Download Topic 4. The First Theorem of Welfare Economics

MSc Economic Evaluation in Health Care
WELFARE ECONOMICS
Topic 4. The First Theorem of Welfare Economics
The First Theorem of Welfare Economics
The most important and useful aspect of the Pareto Principle is the relationship between
Pareto optimality and the equilibrium of an economy in which resources are allocated by
an ideal market mechanism.
In a system of competitive markets a competitive equilibrium is a situation where a set of
relative commodity and factor prices is established such that all markets clear (i.e. the
quantity supplied equals the quantity demanded).
Given certain conditions three important theorems can be proved. The first is that a
competitive general equilibrium exists under the conditions stated. The next two
theorems are referred to as the fundamental theorems of welfare economics and are
welfare statements about a competitive general equilibrium based on the Pareto Principle:
1. The first fundamental theorem of welfare economics (the direct theorem) states that
under certain assumptions a state (i.e. an allocation of goods and factors) resulting
from a competitive equilibrium is Pareto optimal; and,
2. The second fundamental theorem of welfare economics (the converse theorem) states
that under certain assumptions, every Pareto optimum state (i.e. allocation of goods
and factors) can be realised as the outcome of a competitive equilibrium given the
distribution of claims on income.
The first fundamental theorem of welfare economics requires:
1. Efficient exchange of goods and services (economic efficiency in an exchange
economy);
2. Efficient allocation of the factors of production (economic efficiency in a production
economy); and,
3. Efficient output choice (overall efficiency).
Economic efficiency in an exchange economy
For simplicity, suppose an economy in which there are two consumers (households),
Alice (a) and Bob (b), and there are two commodities (good 1 and good 2).
Alice and Bob both wish to maximise their utility subject to their budget constraint.
Individually, the solution to this problem requires that both Alice and Bob are on their
highest attainable indifference curve given their budget constraint. This occurs at the
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point where their indifference curves and budget line are tangential (i.e. the slope of the
indifference curve and the slope of the budget line are equal). This means that the
marginal rate of substitution (MRS) between the two goods equals their price ratio,
(p1/p2).
We can combine these two separate equilibria and construct an Edgeworth box. This is
constructed by turning the indifference curve map and budget constraint of Bob upside
down and connecting them to those of Alice.
The dimensions of the Edgeworth box are equal to the maximum endowments of good 1
and good 2. The distribution of the two goods between Alice and Bob can be described
by any point in the Edgeworth box.
The contract curve is the locus of all allocations of good 1 and good 2 such that the
indifference curves of Alice are tangential to those of Bob. In other words, at each point
on the contract curve the MRS between good 1 and good 2 (i.e. the slope of the
indifference curve) for Alice is equal to that of Bob.
At any point that is not on the contract curve, the MRS of Alice and Bob will be different.
This opens up the possibility of mutually beneficial trade: the indifference curves reached
by Alice and Bob when they consume distributions of the two goods not on the contract
curve form a lens-shaped area within which lie points that are Pareto superior to the
initial distribution and which can be reached by Alice and Bob if they trade quantities of
the two commodities. At any point within this lens that is not on the contract curve there
exists still further Pareto-improving trades.
Once Alice and Bob are on the contract curve no further improvements are possible, i.e.
one household can increase their utility only at the expense of the other. Therefore, any
point on the contract curve is a Pareto optimal allocation of the endowments.
Mathematically this is given by:
MRS a  MRS b 
p1 *
p2 *
[1]
In other words, a Pareto optimal allocation of two goods across two households requires
that the marginal rates of substitution for each household must be equal and this must be
equal to the relative (equilibrium) prices of the two goods.
These conditions can be generalised to an economy with many households and goods and
are referred to as the exchange efficiency conditions.
The exchange efficiency conditions characterise the allocation of a given bundle of
commodities among the households of the economy such that it would not be possible to
make one household better off without making another household worse off (i.e. so that
Pareto optimality is achieved in exchange).
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We can now find the set of all aggregate endowments of goods 1 and 2 which can be
allocated across households to achieve that distribution of utilities or any Pareto superior
distribution of utilities. This set is called the Scitovsky set.
The Scitovsky set is constructed for a particular allocation of goods across households. In
general let xe denote an N*H vector whose element is xhi is the consumption of good i by
household h. Therefore, in the two-person, two-good case discussed up until now, xe =
(xa1, xa2, xb1, xb2).
The boundary of the Scitovsky set is called a community indifference curve (CIC), which
represents all commodity endowments that achieve an equal utility distribution on
aggregate in the community. Two points about CICs should be stressed:
1. All points on the CIC represent exchange efficient allocations of the specified
aggregate endowment bundles; and,
2. The slope of the CIC is equal to the common MRS of the households at the Pareto
optimal allocation.
Economic efficiency in an production economy
For simplicity suppose there are two factors of production, labour L and capital (or
materials) K which are used to produce output of two different goods, good 1 and good 2,
by two firms owned by Alice (a) and Bob (b). The factors of production are turned into
the two final goods using the production function, which gives the maximum quantity of
output that can be produced from a specified set of inputs.
The production function of a good can be represented diagramatically by an isoquant
which represents the different combinations of labour and capital needed to produce the
same level of output. (Note that the isoquant is similar to the indifference curve which we
discussed previously).
The slope of the isoquant is called the marginal rate of technical substitution (MRTS).
The MRTS shows the rate at which one factor of production must be substituted for
another factor of production to produce the same level of output.
MRTS = slope of the isoquant at any one point = 
dL
dK
[2]
(Note that the MRTS is similar to the MRS which we discussed previously).
The firm is assumed to follow of goal of profit maximisation. This means that each firm i
maximises profit , where
i = pixi – wLi – rKI
[3]
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and p is the price of the good that the factors of production are used to produce, x is the
quantity of the good sold, w is the cost of labour (i.e. the wage rate), L is the quantity of
labour used to produce the good, r is the cost of capital (i.e. the interest rate) and K is the
quantity of capital used to produce the good.
The profit-maximising firm will seek to produce its output by methods that minimise its
costs of production. That is, the problem maximisation problem is analytically separable
into two sub-problems:
1. Find the cost-minimising combination of factors of production for producing any
given output level; and,
2. Produce the output level that maximises profit
To find the cost-minimising combination of factors of production for producing any
given output level, we can construct an isocost line for each firm that represents all the
different input combinations that the firm could buy by spending a fixed budget b on
inputs. This isocost line may be thought of as a budget constraint for the firm and is given
by the expression:
b = wL + rK
[4]
The slope of the isocost line is given by:
Slope of isocost line =
r
w
[5]
(Note that the isocost line is similar to the budget constraint which we discussed
previously).
The cost-minimising factor combination for a firm facing given factor prices is given
diagrammatically at the point where the isoquant is tangential to the lowest possible
isocost line.
At any other factor combination the costs of producing the given output are higher (and
so costs are not minimised). Therefore, the condition for cost minimisation is that the
slope of the isoquant is equal to the slope of the isocost line, or
MRTS  
dL
r

dK w
[6]
To produce the output level that maximises profit (the second stage of the firm’s
maximisation problem), the firm must choose an output level where the marginal cost of
producing the good is equal to the output price, i.e,
pi = MCi
[7]
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We can now analyse the equilibrium conditions for efficient production in a situation
where there are two households (firms), [Alice (a) and Bob (b)], two goods (1 and 2) and
two factors of production (L and K).
Alice and Bob both wish to maximise their profits. Individually, this requires that both
Alice and Bob are on their highest attainable isoquant given their isocost line (or
similarly, that they are on the lowest attainable isocost line given their isoquant). This
occurs at the point where their isoquants and isocost lines are tangential (i.e., the slope of
the isoquant and the slope of the isocost line are equal, which means that the marginal
rate of technical substitution (MRS) between the two factors of production equals the
factor price ratio [r/w]).
We can combine these two separate equilibria and construct an Edgeworth box. This is
done by turning the isoquant map and isocost line of Bob upside down and connecting
them to those of Alice. The dimensions of the Edgeworth box are equal to the total factor
endowments owned by the households.
Any point in the Edgeworth box represents an allocation of the factors of production to
the two goods. Efficient factor allocations are those in which it is not possible to increase
the output of one good without reducing the output of the other good.
We can again construct a contract curve which is the locus of points which are efficient.
At any point that is not on the contract curve, the MRTS of Alice and Bob will be
different. This opens up the possibility of mutually beneficial reallocations of the factors
of production: the isoquants reached by Alice and Bob when they use combinations of
factors of production not on the contract curve form a lens-shaped area within which lie
points that are Pareto superior to the initial allocation and which can be reached by Alice
and Bob if they trade quantities of the two factors. At any point within this lens that is not
on the contract curve there exists still further Pareto-improving trades.
Once Alice and Bob are on the contract curve no further improvements are possible, i.e.,
one household can increase their output only at the expense of the other. Therefore, any
point on the contract curve is a Pareto optimal allocation of the factors of production.
Mathematically this is given by:
MRTS a  MRTS b 
r*
w*
[8]
In other words, a Pareto optimal production of two goods across two households using
two factors of production requires that the marginal rates of technical substitution for
each household must be equal and this must be equal to the relative (equilibrium) prices
of the two factors of production.
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These conditions can be generalised to an economy with many factors of production and
goods and are referred to as the production efficiency conditions.
The production efficiency conditions characterise the allocation of the economy’s factors
of production in producing output. The economy will be producing efficiently if factors
of production are allocated in such a way so that it is not possible to produce more of one
good without producing less of another (i.e. so that Pareto optimality is achieved in
production).
As can be seen from the contract curve a given endowment of factors can produce a large
number of Pareto optimal output combinations. This can be translated into the production
possibilities curve (PPC) which bounds the set of all feasible output combinations that
can be produced from the given factor endowments.
The points on the PPC boundary are Pareto optimal because more of one good cannot be
produced without producing less of another good. These therefore dominate interior
points.
The slope of the PPC is given by the marginal rate of transformation (MRT), which
represents the rate at which one good can be transformed into another by reallocating the
factors of production between the two goods.
MRT = slope of the PPC at any one point = 
dx 2
dx 1
[9]
To find the combination of outputs that maximises profits (the goal of the firm) we can
set up an isorevenue line that represents all the different output combinations that achieve
the same level of revenue, R. This isorevenue line is given by the expression:
R = p1x1 + p2x2
[10]
The slope of the isorevenue line is given by:
Slope of isorevenue line =
p1
p2
[11]
(Note that the isorevenue line is similar to the budget constraint and isocost line which
we discussed previously).
The aim of the firm is to maximise their profits. Diagrammatically this is obtained by
achieving the highest isorevenue line given the production possibilities determined by the
factors of production. This occurs where the PPC is tangential to the highest possible
isorevenue line.
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At any other factor point on the PPC profits are not maximised. Therefore, the condition
for profit maximisation is that the slope of the isorevenue line is equal to the slope of the
PPC, or
MRT =
p1 *
p2 *
[12]
Note that for this condition to hold it is necessary to be on the PPC which in turn requires
that
MRTS a  MRTS b 
r*
w*
[13]
Overall efficiency
A large number of allocations exist which satisfy both the exchange efficiency and
production efficiency conditions. These can be reduced by invoking the overall efficiency
conditions. An allocation will be Pareto optimal overall if it is not possible to reallocate
production and distribution so as to make one person better off without making another
person worse off.
Many allocations are exchange efficient (requiring that allocations occur on the CIC) and
production efficient (requiring that production occurs on the PPC).
Overall efficiency is achieved only if the allocation is such that there does not exist any
feasible production possibility that is in the interior of the Scitovsky set (i.e. that the CIC
and PPC are tangential).
This requires that:
MRS = MRT
[14]
These are the overall efficiency conditions.
Summary
The first fundamental theorem of welfare economics states that under certain assumptions
a state (i.e. an allocation of goods and factors) resulting from a competitive equilibrium is
Pareto optimal. At a competitive general equilibrium there will exist a set of equilibrium
prices in factor and product markets such that markets clear. All households face the
same equilibrium prices (p1*, p2*) and, in maximising their utility, equate their MRS to
the common equilibrium prices (equation [1]). Therefore, the exchange efficiency
conditions are met. Profit maximising firms, which face the same equilibrium factor
prices (r*, w*), hire factors of production so as to minimise production costs by equating
their MRTS to the relative equilibrium factor prices (equation [8]). Therefore, the
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production efficiency conditions are met. Finally, since households and firms face the
same equilibrium prices, the overall efficiency conditions are met (equation [14]).
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