2    ∂ x∗
                                                ⇒              = − 1 ,                   26)
                                        ∗         ∂ u B ∂ x B1    ∂ x2
         ∂L         ∂ uB       ∂       x1
               = −µ        + θ              = 0
        ∂ x B2      ∂ x B2     ∂ x2            
                                               


and Lθ = Lµ = 0. Equations 25) and 26) imply

                ∂ u A ∂ x A2   ∂ uB ∂ x B 2    ∂ x∗
                             =              = − 1.                                        27)
                ∂ u A ∂ x A1   ∂ u B ∂ x B1    ∂ x2

Equation 27) states that at a Pareto efficient allocation, the marginal rates of substitution
between any two goods is equal for every consumer, and in turn equal to the slope of the
production possibilities frontier.

                     x1
                                                                        ∗
                                                                     ∂ x1
                                                             slope =
                                                                     ∂ x2

                                                                    ∂ u A ∂ x A2   ∂ uB ∂ xB 2
                                  u0                                             =
                                   A
                                                                    ∂ uA ∂ x A1    ∂ uB ∂ xB1
                           0
                          uB


                                                                   PPF
                                                                           x2



                                             3.26
                                                                           Welfare economics




Competitive behavior
Profit maximization: Let prices in the economy be (p1, p2, pL, pK). For the production of
good 1 we choose (L1, K1) to solve

              max      p1f(L1, K1) - pLL1 - pKK1.

The necessary conditions are

                       p1 f L − p L = 0                fL   p
                                             ⇒            = L                     28)
                       p1 f K − p K = 0                fK   pK

Similarly, the necessary conditions for a profit maximizing plan to produce good 2 imply

                       gL   p
                          = L                                                      29)
                       gK   pK
28) and 29) imply
                        fL   g   p
                           = L = L                                                 30)
                        fK   gK  pK

Compare 30) and 24) to note that competitive, profit maximizing behavior induces a
technically efficient allocation of L and K to the production of x1 and x2.


Utility maximization: Each consumer chooses (xi1, xi2) to solve
       max    ui(xi1, xi2)
               s.t.    a budget constraint, i = A, B.

The necessary conditions imply

               ∂ u A ∂ x A2   ∂ uB ∂ x B 2   p
                            =              = 2                                     31)
               ∂ u A ∂ x A1   ∂ u B ∂ x B1   p1

Compare 31) and 27) to verify that competitive behavior by consumers induces a Pareto
efficient consumption allocation. Also note that the ratio of the goods prices is equal to
the slope of the production possibilities frontier.

Notes
   1) Obviously, all the marginal conditions are not enough to make the statements we
      have been making. We also need the resource constraints.
   2) These marginal relationships are not the only ones that can be inferred. See
      Silberberg for more.


                                            3.27
                                                                               Welfare economics



The compensation criterion
As a criterion for evaluating public policy proposals, the concept of Pareto efficiency is
considered by most to be too restrictive. Specifically, the criterion is said to be
incomplete in the sense that it does not allow us to rank every possible allocation (or,
more generally, social outcome). For example, consider the following graph. According
to the Pareto criterion, a social planner will not be able to rank bundles R and T. Even
worse, the Pareto criterion does not tell us that society prefers R and T to S even though
S is inefficient.

                   2
                                           contract curve             Β
              1

                                                        T




                                                       S
                          R                                      0
                                                                uB
                                                           u0
                                                            A
              Α                                                            1
                                                                      2

In this lecture we will consider a modification of the Pareto criterion called the
compensation criterion, which is commonly used in applied welfare economics. It is
similar to the Pareto criterion but it allows us to compare more outcomes. That is, it is
more complete than the Pareto criterion. Unfortunately, as you will see, it is also
incomplete and it can provide inconsistent comparisons.


The Pareto frontier
It will be useful to use Pareto (sometimes called utility possibility) frontiers. A Pareto
frontier collects all bundles of utility levels that are generated by efficient allocations.

Consider a society with two people (A, B) and two goods (1, 2) and the problem of
finding the Pareto efficient allocations:

       max     uA(xA1, xA2)
                                       0
               s.t.    uB(xB1, xB2) = uB
                       xA1 + xB1 = wA1 + wB1
                       xA2 + xB2 = wA2 + wB2.                                          32)




                                           3.28
                                                                                       Welfare economics



The Lagrange equation for 32) is

               L = uA(xA1, xA2) + µ[uB(xB1, xB2) - uB ]
                                                    0



                        + λ1[wA1 + wB1 - xA1 - xB1] + λ2[wA2 + wB2 - xA2 - xB2].

The necessary conditions for an interior solution to 32) are

               ∂L            ∂ uA
                    ∂ x A1 =      ∂ x A1 − λ 1 = 0                                             33)

               ∂L            ∂ uA
                    ∂ x A2 =      ∂ x A2 − λ 2 = 0                                             34)

               ∂L              ∂ uB
                    ∂ x B1 = µ      ∂ x B1 − λ 1 = 0                                           35)

               ∂L              ∂ uB
                    ∂ x B2 = µ      ∂ xB2 − λ 2 = 0                                            36)

               ∂L        ∂L        ∂L
                    ∂µ =    ∂ λ1 =    ∂ λ2 = 0                                                 37)


As usual, these first-order conditions imply that at an efficient allocation with xij > 0
for i ∈ (A, B) and j ∈ (1, 2),

                        ∂ u A ∂ x A1   ∂ uB ∂ x B1
                                     =
                        ∂ u A ∂ x A2   ∂ uB ∂ x B 2
                                        0
                        uB(xB1, xB2) = uB
                        xA1 + xB1 = wA1 + wB1
                        xA2 + xB2 = wA2 + wB2                                                  38)

[Note that the collection of conditions 38) are identical to 33) through 37)]. Assuming
that a solution to 32) exists and is unique, conditions 38) implicitly define


               xij = x ij (u B , w A1 + w B1 , w A2 + w B 2 ) , i ∈ (A, B) and j ∈ (1, 2)
                       ∗



               λ j = λ∗j ( u B , w A1 + w B1 , w A2 + w B 2 ) , j ∈ (1, 2)

               µ = µ ∗ (u B , w A1 + w B1 , w A 2 + w B 2 ) .


The indirect objective function is
               u A = u ∗ (u B , w A1 + w B1 , w A 2 + w B 2 ) .
                       A




                                                  3.29
                                                                                       Welfare economics



So that there is no confusion later on, let u ∗ (⋅) = v( u B , w A1 + w B1 , w A 2 + w B 2 ) . This is
                                              A

the Pareto frontier. From the Envelope Theorem

                   ∂ u∗    ∂v
                      A
                        =      = − µ ∗ < 0.
                   ∂ uB   ∂ uB

Hence, the Pareto frontier is downward sloping. In the graph below, I have drawn the
Pareto frontier as concave, although we cannot guarantee this. Instead of thinking about
the Pareto frontier as a function , it will be convenient sometimes to describe utility
possibilities as a set:

               [        [                  ]                                                       ]
        U = u( X ) = u A ( x A ), u B ( x B ) such that X = ( x A , x B ) is an efficient allocation .

In the graph below, each point in this space is a utility vector (uA, uB). Points on the
frontier are utility vectors generated by efficient allocations. Points under the frontier are
feasible utility vectors, but they are generated by inefficient allocations.


                            2            contract curve
                                                                         Β
                        1

                                                             T




                                  R                        S
                                                                     0
                                                                    uB
                                                               u0
                                                                A
                        Α                                                    1
                                                                         2
                            uA
                                        u(T)
                                                           Pareto frontier



                                       u(S)
                                                                 u(R)


                                                                             uB




                                                 3.30
                                                                                      Welfare economics



The compensation criterion
In order to define the compensation criterion, we first define the Pareto criterion.

Definition: A feasible allocation X0 is said to Pareto dominate another feasible
   allocation X1 if

               ui(X0) ≥ ui(X1) ∀ i ∈ N
      and      ui(X0) > ui(X1) for some i ∈ N.


In the graph below X0 Pareto dominates only those allocations that induce utility vectors
in the shaded area. For example, X0 dominates X1.

However, X0 and X2 are not comparable by the Pareto criterion. That is, X0 does not
dominate X2 and X2 does not dominate X0. This is what we mean by incompleteness.
The Pareto criterion does not give us a basis to judge the desirability of all allocations
relative to all others.
                     uA




                                            u (X 2 )

                                                                   u (X 0 )

                                                  u( X 1)

                                                                                         uB



Definition: A feasible allocation X0 is said to be potentially Pareto preferred to another
            feasible allocation X if there is some reallocation of X0, say X1, such that

               ∑x =∑x
               i ∈N
                      1
                      ij
                               i ∈N
                                      0
                                      ij   ∀j∈M               (X1 is a reallocation of X0)


and
                 ( )
               ui X 1 ≥ ui ( X ) ∀i ∈ N                  
                                                         
                                                          . (X1 Pareto dominates X)
               u (X )
                 i
                           1
                               > ui ( X ) for some i ∈ N 
                                                         




                                                       3.31
                                                                            Welfare economics



This is the essence of the compensation criterion. A social outcome X0 is preferred to X
if there is a third outcome X1 attainable from X0 that Pareto dominates X. Note that the
definition does not require that we actually move to X1, it only requires its existence.

Put another way: Almost any public policy change is likely to produce winners and
losers (i.e., some will benefit by the change and others will lose). The change is
potentially Pareto preferred if the winners could compensate the losers (a reallocation) so
that if the compensation takes place, no one is harmed by the change and some are
strictly better off. The catch is that the compensation need not take place.

To illustrate, consider two mutually exclusive outcomes: R ≡ (reduce emissions of some
industrial pollutant) and NR ≡ (no environmental regulation). Relative to NR, option R
will hurt the owners of some firms but will provide a cleaner environment for the
enjoyment of others.

Consider our two-person society, and suppose that the Pareto frontiers in the graph below
correspond to the two policy options. Suppose further that a policy option to achieve R
results in allocation X0 and utility vector u(X0), while policy option NR will result in
allocation X and utility vector u(X). Note that X0 does not Pareto dominate X. However,
there is a third allocation X1 that is attainable with policy option R that does Pareto
dominate X. Therefore, X0 is potentially Pareto preferred to X. So, according to the
compensation criterion emissions of the industrial pollutant should be reduced even
though person A is harmed.


               uA




                                                   u ( X 1)
                                   u( X )

                                                              u ( X 0)



                                                         NR       R            uB




                                            3.32
                                                                             Welfare economics



Incompleteness
The graph above illustrates the incompleteness of the compensation criterion. Here X0 is
not potentially preferred to X1, and X1 is not potentially preferred to X0. However,
relative to the Pareto criterion we can compare more outcomes using the compensation
criterion.

Inconsistency
A major shortcoming of the compensation criterion is that it will generate inconsistent
results in some situations. Suppose that the Pareto frontiers for the policy options we
considered above are as in the graph below. Here, allocation X0 is potentially Pareto
preferred to X through the reallocation X1. So, according to the compensation criterion,
emissions should be regulated. However, allocation X is potentially Pareto preferred to
X0 through the reallocation X2. So, according to the compensation criterion, emissions
should not be regulated. In such a case, the compensation criterion gives us an
inconsistent ranking. It tells us to regulate emissions, but also do not regulate emissions.

                 uA



                                   u( X 2)

                        u ( X 0)


                                                           u (X 1 )
                                             u( X )

                                                      NR              R           uB




The compensation criterion and transferable utility
It is often assumed in economics and game theory that utility can be transferred from one
person to others. Utility is transferable only if there exists some commodity that enters
each individual's utility function linearly and separately from all other commodities. For
example, utility is transferable in our two-person, two-good society only if the utility
functions have the form

               ui(xi1, xi2) = vi(xi1) + xi2, i ∈ (A, B).                             39)




                                              3.33
                                                                                      Welfare economics



Utility functions of this form are called quasi-linear utility functions. Given quasi-linear
utility functions, utility is transferable if individuals can make payments to each other
using good 2. In the language of game theory, we say that side-payments are allowed.
(Sometimes we assume that good 2 is money). Note that the cost to an individual of
increasing the utility of the other by one unit is one unit of utility.

Now suppose that we want to choose an allocation that maximizes the sum of individual
utilities subject to the resource constraints:

         max        V = uA(xA1, xA2) + uB(xB1, xB2) = vA(xA1) + xA2 + vB(xB1) + xB2,
         s.t.       xA1 + xB1 = wA1 + wB1
                    xA2 + xB2 = wA2 + wB2,                                                    40)


[Note that V is a social welfare function]. If vA(xA1) and vB(xB1) are both monotonically
increasing and strictly concave then a solution to 40) exists and it is unique. Given a
solution to 40), we can define the indirect objective function

                    V* = V ( w A1 + w B1 , w A2 + w B 2 ) .

This function gives us the maximal attainable utility for the two-person society. Now, if
side-payments are allowed (utility is transferable), V* can be allocated in any way so that

                    uA + uB = V*.                                                             41)

Recall that an allocation that maximizes a social welfare function is Pareto efficient.
Define the set of utility vectors

                    [( )
         US = u V ∗ = (u A , u B ) such that u A + u B = V ∗ .  ]                             42)

This set describes the Pareto frontier when utility is transferable. The function describing
the frontier in the (uB, uA) space is

                    uA = V* - uB                                                              42')

Note that this is a linearly decreasing function with slope equal to -1 and horizontal and
vertical intercepts equal to V*.

Recall that before we described the Pareto frontier as the set

     [          [                    ]                                                   ]
U = u( X ) = u A ( x A ), u B ( x B ) such that X = ( x A , x B ) is an efficient allocation , 43)




                                                    3.34
                                                                             Welfare economics



and the function

               u A = v( u B , w A1 + w B1 , w A 2 + w B 2 ) .                        43')

Now, we did not allow side-payments when we generated 43) and 43'). In the case of the
quasi-linear utility functions 39) with vA(xA1) and vB(xB1) monotonically increasing and
strictly concave, the function 43') is decreasing and strictly concave in uB. The graph
below is of the two frontiers. Note that both frontiers are derived from the same
individual utility functions and society's endowments of the two goods, wAj + wBj, j ∈ (1,
2). The only difference between them is that side-payments are allowed along US and are
not allowed along U.

                  uA
             V*




                          u (X* )


                                                        u1
                                      u ( X 0)
                                                  U              US
                                                                               uB
                                                                V*

Suppose that X* is the allocation that maximizes social welfare as defined by 40).
Without a transfer of utility between the individuals, the utility vector at this allocation
u(X*) is the point of tangency between US and U. Note that even though X0 efficient in
the absence of side-payments, X* does not Pareto dominate X0 and vice-versa.

However, suppose that the two players agree to the allocation X* and a transfer of utility
from A to B so that the utility vector u1 is achieved. [Actually, the players can agree to
this scheme or it can be imposed by a social planner]. Since both players are better off
with X* and the transfer than they would be at X0, we can say that X* is potentially Pareto
preferred to X0.

Note the difference between what we have just done and our definition of "potentially
Pareto preferred". There an allocation X* was preferred to another X0 if there was a
reallocation of consumption bundles that Pareto dominated X0. With transferable utility,
we do not reallocate the consumption allocation to reach a dominant outcome -- we
choose the social welfare maximizing consumption allocation and then reallocate utility.


                                                 3.35
                                                                             Welfare economics



                                       Excercises

[1] We know that a Pareto efficient allocation is one from which there is no move that
can make at least one person better off without harming another. Define a Pareto
improving move as one that makes one person better off without harming another. Note
that a Pareto improving move does not necessarily result in a Pareto efficient allocation.
Define a Pareto superior allocation as one that results from a Pareto improving move.
Note that this definition does not require a superior allocation to be an efficient one. In
the following graph, the point W is the endowment point, while R, S and T are alternative
allocations. Consider moves to and from these points to illustrate the concepts of Pareto
efficiency, Pareto improving moves and Pareto superior allocations.
                        2
                                                             Β
                   1



                                       R
                                                                 contract curve

                             T                    S


                    Α                                 W                u0
                                                                        A
                                                                   1
                                                       0
                                                      uB     2
[2] Consider a two-person pure exchange economy with two goods. Suppose that A
does not value good 2 at all. That is, only increases in the consumption of good 1 will
increase A's utility. Person B's indifference curves have the usual shape. In an
Edgeworth Box locate the contract curve and the core.

[3] "Jack Sprat can eat no fat, his wife can eat no lean." In an Edgeworth Box find the
contract 'curve'. What about the core?

[4] Discuss the relationship between:
    [a] efficient and core allocations
    [b] efficient allocations and endowments
    [c] core allocations and endowment

[5] In a pure exchange economy with two people (A and B) and two goods (1 and 2),
suppose that
               uA(xA1, xA2) = 2xA1 + xA2
               uB(xB1, xB2) = xB1xB2
               wA = (10, 0)
               wB = (0, 10)

   [a] Draw an Edgeworth box to illustrate the economy. Draw a few indifference
       curves and the point W = (wA, wB).
   [b] Solve for the Pareto efficient allocations. Illustrate them graphically. Illustrate
       the core allocations graphically.


                                           3.36
                                                                              Welfare economics



[6] Explain how the Pareto efficient allocations in a two-person, two-good exchange
economy can be characterized with the appropriate constrained optimization problem.
You don't have to go as far as deriving first-order conditions. Just set the problem up to
fit the definition of Pareto efficient allocations.

[7] Consider a two-person (A and B), two-good (1 and 2), pure exchange economy. The
consumers have identical utility functions,

                       ui(xi1, xi2) = ln(xi1) + xi2, i = A, B.

Their endowments are (wA1, wA2) = (2, 0) and (wB1, wB2) = (0, 2).
   [a] Find the contract curve and draw it in an Edgeworth box.
   [b] Can the allocation [(xA1, xA2), (xB1, xB2)] = [(1, 1), (1, 1)] be a competitive
       equilibrium allocation?

[8] In a two-person, two-good, exchange economy, the consumers have utility functions

                       ui(xi1, xi2) = xi1xi2, i = A, B,

and endowments (wA1, wA2) = (10, 0) and (wB1, wB2) = (0, 10). Find the competitive
equilibrium allocation for prices (p1, p2) = (1, 1). Is this allocation Pareto efficient?

[9] In a pure exchange economy with two people (A and B) and two goods (1 and 2),
suppose that
               uA(xA1, xA2) = 2xA1 + xA2
               uB(xB1, xB2) = xB1xB2
               wA = (10, 0)
               wB = (0, 10).

Denote the prices of the two goods as p1 and p2.
   [a] Find the competitive equilibrium (or equilibria). Verify the first welfare theorem.
   [b] Assume that (p1, p2) = (4, 1). Show that these prices cannot support a competitive
       equilibrium. Do the same for (p1, p2) = (4, 3).
   [c] Verify that the allocation X = [(xA1, xA2) = (25/3, 20/3), (xB1, xB2) = (5/3, 10/3)] is
       an efficient allocation. Find the set of prices and endowments such that X is a
       competitive equilibrium allocation. Use this exercise to illustrate the second
       welfare theorem.

[10] Waldo and Penelope have preferences over liver and onions. In fact, both view liver
and onions as perfect complements to be consumed in one-to-one proportions -- one
onion for every pound of liver. Assume that Waldo is endowed with 5 onions and no
liver, and Penelope is endowed with 5 onions and 20 pounds of liver.
    [a] Find the contract curve (or space) and the core.
    [b] How does the contract curve and the core change if 5 pounds of liver is
        transferred from Penelope to Waldo.



                                               3.37
                                                                           Welfare economics



   [c] Find the set of competitive equilibria for the original endowment. How does this
       set change if the endowments change as in b)?
   [d] Comment on the necessity of equating marginal rates of substitution to find
       efficient allocations and competitive equilibria in this case and in general.

[11] Use the welfare theorems as applied to exchange economies to discuss the
relationships among core allocations, efficient allocations, and competitive equilibria.
Use words, not math.

[12] Ken and Barbie have preferences over quiche and Perrier. Ken only consumes
quiche and Perrier in one-to-one proportions -- one bottle of Perrier for every slice of
quiche. Barbie has indifference curves that have the normal convex shape. Ken is
endowed with two slices of quiche and two bottles of Perrier, while Barbie is endowed
with four slices of quiche and 4 bottles of Perrier.
   [a] Find the contract curve and the core. How does the contract curve and the core
       change if we transfer one slice of quiche from Ken to Barbie.
   [b] For the original endowments, find the competitive equilibrium.
   [c] Verify that the allocation in which each person has three units of each good is an
       efficient allocation. Use this allocation to illustrate the second welfare theorem.

[13] Any complaints about the operation of a competitive market system can be reduced
to complaints about equity, and such complaints can be addressed by lump-sum
transfers. State the welfare theorems in the context of an exchange economy and use
them to evaluate this statement.

[14] Bennie and Joon have preference over Macadamia nuts and Listerine. Each
considers Macadamia nuts and Listerine as complements to be consumed in one-to-one
proportions -- one pound of Macadamia nuts for every gallon of Listerine. Bennie is
endowed with two pounds of Macadamia nuts and one gallon of Listerine, while Joon is
endowed with one pound of Macadamia nuts and two gallons of Listerine.
   [a] Find the contract curve and the core.
   [b] Use the allocation in which both individuals consume 1.5 pounds of Macadamia
       nuts and 1.5 gallons of Listerine to illustrate the second welfare theorem.

[15] If two allocations are not comparable by the Pareto criterion, they are not
comparable by the compensation criterion. State the Pareto criterion and the
compensation criterion, then tell me whether the statement is true or false and prove your
conclusion.

[16] Show that every Pareto efficient allocation is potentially Pareto preferred to every
Pareto inefficient allocation.

[17] The compensation criterion and the Pareto criterion are normative concepts that are
used to examine the 'desirability' of social states. Compare and contrast these two
criteria.



                                          3.38

								
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