Chapter 5:
Welfare economics and the
environment
Part 1 Efficiency and optimality
The setting
• At any point in time, an economy will have access to particular
quantities of productive resources.
• Individuals have preferences about the various goods that it is
feasible to produce using the available resources.
• An allocation of resources describes what goods are produced and
in what quantities they are produced, which combinations of
resource inputs are used in producing those goods, and how the
outputs of those goods are distributed between persons.
• At first, we make two assumptions:
1.
2.
No externalities exist in either consumption or production
All produced goods and services are private (not public) goods
• For simplicity, strip the problem down to its barest essentials:
– economy consists of two persons (A and B);
– two goods (X and Y) are produced;
– production of each good uses two inputs (K for capital and L
for labour) each available in a fixed quantity.
Utility functions
The utility functions for A and B:
A
A
A
A
U = U (X , Y )
B
B
B
B
U = U (X , Y )
Marginal utility written as and defined by
U AX  U A /X A
with equivalent notation for the other three marginal utilities.
Production functions
The two production functions for goods X and Y:
X  X(K X , LX )
Y  Y(K Y , LY )
Marginal product written as and defined by
MP  Y/ L
Y
L
Y
with equivalent notation for the other three marginal products.
Marginal rate of utility substitution for A:
• the rate at which X can be substituted for Y at the margin, or vice versa,
while holding the level of A's utility constant
• It varies with the levels of consumption of X and Y and is given by the slope
of the indifference curve
• Denote A's marginal rate of substitution as MRUSA
• Similarly for B
The marginal rate of technical substitution in the production of X:
• the rate at which K can be substituted for L at the margin, or vice versa,
while holding the level output of X constant
• It varies with the input levels for K and L and is given by the slope of the
isoquant
• Denote the marginal rate of substitution in the production of X as MRTSX
• Similarly for Y
The marginal rates of transformation for the commodities X and Y:
The marginal rate of transformation shows how much it costs to produce
one good in terms of the forgone production of the other good
the rates at which the output of one can be transformed into the
other by marginally shifting capital or labour from one line of
production to the other
MRTL is the increase in the output of Y obtained by shifting a
small amount of labour from use in the production of X to use in
the production of Y, or vice versa
MRTK is the increase in the output of Y obtained by shifting a
small, amount of capital from use in the production of X to use in
the production of Y, or vice versa
Economic efficiency
• An allocation of resources is efficient if it is not possible to make
one or more persons better off without making at least one other
person worse off.
• A gain by one or more persons without anyone else suffering is a
Pareto improvement.
• When all such gains have been made, the resulting allocation is
Pareto optimal (or Pareto efficient).
• Efficiency in allocation requires that three efficiency conditions are
fulfilled
1.
2.
3.
efficiency in consumption
efficiency in production
product-mix efficiency
Efficiency in consumption
•
Consumption efficiency requires that the marginal rates of
utility substitution for the two individuals are equal:
MRUSA = MRUSB
(4.3)
• If this condition were not satisfied, it would be possible to rearrange the allocation as between A and B of whatever is
being produced so as to make one better-off without making
the other worse-off.
Figure 4.1 Efficiency in consumption.
BY
AX
AXa
S
AXb
A0
IB1
IB0
BYa
IA
.a
AYa
.b
BYb
AYb
IA
B0
BXa
IB0
IB1
BX
BXb
T
AY
Efficiency in production
• Efficiency in production requires that the marginal rate of
technical substitution be the same in the production of both
commodities. That is
MRTSX = MRTSY
(4.4)
• If this condition were not satisfied, it would be possible to reallocate inputs to production so as to produce more of one of
the commodities without producing less of the other.
KY
LXa
LX
LXb
X0
IY1
IY0
KYa
IX
.a
KXa
.b
KYb
KXb
IY1
IX
Y0
LYa
IY0
LY
LYb
KX
Product-mix efficiency
Figure 4.3 Product-mix efficiency.
Y
I
YM
a
Ya

Yb

b
c
Yc
I
XM
0
Xa
Xb
XC
X
All three conditions must be satisfied
• An economy attains a fully efficient static allocation of
resources if the conditions given by equations 4.3, 4.4
and 4.5 are satisfied simultaneously.
• The results readily generalise to economies with many
inputs, many goods and many individuals.
– The only difference will be that the three efficiency
conditions will have to hold for each possible pairwise
comparison that one could make.
An efficient allocation of resources is not unique
• For an economy with given quantities of available
resources, and given production functions and utility
functions, there will be many efficient allocations of
resources.
Figure 4.4 The set of allocations for consumption efficiency.
BY
AX
A0
S
A
B
B
A
B
C
A
B
A
B
A
.
A
A
A
C
A
A
B
B
B
B
B
BX
B0
T
The set of allocations for consumption efficiency
AY
Continuing the reasoning ...
•
Now consider the efficiency in production condition, and Figure 4.2 again. Here
we are looking at variations in the amounts of X and Y that are produced.
•
Clearly, in the same way as for Figures 4.1 and 4.4, we could introduce larger
subsets of all the possible isoquants for the production of X and Y to show that
there are many X and Y combinations that satisfy equation 4.4, combinations
representing uses of capital and labour in each line of production such that the
slopes of the isoquants are equal, MRTSX = MRTSY.
•
So, there are many combinations of X and Y output levels that are consistent with
allocative efficiency ...
... and for any particular combination there are many allocations as between A
and B that are consistent with allocative efficiency.
•
•
These two considerations can be brought together in a single diagram, as in Figure
4.5, where the vertical axis measures A's utility and the horizontal B's.
The utility possibility frontier: The utility possibility frontier is the locus of all
possible
combinations
of
UA
and
UB
that
correspond
to efficiency in allocation.
UB
U Bmax
T

U BR
R
Z
S
UA
0
U AR
U Amax
The social welfare function and optimality
•
In order to consider such choices we need the concept of a social welfare
function, SWF.
• A SWF can be used to rank alternative allocations.
• For the two person economy that we are examining, a SWF will be of the
general form:
W = W( UA , UB)
(4.6)
• The only assumption that we make here regarding the form of the SWF is
that welfare is non-decreasing in UA and UB.
• A utility function associates numbers for utility with combinations of
consumption levels X and Y
• A SWF associates numbers for social welfare with combinations of utility
levels UA and UB.
• Just as we can depict a utility function in terms of indifference curves, so
we can depict a SWF in terms of social welfare indifference curves.
Figure 4.6 Maximised social welfare.
UB
Figure 4.6 shows a social welfare indifference curve
WW, which has the same slope as the utility
possibility frontier at b, which point identifies the
combination of UA and UB that maximises the SWF.
W
U aB
a

U Bb

b
c
U cB
0
The fact that the optimum lies on the utility
possibility frontier means that all of the
necessary conditions for efficiency must
hold at the optimum. Conditions 4.3, 4.4
and 4.5 must be satisfied for the
maximisation of welfare.
U aA
U Ab
U cA
W
UA
An additional condition
• Also, an additional condition, the equality of the slopes of a social
indifference curve and the utility possibility frontier, must be satisfied.
• This condition can be stated as
WA U BX U BY
 A  A
WB U X U Y
(4.7)
– The left-hand side here is the slope of the social welfare indifference curve.
– The two right-hand side terms are alternative expressions for the slope of the
utility possibility frontier.
– At a social welfare maximum, the slopes of the indifference curve and the
frontier must be equal, so that it is not possible to increase social welfare by
transferring goods, and hence utility, between persons.
Figure 4.7 Welfare and efficiency.
UB

Clearly, any change which is a Pareto improvement must
increase social welfare as defined here. Given that the SWF
is non-decreasing in UA and UB, increasing UA/UB without
reducing UB/UA must increase social welfare. For the kind of
SWF employed here, a Pareto improvement is an unambiguously good thing.
D
C


E
W2
W1
0
UA
Figure 4.7 Welfare and efficiency.
UB

While allocative efficiency is a necessary condition
for optimality, it is not generally true that moving
from an allocation that is not efficient to one that is
efficient must represent a welfare improvement.
Such a move might result in a lower level of social
welfare.
D
C


At C the allocation is not efficient,
at D it is. However, the allocation
at C gives a higher level of social
welfare than does that at D.
E
W2
W1
0
UA
Figure 4.7 Welfare and efficiency.
UB

Nevertheless, whenever there is an inefficient
allocation, there is always some other allocation
which is both efficient and superior in welfare
terms. Compare points C and E. E is allocatively
efficient while C is not, and E is on a higher social
welfare indifference curve. The move from C to E is
a Pareto improvement where both A and B gain,
and hence involves higher social welfare.
D
C


E
W2
W1
0
UA
Figure 4.7 Welfare and efficiency.
UB
On the other hand, going from C to D replaces an
inefficient allocation with an efficient one, but the
change is not a Pareto improvement - B gains but A
suffers - and involves a reduction in social welfare.

D
C


E
W2
W1
0
UA
Part 3 Market failure, public policy
and the environment
Public Goods
• Two characteristics of goods and services are relevant to the public/private
question.
• These are rivalry and excludability. What we call rivalry is sometimes referred to in
the literature as divisibility.
– Rivalry refers to whether one agent's consumption is at the expense of
another's consumption.
– Excludability refers to whether agents can be prevented from consuming. The
question of excludability is a matter of law and convention, as well as physical
characteristics.
• Pure private goods exhibit both rivalry and excludability. These are 'ordinary' goods
and services, such as ice cream. For a given amount of ice cream available, any
increase in consumption by A must be at the expense of consumption by others, is
rival. Any individual can be excluded from ice cream consumption.
• Pure public goods exhibit neither rivalry nor excludability. An example is the
services of the national defence force.
Open access natural resources exhibit rivalry but not excludability. An
example would be ocean fisheries outside the territorial waters of any nation.
In that case, no fishing boat can be prevented from exploiting the fishery,
since it is not subject to private property rights and there is no government
that has the power to treat it as common property and regulate its
exploitation. However, exploitation is definitely rivalrous. An increase in the
catch by one fishing boat means that there is less for other boats to take.
Congestible resources exhibit excludability but not, up to the point at which
congestion sets in, rivalry. An example is the services to visitors provided by
a wilderness area.